# Patent

## Title
2D arrays of diamond shaped cells having multiple josephson junctions

## Abstract

A two-dimensional SQIF array and methods for manufacture can include at least two bi-SQUIDs that share an inductance. The bi-SQUIDs can be combined to establish a diamond-shaped cell. A plurality of the diamond shaped cells can be packed tightly together so that each cell shares at least three cell junctions with adjacent cells to establish the SQIF array. Because of the close proximity of the cells, the effect that the mutual inductances each cell has on adjacent cells can be accounted for, as well as the SQIF array boundary conditions along the array edges. To do this, a matrix of differential equations can be solved to provide for the recommended inductance of each bi-SQUID in the SQIF array. Each bi-SQUID can be manufactured with the recommended inductance to result in a SQIF having an increased strength of anti-peak response, but without sacrificing the linearity of the response.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation-in-part of U.S. application Ser. No. 13/686,994, filed Nov. 28, 2012, and entitled “Linear Voltage Response of Non-Uniform Arrays of Bi-SQUIDs”. The '994 application is hereby incorporated by reference herein in its entirety


FEDERALLY-SPONSORED RESEARCH AND DEVELOPMENT
This invention (Navy Case No. 102297) is assigned to the United States Government and is available for licensing for commercial purposes. Licensing and technical inquires may be directed to the Office of Research and Technical Applications, Space and Naval Warfare Systems Center, Pacific, Code 72120, San Diego, Calif. 92152; voice (619) 553-5118; email ssc pac T2@navy.mil.


FIELD OF THE INVENTION
The present invention pertains generally to mechanisms and methods for improving signal detection and amplification. More specifically, the present invention pertains to Superconducting Quantum Interference Filters (SQIFs). The invention is particularly, but not exclusively, useful as a SQIF having multiple Superconductive Quantum Interference Devices (SQUIDs), which can be arranged in a two-dimensional array so that adjacent bi-SQUIDs share inductances.
BACKGROUND OF THE INVENTION
The SQUID is one of the most sensitive magnetic field devices in the prior art, and it can be used for wide range of applications, including biology, medicine, geology, systems for semiconductor circuit diagnostics, security, magnetic resonance imaging (MRI) and even cosmology research. In recent years, arrays of coupled oscillators have been considered as a general mechanism for improving signal detection and amplification. Theoretical and experimental studies can be interpreted to show that arrays of SQUIDs can yield comparable improvements in signal output relative to background noise, over those of a single SQUID device.
A peculiar configuration that has gained considerable attention among the physics and engineering community is that of multi-loop arrays of Josephson Junctions (JJs) with non-uniformly distributed loop areas. Typically, each loop contains two JJs, i.e., a standard DC-SQUID, but the loop size varies from loop to loop. These types of unconventional geometric structures of JJs are known to exhibit a magnetic flux dependent voltage response V(φe), where φe denotes an external magnetic flux normalized by the quantum flux, that has a pronounced single peak with a large voltage swing at zero magnetic field. The potential high dynamic range and linearity of the “anti-peak” voltage response render the array an ideal detector of absolute strength of external magnetic fields. These arrays are also commonly known as Superconducting Quantum Interference Filters (SQIFs).
Improving the linearity of SQIFs is critical for developing advanced technologies, such as low noise amplifiers (LNA's) that can further increase link margins and affect an entire communication system. SQIFs can also be used in unmanned aerial vehicles (UAVs), where size, weight and power are limited, and as “electrically small” antennas to provide acceptable gain for the antenna. SQIFs can also be used in land mine detection applications. But for all of these applications, it is desired to improve the linear response of the SQIF device.
The quest to increase the linearity of SQUID and SQIF arrays was boosted by the introduction of the bi-SQUID, which has three 3 JJ's. A non-linear inductance of the additional third JJ can provide a desired linearizing effect for the orverall SQIF. These bi-SQUIDs are now being used in uniform and non-uniform (SQIF) arrays in place of conventional dc SQUIDs with a goal of achieving higher linearity. However, most of the design efforts to date that use SQUIDs and/or bi-SQUIDs are directed at the optimization of one-dimensional (1D) serial or parallel arrays and their combinations. This is due to the higher complexity analysis and modeling required for 2D arrays to account for mutual coupling that occurs between neighboring cells and complex current distribution in arrays. What is desired is a two-dimensnional array of bi-SQUIDs.
In view of the above, it is an object of the present invention to provide a two-dimensional (2D) SQIF array having a tightly coupled 2D network of bi-SQUID cells, in which junctions and inductances are shared between adjacent bi-SQUID cells. Another object of the present invention is to provide a two-dimensional (2D) SQIF array having diamond shaped, dual bi-SQUID cells. Still another object of the present invention is to provide a 2D SQIF that maximizes the number of bi-SQUID cells which can be placed with a given SQIF array area. Yet another object of the present invention is to maximize the voltage response and dynamic range of a SQIF by manipulating the critical current, inductive coupling between loops, number of loops, bias current, and distribution of loop areas of the array cells of diamond bi-SQUIDs. Still another object of the present invention is to provide a 2D SQIF array with increased high dynamic range and linearity of the “anti-peak” response, for use in multiple applications such as magnetic field detectors, magnetic antennas and wide-band low-noise amplifiers. Another object of the present invention is to provide a SQIF and methods for manufacture that can be easily tailored in a cost-effective manner to result in a SQIF having bi-SQUID array cells that has been optimized according to the user's needs.
SUMMARY OF THE INVENTION
A two-dimensional SQIF array and methods for manufacture according to several embodiments of the present invention can include at least two bi-SQUIDs that share an inductance. The two bi-SQUIDs can be combined to establish a cell that is diamond shaped when viewed in top plan. A plurality of the diamond shaped cells can be packed tightly together to establish the SQIF array. With this configuration, each cell shares at least three cell junctions with adjacent cells.
Because of the close proximity of the cells, the effect that the mutual inductances each cell has on adjacent cells can be accounted for. Additionally, the boundary conditions along the edges can be determined. To do this, a matrix of sixty-six 66 differential equations can be solved to provide for the recommended inductance of each bi-SQUID in the 2D SQIF array. Each bi-SQUID can be manufactured with the recommended inductance to result in a SQIF having an increased strength of anti-peak response, but without sacrificing the linearity of the response.



BRIEF DESCRIPTION OF THE DRAWINGS
The novel features of the present invention will be best understood from the accompanying drawings, taken in conjunction with the accompanying description, in which similarly-referenced characters refer to similarly-referenced parts, and in which:
FIG. 1 is a schematic diagram of a direct current Superconductive Quantum Interference Devices (DC-SQUID) as known in the prior art;
FIG. 2 is a schematic diagram of a prior art bi-SQUID;
FIG. 3 is a graphical representation of the voltage response of the DC-SQUID and bi-SQUID of FIGS. 1 and 2, as a function of magnetic field strength;
FIG. 4 is a schematic diagram of a single cell dual bi-SQUID of the 2D Array of the present invention according to several embodiments;
FIG. 5 is a graphical representation of the voltage response of the single cell dual bi-SQUID of FIG. 4 as a function of magnetic field strength;
FIG. 6 is a top plan view of the single cell dual bi-SQUID of FIG. 4;
FIG. 7 is a schematic view of the 2D SQIF array of the present invention according to several embodiments, which illustrate a plurality of the cells of FIGS. 4 and 5;
FIGS. 8a-8i are schematic views of the various boundary conditions for the 2D SQIF array of FIG. 7;
FIG. 9 is a top plan view of a portion of the 2D SQIF array of FIG. 7;
FIG. 10 is a graphical representation of the voltage response of the 2D SQIF array of FIG. 7 as a function of magnetic field strength;
FIG. 11 is a graphical representation of graphical representation of the flux noise spectral density versus frequency at the mid-point of the positive slope of the anti-peak in FIG. 10;
FIG. 12 is a graphical representation of energy sensitivity versus frequency at the mid-point of the positive slope of the anti-peak in FIG. 10;
FIG. 13 is a graphical representation of noise temperature at the mid-point of the positive slope of the anti-peak in FIG. 10; and,
FIG. 14 is a block diagram, which illustrates steps that can be taken to accomplish the methods of the present invention according to several embodiments.



DETAILED DESCRIPTION OF THE EMBODIMENTS
A. Prior Art.
In brief overview, and referring initially to FIGS. 1 and 2, a DC SQUID 10 and DC bi-SQUID 20 are shown. FIG. 1 is a schematic diagram of a single DC SQUID 10. DC SQUID 10 can have two Josephson Junctions (JJ's) 12a, 12b arranged in parallel, connected with superconducting material, as represented by the schematic diagram in FIG. 1. A DC bi-SQUID 20, which is a SQUID with an additional Josephson junction 12c bisecting the superconducting loop, was introduced as an alternative to traditional SQUIDs. DC bi-SQUIDs have shown superior linearity in the average voltage response anti-peak feature. In FIG. 2, (ib, i1, i2, i3, i4, i5) can be the normalized currents through bi-SQUID 20, (φ1, φ2, φ3) can represent the phases across the Josephson junctions 12a-12c, β/2 is the parameter related to the inductance values and xe (point 14) can represent the point in the equations where the external fields (not shown) are included.
The equations for modeling the JJ's for the DC SQUID of FIG. 1 can be derived using Kirchhoff's current law, which is the principle of conservation of electric charge and implies that at any junction in an electrical circuit the sum of currents flowing into that node is equal to the sum of currents flowing out of that node, along with a resistively shunted junction (RSJ) model of the over-damped JJ. The JJ's can be assumed to be symmetric such that for the normalized critical currents ic1=ic2=1.0. The system of equations that models a single DC SQUID dynamics is:




  
    
      
        
          
            
              φ
              .
            
            1
          
          =
          
            J
            -
            
              
                1
                β
              
              ⁢
              
                (
                
                  
                    φ
                    1
                  
                  -
                  
                    φ
                    2
                  
                  -
                  
                    φ
                    e
                  
                
                )
              
            
            -
            
              sin
              ⁢
              
                
              
              ⁢
              
                φ
                1
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                φ
                .
              
              2
            
            =
            
              J
              +
              
                
                  1
                  β
                
                ⁢
                
                  (
                  
                    
                      φ
                      1
                    
                    -
                    
                      φ
                      2
                    
                    -
                    
                      φ
                      e
                    
                  
                  )
                
              
              -
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  2
                
              
            
          
          ,
        
      
    
    
      
        (
        1
        )
      
    
  




where φ1 and φ2 are the phases across each of the Josephson junctions and the dots denote the time differentiation with normalized time





  τ
  =
  
    
      
        ω
        c
      
      ⁢
      t
    
    =
    
      
        
          2
          ⁢
          
            eI
            0
          
          ⁢
          
            R
            N
          
        
              
      ⁢
      
        t
        .
      
    
  




The parameter ωc in τ=2π/(Φ0)(I0)(RN). The parameter RN in τ is the normal state resistance of the Josephson junctions and I0 is the critical current of the Josephson junctions.





  β
  =
  
    2
    ⁢
    π
    ⁢
    
      
    
    ⁢
    
      
        LI
        0
      
      
        Φ
        0
      
    
  




is the nonlinear parameter related to the SQUID inductance





  L
  ,
  
    J
    =
    
      
        I
        b
      
      
        2
        ⁢
        
          I
          0
        
      
    
  
  ,




where Ib is bias current, and φe=2πaxe, where xe is the normalized external magnetic field and





  a
  =
  
    
      β
      .
      
        
      
      ⁢
      
        Φ
        0
      
    
    ≡
    
      h
      
        2
        ⁢
        e
      
    
    ≈
    
      2.07
      ×
      
        10
        
          -
          15
        
      
    
  




tesla meter squared is the flux quantum, where h is Plank's constant and 2e is the charge on the Cooper pair.

The phase equations for the single DC bi-SQUID schematic in FIG. 2 can be derived in a similar way to those of the single DC SQUID. In this case there is a third junction J3 that is related to the first and second junctions through the phases: φ1+φ3=φ2. Using this relationship, all the terms that include φ3 can be replaced with φ2−φ1, thereby reducing the number of phase equations needed to model the system from three to two. The governing equations for a single bi-SQUID are




  
    
      
        
          
            
              φ
              .
            
            1
          
          =
          
            
              
                i
                b
              
              2
            
            -
            
              
                1
                
                  3
                  ⁢
                  β
                
              
              ⁢
              
                (
                
                  
                    φ
                    1
                  
                  -
                  
                    φ
                    2
                  
                  -
                  
                    φ
                    e
                  
                
                )
              
            
            +
            
              
                1
                3
              
              ⁢
              
                i
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    
                      φ
                      2
                    
                    -
                    
                      φ
                      1
                    
                  
                  )
                
              
            
            -
            
              
                2
                3
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              
                φ
                1
              
            
            -
            
              
                1
                3
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              
                φ
                2
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                φ
                .
              
              2
            
            =
            
              
                
                  i
                  b
                
                2
              
              +
              
                
                  1
                  
                    3
                    ⁢
                    β
                  
                
                ⁢
                
                  (
                  
                    
                      φ
                      1
                    
                    -
                    
                      φ
                      2
                    
                    -
                    
                      φ
                      e
                    
                  
                  )
                
              
              -
              
                
                  1
                  3
                
                ⁢
                
                  i
                  
                    c
                    ⁢
                    
                      
                    
                    ⁢
                    3
                  
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        φ
                        2
                      
                      -
                      
                        φ
                        1
                      
                    
                    )
                  
                
              
              -
              
                
                  1
                  3
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  1
                
              
              -
              
                
                  2
                  3
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  2
                
              
            
          
          ,
        
      
    
    
      
        (
        2
        )
      
    
  




where





  
    i
    
      c
      ⁢
      
        
      
      ⁢
      3
    
  
  =
  
    
      I
      
        c
        ⁢
        
          
        
        ⁢
        3
      
    
    
      I
      0
    
  




is the normalized critical current on the third junction and all other parameters are defined as the single DC SQUID.

The voltage response of both the single DC SQUID 10 of FIG. 1 and the DC bi-SQUID 20 of FIG. 2 can be simulated by integrating the systems of equations that model the system dynamics, Eq (1) and Eq (2) respectively. After the phases φ1 and φ2 have been determined the derivatives {dot over (φ)}1 and {dot over (φ)}2 are evaluated and the time-dependent voltage




  
    V
    ⁡
    
      (
      t
      )
    
  
  =
  
    
      
        
          φ
          .
        
        1
      
      +
      
        
          φ
          .
        
        2
      
    
    2
  




is calculated. The average voltage, <V>, of a SQUID (or bi-SQUID) at a point in xe is the mean value of the voltage over time. FIG. 3 illustrates the results of this process. FIG. 3 is an average voltage response of a single DC SQUID compared with that of a single DC bi-SQUID. The average voltage response of the bi-SQUID (line 32), with the proper selection of parameters, has a more linear average voltage response than the conventional DC SQUIDs (line 34 in FIG. 3) . The simulations were performed with J=1.0, β=1.0 and ic3=1.0. The higher linearity of the average voltage response of the bi-SQUID can increase the overall utility of the device that incorporates the bi-SQUID as a linear amplifier, provided the bi-SQUIDs are incorporated to optimize the overal linearity of the device.

The structure and cooperation of structure of the SQUID and bi-SQUID, and the design and performance of one-dimensional SQIF's using such components are discused in more detail in the aforementioned '994 application, which has been incorporated in its entirety into this specification.
B. Dual Bi-SQUID Cell.
Referring now to FIG. 4, FIG. 4 depicts a circuit of a dual bi-SQUID cell structure 40 of the present invention according to several embodiments. As shown in FIG. 4, cell 40 can further include at least two bi-SQUIDs 20a, 20b (indicated by the dotted lines in FIG. 4), which can be merged so that the bi-SQUIDs share at least one two bi-SQUID junctions 44a, 44b and at least one inductance 46. With this configuration, cell 40 can present a diamond shaped appearance when viewed in top plan. It should be appreciated, however, that other geometric views (such as hexangonal, for example) are possible, provided the equations discussed below are modified and derived using Kirchoff's law to account for the conservation of currents in such alternative embodiments.
In FIG. 4, ib can be taken to represent an input base current into cell 40, (i1, . . . , i11) can represent the normalized currents through the various bi-SQUID junctions 44, and (φ1 . . . φ6) can represent the phases across the JJ's 42 in cell 40 through bi-SQUIDs 20, (L1, L2a, . . . , L6a, L2b, . . . , L6b) represent the parameters related to the inductances and xe (points 48a, 48b) can represent the points in the equations where the external fields are included. Using these representations, six governing equations are needed to simulate the average voltage response for this device, which contains 12 currents, 6 Josephson junctions and 9 inductors.
Kirchhoff's current law results in the following relations for the currents and phases in the diamond-shaped dual bi-SQUID




  
    
      
        
          
            
              
                i
                b
              
              =
              
                
                  i
                  1
                
                +
                
                  i
                  2
                
              
            
          
          
            
              
                i
                1
              
              =
              
                
                  i
                  3
                
                +
                
                  i
                  8
                
              
            
          
        
        
          
            
              
                
                  i
                  2
                
                +
                
                  i
                  3
                
              
              =
              
                i
                9
              
            
          
          
            
              
                i
                8
              
              =
              
                
                  i
                  7
                
                +
                
                  i
                  10
                
              
            
          
        
        
          
            
              
                
                  i
                  9
                
                +
                
                  i
                  7
                
              
              =
              
                i
                11
              
            
          
          
            
              
                
                  i
                  10
                
                +
                
                  i
                  6
                
              
              =
              
                i
                4
              
            
          
        
        
          
            
              
                i
                11
              
              =
              
                
                  i
                  6
                
                +
                
                  i
                  5
                
              
            
          
          
            
              
                
                  i
                  4
                
                +
                
                  i
                  5
                
              
              =
              
                i
                b
              
            
          
        
        
          
            
              
                i
                1
              
              =
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  1
                
              
            
          
          
            
              
                i
                2
              
              =
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    2
                  
                
                +
                
                  
                    φ
                    .
                  
                  2
                
              
            
          
        
        
          
            
              
                i
                4
              
              =
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    4
                  
                
                +
                
                  
                    φ
                    .
                  
                  4
                
              
            
          
          
            
              
                i
                5
              
              =
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    5
                  
                
                +
                
                  
                    φ
                    .
                  
                  5
                
              
            
          
        
        
          
            
              
                i
                3
              
              =
              
                
                  
                    i
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      3
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    3
                  
                
                +
                
                  
                    φ
                    .
                  
                  3
                
              
            
          
          
            
              
                
                  i
                  6
                
                =
                
                  
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      6
                    
                  
                  +
                  
                    
                      φ
                      .
                    
                    6
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




where ic3=Ic3/I0 is the normalized critical current of the third junction, ic6=Ic6/I0 is the normalized critical current of the sixth junction, Ic1=Ic2=Ic4=Ic5=I0, and φi are the phases across of the Josephson junctions for i=1, . . . ,6. The dots on the equations in this specficiation denote time differentiation with respect to normalized time τ=ωct, where ωc and t remain defined as above. The resistively-shunted junction (RSJ) model provides a current-phase relation around the top half of the diamond of dual bi-SQUID cell 40 (bi-SQUID 20a)

φ1+L1i7+L2ai8=2πxea1+φ2+L2bi9,

where φe=2πa1xe, where xe is the normalized external magnetic flux and a1=L1+L2a+L2b. Combining i7=i8−i10 and i7=i11−i9 such that





  
    
      i
      7
    
    =
    
      
        
          
            i
            8
          
          -
          
            i
            10
          
        
        2
      
      +
      
        
          
            i
            11
          
          -
          
            i
            9
          
        
        2
      
    
  
  ,




and then substituting the current relations for i7, i10, i9, and i8 yields





  
    
      
        
          
            
              (
              
                
                  
                    L
                    1
                  
                  2
                
                +
                
                  L
                  
                    2
                    ⁢
                    a
                  
                
              
              )
            
            ⁢
            
              i
              1
            
          
          -
          
            
              
                L
                1
              
              2
            
            ⁢
            
              i
              4
            
          
          +
          
            
              
                L
                1
              
              2
            
            ⁢
            
              i
              5
            
          
          +
          
            
              L
              1
            
            ⁢
            
              i
              6
            
          
        
        =
        
          
            φ
            2
          
          -
          
            φ
            1
          
          +
          
            
              (
              
                
                  
                    L
                    1
                  
                  2
                
                +
                
                  L
                  
                    2
                    ⁢
                    b
                  
                
              
              )
            
            ⁢
            
              i
              2
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              1
            
          
          +
          
            
              (
              
                
                  L
                  1
                
                +
                
                  L
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    a
                  
                
                +
                
                  L
                  
                    2
                    ⁢
                    b
                  
                
              
              )
            
            ⁢
            
              
                i
                3
              
              .
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




In order to get the first of the six equations that is needed to describe the dynamics of the diamond shape, the bias current relation i2=ib−i1 and Josephson junction relations from Eq. (3) are substituted into Eq. (4) to become




  
    
      
        
          
            
              
                L
                12
              
              ⁡
              
                (
                
                  
                    
                      φ
                      .
                    
                    1
                  
                  -
                  
                    
                      φ
                      .
                    
                    3
                  
                
                )
              
            
            +
            
              
                
                  L
                  1
                
                2
              
              ⁢
              
                (
                
                  
                    2
                    ⁢
                    
                      
                        φ
                        .
                      
                      6
                    
                  
                  +
                  
                    
                      φ
                      .
                    
                    5
                  
                  -
                  
                    
                      φ
                      .
                    
                    4
                  
                
                )
              
            
          
          =
          
            
              
                (
                
                  
                    
                      L
                      1
                    
                    2
                  
                  +
                  
                    L
                    
                      2
                      ⁢
                      b
                    
                  
                
                )
              
              ⁢
              
                i
                b
              
            
            +
            
              φ
              2
            
            -
            
              φ
              1
            
            +
            
              2
              ⁢
              π
              ⁢
              
                
              
              ⁢
              
                x
                e
              
              ⁢
              
                a
                1
              
            
            +
            
              
                L
                12
              
              ⁡
              
                (
                
                  
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      3
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                
                )
              
            
            +
            
              
                
                  L
                  1
                
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      4
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      5
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      6
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        5
        )
      
    
  




where L12=L1+L2a+L2b.

To solve for the second of the six equations that governs the dynamics of the single diamond the current relations for the bias current i1=ib−i2 and Josephson junctions from Eq. (3) are substituted into Eq. (4) to give




  
    
      
        
          
            
              
                
                  L
                  1
                
                2
              
              ⁢
              
                (
                
                  
                    2
                    ⁢
                    
                      
                        φ
                        .
                      
                      6
                    
                  
                  +
                  
                    
                      φ
                      .
                    
                    5
                  
                  -
                  
                    
                      φ
                      .
                    
                    4
                  
                
                )
              
            
            -
            
              
                L
                12
              
              ⁡
              
                (
                
                  
                    
                      φ
                      .
                    
                    2
                  
                  +
                  
                    
                      φ
                      .
                    
                    3
                  
                
                )
              
            
          
          =
          
            
              
                -
                
                  (
                  
                    
                      
                        L
                        1
                      
                      2
                    
                    +
                    
                      L
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        a
                      
                    
                  
                  )
                
              
              ⁢
              
                i
                b
              
            
            +
            
              φ
              2
            
            -
            
              φ
              1
            
            +
            
              2
              ⁢
              
                
              
              ⁢
              π
              ⁢
              
                
              
              ⁢
              
                x
                e
              
              ⁢
              
                a
                1
              
            
            +
            
              
                L
                12
              
              ⁡
              
                (
                
                  
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      3
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      2
                    
                  
                
                )
              
            
            +
            
              
                
                  L
                  1
                
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      4
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      5
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      6
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        6
        )
      
    
  




where L12=L1+L2a+L2b.

The RSJ model also provides a current-phase relation around the upper loop in the single diamond according to the equation φ1+φ3+L3i3=φ2, where L3=L3a+L3b. Substituting the Josephson junction relations from Eq. (3) and reorganizing yields the third of the six equations for the dynamics of the single diamond.

L3{dot over (φ)}3=φ2−φ1−φ3−L3ic3 sin φ3.  (7)

A current-phase relation around the bottom half (bi-SQUID 20b) of the single diamond cell 40 can be determined to be φ4+L5ai10=2πxea2+φ5+L5bi11+L1i7, where φe=2πa2xe and a2=L1+L5a+L5b. Substituting the current relations




  
    
      i
      7
    
    =
    
      
        
          
            i
            8
          
          -
          
            i
            10
          
        
        2
      
      +
      
        
          
            i
            11
          
          -
          
            i
            9
          
        
        2
      
    
  
  ,




i10=i4−i6, i11=i5+i6, i9=i2+i3 and i8=i1−i3 from Eq. (3) and then reorganizing gives





  
    
      
        
          
            
              (
              
                
                  L
                  
                    5
                    ⁢
                    
                      
                    
                    ⁢
                    a
                  
                
                +
                
                  
                    L
                    1
                  
                  2
                
              
              )
            
            ⁢
            
              i
              4
            
          
          -
          
            
              (
              
                
                  L
                  1
                
                +
                
                  L
                  
                    5
                    ⁢
                    
                      
                    
                    ⁢
                    a
                  
                
                +
                
                  L
                  
                    5
                    ⁢
                    
                      
                    
                    ⁢
                    b
                  
                
              
              )
            
            ⁢
            
              i
              6
            
          
        
        =
        
          
            φ
            5
          
          -
          
            φ
            4
          
          +
          
            2
            ⁢
            
              
            
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              2
            
          
          +
          
            
              (
              
                
                  L
                  
                    5
                    ⁢
                    
                      
                    
                    ⁢
                    b
                  
                
                +
                
                  
                    L
                    1
                  
                  2
                
              
              )
            
            ⁢
            
              i
              5
            
          
          +
          
            
              
                L
                1
              
              2
            
            ⁢
            
              i
              1
            
          
          -
          
            
              
                L
                1
              
              2
            
            ⁢
            
              i
              2
            
          
          -
          
            
              L
              1
            
            ⁢
            
              
                i
                3
              
              .
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




Substituting the current relation for the bias current i5=ib−i4 from Eq. (3) into Eq. (8) and then substituting the Josephson junction relations from Eq. (3) and reorganizing leads to the fourth governing equation





  
    
      
        
          
            
              
                L
                15
              
              ⁡
              
                (
                
                  
                    
                      φ
                      .
                    
                    4
                  
                  -
                  
                    
                      φ
                      .
                    
                    6
                  
                
                )
              
            
            +
            
              
                
                  L
                  1
                
                2
              
              ⁢
              
                (
                
                  
                    2
                    ⁢
                    
                      
                        φ
                        .
                      
                      3
                    
                  
                  +
                  
                    
                      φ
                      .
                    
                    2
                  
                  -
                  
                    
                      φ
                      .
                    
                    1
                  
                
                )
              
            
          
          =
          
            
              
                (
                
                  
                    L
                    
                      5
                      ⁢
                      
                        
                      
                      ⁢
                      b
                    
                  
                  +
                  
                    
                      L
                      1
                    
                    2
                  
                
                )
              
              ⁢
              
                i
                b
              
            
            +
            
              φ
              5
            
            -
            
              φ
              4
            
            +
            
              2
              ⁢
              
                
              
              ⁢
              π
              ⁢
              
                
              
              ⁢
              
                x
                e
              
              ⁢
              
                a
                2
              
            
            +
            
              
                L
                15
              
              ⁡
              
                (
                
                  
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      6
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      4
                    
                  
                
                )
              
            
            +
            
              
                
                  L
                  1
                
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      2
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      3
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  




where L15=L1+L5a+L5b.

To solve for the fifth of the six equations for the dynamics of the single diamond shape the current relations for the bias current i4=ib−i5 and Josephson junctions from Eq. (3) is substituted into Eq. (8) to become




  
    
      
        
          
            
              
                
                  L
                  1
                
                2
              
              ⁢
              
                (
                
                  
                    2
                    ⁢
                    
                      
                        φ
                        .
                      
                      3
                    
                  
                  +
                  
                    
                      φ
                      .
                    
                    2
                  
                  -
                  
                    
                      φ
                      .
                    
                    1
                  
                
                )
              
            
            -
            
              
                L
                15
              
              ⁡
              
                (
                
                  
                    
                      φ
                      .
                    
                    5
                  
                  +
                  
                    
                      φ
                      .
                    
                    6
                  
                
                )
              
            
          
          =
          
            
              
                -
                
                  (
                  
                    
                      L
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        b
                      
                    
                    +
                    
                      
                        L
                        1
                      
                      2
                    
                  
                  )
                
              
              ⁢
              
                i
                b
              
            
            +
            
              φ
              5
            
            -
            
              φ
              4
            
            +
            
              2
              ⁢
              π
              ⁢
              
                
              
              ⁢
              
                x
                e
              
              ⁢
              
                a
                2
              
            
            +
            
              
                L
                15
              
              ⁡
              
                (
                
                  
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      6
                    
                  
                  +
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      5
                    
                  
                
                )
              
            
            +
            
              
                
                  L
                  1
                
                2
              
              ⁢
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      1
                    
                  
                  -
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      2
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      i
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      3
                    
                  
                
                )
              
            
          
        
        ,
      
    
    
      
        (
        10
        )
      
    
  




where L15=L1+L5a+L5b.

For the sixth governing equations a current-phase relation can be found for the lower loop in the diamond

φ4+φ6+L6i6=φ5,

where L6=L6a+L6b. Substituting the Josephson junction relations from Eq. (3) into the relation and reorganizing yields

L6{dot over (φ)}6=φ5−φ4−φ6−L6ic3 sin φ3.  (11)

The equations from Eq. (5)-(7) and Eq. (9)-(11) can be combined to obtain the full system of equations that governs the phase dynamics of the diamond-shaped bi-SQUID:




  
    
      
        
          
            
              
                
                  L
                  12
                
                ⁡
                
                  (
                  
                    
                      
                        φ
                        .
                      
                      1
                    
                    -
                    
                      
                        φ
                        .
                      
                      3
                    
                  
                  )
                
              
              +
              
                
                  
                    L
                    1
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      2
                      ⁢
                      
                        
                          φ
                          .
                        
                        6
                      
                    
                    +
                    
                      
                        φ
                        .
                      
                      5
                    
                    -
                    
                      
                        φ
                        .
                      
                      4
                    
                  
                  )
                
              
            
            =
            
              
                
                  (
                  
                    
                      
                        L
                        1
                      
                      2
                    
                    +
                    
                      L
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        b
                      
                    
                  
                  )
                
                ⁢
                
                  i
                  b
                
              
              +
              
                φ
                2
              
              -
              
                φ
                1
              
              +
              
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  x
                  e
                
                ⁢
                
                  a
                  1
                
              
              +
              
                
                  L
                  12
                
                ⁡
                
                  (
                  
                    
                      
                        i
                        
                          c
                          ⁢
                          
                            
                          
                          ⁢
                          3
                        
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        3
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        1
                      
                    
                  
                  )
                
              
              +
              
                
                  
                    L
                    1
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        4
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        5
                      
                    
                    -
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        i
                        
                          c
                          ⁢
                          
                            
                          
                          ⁢
                          6
                        
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        6
                      
                    
                  
                  )
                
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                    L
                    1
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      2
                      ⁢
                      
                        
                          φ
                          .
                        
                        6
                      
                    
                    +
                    
                      
                        φ
                        .
                      
                      5
                    
                    -
                    
                      
                        φ
                        .
                      
                      4
                    
                  
                  )
                
              
              -
              
                
                  L
                  12
                
                ⁡
                
                  (
                  
                    
                      
                        φ
                        .
                      
                      2
                    
                    +
                    
                      
                        φ
                        .
                      
                      3
                    
                  
                  )
                
              
            
            =
            
              
                
                  -
                  
                    (
                    
                      
                        
                          L
                          1
                        
                        2
                      
                      +
                      
                        L
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          a
                        
                      
                    
                    )
                  
                
                ⁢
                
                  i
                  b
                
              
              +
              
                φ
                2
              
              -
              
                φ
                1
              
              +
              
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  x
                  e
                
                ⁢
                
                  a
                  1
                
              
              +
              
                
                  L
                  12
                
                ⁡
                
                  (
                  
                    
                      
                        i
                        
                          c
                          ⁢
                          
                            
                          
                          ⁢
                          3
                        
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        3
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        2
                      
                    
                  
                  )
                
              
              +
              
                
                  
                    L
                    1
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        4
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        5
                      
                    
                    -
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        i
                        
                          c
                          ⁢
                          
                            
                          
                          ⁢
                          6
                        
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        6
                      
                    
                  
                  )
                
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                L
                3
              
              ⁢
              
                
                  φ
                  .
                
                3
              
            
            =
            
              
                φ
                2
              
              -
              
                φ
                1
              
              -
              
                φ
                3
              
              -
              
                
                  L
                  3
                
                ⁢
                
                  i
                  
                    c
                    ⁢
                    
                      
                    
                    ⁢
                    3
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  3
                
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  L
                  15
                
                ⁡
                
                  (
                  
                    
                      
                        φ
                        .
                      
                      4
                    
                    -
                    
                      
                        φ
                        .
                      
                      6
                    
                  
                  )
                
              
              +
              
                
                  
                    L
                    1
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      2
                      ⁢
                      
                        
                          φ
                          .
                        
                        3
                      
                    
                    +
                    
                      
                        φ
                        .
                      
                      2
                    
                    -
                    
                      
                        φ
                        .
                      
                      1
                    
                  
                  )
                
              
            
            =
            
              
                
                  (
                  
                    
                      L
                      
                        5
                        ⁢
                        
                          
                        
                        ⁢
                        b
                      
                    
                    +
                    
                      
                        L
                        1
                      
                      2
                    
                  
                  )
                
                ⁢
                
                  i
                  b
                
              
              +
              
                φ
                5
              
              -
              
                φ
                4
              
              +
              
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  x
                  e
                
                ⁢
                
                  a
                  2
                
              
              +
              
                
                  L
                  15
                
                ⁡
                
                  (
                  
                    
                      
                        i
                        
                          c
                          ⁢
                          
                            
                          
                          ⁢
                          6
                        
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        6
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        4
                      
                    
                  
                  )
                
              
              +
              
                
                  
                    L
                    1
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        1
                      
                    
                    -
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        2
                      
                    
                    -
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        i
                        
                          c
                          ⁢
                          
                            
                          
                          ⁢
                          3
                        
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        3
                      
                    
                  
                  )
                
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                    
                      L
                      1
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        2
                        ⁢
                        
                          
                            φ
                            .
                          
                          3
                        
                      
                      +
                      
                        
                          φ
                          .
                        
                        2
                      
                      -
                      
                        
                          φ
                          .
                        
                        1
                      
                    
                    )
                  
                
                -
                
                  
                    L
                    15
                  
                  ⁡
                  
                    (
                    
                      
                        
                          φ
                          .
                        
                        5
                      
                      +
                      
                        
                          φ
                          .
                        
                        6
                      
                    
                    )
                  
                
              
              =
              
                
                  
                    -
                    
                      (
                      
                        
                          L
                          
                            5
                            ⁢
                            b
                          
                        
                        +
                        
                          
                            L
                            1
                          
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    i
                    b
                  
                
                +
                
                  φ
                  5
                
                -
                
                  φ
                  4
                
                +
                
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    x
                    e
                  
                  ⁢
                  
                    a
                    2
                  
                
                +
                
                  
                    L
                    15
                  
                  ⁡
                  
                    (
                    
                      
                        
                          i
                          
                            c
                            ⁢
                            
                              
                            
                            ⁢
                            6
                          
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          6
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          5
                        
                      
                    
                    )
                  
                
                +
                
                  
                    
                      L
                      1
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          1
                        
                      
                      -
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          2
                        
                      
                      -
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          i
                          
                            c
                            ⁢
                            
                              
                            
                            ⁢
                            3
                          
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          3
                        
                      
                    
                    )
                  
                
              
            
            ;
            and
          
          ,
          
            

          
          ⁢
          
            
              
                L
                6
              
              ⁢
              
                
                  φ
                  .
                
                6
              
            
            =
            
              
                φ
                5
              
              -
              
                φ
                4
              
              -
              
                φ
                6
              
              -
              
                
                  L
                  6
                
                ⁢
                
                  i
                  
                    c
                    ⁢
                    
                      
                    
                    ⁢
                    3
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  3
                
              
            
          
          ,
        
      
    
    
      
        (
        12
        )
      
    
  




where L12=L1+L2a+L2b, L3=L3a+L3b, L15=L1+L5a+L5b, L6=L6a+L6b and φi are the phases on each of the Josephson junctions, i=1, . . . ,6. The normalized critical current on the third junction is





  
    
      i
      
        c
        ⁢
        
          
        
        ⁢
        3
      
    
    =
    
      
        I
        
          c
          ⁢
          
            
          
          ⁢
          3
        
      
      
        I
        0
      
    
  
  ,




the normalized critical current of the sixth junction is





  
    
      i
      
        c
        ⁢
        
          
        
        ⁢
        6
      
    
    =
    
      
        I
        
          c
          ⁢
          
            
          
          ⁢
          6
        
      
      
        I
        0
      
    
  
  ,




Ic1=Ic2=Ic4=Ic5=I0, xe is the normalized external magnetic flux, a1=L1+L2a+L2b and a2=L1+L5a+L5b.

Once the equations above are solved for inductances, values can be substituted there for the inductance and the average voltage response for the diamond shaped bi-SQUID cell 40 of the present invention according to several embodiments is shown in FIG. 5. The response can be plottted for J=1.001, L1=0.54, L2a=L2b=L5a=L5b=0.24, L3a=L3b=L6a=L6b=0.3, and ic3=0.5 These values are all normalized values, without units. As shown in FIG. 5, this average voltage response retains its linearity, with proper choice of parameters, such as varying the inductances instead of the loop sizes the triangle-shaped bi-SQUIDs 20 in the cell. The actual size of the triangle could also be varied.
Referring now to FIG. 6, FIG. 6 is a microphotograph of the single diamond bi-SQUID of FIG. 4. FIG. 6 shows the implementation of the diamond-shaped dual bi-SQUID cell once fabricated. One such fabrication method that can be used is the HYPRES Nb fabrication process, as described in Niobium (Nb) process design rule, revision #24, Jan. 11, 2008, 11, available at http://www.hypres.com. Rshunt is the shunting resistance and Vc is the critical voltage across each of the JJ's. For the fabrication shown in FIG. 6, J1=J2=J3=0.25 mA, Rshunt=2.4 Ω, Vc=IcRsh=600 μV.
The cell 40 shown in FIG. 6 can be manufactured using all four Nb layers: a ground plane layer, two layers for junctions and inductors, and a top layer to implement a flux bias line 50 overlaying the bi-SQUID cells. The ground plane was used only under Nb layers forming bi-SQUID inductors and junctions in order to maintain their low specific inductance. However, ground plane can be partially removed from under inductor 46 to increase its inductance value, if desired. The ground plane can also be removed from the central area of the bi-SQUID loops (areas 52 in FIG. 6) to allow an external magnetic field to thread through the cell.
C. Implementation of 2D Diamond bi-SQUID SQIF Arrays.
1. Construction of 2D SQIF Array.
To construct the 2D SQIF array of the present invention according to several embodiments, and referring now to FIGS. 7-9, building blocks of dual bi-SQUID cells 40 like those shown in FIG. 6 can be used. The repeating pattern in the 2D array that is illustrated in FIGS. 7-9 can be a series of cells 40 having a diamond shape created by two bi-SQUIDs 20a, 20b, which can have the structure as recited above for cell 40 (See FIG. 4). An example of such as array is shown electrically and in top plan in FIG. 7. The phase equations that model the 2D diamond arrays can be derived in a similar way as the single diamond structure of bi-SQUIDs. But the above analysis in the previous section and Equations (5)-(7) and (9)-(11) only models the response for one cell 40. To model the effects for the SQIF array as a whole, the analysis can now be extended to the full 2D diamond array. But because of the tightly packed arrangement of diamond-shaped cells 40, each bi-SQUID shares its JJs and inductances with neighboring bi-SQUIDs. Stated differently, each bi-SQUID cell 40 affect adjacent bi-SQUID cells 40 as well of the bi-SQUIDs of other cells 40 in the array 70. Therefore to model the entire array, Kirchoff's law can be used to sum current thorugh each and every junciton for a network of cells 40 that share inductances, instead of a single cell. Also, the effects that each cell 40 has on every other cell in the 2D array, and the effects that occur at boundary conditions can be accounted for (again, using Kirchoff's law) in the methods of the present invention.
At the corners of the 2D SQIF array 70, each dual bi-SQUID cell 40 has a contact to three neighboring cells at cell junctions 80. FIGS. 8b, 8d, 8g and 8i illustrate this boundary condition at the respective upper left, upper right, lower left and lower right corners of the 2D SQIF array. FIGS. 8e and 8f illustrate the boundary conditions on the respective left side and right side of the 2D SQIF array. FIGS. 8c and 8h illustrate the respective top and bottom conditions on the 2D SQIF array. The array dc bias can be fed uniformly from the top bi-SQUID row (cells 401,1 through 401,N of the 2D SQIF array 70). Similarly, the array can be grounded to the bottom bi-SQUID row (cells 40M,1 through 40M,N or array 70 in FIG. 7). The inductively coupled flux bias line is overlaid on the top of the array forming loops for each column. The direction of the de flux bias control current is from top to bottom of array 70 in FIG. 7. The 2-D array design in FIGS. 7-8 avoids the use of long parasitic wires, as every component of the array is an essential element of a bi-SQUID, which results in an efficient use of the available area.
To account for the boundary conditions in an M×N 2D array, and referring again to FIG. 7, The number of diamonds cells 40 in the vertical direction can be 2M and the number of bi-SQUIDs in the horizontal direction can be 2N-1. This means that, for example a 10×8 array can have 20×15=300 bi-SQUIDs. The segmenting of the array in the manner allows for the smallest repeated pattern grouping, which can further result in the simplest set of modeling equations for computational purposes. The 2D array equations are derived in pieces that include the diamond structure and two transition areas, as shown by area 82 in FIG. 8a. This segmentation of the array is the smallest repeatable pattern.
Using the above methodology, the equation for the 2D array cells in the interior of the array, i.e. cells 40j,i where j=2 to M-1 and i=2 to N-1 as shown in FIG. 8a results in the following equations (13) through (20) in Appendix found at the end of this Specification. Next, the methodology was used for the top row of cells 40j,i for j=1 and i=2, . . . , N-1 (FIG. 8c) resulting in the development of equations (21) through (28) of the Appendix. Still further, the methodology was used to develop the equations (29) through (36), which model the bottom row of cells 40j,i for j=M and i=2, . . . ,N-1 (FIG. 8h). In similar fashion, the equations that model the left side cells 40j,i for j=2, . . . , M-1 and i=1 (Equations (37) through (44), FIG. 8e), the top left hand corner cell 401,1 (FIG. 8b and Equations (45) through (52)), the bottom left hand corner cell 40M,1 (FIG. 8g, Appendix Equations (53) through (60)) were developed. Finally, the equations for the rights side of array 70, cells 40j,i for j=2, . . . , M-1 and i=N of the array (FIG. 8f) were developed and listed as Equations (61) through (66) in the equations Appendix, and the equations for the top right hand corner cell 401,N (FIG. 8d, Equations (67) through (72)), and bottom right hand corner cell 40M,N (FIG. 8i, Equations (73) through (78)) were developed. The net result is a matrix of sixty-six equations, Equations (13) through (78), which can accurately the behavior of the 2D SQIF array.
As described above for a single cell 40, once Equations (13)-(78) are developed and solved for phases φ and phase rates (derivatives of φ) for each JJ in each cell 40. These derivatives are used to calculate the average voltage response of the bi-SQUID array, which is the output measured from the fabricated bi-SQUID chips Next, the cell can be manufactured with the cell inductances that achieve the desired phase and phase rate (and by extension, the desired voltage response). The inductances, critical currents, normal state resistance, bias currents and external field are known or controllable. One way this has been accomplished in the prior art was by varying the loop size of bi-SQUID 20. But for SQIFarray 70, uniform loop sizes are needed for ease of manufacture of diamon cells 40 and 2D SQIF array 70. So, in order to vary the inductances in 2D SQIF array 70 while the a uniform loop size (area of the cell 40), the inductances, critical currents, normal state resistance, bias currents and external field are known or controllable. The arrays can be constructed by varying the area of triangle array cells 20 and ground plane hole sizes in a manner that reults in a normal Gaussian distribution of cell inductances.
FIG. 9 shows layout implementation of a portion of the 80×15 2D array of FIG. 7, using diamond dual bi-SQUID cells. The total area of the 2D array of dual-bi-SQUID cells shown in for an 80×15 2D array can be 1623 μm2. The central area of each cell 40 contains ground plane openings to facilitate RF signal reception.
2. Experimental Investigation 2D SQIF Noise Properties
FIG. 10 illustrates the measured flux-to-voltage characteristics of a 15×80 2D SQIF array of cell 40 of FIGS. 7-9, with a 70% Gaussian spread (σ=70%) of inductance values. Like FIG. 3 for SQUIDs and bi-SQUIDs, FIG. 10 is a graph of the average voltage, <V>, of the 2D SQIF array at a point in xe, which is the mean value of the voltage over time. As shown in FIG. 10, the measured flux/voltage characteristic for σ=70% is 50 mV/div 0.5 mA/div, max voltage=295 mV, ΔV/ΔI (flux bias)  ̃735 V/A. FIGS. 11-13 show the corresponding flux noise measurement at the mid-point of the positive slope of the anti-peak (point 1002 of peak 1000 in FIG. 10). As can be seen from FIG. 11, the flux noise spectral density can approach 2×10−6Φ0/√{square root over (Hz)}, which is the expected value for this size array. FIGS. 12 and 13 show the corresponding energy characteristics of the (15×80) bi-SQUID SQIF array, which were calculated as follows:
Noise energy per unit bandwidth via flux noise in a SQUID:




  
    
      
        
          
            ɛ
            ⁡
            
              (
              f
              )
            
          
          =
          
            
              
                S
                Φ
              
              ⁡
              
                (
                f
                )
              
            
            
              2
              ⁢
              
                
              
              ⁢
              L
            
          
        
        ,
      
    
    
      
        (
        79
        )
      
    
  




where f is frequency, L−inductance of bi-SQUID calculated from the measured seperately ΔIc modulations of the curve defined as





  
    L
    =
    
      
        Φ
        0
      
      
        Δ
        ⁢
        
          
        
        ⁢
        
          I
          c
        
      
    
  
  ;




and,

Noise temperature:




  
    
      
        
          T
          N
        
        =
        
          
            π
            ⁢
            
              
            
            ⁢
            f
            ⁢
            
              
            
            ⁢
            
              ɛ
              ⁡
              
                (
                f
                )
              
            
          
          
            k
            B
          
        
      
    
    
      
        (
        80
        )
      
    
  




where kB is Boltzmann's constant.

As shown by FIGS. 10-13, a well-defined zero-field anti-peak for a 2D SQIF array of cells 40 can be maintained if the mutual inductances between cells and the boundary conditions are accounted for. Stated differently, more bi-SQUID's can be packed into a corresponding areas using the methods of the present invention without sacrificing the overall linearity anti-peak response of the 2D SQIF array, when considered as a whole.
3. 2D SQIF Array Antenna Sensitivity Analysis
The sensitivities illustrated in FIGS. 10-13 were based on a SQIF-based antenna, by assuming that the area for a single diamond shaped (double) bi-SQUID cell 40 is 1.62×10−9 m2. Using that as the effective area for cell 40, multiplying by 15×80/2 to account for the number of individual bi-SQUIDs in the array, and taking the flux noise from FIG. 8(b) as 2×10−6Φ0/√{square root over (Hz)}, gives a field sensitivity of 4.25 fT/√{square root over (Hz)}. Assuming a scaling as a function of √{square root over (N)}, the field noise for a 1000×2000 array would be 0.104 fT/√{square root over (Hz)} at 100 kHz. Approximating the physical dimension of the diamond cell 40 as 71 μm×71 μm means that the diamond occupies an area of 5×10−9 m2 (consistent with the effective area of the diamond shaped bi-SQUID). Thus a 1000×2000 array would occupy an area of 50 cm2, since each diamond contains two bi-SQUIDs. This corresponds to a square with 7.1 cm on a side. A corresponding linear (1D) array of the prior art, which would essentially be 2,000,000 bi-SQUIDs that are meandered in series, would require a much larger area, an area of 10,562 cm2.
Referring now to FIG. 14, a block diagram 100 that illustrates steps that can be taken to practice the methods of the present invention according to several embodiments is shown. As shown method 100 can include the initial step 102 of providing a plurality of bi-SQUID's 20. The bi-SQUIDs 20 can have the structure as described above. The methods can further include the step 104 of merging pairs of bi-SQUID's 20 into diamond-shaped cells 40 and connecting the ells so that each cell 40 shares at least three cell junctions 80 (step 106). Next, the methods according to several embodiments can include the step 108 of modeling the phases of the 2D SQIF array. Step 108 can further be accomplished by accounting for mutual inductances between cells 40 (step 110 in FIG. 14) and by using the equations (13) through (78) of the Appendix to model the boundarys conditions of array 70 (step 112 in FIG. 14). As shown by step 114 in FIG. 14, the methods according to several embodiments can further include the step of manufacturing cells 40 with inductances that achieve the phases and phase behavior resulting form the accomplishment of step 108, in order the achieve the anti-peak response shown in FIG. 10. For embodiments the loop size of cell 40 is uniform, one way to do this can be to vary the ground plane hole size, as discussed above.
The use of the terms “a” and “an” and “the” and similar references in the context of describing the invention (especially in the context of the following claims) is to be construed to cover both the singular and the plural, unless otherwise indicated herein or clearly contradicted by context. The terms “comprising,” “having,” “including,” and “containing” are to be construed as open-ended terms (i.e., meaning “including, but not limited to,”) unless otherwise noted. Recitation of ranges of values herein are merely intended to serve as a shorthand method of referring individually to each separate value falling within the range, unless otherwise indicated herein, and each separate value is incorporated into the specification as if it were individually recited herein. All methods described herein can be performed in any suitable order unless otherwise indicated herein or otherwise clearly contradicted by context. The use of any and all examples, or exemplary language (e.g., “such as”) provided herein, is intended merely to better illuminate the invention and does not pose a limitation on the scope of the invention unless otherwise claimed. No language in the specification should be construed as indicating any non-claimed element as essential to the practice of the invention.
Preferred embodiments of this invention are described herein, including the best mode known to the inventors for carrying out the invention. Variations of those preferred embodiments may become apparent to those of ordinary skill in the art upon reading the foregoing description. The inventors expect skilled artisans to employ such variations as appropriate, and the inventors intend for the invention to be practiced otherwise than as specifically described herein. Accordingly, this invention includes all modifications and equivalents of the subject matter recited in the claims appended hereto as permitted by applicable law. Moreover, any combination of the above-described elements in all possible variations thereof is encompassed by the invention unless otherwise indicated herein or otherwise clearly contradicted by context.
APPENDIX
2D Diamond Array Governing Equations
For the model equations below, the variables φm,i,j, m=1, . . . ,8, i=1, . . . , N, j=1, . . . , M are the phases on each of the Josephson junctions (m=1, . . . ,8). The dots denote the time differentiation with normalized time τ=Ωct, where t is time and




  
    ω
    c
  
  =
  
    
      2
      ⁢
      
        eI
        c
      
      ⁢
      
        R
        N
      
    
      




is the normalized time constant. The parameter RN in Ωc is the normal state resistance of the Josephson junctions, Ic is the critical current of the Josephson junctions, e is the charge of an electron, and h is the reduced Planck constant.





  
    i
    b
  
  =
  
    
      I
      b
    
    
      I
      c
    
  




is the normalized bias current, where Ib is the bias current.





  
    Φ
    0
  
  ≡
  
    h
    
      2
      ⁢
      e
    
  
  ≈
  
    2.07
    ×
    
      10
      
        -
        15
      
    
  




tesla meter squared is the flux quantum, where h is Plank's constant and 2e is the charge on the Cooper pair. The normalized critical current on the third junction is





  
    
      i
      
        
          c
          ⁢
          
            
          
          ⁢
          3
        
        ,
        i
        ,
        j
      
    
    =
    
      
        I
        
          
            c
            ⁢
            
              
            
            ⁢
            3
          
          ,
          i
          ,
          j
        
      
      
        I
        c
      
    
  
  ,




the normalized critical current of the sixth junction is





  
    
      i
      
        
          c
          ⁢
          
            
          
          ⁢
          6
        
        ,
        i
        ,
        j
      
    
    =
    
      
        I
        
          
            c
            ⁢
            
              
            
            ⁢
            6
          
          ,
          i
          ,
          j
        
      
      
        I
        c
      
    
  
  ,




the normalized critical current of the seventh junction is





  
    
      i
      
        
          c
          ⁢
          
            
          
          ⁢
          7
        
        ,
        i
        ,
        j
      
    
    =
    
      
        I
        
          
            c
            ⁢
            
              
            
            ⁢
            7
          
          ,
          i
          ,
          j
        
      
      
        I
        c
      
    
  
  ,




the normalized critical current of the eighth junction is





  
    
      i
      
        
          c
          ⁢
          
            
          
          ⁢
          8
        
        ,
        i
        ,
        j
      
    
    =
    
      
        I
        
          
            c
            ⁢
            
              
            
            ⁢
            8
          
          ,
          i
          ,
          j
        
      
      
        I
        c
      
    
  
  ,




Ic1,i,j=Ic2,i,j=Ic4,i,j=Ic5,i,j=I0,





  
    x
    
      e
      ,
      i
      ,
      j
    
  
  =
  
    
      B
      e
    
    
      Φ
      0
    
  




is the normalized external magnetic flux per unit area and an,i,j, n=1, . . . ,4 is the bi-SQUID area. The approximate assumptions that a1,i,j=L1,i,j+L2a,i,j+L2b,i,j, a2=L1,i,j+L5a,i,j+L5b,i,j, a3,i,j=L4,i,j+L2a,i+1,j+L2b,i,j, a4=L4,i,j+1+L5a,i+1,j+L5b,i,j are used.

Using the assumptions and variables able, the equations for the array 70 interior cells, i.e, Cells 40j,i for j=2, . . . ,M-1 and i=2, . . . ,N-1 become (8 equations):




  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    i
                    ,
                    j
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    j
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    2
                    ,
                    i
                    ,
                    j
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    j
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
          +
          
            
              L
              
                1
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      j
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    j
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
        
        =
        
          
            φ
            
              2
              ,
              i
              ,
              j
            
          
          -
          
            φ
            
              1
              ,
              i
              ,
              j
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                i
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      1
                      ,
                      i
                      ,
                      j
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      
                        i
                        -
                        1
                      
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      
                        i
                        -
                        1
                      
                      ,
                      j
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      j
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      2
                      ,
                      i
                      ,
                      j
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      j
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      i
                      ,
                      j
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                1
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      j
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          13
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    2
                    ,
                    i
                    ,
                    j
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    j
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    j
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    i
                    ,
                    j
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    j
                  
                
              
              )
            
          
          -
          
            
              L
              
                4
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      
                        j
                        -
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      
                        j
                        -
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      j
                    
                  
                
              
              )
            
          
        
        =
        
          
            φ
            
              1
              ,
              
                i
                +
                1
              
              ,
              j
            
          
          -
          
            φ
            
              2
              ,
              i
              ,
              j
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                3
                ,
                i
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      2
                      ,
                      i
                      ,
                      j
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      j
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      i
                      ,
                      j
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                j
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      1
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  
                    sin
                    
                      7
                      ,
                      i
                      ,
                      j
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                4
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        
                          j
                          -
                          1
                        
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        
                          j
                          -
                          1
                        
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          14
          )
        
      
    
  




(L3a,i,j+L3b,i,j){dot over (φ)}3,i,j=φ2,i,j−φ1,i,j−φ3,i,j−(L3a,i,j+L3b,i,j)ic3,i,j sin φ3,i,j  Equation (15)





  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    i
                    ,
                    j
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    j
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    j
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    j
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    i
                    ,
                    j
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
          -
          
            
              L
              
                1
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      j
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    j
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
        
        =
        
          
            φ
            
              5
              ,
              i
              ,
              j
            
          
          -
          
            φ
            
              4
              ,
              i
              ,
              j
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                i
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      4
                      ,
                      i
                      ,
                      j
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      j
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      
                        i
                        -
                        1
                      
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      
                        i
                        -
                        1
                      
                      ,
                      j
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      5
                      ,
                      i
                      ,
                      j
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      
                        8.
                        ⁢
                        i
                      
                      ,
                      j
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      j
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                1
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      
                        
                      
                      ⁢
                      i
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        
                          k
                          .
                          j
                        
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      j
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          16
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    i
                    ,
                    j
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    j
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    j
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    j
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
          +
          
            
              L
              
                4
                ,
                i
                ,
                
                  j
                  +
                  1
                
              
            
            (
            
              
            
            ⁢
            
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  i
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    k
                    ,
                    j
                  
                
              
              +
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  i
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    k
                    ,
                    j
                  
                
              
              -
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  i
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    k
                    ,
                    
                      j
                      +
                      1
                    
                  
                
              
              -
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  i
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    2
                    ,
                    k
                    ,
                    
                      j
                      +
                      1
                    
                  
                
              
            
            )
          
        
        =
        
          
            φ
            
              4
              ,
              
                i
                +
                1
              
              ,
              j
            
          
          -
          
            φ
            
              5
              ,
              i
              ,
              j
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                4
                ,
                i
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      5
                      ,
                      i
                      ,
                      j
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      i
                      ,
                      j
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      j
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                j
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      4
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      j
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      j
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      i
                      ,
                      j
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                4
                ,
                i
                ,
                
                  j
                  +
                  1
                
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        
                          j
                          +
                          1
                        
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        
                          j
                          +
                          1
                        
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          17
          )
        
      
    
  




(L6a,i,j+L6b,i,j){dot over (φ)}6,i,j=φ4,i,j−φ5,i,j−φ6,i,j−(L6a,i,j+L6b,i,j)ic3,i,j sin φ3,i,j  Equation (18)

(L7a,i,j+L7b,i,j){dot over (φ)}7,i,j+L2a,i+1,j({dot over (φ)}1,i+1,j+{dot over (φ)}7,i,j−{dot over (φ)}3,i+1,j)−L2b,i,j({dot over (φ)}2,i,j+{dot over (φ)}3,i,j−{dot over (φ)}7,i,j)=−φ7,i,j−(L7a,i,j+L7b,i,j)ic7,i,j sin φ7,i,j−L2a,i+1,j(sin φ1,i+1,j+ic7,i,j sin φ7,i,j−ic3,i+1,j sin φ3,i+1,j)+L2b,i,j(sin φ2,i,j+ic3,i,j sin φ3,i,j−ic7,i,j sin φ7,i,j)  Equation (19)

(L8a,i,j+L8b,i,j){dot over (φ)}8,i,j+L5b,i,j({dot over (φ)}5,i,j+{dot over (φ)}8,i,j−{dot over (φ)}6,i,j)−L5b,i+1,j({dot over (φ)}4,i+1,j+{dot over (φ)}6,i+1,j−{dot over (φ)}8,i,j)=−φ8,i,j−(L8a,i,j+L8b,i,j)ic8,i,j sin φ7,i,j−L5b,i,j(sin φ5,i,j+ic8,i,j sin φ8,i,j−ic6,i,j sin φ6,i,j)+L5a,i+1,j(sin φ4,i+1,j+ic6,i+1,j sin φ6,i+1,j−ic8,i,j sin φ8,i,j)  Equation (20)

For the cells in the top row of array 70, i.e. for cells 40j,i for j=1 and i=2, . . . , N-1 (8 equations):




  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    i
                    ,
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    1
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    1
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    2
                    ,
                    i
                    ,
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    1
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    i
                    ,
                    1
                  
                
              
              )
            
          
          +
          
            
              L
              
                1
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      1
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    1
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    1
                  
                
              
              )
            
          
        
        =
        
          
            φ
            
              2
              ,
              i
              ,
              1
            
          
          -
          
            φ
            
              1
              ,
              i
              ,
              1
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                i
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      1
                      ,
                      i
                      ,
                      1
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      
                        i
                        -
                        1
                      
                      ,
                      i
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      
                        i
                        -
                        1
                      
                      ,
                      1
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      1
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      2
                      ,
                      i
                      ,
                      1
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      1
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      i
                      ,
                      1
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                1
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ⁢
                        
                          
                        
                        ,
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ⁢
                      
                        
                      
                      ,
                      1
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          21
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    2
                    ,
                    i
                    ,
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    1
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    i
                    ,
                    1
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    i
                    ,
                    1
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    1
                  
                
              
              )
            
          
          +
          
            
              L
              
                4
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      1
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      1
                    
                  
                
              
              )
            
          
        
        =
        
          
            i
            ×
            
              L
              
                4
                ,
                i
                ,
                1
              
            
            ⁢
            
              i
              b
            
          
          +
          
            φ
            
              1
              ,
              
                i
                +
                1
              
              ,
              1
            
          
          -
          
            φ
            
              2
              ,
              i
              ,
              1
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                
                  3
                  ⁢
                  f
                
                ,
                i
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      2
                      ,
                      i
                      ,
                      1
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      1
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      i
                      ,
                      1
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                1
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      1
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      1
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      i
                      ,
                      1
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      1
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                4
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          22
          )
        
      
    
  




(L3a,i,1+L3b,i,1){dot over (φ)}3,i,1=φ2,i,1−φ1,i,1−φ3,i,1−(L3a,i,1+L3b,i,1)ic3,i,1 sin φ3,i,1  Equation (23)





  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    i
                    ,
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    1
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    
                      1
                      -
                      i
                    
                    ,
                    1
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    i
                    ,
                    1
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    j
                  
                
              
              )
            
          
          -
          
            
              L
              
                1
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      1
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    1
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    1
                  
                
              
              )
            
          
        
        =
        
          
            φ
            
              5
              ,
              i
              ,
              1
            
          
          -
          
            φ
            
              4
              ,
              i
              ,
              1
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                i
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      4
                      ,
                      i
                      ,
                      1
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      1
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      
                        i
                        -
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      
                        i
                        -
                        1
                      
                      ,
                      1
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      5
                      ,
                      i
                      ,
                      1
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      i
                      ,
                      1
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      1
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                1
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      
                        
                      
                      ⁢
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      1
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          24
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    i
                    ,
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    1
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    1
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    1
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    1
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    1
                  
                
              
              )
            
          
          +
          
            
              L
              
                4
                ,
                i
                ,
                2
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      1
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      2
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      2
                    
                  
                
              
              )
            
          
        
        =
        
          
            φ
            
              4
              ,
              
                i
                +
                1
              
              ,
              1
            
          
          -
          
            φ
            
              5
              ,
              i
              ,
              1
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                4
                ,
                i
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                1
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      5
                      ,
                      i
                      ,
                      1
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      i
                      ,
                      1
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      1
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                1
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      4
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      1
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      1
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      1
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      i
                      ,
                      1
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                4
                ,
                i
                ,
                2
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        2
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        2
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          25
          )
        
      
    
  




(L6a,i,1+L6b,i,1){dot over (φ)}6,i,1=φ4,i,1−φ5,i,1−φ6,i,1−(L6a,i,1+L6b,i,1)ic3,i,1 sin φ3,i,1  Equation (26)

(L7a,i,1+L7b,i,1){dot over (φ)}7,i,1+L2a,i+1,1({dot over (φ)}1,i+1,1+{dot over (φ)}7,i,1−{dot over (φ)}3,i+1,1)−L2b,i,1({dot over (φ)}2,i,1+{dot over (φ)}3,i,1−{dot over (φ)}7,i,1)=−φ7,i,1−(L7a,i,1+L7b,i,1)ic7,i,1 sin φ7,i,1−L2a,i+1,1(sin φ1,i+1,1+ic7,i,1 sin φ7,i,1−ic3,i+1,1 sin φ3,i+1,1)+L2b,i,1(sin φ2,i,1+ic3,i,1 sin φ3,i,1−ic7,i,1 sin φ7,i,1)  Equation (27)

(L8a,i,1+L8b,i,1){dot over (φ)}8,i,1+L5b,i,1({dot over (φ)}5,i,1+{dot over (φ)}8,i,1−{dot over (φ)}6,i,1)−L5a,i+1,1({dot over (φ)}4,i+1,1+{dot over (φ)}6,i+1,1−{dot over (φ)}8,i,1)=−φ8,i,1−(L8a,i,1+L8b,i,1)ic8,i,1 sin φ7,i,1−L5b,i,1(sin φ5,i,1+ic8,i,1 sin φ8,i,1−ic6,i,1 sin φ6,i,1)+L5a,i+1,1(sin φ4,i+1,1+ic6,i+1,1 sin φ6,i+1,1−ic8,i,1 sin φ8,i,1)  Equation (28)

In similar fashion, the phases for the bottom row cells in array 70, i.e., cells 40j,i for j=M and i=2, . . . , N-1 (8 equations):




  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    
                      φ
                      .
                    
                    ⁢
                    
                      
                    
                  
                  
                    1
                    ,
                    i
                    ,
                    M
                  
                
                +
                
                  
                    
                      φ
                      .
                    
                    ⁢
                    
                      
                    
                  
                  
                    7
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    M
                  
                
                -
                
                  
                    
                      φ
                      .
                    
                    ⁢
                    
                      
                    
                  
                  
                    3
                    ,
                    i
                    ,
                    M
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    
                      φ
                      .
                    
                    ⁢
                    
                      
                    
                  
                  
                    2
                    ,
                    i
                    ,
                    M
                  
                
                +
                
                  
                    
                      φ
                      .
                    
                    ⁢
                    
                      
                    
                  
                  
                    3
                    ,
                    i
                    ,
                    M
                  
                
                -
                
                  
                    
                      φ
                      .
                    
                    ⁢
                    
                      
                    
                  
                  
                    7
                    ,
                    i
                    ,
                    M
                  
                
              
              )
            
          
          +
          
            
              L
              
                1
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      M
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    M
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      
                        φ
                        .
                      
                      ⁢
                      
                        
                      
                    
                    
                      4
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        φ
                        .
                      
                      ⁢
                      
                        
                      
                    
                    
                      5
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    
                      φ
                      .
                    
                    ⁢
                    
                      
                    
                  
                  
                    6
                    ,
                    i
                    ,
                    M
                  
                
              
              )
            
          
        
        =
        
          
            φ
            
              2
              ,
              i
              ,
              M
            
          
          -
          
            φ
            
              1
              ,
              i
              ,
              M
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                i
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                M
              
            
            (
            
              
            
            ⁢
            
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    1
                    ,
                    i
                    ,
                    M
                  
                
              
              +
              
                
                  i
                  
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      7
                    
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    M
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    7
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    M
                  
                
              
              -
              
                
                  i
                  
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      3
                    
                    ,
                    i
                    ,
                    M
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    3
                    ,
                    i
                    ,
                    M
                  
                
              
            
            )
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      2
                      ,
                      i
                      ,
                      M
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      M
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      i
                      ,
                      M
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                1
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      M
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          29
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    2
                    ,
                    i
                    ,
                    M
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    M
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    i
                    ,
                    M
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    M
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    7
                    ,
                    i
                    ,
                    M
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    M
                  
                
              
              )
            
          
          -
          
            
              L
              
                4
                ,
                i
                ,
                M
              
            
            (
            
              
            
            ⁢
            
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  i
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    k
                    ,
                    
                      M
                      -
                      1
                    
                  
                
              
              +
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  i
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    k
                    ,
                    
                      M
                      -
                      1
                    
                  
                
              
              -
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  i
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    k
                    ,
                    M
                  
                
              
              -
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  i
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    2
                    ,
                    k
                    ,
                    M
                  
                
              
            
            )
          
        
        =
        
          
            φ
            
              1
              ,
              
                i
                +
                1
              
              ,
              M
            
          
          -
          
            φ
            
              2
              ,
              i
              ,
              M
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                3
                ,
                i
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  b
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      2
                      ,
                      i
                      ,
                      M
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      M
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        7
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      7
                      ,
                      i
                      ,
                      M
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                
                  M
                  i
                
              
            
            (
            
              
            
            ⁢
            
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    1
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    M
                  
                
              
              +
              
                
                  i
                  
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      7
                    
                    ,
                    i
                    ,
                    M
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    7
                    ,
                    i
                    ,
                    M
                  
                
              
              -
              
                
                  i
                  
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      3
                    
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    M
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    3
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    M
                  
                
              
            
            )
          
          +
          
            
              L
              
                4
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        
                          M
                          -
                          1
                        
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        
                          M
                          -
                          1
                        
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        i
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          30
          )
        
      
    
  




(L3a,i,M+L3b,i,M){dot over (φ)}3,i,M=φ2,i,M−φ1,i,M−φ3,i,M−(L3a,i,M+L3b,i,M)ic3,i,M sin φ3,i,M  Equation (31)





  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    i
                    ,
                    M
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    M
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    M
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    M
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    i
                    ,
                    M
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    M
                  
                
              
              )
            
          
          -
          
            
              L
              
                1
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      M
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    3
                    ,
                    i
                    ,
                    M
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    M
                  
                
              
              )
            
          
        
        =
        
          
            φ
            
              5
              ,
              i
              ,
              M
            
          
          -
          
            φ
            
              4
              ,
              i
              ,
              M
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                i
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                i
                ,
                M
              
            
            (
            
              
            
            ⁢
            
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    4
                    ,
                    i
                    ,
                    M
                  
                
              
              +
              
                
                  i
                  
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      6
                    
                    ,
                    i
                    ,
                    M
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    6
                    ,
                    i
                    ,
                    M
                  
                
              
              -
              
                
                  i
                  
                    
                      c
                      ⁢
                      
                        
                      
                      ⁢
                      8
                    
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    M
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                
                  φ
                  
                    8
                    ,
                    
                      i
                      -
                      1
                    
                    ,
                    M
                  
                
              
            
            )
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      5
                      ,
                      i
                      ,
                      M
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      i
                      ,
                      M
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      M
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                1
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        3
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      3
                      ,
                      i
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      i
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      M
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          32
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    i
                    ,
                    M
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    M
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    i
                    ,
                    M
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    M
                  
                
                +
                
                  
                    φ
                    .
                  
                  
                    6
                    ,
                    
                      i
                      +
                      1
                    
                    ,
                    M
                  
                
                -
                
                  
                    φ
                    .
                  
                  
                    8
                    ,
                    i
                    ,
                    M
                  
                
              
              )
            
          
          +
          
            
              L
              
                4
                ,
                i
                ,
                
                  M
                  +
                  1
                
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      M
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      M
                    
                  
                
              
              )
            
          
        
        =
        
          
            i
            ×
            
              i
              b
            
            ⁢
            
              L
              
                4
                ,
                i
                ,
                
                  M
                  +
                  1
                
              
            
          
          +
          
            φ
            
              4
              ,
              
                i
                +
                1
              
              ,
              M
            
          
          -
          
            φ
            
              5
              ,
              i
              ,
              M
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                4
                ,
                i
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  b
                
                ,
                i
                ,
                M
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      5
                      ,
                      i
                      ,
                      M
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      i
                      ,
                      M
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      i
                      ,
                      M
                    
                  
                
              
              )
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                
                  i
                  +
                  1
                
                ,
                M
              
            
            ⁡
            
              (
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      4
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      M
                    
                  
                
                +
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        6
                      
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      6
                      ,
                      
                        i
                        +
                        1
                      
                      ,
                      M
                    
                  
                
                -
                
                  
                    i
                    
                      
                        c
                        ⁢
                        
                          
                        
                        ⁢
                        8
                      
                      ,
                      i
                      ,
                      M
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    
                      8
                      ,
                      i
                      ,
                      M
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                4
                ,
                i
                ,
                
                  M
                  +
                  1
                
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    i
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          33
          )
        
      
    
  




(L6a,i,M+L6b,i,M){dot over (φ)}6,i,M=φ4,i,M−φ5,i,M−φ6,i,M−(L6a,i,M+L6b,i,M)ic3,i,M sin φ3,i,M  Equation (34)

(L7a,i,M+L7b,i,M){dot over (φ)}7,i,M+L2a,i+1,M({dot over (φ)}1,i+1,M+{dot over (φ)}7,i,M−{dot over (φ)}3,i+1,M)−L2b,i,M({dot over (φ)}2,i,M+{dot over (φ)}3,i,M−{dot over (φ)}7,i,M)=−φ7,i,M−(L7a,i,M+L7b,i,M)ic7,i,M sin φ7,i,M−L2a,i+1,M(sin φ1,i+1,M+ic7,i,M sin φ7,i,M−ic3,i+1,M sin φ3,i+1,M)+L2b,i,M(sin φ2,i,M+ic3,i,M sin φ3,i,M−ic7,i,M sin φ7,i,M)  Equation (35)

(L8a,i,M+L8b,i,M){dot over (φ)}8,i,M+L5b,i,M({dot over (φ)}5,i,M+{dot over (φ)}8,i,M−{dot over (φ)}6,i,M)−L5a,i+1,M({dot over (φ)}4,i+1,M+{dot over (φ)}6,i+1,M−{dot over (φ)}8,i,M)=−φ8,i,M−(L8a,i,M+L8b,i,M)ic8,i,M sin φ7,i,M−L5b,i,M(sin φ5,i,M+ic8,i,M sin φ8,i,M−ic6,i,M sin φ6,i,M)+L5a,i+1,M(sin φ4,i+1,M+ic6,i+1,M sin φ6,i+1,M−ic8,i,M sin φ8,i,M)  Equation (36)

For the cells 40 on the left side cells of 2D SQIF array 70, i.e. for cells 40j,i for j=2, . . . , M-1 and i=1 (8 equations):

L2a,1,j({dot over (φ)}1,1,j−{dot over (φ)}3,1,j)−L2b,1,j({dot over (φ)}2,1,j+{dot over (φ)}3,1,j−{dot over (φ)}7,1,j)+L1,1,j({dot over (φ)}1,1,j−{dot over (φ)}3,1,j−{dot over (φ)}4,1,j−{dot over (φ)}6,1,j)=φ2,1,j−φ1,1,j+2πxea1,1,j−L2a,1,j(sin φ1,1,j−ic3,1,j sin φ3,1,j)+L2b,1,j(sin φ2,1,j+ic3,1,j sin φ3,1,j−ic7,1,j sin φ7,1,j)−L1,1,j(sin φ1,1,j−ic3,1,j sin φ3,1,j−sin φ4,1,j−ic6,1,j sin φ6,1,j)  Equation (37)

L2b,1,j({dot over (φ)}2,1,j+{dot over (φ)}3,1,j−{dot over (φ)}7,1,j)−L2a,2,j({dot over (φ)}1,2,j+{dot over (φ)}7,1,j−{dot over (φ)}3,2,j)−L4,1,j({dot over (φ)}4,1,j−1+{dot over (φ)}5,1,j−1−{dot over (φ)}1,1,j−{dot over (φ)}2,1,j)=φ1,2,j−φ2,1,j+2πxea3,1,j−L2b,1,j(sin φ2,1,j+ic3,1,j sin φ3,1,j−ic7,1,j sin φ7,1,j)+L2a,2,j(sin φ1,2,j+ic7,1,j sin φ7,1,j−ic3,2,j sin φ3,2,j)+L4,1,j(sin φ4,1,j−1+sin φ5,1,j−1−sin φ1,1,j−sin φ2,1,j)  Equation (38)

(L3a,1,j+L3b,1,j){dot over (φ)}3,i,j=φ2,1,j−φ1,1,j−φ3,1,j−(L3a,1,j+L3b,1,j)ic3,1,j sin φ3,1,j  Equation (39)

L5a,1,j({dot over (φ)}4,1,j+{dot over (φ)}6,1,j)−L5b,1,j({dot over (φ)}8,1,j+{dot over (φ)}5,1,j−{dot over (φ)}6,1,j)−L1,1,j({dot over (φ)}1,1,j−{dot over (φ)}3,1,j−{dot over (φ)}4,1,j−{dot over (φ)}6,1,j)=φ5,1,j−φ4,1,j+2πxea2,1,j−L2a,1,j(sin φ4,1,j+ic6,1,j sin φ6,1,j)+L5b,1,j(sin φ5,1,j+ic8,1,j sin φ8,1,j−ic6,1,j sin φ6,1,j)+L1,1,j(sin φ1,1,j−ic3,1,j+sin φ3,1,j−sin φ4,1,j−ic6,1,j sin φ6,1,j)  Equation (40)

L5b,1,j({dot over (φ)}5,1,j+{dot over (φ)}8,1,j−{dot over (φ)}6,1,j)−L5a,2,j({dot over (φ)}4,2,j+{dot over (φ)}6,2,j−{dot over (φ)}8,1,j)+L4,1,j+1({dot over (φ)}4,1,j+{dot over (φ)}5,1,j−{dot over (φ)}1,1,j+1−{dot over (φ)}2,1,j+1)=φ4,2,j−φ5,1,j+2πxea4,1,j−L5b,1,j(sin φ5,1,j+ic8,1,j sin φ8,1,j−ic6,1,j sin φ6,1,j)+L5a,2,j(sin φ4,2,j+ic6,2,j sin φ6,2,j−ic8,1,j sin φ8,1,j)−L4,1,j+1(sin φ4,1,j+sin φ5,1,j−sin φ1,1,j+1−sin φ2,1,j+1)  Equation (41)

(L6a,1,j+L6b,1,j){dot over (φ)}6,1,j=φ4,1,j−φ5,1,j−φ6,1,j−(L6a,1,j+L6b,1,j)ic3,1,j sin φ3,1,j  Equation (42)

(L7a,1,j+L7b,1,j){dot over (φ)}7,1,j+L2a,2,j({dot over (φ)}1,2,j+{dot over (φ)}7,1,j−{dot over (φ)}3,2,j)−L2b,1,j({dot over (φ)}2,1,j+{dot over (φ)}3,1,j−{dot over (φ)}7,1,j)=−φ7,1,j−(L7a,1,j+L7b,1,j)ic7,1,j sin φ7,1,j−L2a,2,j(sin φ1,2,j+ic7,1,j sin φ7,1,j−ic3,2,j sin φ3,2,j)+L2b,1,j(sin φ2,1,j+ic3,1,j sin φ3,1,j−ic7,1,j sin φ7,1,j)  Equation (43)

(L8a,1,j+L8b,1,j){dot over (φ)}8,1,j+L5b,1,j({dot over (φ)}5,1,j+{dot over (φ)}8,1,j−{dot over (φ)}6,1,j)−L5a,2,j({dot over (φ)}4,2,j+{dot over (φ)}6,2,j−{dot over (φ)}8,1,j)=−φ8,1,j−(L8a,1,j+L8b,1,j)ic8,1,j sin φ7,1,j−L5b,1,j(sin φ5,1,j+ic8,1,j sin φ8,1,j−ic6,1,j sin φ6,1,j)+L5a,2,j(sin φ4,2,j+ic6,2,j sin φ6,2,j−ic8,1,j sin φ8,1,j)  Equation (44)

To model the boundary conditions at the corners of the 2D SQIF array 70, and startin with top left-hand corner cell 40j,i for j=1 and i=1 (8 equations):

L2a,1,1({dot over (φ)}1,1,1−{dot over (φ)}3,1,1)−L2b,1,1({dot over (φ)}2,1,1+{dot over (φ)}3,1,1−{dot over (φ)}7,1,1)+L1,1,1({dot over (φ)}1,1,1−{dot over (φ)}3,1,1−{dot over (φ)}4,1,1−{dot over (φ)}6,1,1)=φ2,1,1−φ1,1,1+2πxea1,1,1−L2a,1,1(sin φ1,1,1−ic3,1,1 sin φ3,1,1)+L2b,1,1(sin φ2,1,1+ic3,1,1 sin φ3,1,1−ic7,1,1 sin φ7,1,1)−L1,1,1(sin φ1,1,1−ic3,1,1 sin φ3,1,1−sin φ4,1,1−ic6,1,1 sin φ6,1,1)  Equation (45)

L2b,1,1({dot over (φ)}2,1,1+{dot over (φ)}3,1,1−{dot over (φ)}7,1,1)−L2a,2,1({dot over (φ)}1,2,1+{dot over (φ)}7,1,1−{dot over (φ)}3,2,1)+L4,1,1({dot over (φ)}1,1,1+{dot over (φ)}2,1,1)=L4,1,1ib+φ1,2,1−φ2,1,1+2πxea3,1,1−L2b,1,1(sin φ2,1,1+ic3,1,1 sin φ3,1,1−ic7,1,1 sin φ7,1,1)+L2a,2,1(sin φ1,2,1+ic7,1,1 sin φ7,1,1−ic3,2,1 sin φ3,2,1)−L4,1,1(sin φ1,1,1+sin φ2,1,1)  Equation (46)

(L3a,1,1+L3b,1,1){dot over (φ)}3,1,1=φ2,1,1−φ1,1,−φ3,1,1−(L3a,1,1+L3b,1,1)ic3,1,1 sin φ3,1,1  Equation (47)

L5a,1,1({dot over (φ)}4,1,1+{dot over (φ)}6,1,1)−L5b,1,1({dot over (φ)}8,1,1+{dot over (φ)}5,1,1−{dot over (φ)}6,1,1)−L1,1,1({dot over (φ)}1,1,1−{dot over (φ)}3,1,1−{dot over (φ)}4,1,1−{dot over (φ)}6,1,1)=φ5,1,1−φ4,1,1+2πxea2,1,1−L2a,1,1(sin φ4,1,1+ic6,1,1 sin φ6,1,1)+L5b,1,1(sin φ5,1,1+ic8,1,1 sin φ8,1,1−ic6,1,1 sin φ6,1,1)+L1,1,1(sin φ1,1,1−ic3,1,1 sin φ3,1,1−sin φ4,1,1−ic6,1,1 sin φ6,1,)  Equation (48)

L5b,1,1({dot over (φ)}5,1,1+{dot over (φ)}8,1,1−{dot over (φ)}6,1,1)−L5a,2,1({dot over (φ)}4,2,1+{dot over (φ)}6,2,1−{dot over (φ)}8,1,1)+L4,1,2({dot over (φ)}4,1,1+{dot over (φ)}5,1,1−{dot over (φ)}1,1,2−{dot over (φ)}2,1,2)=φ4,2,1−φ5,1,1+2πxea4,1,1−L5b,1,1(sin φ5,1,1+ic8,1,1 sin φ8,1,1−ic6,1,1 sin φ6,1,1)+L5a,2,1(sin φ4,2,1+ic6,2,1 sin φ6,2,1−ic8,1,1 sin φ8,1,1)−L4,1,2(sin φ4,1,1+sin φ5,1,1−sin φ1,1,2−sin φ2,1,2)  Equation (49)

(L6a,1,1+L6b,1,1){dot over (φ)}6,1,1=φ4,1,1−φ5,1,1−φ6,1,1−(L6a,1,1+L6b,1,1)ic3,1,1 sin φ3,1,1  Equation (50)

(L7a,1,1+L7b,1,1){dot over (φ)}7,1,1+L2a,2,1({dot over (φ)}1,2,1+{dot over (φ)}7,1,1−{dot over (φ)}3,2,1)−L2b,1,1({dot over (φ)}2,1,1+{dot over (φ)}3,1,1−{dot over (φ)}7,1,1)=−φ7,1,1−(L7a,1,1+L7b,1,1)ic7,1,1 sin φ7,1,1−L2a,2,1(sin φ1,2,1+ic7,1,1 sin φ7,1,1−ic3,2,1 sin φ3,2,1)+L2b,1,1(sin φ2,1,1+ic3,1,1 sin φ3,1,1−ic7,1,1 sin φ7,1,1)  Equation (51)

(L8a,1,1+L8b,1,1){dot over (φ)}8,1,1+L5b,1,1({dot over (φ)}5,1,1+{dot over (φ)}8,1,1−{dot over (φ)}6,1,1)−L5a,2,1({dot over (φ)}4,2,1+{dot over (φ)}6,2,1−{dot over (φ)}8,1,1)=−φ8,1,1−(L8a,1,1+L8b,1,1)ic8,1,1 sin φ7,1,1−L5b,1,1(sin φ5,1,1+ic8,1,1 sin φ8,1,1−ic6,1,1 sin φ6,1,1)+L5a,2,1(sin φ4,2,1+ic6,2,1 sin φ6,2,1−ic8,1,1 sin φ8,1,1)  Equation (52)

For the bottom left-hand corner cell 40j,i for j=M and i=1:

L2a,1,M({dot over (φ)}1,1,M−{dot over (φ)}3,1,M)−L2b,1,M({dot over (φ)}2,1,M+{dot over (φ)}3,1,M−{dot over (φ)}7,1,M)+L1,1,M({dot over (φ)}1,1,M−{dot over (φ)}3,1,M−{dot over (φ)}4,1,M−{dot over (φ)}6,1,M)=φ2,1,M−φ1,1,M+2πxea1,1,M−L2a,1,M(sin φ1,1,M−ic3,1,M sin φ3,1,M)+L2b,1,M(sin φ2,1,M+ic3,1,M sin φ3,1,M−ic7,1,M sin φ7,1,M)−L1,1,M(sin φ1,1,M−ic3,1,M sin φ3,1,M−sin φ4,1,M−ic6,1,M sin φ6,1,M)  Equation (53)

L2b,1,M({dot over (φ)}2,1,M+{dot over (φ)}3,1,M−{dot over (φ)}7,1,M)−L2a,2,M({dot over (φ)}1,2,M+{dot over (φ)}7,1,M−{dot over (φ)}3,2,M)−L4,1,M+1({dot over (φ)}4,1,M−1+{dot over (φ)}5,1,M−1−{dot over (φ)}1,1,M−{dot over (φ)}2,1,M)=φ1,2,M−φ2,1,M+2πxea3,1,M−L2b,1,M(sin φ2,1,M+ic3,1,M sin φ3,1,M−ic7,1,M sin φ7,1,M)+L2a,2,M(sin φ1,2,M+ic7,1,M sin φ7,1,M−ic3,2,M sin φ3,2,M)+L4,1,M(sin φ4,1,M−1+sin φ5,1,M−1−sin φ1,1,M−sin φ2,1,M)  Equation (54)

(L3a,1,M+L3b,1,M){dot over (φ)}3,1,M=φ2,1,M−φ1,1,M−φ3,1,M−(L3a,1,M+L3b,1,M)ic3,1,M sin φ3,1,M  Equation (55)

L5a,1,M({dot over (φ)}4,1,M+{dot over (φ)}6,1,M)−L5b,1,M({dot over (φ)}8,1,M+{dot over (φ)}5,1,M−{dot over (φ)}6,1,M)−L1,1,M({dot over (φ)}1,1,M−{dot over (φ)}3,1,M−{dot over (φ)}4,1,M−{dot over (φ)}6,1,M)=φ5,1,M−φ4,1,M+2πxea2,1,M−L2a,1,M(sin φ4,1,M+ic6,1,M sin φ8,1,M)+L5b,1,M(sin φ5,1,M+ic8,1,M sin φ8,1,M−ic6,1,M sin φ6,1,M)+L1,1,M(sin φ1,1,M−ic3,1,M sin φ3,1,M−sin φ4,1,M−i c6,1,M sin φ6,1,M)  Equation (56)

L5b,1,M({dot over (φ)}5,1,M+{dot over (φ)}8,1,M−{dot over (φ)}6,1,M)−L5a,2,M({dot over (φ)}4,2,M+{dot over (φ)}6,2,M−{dot over (φ)}8,1,M)+L4,1,M+1({dot over (φ)}4,1,M+{dot over (φ)}5,1,M)=φ4,2,M−φ5,1,M+2πxea4,1,M−L5b,1,M(sin φ5,1,M+ic8,1,M sin φ8,1,M−ic6,1,M sin φ6,1,M)+L5a,2,M(sin φ4,2,M+ic6,2,M sin φ6,2,M−ic8,1,M sin φ8,1,M)−L4,1,M+1(sin φ4,1,M+sin φ5,1,M)  Equation (57)

(L6a,1,M+L6b,1,M){dot over (φ)}6,1,M=φ4,1,M−φ5,1,M−φ6,1,M−(L6a,1,M+L6b,1,M)ic3,1,M sin φ3,1,M  Equation (58)

(L7a,1,M+L7b,1,M){dot over (φ)}7,1,M+L2a,2,M({dot over (φ)}1,2,M+{dot over (φ)}7,1,M−{dot over (φ)}3,2,M)−L2b,1,M({dot over (φ)}2,1,M+{dot over (φ)}3,1,M−{dot over (φ)}7,1,M)=−φ7,1,M−(L7a,1,M+L7b,1,M)ic7,1,M sin φ7,1,M−L2a,2,M(sin φ1,2,M+ic7,1,M sin φ7,1,M−ic3,2,M sin φ3,2,M)+L2b,1,M(sin φ2,1,M+ic3,1,M sin φ3,1,M−ic7,1,M sin φ7,1,M)  Equation (59)

(L8a,1,M+L8b,1,M){dot over (φ)}8,1,M+L5b,1,M({dot over (φ)}5,1,M+{dot over (φ)}8,1,M−{dot over (φ)}6,1,M)−L5a,2,M({dot over (φ)}4,2,M+{dot over (φ)}6,2,M−{dot over (φ)}8,1,M)=−φ8,1,M−(L8a,1,M+L8b,1,M)ic8,1,M sin φ7,1,M−L5b,1,M(sin φ5,1,M+ic8,1,M sin φ8,1,M−ic6,1,M sin φ6,1,M)+L5a,2,M(sin φ4,2,M+ic6,2,M sin φ6,2,M−ic8,1,M sin φ8,1,M)  Equation (60)

For the right side cells of 2D SQIF array 70, i.e. for cells 40j,i for j=2, . . . , M-1 and i=N (6 equations):




  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                1
                ,
                N
                ,
                j
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      
                        j
                        -
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      
                        j
                        -
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      j
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
          
        
        =
        
          
            φ
            
              2
              ,
              N
              ,
              j
            
          
          -
          
            φ
            
              1
              ,
              N
              ,
              j
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                1
                ,
                N
                ,
                j
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        
                          j
                          -
                          1
                        
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        
                          j
                          -
                          1
                        
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  7
                
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          61
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      
                        j
                        -
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      
                        j
                        -
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      j
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                j
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
          
        
        =
        
          
            φ
            
              2
              ,
              N
              ,
              j
            
          
          -
          
            φ
            
              1
              ,
              N
              ,
              j
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        
                          j
                          -
                          1
                        
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        
                          j
                          -
                          1
                        
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                j
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                
                  
                
                ⁢
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  7
                
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          62
          )
        
      
    
  




(L3a,N,j+L3b,N,j){dot over (φ)}3,N,j=φ2,N,j−φ1,N,j−φ3,N,j−(L3a,N,j+L3b,N,j)ic3,N,j sin φ3,N,j  Equation (63)





  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                4
                ,
                N
                ,
                j
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      j
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      j
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
          
        
        =
        
          
            φ
            
              5
              ,
              N
              ,
              j
            
          
          -
          
            φ
            
              4
              ,
              N
              ,
              j
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                4
                ,
                N
                ,
                j
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  8
                
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          64
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      j
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      j
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      j
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                j
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              
                φ
                .
              
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
          
        
        =
        
          
            φ
            
              5
              ,
              N
              ,
              j
            
          
          -
          
            φ
            
              4
              ,
              N
              ,
              j
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        j
                      
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                j
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      
                        
                      
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    j
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    j
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                j
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                j
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                j
              
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                j
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  8
                
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                j
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          65
          )
        
      
    
  




(L6a,N,j+L6b,N,j){dot over (φ)}6,N,j=φ4,N,j−φ5,N,j−φ6,N,j−(L6a,N,j+L6b,N,j)ic3,N,j sin φ3,N,j  Equation (66)

To model the top right-hand corner cell 40 of array 70, cell 40j,i for j=1 and i=N (6 equations):




  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                1
                ,
                N
                ,
                1
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      1
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      1
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
          
        
        =
        
          
            N
            ×
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              i
              b
            
          
          +
          
            φ
            
              2
              ,
              N
              ,
              1
            
          
          -
          
            φ
            
              1
              ,
              N
              ,
              1
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                1
                ,
                N
                ,
                1
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                1
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  7
                
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          67
          )
        
      
    
  








  
    
      
        
          
            -
            
              
                L
                
                  
                    2
                    ⁢
                    a
                  
                  ,
                  N
                  ,
                  1
                
              
              ⁡
              
                (
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        φ
                        .
                      
                      
                        1
                        ,
                        k
                        ,
                        1
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          k
                          =
                          1
                        
                        N
                      
                      ⁢
                      
                        
                          φ
                          .
                        
                        
                          2
                          ,
                          k
                          ,
                          1
                        
                      
                    
                  
                
                )
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                1
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
          
        
        =
        
          
            
              -
              N
            
            ×
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              b
            
          
          +
          
            φ
            
              2
              ,
              N
              ,
              1
            
          
          -
          
            φ
            
              1
              ,
              N
              ,
              1
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                1
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                i
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  7
                
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          68
          )
        
      
    
  




(L3a,N,1+L3b,N,1){dot over (φ)}3,N,1=φ2,N,1−φ1,N,1−φ3,N,1−(L3a,N,1+L3b,N,1)ic3,N,1 sin φ3,N,1  Equation (69)





  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                4
                ,
                N
                ,
                1
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      1
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      1
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
          
        
        =
        
          
            φ
            
              5
              ,
              N
              ,
              1
            
          
          -
          
            φ
            
              4
              ,
              N
              ,
              1
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                4
                ,
                N
                ,
                1
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  8
                
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          70
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      1
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      1
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      1
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                1
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              
                φ
                .
              
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
          
        
        =
        
          
            φ
            
              5
              ,
              N
              ,
              1
            
          
          -
          
            φ
            
              4
              ,
              N
              ,
              1
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        1
                      
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                1
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    1
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    1
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                1
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                1
              
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                1
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  8
                
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                1
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          71
          )
        
      
    
  




(L6a,N,1+L6b,N,1){dot over (φ)}6,N,1=φ4,N,1−φ5,N,1−φ6,N,1−(L6a,N,1+L6b,N,1)ic3,N,1 sin φ3,N,1  Equation (72)

In similar fashion of the above, in order to model the bottom right-hand corner cell of 2D SQIF array 70, cell 40j,i for j=M and i=N (6 equations):




  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                1
                ,
                N
                ,
                M
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      
                        M
                        -
                        1
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      
                        M
                        -
                        1
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      M
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
          
        
        =
        
          
            φ
            
              2
              ,
              N
              ,
              M
            
          
          -
          
            φ
            
              1
              ,
              N
              ,
              M
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                N
                ,
                M
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        
                          M
                          -
                          1
                        
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        
                          M
                          -
                          1
                        
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
              
              )
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                1
                ,
                N
                ,
                M
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  7
                
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          73
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            (
            
              
            
            ⁢
            
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    4
                    ,
                    k
                    ,
                    
                      M
                      -
                      1
                    
                  
                
              
              +
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    5
                    ,
                    k
                    ,
                    
                      M
                      -
                      1
                    
                  
                
              
              -
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    1
                    ,
                    k
                    ,
                    M
                  
                
              
              -
              
                
                  ∑
                  
                    k
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    φ
                    .
                  
                  
                    2
                    ,
                    k
                    ,
                    M
                  
                
              
            
            )
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                M
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
          
        
        =
        
          
            φ
            
              2
              ,
              N
              ,
              M
            
          
          -
          
            φ
            
              1
              ,
              N
              ,
              M
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                1
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        
                          M
                          -
                          1
                        
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        
                          M
                          -
                          1
                        
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                M
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      2
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                
                  
                
                ⁢
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  2
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  7
                
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                7
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          74
          )
        
      
    
  




(L3a,N,M+L3b,N,M){dot over (φ)}3,N,M=φ2,N,M−φ1,N,M−φ3,N,M−(L3a,N,M+L3b,N,M)ic3,N,M sin φ3,N,M  Equation (75)





  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                4
                ,
                N
                ,
                M
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      M
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      M
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
          
        
        =
        
          
            φ
            
              5
              ,
              N
              ,
              M
            
          
          -
          
            φ
            
              4
              ,
              N
              ,
              M
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                4
                ,
                N
                ,
                M
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  8
                
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          76
          )
        
      
    
  








  
    
      
        
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                
                  
                
                ⁢
                N
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      1
                      ,
                      k
                      ,
                      M
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      2
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      4
                      ,
                      k
                      ,
                      M
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    
                      φ
                      .
                    
                    
                      5
                      ,
                      k
                      ,
                      M
                    
                  
                
              
              )
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                5
                ,
                N
                ,
                M
              
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              
                φ
                .
              
              
                6
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                2
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                3
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              
                φ
                .
              
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
          
        
        =
        
          
            φ
            
              5
              ,
              N
              ,
              M
            
          
          -
          
            φ
            
              4
              ,
              N
              ,
              M
            
          
          +
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              x
              e
            
            ⁢
            
              a
              
                2
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁡
            
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        1
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        2
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      N
                      -
                      1
                    
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        4
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
                -
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    N
                  
                  ⁢
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      φ
                      
                        5
                        ,
                        k
                        ,
                        M
                      
                    
                  
                
              
              )
            
          
          +
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                5
                ,
                N
                ,
                M
              
            
          
          -
          
            
              (
              
                
                  L
                  
                    1
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      a
                    
                    ,
                    N
                    ,
                    M
                  
                
                +
                
                  L
                  
                    
                      5
                      ⁢
                      b
                    
                    ,
                    N
                    ,
                    M
                  
                
              
              )
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  6
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                6
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                2
                ,
                N
                ,
                M
              
            
          
          -
          
            
              L
              
                1
                ,
                N
                ,
                M
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                ,
                N
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                3
                ,
                N
                ,
                M
              
            
          
          +
          
            
              L
              
                
                  5
                  ⁢
                  a
                
                ,
                N
                ,
                M
              
            
            ⁢
            
              i
              
                
                  c
                  ⁢
                  
                    
                  
                  ⁢
                  8
                
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              φ
              
                8
                ,
                
                  N
                  -
                  1
                
                ,
                M
              
            
          
        
      
    
    
      
        Equation
        ⁢
        
          
        
        ⁢
        
          (
          77
          )
        
      
    
  




(L6a,N,M+L6b,N,M){dot over (φ)}6,N,M=φ4,N,M−φ5,N,M−φ6,N,M−(L6a,N,M+L6b,N,M)ic3,N,M sin φ3,N,M  Equation (78)

## Claims

1. A two-dimensional Superconducting Quantum Interference Filter (SQIF) array comprising:
at least three cells arranged in two dimensions;
each of said cells having at least two bi-Superconducting Quantum Interference Devices (bi-SQUIDs);
said at least two bi-SQUIDs being merged together so that said bi-SQUIDs share at least two bi-SQUID junctions and at least one inductance, wherein said at least one inductance is varied to maintain an over-all anti-peak response by adjusting a hole size of a ground plane under said inductance.

2. The SQIF array of claim 1 wherein said cells are diamond-shaped or hexagonal-shaped when viewed in top plan.

3. The SQIF array of claim 1 wherein each of said cells has at least four cell junctions, and further wherein each of said cells shares at least three cell junctions with an adjacent cell.

4. The SQIF array of claim 1 wherein each said bi-SQUID has a loop size, wherein said loop sizes are uniform, and wherein said bi-SQUIDs have non-uniform inductances, and further wherein said non-uniform inductances are modeled to maintain said overall anti-peak response for said SQIF array that is linear.

5. A method for establishing a two-dimensional Superconducting Quantum Interference Filter (SQIF) array, comprising the steps of:
A) providing a plurality of bi-Superconducting Quantum Interference Devices (bi-SQUIDs);
B) merging at least two bi-SQUIDs to establish a plurality of array cells; said step B) being accomplished so that said at least two bi-SQUIDs share at least two bi-SQUID junctions and at least one inductance, and so that said step B) establishes at least four cell junctions, wherein said at least one inductance is varied to maintain an overall anti-peak response by adjusting a hole size of a ground plane under said inductance; and,
C) connecting said array cells in a two-dimensional manner so that each array cell shares at least three cell junctions with another of said array cells.

6. The method of claim 5, wherein said array cells are diamond shaped or hexagonal shaped when viewed in top plan.

7. The method of claim 5, wherein each said bi-SQUID has a loop size, wherein said loop sizes are uniform, and wherein said step B) is accomplished using bi-SQUIDs having non-uniform inductances, and further comprising the step of:
D) modeling said non-uniform inductances to maintain said overall anti-peak response for said SQIF array that is linear.

8. The method of claim 7, wherein said non-uniform inductances have a distribution σ of at least 30 percent.