Abstract:
A new surface impedance model for extraction in lossy two-dimensional (2D) interconnects of rectangular cross-section is presented. The model is derived directly from the volumetric electric field integral equation (EFiE) under the approximation of the unknown volumetric current density as a product of the exponential factor describing the skin-effect and the unknown surface current density on the conductor&#39;s periphery. By proper accounting for the coupling between the boundary elements situated on the top and bottom surfaces of conductor with the elements located on the side-walls, the model maintains accuracy from DC to multi-GHz frequencies as well as for conductors with both large and small thickness/width ratios. A generalization of the full-periphery surface impedance model to the three-dimensional electric field integral equation is also described.

Description:
[0001]    This application claims the benefit under 35 U.S.C. 119(e) of U.S. provisional application Ser. No. 61/041,262, filed Apr. 1, 2008. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present invention relates to a method of modelling surface impedance of a conductor having a current flow, and furthermore relates to a resulting surface impedance model and its implementation using a suitable method of moment discretization scheme. The surface impedance model is derived directly from the volumetric electric field integral equation under the approximation of the unknown volumetric current density as a product of the exponential factor describing the skin-effect and the unknown surface current density on the conductor&#39;s periphery. 
       BACKGROUND 
       [0003]    Rigorous and efficient electromagnetic modeling of interconnects can be achieved through incorporation of appropriate surface impedance models into the boundary-element discretization techniques such as MoM. To be useful the model must be accurate in the broad range of frequencies spanning from DC to tens of GHz, applicable to the conductor cross-sections with both small and large thickness-to-width ratios, computationally efficient, and also easy to retrofit into existing MoM solvers. 
         [0004]    A variety of surface impedance models have been proposed in the past as described in the following documents: 
         [0005]    1. J. D. Morsey, et. al., “Finite-Thickness Conductor Models for Full-Wave Analysis of Interconnects With a Fast Integral Equation Method,”  IEEE Trans. on Advanced Packaging , vol. 27, no. I, pp. 24-33, February 2004. 
         [0006]    2. J. Rautio, et al., “Microstrip Conductor Loss Models for Electromagnetic Analysis,”  IEEE Trans. on Microwave Theory Tech ., vol. 51, no. 3, pp. 915-921, March 2003. 
         [0007]    3. F. Ling, et. al., “Large-Scale Broad-Band Parasitic Extraction for Fast Layout Verification of 3-D RF and Mixed-Signal On-Chip Structures,”  IEEE Trans. Microwave Theory Tech ., vol. 53, no. 1, pp. 264-273, January 2005. 
         [0008]    4. A. W. Glisson, “Electromagnetic scattering by arbitrarily shaped surfaces with impedance boundary conditions,”  Radio Science, vol.  27, no. 6, pp. 935-943, November-December 1992, 
         [0009]    The above mentioned requirements have not been satisfactorily met thus far by the above noted documents and this consequently motivates the on-going quest for more efficient and accurate models. 
         [0010]    Other prior art references relevant to the present invention include the following. 
         [0011]    5. C. R. Paul,  Analysis of Mulliconduclor Transmission Lines , Ch. 3, John Wiley &amp; Sons, Inc., Toronto, CA, 1994. 
         [0012]    6. D. DeZutter, et al., “Skin Effect Modeling Based on a Differential Surface Admittance Operator,”  IEEE Trans. Microwave Theory Tech ., vol. 53, no. 8, pp. 2526-2538, August. 2005. 
         [0013]    7. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shapes,”  IEEE Trans. Antennas Propag., Vol.  30, pp. 409-418, May 1982. 
         [0014]    8. K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part I: Theory,”  IEEE Trans. Antennas Propag ., Vol. 38, pp. 335-344, March 1990. 
         [0015]    Any documents referred to in the accompanying specification are hereby incorporated by reference. 
       SUMMARY OF THE INVENTION 
       [0016]    In this document a surface impedance model is derived as a result of volumetric electric field integral equation (EFIE) reduction to the surface EFIE, via approximation of the unknown cross-sectional volumetric current as a product of the exponential factor describing the skin-effect and the surface current density on the periphery of the conductor. Thus, the derived surface EFIE allows for both the flexibility in current variation on the conductor&#39;s surface, which is essential for adequate capturing of the proximity effects, and the proper exponential attenuation of the current inside of the conductor, which is critical for accurate skin-effect description. The numerically extracted per-unit-length (p.u.l) resistance and inductance matrices demonstrate the model to be accurate in a wide range of frequencies, as well as for both large and small thickness-to-width cross-sectional aspect ratios. 
         [0017]    According to one aspect of the invention there is provided a method of modelling surface impedance of a conductor having a current flow, the method comprising: 
         [0018]    formulating a volumetric electric field integral equation with respect to an unknown volumetric current density of the current flow in the conductor; 
         [0019]    representing the unknown volumetric current density in the volumetric electric field integral equation as a product of a current density of a peripheral surface of the conductor and an exponential factor describing a cross-sectional distribution of the current according to skin effect; 
         [0020]    adopting an approximation of Green&#39;s function across a cross-section of the conductor; and 
         [0021]    reducing the volumetric electric field integral equation including the unknown volumetric current density representation to a surface integral equation using the approximation of Green&#39;s function. 
         [0022]    The method preferably includes adopting an approximation of Green&#39;s function by fixing Green&#39;s function across the cross-section of the conductor. 
         [0023]    The method is suitable for conductors having either a rectangular cross-section or a polygonal, non-rectangular cross-section. 
         [0024]    The volumetric electric field integral equation is preferably formulated with respect to the unknown volumetric current density j z  in a cross-section of the conductor due to a vector of excitation V p.u.l.  by enforcing Ohm&#39;s law E z (ρ)=σ −1 j z (ρ) inside the conductor (ρ∈S) as follows: 
         [0000]    
       
         
           
             
               
                 
                   σ 
                   
                     - 
                     1 
                   
                 
                  
                 
                   
                     j 
                     z 
                   
                    
                   
                     ( 
                     ρ 
                     ) 
                   
                 
               
               + 
               
                 
                   ωμ 
                   0 
                 
                  
                 
                   
                     ∫ 
                     S 
                   
                    
                   
                     
                       G 
                        
                       
                         ( 
                         
                           ρ 
                            
                           
                             ρ 
                             ′ 
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         j 
                         z 
                       
                        
                       
                         ( 
                         
                           ρ 
                           ′ 
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         S 
                         ′ 
                       
                     
                   
                 
               
             
             = 
             
               - 
               
                 
                   
                     V 
                     
                       p 
                       . 
                       u 
                       . 
                       l 
                       . 
                     
                   
                    
                   
                     ( 
                     ρ 
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
         [0025]    For a conductor which extends in a z-direction and has a rectangular cross-section, the method preferably includes approximating the unknown volumetric current density, represented as j z , across the conductor according to skin-effect of a plane-wave incident on a conducting plane with infinite extension as follows: 
         [0000]    
       
         
           
             
               
                 j 
                 z 
               
                
               
                 ( 
                 ρ 
                 ) 
               
             
             ≅ 
             
               
                 
                    
                    
                   
                       
                   
                    
                   
                     
                       k 
                       σ 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               J 
                               z 
                               t 
                             
                              
                             
                               ( 
                               y 
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   k 
                                   σ 
                                 
                                  
                                 
                                   ( 
                                   
                                     X 
                                     - 
                                     x 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               J 
                               z 
                               b 
                             
                              
                             
                               ( 
                               y 
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 σ 
                               
                                
                               x 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   1 
                   - 
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         k 
                         σ 
                       
                        
                       X 
                     
                   
                 
               
               + 
               
                 
                    
                    
                   
                       
                   
                    
                   
                     
                       k 
                       σ 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               J 
                               z 
                               l 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 σ 
                               
                                
                               y 
                             
                           
                         
                         + 
                         
                           
                             
                               J 
                               z 
                               r 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   k 
                                   σ 
                                 
                                  
                                 
                                   ( 
                                   
                                     Y 
                                     - 
                                     y 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   1 
                   - 
                   
                      
                     
                       
                         - 
                          
                       
                        
                       
                           
                       
                        
                       
                         k 
                         σ 
                       
                        
                       Y 
                     
                   
                 
               
             
           
         
       
     
         [0026]    where J z   t (y), J z   b (y), J z   l (x), J z   r (x), are unknown surface current densities at points of radius-vector ρ projections onto respective top, bottom, left, and right walls of the conductor. 
         [0027]    Reducing the volumetric electric field integral equation to a surface electric field integral equation with respect to the current density of the peripheral surface of the conductor is preferably accomplished by restricting an observation point ρ to a periphery of the conductor as follows: 
         [0000]    
       
         
           
             
               
                 
                   Z 
                   σ 
                 
                  
                 
                   { 
                   
                     
                       J 
                       z 
                     
                      
                     
                       ( 
                       ρ 
                       ) 
                     
                   
                   } 
                 
               
               + 
               
                 
                   ωμ 
                   0 
                 
                  
                 
                   
                     ∫ 
                     C 
                   
                    
                   
                     
                       G 
                        
                       
                         ( 
                         
                           ρ 
                            
                           
                             ρ 
                             ′ 
                           
                         
                         ) 
                       
                     
                      
                     
                       
                         J 
                         z 
                       
                        
                       
                         ( 
                         
                           ρ 
                           ′ 
                         
                         ) 
                       
                     
                      
                     
                        
                       
                         c 
                         ′ 
                       
                     
                   
                 
               
             
             = 
             
               - 
               
                 
                   V 
                   
                     p 
                     . 
                     u 
                     . 
                     l 
                     . 
                   
                 
                  
                 
                   ( 
                   ρ 
                   ) 
                 
               
             
           
         
       
     
         [0028]    The method preferably further includes combining the unknown surface current densities over the top, bottom, left and right sides and evaluating integrals thereof over a conductor thickness T and conductor width W as follows: 
         [0000]    
       
         
           
             
               
                 ∫ 
                 V 
               
                
               
                 
                   
                     
                       
                         G 
                         _ 
                       
                       A 
                     
                      
                     
                       ( 
                       
                         r 
                         , 
                         
                           r 
                           ′ 
                         
                       
                       ) 
                     
                   
                   · 
                   
                     j 
                      
                     
                       ( 
                       
                         r 
                         ′ 
                       
                       ) 
                     
                   
                 
                  
                 
                    
                   
                     v 
                     ′ 
                   
                 
               
             
             ≅ 
             
               
                 ∫ 
                 S 
               
                
               
                 
                   
                     
                       
                         G 
                         _ 
                       
                       A 
                     
                      
                     
                       ( 
                       
                         r 
                         , 
                         
                           r 
                           ′ 
                         
                       
                       ) 
                     
                   
                   · 
                   
                     J 
                      
                     
                       ( 
                       
                         r 
                         ′ 
                       
                       ) 
                     
                   
                 
                  
                 
                   
                      
                     
                       s 
                       ′ 
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0029]    The volumetric electric field integral equation may be formulated with respect to the unknown volumetric current density of the current flow in the conductor for current flows along more than one axis for three dimensional current flows. 
         [0030]    The unknown volumetric current density may be related to unknown surface current densities for each surface of the conductor in which each unknown surface current density is represented as a multidimensional vector. 
         [0031]    The method may further include reducing the volumetric electric field integral equation to the surface integral equation in which a tangential electrical field at any point on a surface of the conductor is related to a surface current density at corresponding points on all segments of a cross-sectional periphery of the conductor and/or the tangential electrical field at any point on a surface of the conductor is related to a tangential magnetic field at corresponding points on all segments of a cross-sectional periphery of the conductor. 
         [0032]    According to a second aspect of the present invention there is provided a surface impedance model for a conductor having a current flow, the model comprising: 
         [0033]    a surface electric field integral equation resulting from a reduction of a volumetric electric field integral equation in which an unknown cross-sectional volumetric current of the volumetric electric field integral equation is approximated as a product of an exponential factor describing a cross-sectional distribution of the current according to skin effect and a current density on a peripheral surface of the conductor. 
         [0034]    According to a further aspect of the present invention there is provided a method of implementing a surface impedance model of a conductor comprising a surface electric field integral equation including a surface impedance term and an integral term resulting from a reduction of a volumetric electric field integral equation in which an unknown cross-sectional volumetric current of the volumetric electric field integral equation is approximated as a product of an exponential factor describing a cross-sectional distribution of the current according to skin effect and a current density on a peripheral surface of the conductor, the method including: 
         [0035]    discretizing the surface electric field integral equation of a periphery of the conductor into a discrete form comprising a sum of a sparse matrix corresponding to the surface impedance term of the surface electric field integral equation and a dense matrix corresponding to the integral term of the surface electric field integral equation; and 
         [0036]    identifying for each discrete element of the discrete form, other discrete elements which are related via the surface impedance model. 
         [0037]    Preferably the method comprises using the surface impedance model in conjunction with a suitable method of moment discretization scheme. 
         [0038]    The dense matrix may be represented as follows: 
         [0000]        Z   mn   A   =iωμ   0 ∫ ΔC     n     G (ρ m |ρ′) dc′.  
 
         [0039]    According to the present invention as described herein, a simple and effective surface impedance model suitable for RL-extraction in 2D interconnects of rectangular cross-section with both small and large thickness/width ratio is demonstrated. The model maintains accuracy from DC to multi-GHz frequencies due to properly captured skin-effect current attenuation off the conductor cross-sectional periphery. The extension of the proposed model to 3D Rao-Wilton-Glisson MoM discretization of EFIE is also demonstrated. 
         [0040]    Some embodiments of the invention will now be described in conjunction with the accompanying drawings in which: 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0041]      FIG. 1  ( a ) is a schematic representation of an interconnect geometry relating to the volumetric current density distribution across the conductor approximated according to skin effect. 
           [0042]      FIG. 1  ( b ) is a schematic representation of the interconnect of  FIG. 1  ( a ) discretized with N=14 piece-wise basis functions. 
           [0043]      FIG. 1  ( c ) is a schematic representation of additional matrix fillings due to the application of a full-peripheral Z s  model according to the present invention. 
           [0044]      FIG. 2  illustrates graphical representations of inductance and resistance computations relative to frequency for an isolated square signal line with σ=57.2 MS/m. 
           [0045]      FIG. 3  illustrates graphical representations of inductance and resistance computations relative to frequency for a differential line pair of conductors with σ=56 MS/m. 
           [0046]      FIG. 4  is a schematic representation of exemplary source triangles on the left, top, and bottom walls yielding non-zero ramp function to ramp function interactions with an observation triangle on the top wall in the discretize surface impedance operator. 
           [0047]      FIG. 5  is a schematic representation of an example of an mth source triangle on the right wall and mth observation triangle on the top-wall yielding non-zero ramp function to ramp function interactions in the discretize surface impedance operator. 
       
    
    
       [0048]    In the drawings like characters of reference indicate corresponding parts in the different figures. 
       DETAILED DESCRIPTION 
       [0049]    The formulation of a surface impedance model as described herein initially considers a generic 2-D interconnect structure of cross-section S and conductivity σ situated in homogeneous nonmagnetic medium of complex relative permittivity {circumflex over (∈)}=∈+σ/(iω∈) where ω represents the angular frequency and i=√{square root over (−1)}. The interconnect is made up of a very good conducting material (i.e. σ&gt;&gt;ω∈). Assuming the current flow in the z-direction, we relate the electric field E z  to volumetric current density j z  through the vector potential A z  as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       z 
                     
                      
                     
                       ( 
                       ρ 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         ω 
                       
                        
                       
                           
                       
                        
                       
                         
                           A 
                           z 
                         
                          
                         
                           ( 
                           ρ 
                           ) 
                         
                       
                     
                     - 
                     
                       
                          
                         
                           φ 
                            
                           
                             ( 
                             ρ 
                             ) 
                           
                         
                       
                       
                          
                         z 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0050]    where Φ is the scalar potential, and ρ=x{circumflex over (x)}+yŷ is the 2-D position vector depicted in the coordinate system ρ(t,w), as shown in  FIG. 1(   a ) and {circumflex over (z)}={circumflex over (x)}×ŷ, and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       z 
                     
                      
                     
                       ( 
                       ρ 
                       ) 
                     
                   
                   = 
                   
                     
                       μ 
                       0 
                     
                      
                     
                       
                         ∫ 
                         S 
                       
                        
                       
                         
                           G 
                            
                           
                             ( 
                             
                               ρ 
                                
                               
                                 ρ 
                                 ′ 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             j 
                             z 
                           
                            
                           
                             ( 
                             
                               ρ 
                               ′ 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             S 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0051]    In equation (2), G is the two-dimensional free-space quasi-static Green&#39;s function 
         [0000]        G (ρ|ρ′)= G ( x,y|x′,y ′)=−1/(2π)ln(|ρ−ρ′|)  (3)
 
         [0000]    where the logarithmic nature of the Green&#39;s function is obtained due to considering only the dominant conduction current σE z  and dropping off the negligible displacement current ω∈E z . The volumetric EFIE with respect to unknown current j z  in the conductor cross-section due to per-unit-length excitation follows from Ohm&#39;s law E z (ρ)=σ −1 j z (ρ) enforced inside the conductor (ρ∈S) 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         σ 
                         
                           - 
                           1 
                         
                       
                        
                       
                         
                           j 
                           z 
                         
                          
                         
                           ( 
                           ρ 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ωμ 
                         0 
                       
                        
                       
                         
                           ∫ 
                           S 
                         
                          
                         
                           
                             G 
                              
                             
                               ( 
                               
                                 ρ 
                                  
                                 
                                   ρ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               j 
                               z 
                             
                              
                             
                               ( 
                               
                                 ρ 
                                 ′ 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               S 
                               ′ 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     - 
                     
                       
                         V 
                         
                           p 
                           . 
                           u 
                           . 
                           l 
                           . 
                         
                       
                        
                       
                         ( 
                         ρ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0052]    The boundary-element method such as MoM seeks to decrease the computational complexity of the numerical solution in volumetric EFIE (4) through its reduction to a surface EFIE where the unknown function is the current distribution J z (ρ) on the conductor surface ρ∈∂S. In order to perform such reduction let us consider a rectangular cross-section wire with focal coordinate system shown in  FIG. 1(   a ). The sought volumetric current density J z  distribution across the conductor can be approximated according to skin-effect as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       j 
                       z 
                     
                      
                     
                       ( 
                       ρ 
                       ) 
                     
                   
                   ≅ 
                   
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           
                             k 
                             σ 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     J 
                                     z 
                                     t 
                                   
                                    
                                   
                                     ( 
                                     y 
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         k 
                                         σ 
                                       
                                        
                                       
                                         ( 
                                         
                                           X 
                                           - 
                                           x 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     J 
                                     z 
                                     b 
                                   
                                    
                                   
                                     ( 
                                     y 
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     x 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             X 
                           
                         
                       
                     
                     + 
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           
                             k 
                             σ 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   
                                     J 
                                     z 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     y 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     J 
                                     z 
                                     r 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         k 
                                         σ 
                                       
                                        
                                       
                                         ( 
                                         
                                           Y 
                                           - 
                                           y 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             Y 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0053]    where J z   t (y), J z   b (y), J z   l (x), J z   r (x) are the unknown surface current densities at the points of radius-vector P projections onto the top, bottom, left, and right walls of the conductor, respectively as shown in  FIG. 1(   a ). In Equation (5) k σ =√{square root over (ωμ 0 σ/2)}(1−i) is the complex wavenumber of the conductor V material, while X and Y denote the cross-sectional thickness and width, respectively. Substitution of formula (5) into volumetric EFIE (4) and ignoring variation of the Green&#39;s function along the coordinate normal to the conductor&#39;s periphery yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       S 
                     
                      
                     
                       
                         G 
                          
                         
                           ( 
                           
                             ρ 
                              
                             
                               ρ 
                               ′ 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           j 
                           z 
                         
                          
                         
                           ( 
                           
                             ρ 
                             ′ 
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           S 
                           ′ 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         ∫ 
                         
                           C 
                           t 
                         
                       
                        
                       
                         
                           G 
                            
                           
                             ( 
                             
                               
                                 ρ 
                                  
                                 X 
                               
                               , 
                               
                                 y 
                                 ′ 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             J 
                             z 
                             t 
                           
                            
                           
                             ( 
                             
                               y 
                               ′ 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             y 
                             ′ 
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           C 
                           b 
                         
                       
                        
                       
                         
                           G 
                            
                           
                             ( 
                             
                               
                                 ρ 
                                  
                                 0 
                               
                               , 
                               
                                 y 
                                 ′ 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             J 
                             z 
                             b 
                           
                            
                           
                             ( 
                             
                               y 
                               ′ 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             y 
                             ′ 
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           C 
                           l 
                         
                       
                        
                       
                         
                           G 
                            
                           
                             ( 
                             
                               
                                 ρ 
                                  
                                 
                                   x 
                                   ′ 
                                 
                               
                               , 
                               0 
                             
                             ) 
                           
                         
                          
                         
                           
                             J 
                             z 
                             l 
                           
                            
                           
                             ( 
                             
                               x 
                               ′ 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             x 
                             ′ 
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           C 
                           r 
                         
                       
                        
                       
                         
                           G 
                            
                           
                             ( 
                             
                               
                                 ρ 
                                  
                                 
                                   x 
                                   ′ 
                                 
                               
                               , 
                               Y 
                             
                             ) 
                           
                         
                          
                         
                           
                             J 
                             z 
                             r 
                           
                            
                           
                             ( 
                             
                               x 
                               ′ 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             x 
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0054]    Note that ignoring Green&#39;s function variation along the normal to the conductor surface is optional. Instead, one may choose to include effect of Green&#39;s function variation by performing integration along this coordinate numerically to given precision. 
         [0055]    Restriction of the observation point ρ to the conductor periphery C in equation (4) together with the approximation in equation (6) reduces the volumetric EFIE (4) to the surface EFIE with respect to the unknown surface current density J z (ρ′) as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         Z 
                         σ 
                       
                        
                       
                         { 
                         
                           
                             J 
                             z 
                           
                            
                           
                             ( 
                             ρ 
                             ) 
                           
                         
                         } 
                       
                     
                     + 
                     
                       
                         ωμ 
                         0 
                       
                        
                       
                         
                           ∫ 
                           C 
                         
                          
                         
                           
                             G 
                              
                             
                               ( 
                               
                                 ρ 
                                  
                                 
                                   ρ 
                                   ′ 
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               J 
                               z 
                             
                              
                             
                               ( 
                               
                                 ρ 
                                 ′ 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               c 
                               ′ 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     - 
                     
                       
                         V 
                         
                           p 
                           . 
                           u 
                           . 
                           l 
                           . 
                         
                       
                        
                       
                         ( 
                         ρ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0056]    where the radius-vectors ρ and ρ′ reside on the conductor surface C=∂S, C being the union of the top, bottom, left, and right conductor sides C t , C b , C l , and C r  respectively. 
         [0057]    In EFIE (7) Z σ {J z (ρ)} denotes the surface impedance operator relating z-directed electric field on the surface of conductor E z (ρ) to the values of the surface current densities J z (ρ) at the corresponding points on the top, bottom, left, and right sides of the conductor ( FIG. 1   a ) according to Ohm&#39;s law and the approximation in equation (5). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                       σ 
                     
                      
                     
                       { 
                       
                         
                           J 
                           z 
                         
                          
                         
                           ( 
                           ρ 
                           ) 
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               E 
                               z 
                               t 
                             
                              
                             
                               ( 
                               y 
                               ) 
                             
                           
                           , 
                           
                             
                               E 
                               z 
                               b 
                             
                              
                             
                               ( 
                               y 
                               ) 
                             
                           
                           , 
                           
                             
                               E 
                               z 
                               l 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                           , 
                           
                             
                               E 
                               z 
                               r 
                             
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                         ] 
                       
                       T 
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                       × 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   1 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         X 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         σ 
                                       
                                        
                                       X 
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         X 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         
                                           σ 
                                            
                                           
                                               
                                           
                                            
                                           y 
                                         
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         Y 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         
                                           σ 
                                            
                                           
                                               
                                           
                                            
                                           
                                             ( 
                                             
                                               Y 
                                               - 
                                               y 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         Y 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         σ 
                                       
                                        
                                       X 
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         X 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   1 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         X 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         
                                           σ 
                                            
                                           
                                               
                                           
                                            
                                           y 
                                         
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         Y 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         
                                           σ 
                                            
                                           
                                               
                                           
                                            
                                           
                                             ( 
                                             
                                               Y 
                                               - 
                                               y 
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         Y 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           k 
                                           σ 
                                         
                                          
                                         
                                           ( 
                                           
                                             X 
                                             - 
                                             x 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         X 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         σ 
                                       
                                        
                                       x 
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         X 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   1 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         Y 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         
                                           σ 
                                            
                                           
                                               
                                           
                                            
                                           Y 
                                         
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         Y 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           k 
                                           σ 
                                         
                                          
                                         
                                           ( 
                                           
                                             X 
                                             - 
                                             x 
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         X 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         σ 
                                       
                                        
                                       x 
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         X 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         k 
                                         
                                           σ 
                                            
                                           
                                               
                                           
                                            
                                           Y 
                                         
                                       
                                     
                                   
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         Y 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   1 
                                   
                                     1 
                                     - 
                                     
                                        
                                       
                                         
                                           - 
                                            
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           k 
                                           σ 
                                         
                                          
                                         Y 
                                       
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         · 
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     J 
                                     z 
                                     t 
                                   
                                    
                                   
                                     ( 
                                     y 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     J 
                                     z 
                                     b 
                                   
                                    
                                   
                                     ( 
                                     y 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     J 
                                     z 
                                     l 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     J 
                                     z 
                                     r 
                                   
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             T 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0058]    The operator behavior at dc is examined via taking the limit of equation (8) as it approaches zero. Using Taylor approximation for very small exponential arguments, i.e. e a ≅a+1, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       lim 
                       
                         ω 
                         → 
                         0 
                       
                     
                      
                     
                       
                         Z 
                         σ 
                       
                        
                       
                         { 
                         
                           
                             J 
                             z 
                           
                            
                           
                             ( 
                             ρ 
                             ) 
                           
                         
                         } 
                       
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               X 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               X 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               Y 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               Y 
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               X 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               X 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               Y 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               Y 
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               X 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               X 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               Y 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               Y 
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               X 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               X 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               Y 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 σ 
                               
                                
                               
                                   
                               
                                
                               Y 
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               
                                 J 
                                 z 
                                 t 
                               
                                
                               
                                 ( 
                                 y 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 J 
                                 z 
                                 b 
                               
                                
                               
                                 ( 
                                 y 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 J 
                                 z 
                                 l 
                               
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 J 
                                 z 
                                 r 
                               
                                
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0059]    which indicates the dc resistance seen by the four surface current densities on the conductor&#39;s periphery. At high frequencies, the exponential terms of the operator become negligible, thus, turning the operator&#39;s off-diagonal elements to zero whereas the diagonal terms tend to (1+i) √{square root over (ωμ 0 /2σ)}, which corresponds to the local surface impedance of the half-space with conductivity σ. From equation (8) it is also observed that lim σ→+∞ Z σ {J z (ρ)}=0, which implies that equation (7) becomes an EFIE for a perfectly conducting interconnect. 
         [0060]    Implementation of Method of Moments 
         [0061]    The proposed surface impedance model is intended for use in conjunction with an appropriate MoM discretization scheme. Below we demonstrate the MoM implementation with N=14 piece-wise basis functions distributed over the conductor periphery as shown in  FIG. 1(   b ). Under the above discretization the surface EFIE (7) is reduced to the linear algebraic equations as follows: 
         [0000]      ( Z   σ+   Z   A )· J=V   (10)
 
         [0000]    where Z mn   A =iωμ 0 ∫ ΔC     n   G(ρ m |ρ′)dc′ is the dense matrix of the vector potential interactions, V m =−V p.u.l. (ρ m ) is the vector of excitation, J m =J z (ρ m ) is the vector of unknown current densities on the conductor&#39;s periphery, and Z mn   σ  is the sparse matrix corresponding to the surface impedance operator in equation (8). Indexes m and n in the above matrices run from 1 to N. The non-zero elements in the sparse matrix Z σ  for this particular discretization are shown in  FIG. 1(   c ) with the patterned squares. The black squares in  FIG. 1(   c ) depict the non-zero entries in Z σ  corresponding to the field E z  tested at the observation point ρ={circumflex over (x)}X+ŷy 3  situated in the middle of the 3rd element on the top peripheral segment C t . In accord with equation (8), this field is related to the surface current density on the top, bottom and side walls as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       z 
                       t 
                     
                      
                     
                       ( 
                       
                         y 
                         3 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         Z 
                         
                           3 
                           , 
                           3 
                         
                         
                           σ 
                           , 
                           tt 
                         
                       
                        
                       
                         
                           J 
                           z 
                           t 
                         
                          
                         
                           ( 
                           
                             y 
                             3 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         Z 
                         
                           3 
                           , 
                           3 
                         
                         
                           σ 
                           , 
                           tb 
                         
                       
                        
                       
                         
                           J 
                           z 
                           b 
                         
                          
                         
                           ( 
                           
                             y 
                             3 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         Z 
                         
                           3 
                           , 
                           1 
                         
                         
                           σ 
                           , 
                           tl 
                         
                       
                        
                       
                         
                           J 
                           z 
                           l 
                         
                          
                         
                           ( 
                           
                             x 
                             1 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         Z 
                         
                           3 
                           , 
                           1 
                         
                         
                           σ 
                           , 
                           tr 
                         
                       
                        
                       
                         
                           J 
                           z 
                           r 
                         
                          
                         
                           ( 
                           
                             x 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0062]    where the matrix Z σ  entries are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                       
                         3 
                         , 
                         3 
                       
                       
                         σ 
                         , 
                         tt 
                       
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                        
                       
                         1 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 σ 
                               
                                
                               X 
                             
                           
                         
                       
                     
                   
                   , 
                   
                       
                   
                    
                   
                     
                       Z 
                       
                         3 
                         , 
                         3 
                       
                       
                         σ 
                         , 
                         tb 
                       
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                        
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             X 
                           
                         
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 σ 
                               
                                
                               X 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       Z 
                       
                         3 
                         , 
                         1 
                       
                       
                         σ 
                         , 
                         tl 
                       
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                        
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             
                               y 
                               3 
                             
                           
                         
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 σ 
                               
                                
                               Y 
                             
                           
                         
                       
                     
                   
                   , 
                   
                       
                   
                    
                   
                     
                       Z 
                       
                         3 
                         , 
                         1 
                       
                       
                         σ 
                         , 
                         tr 
                       
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                        
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   k 
                                   σ 
                                 
                                  
                                 
                                   ( 
                                   
                                     Y 
                                     - 
                                     
                                       y 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             1 
                             - 
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   σ 
                                 
                                  
                                 Y 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0063]    The remaining non-zero element in the surface impedance matrix Z σ  are filled out in a similar manner. 
         [0064]    Numerical Results 
         [0065]    The proposed full-peripheral surface impedance model has been tested for several interconnects with both large and small thickness/width ratios and has shown a reliable performance. A detailed description of p.u.l. resistance and inductance matrices for k conductor system can be found in C. R. Paul,  Analysis of Mulliconduclor Transmission Lines , Ch. 3, John Wiley &amp; Sons, Inc., Toronto, CA, 1994. 
         [0066]    In the first numerical experiment, the p.u.l. resistance and inductance are extracted for the 4.62 mm-wide copper wire of square cross-section represented in  FIG. 2  via the solution of EFIE (7) with surface impedance equation (8). The extracted parameters are compared to the accurate volumetric EFIE solution equation (4) as well as to the extracted p.u.l. resistance presented in D. DeZutter, et al., “Skin Effect Modeling Based on a Differential Surface Admittance Operator,”  IEEE Trans. Microwave Theory Tech ., vol. 53, no. 8, pp. 2526-2538, August. 2005. The MoM discretization at all frequencies consisted of 20 equidistant segments per each side of the conductor. The results from the two-plane surface impedance model utilized in the 2.5D interconnect models, as described in J. D. Morsey, et. al., “Finite-Thickness Conductor Models for Full-Wave Analysis of Interconnects With a Fast Integral Equation Method,”  IEEE Trans. on Advanced Packaging , vol. 27, no. 1, pp. 24-33, February 2004, are also shown in  FIG. 2 . The error in the latter increases at high frequencies because only the top and bottom segments of the conductor&#39;s periphery support the current. Such model becomes inaccurate when the thickness of conductor is comparable to its width. The EM analysis yielded a dc resistance of 0.8191 mΩ/m and the √{square root over (ƒ)} resistance dependence at high frequency is apparent from  FIG. 2 . The p.u.l. inductance is undefined under this scenario. 
         [0067]    The example in  FIG. 3  demonstrates the accuracy of the new model in the presence of proximity effects. The p.u.l. resistance and inductance are extracted for a differential line pair made of two copper wires featuring 2 mm square cross-section and various separation distances d.  FIG. 3  depicts the extracted resistance and inductance of the configuration as a function of frequency. The results obtained using the proposed model are compared with those published in C. R. Paul,  Analysis of Mulliconduclor Transmission Lines , Ch. 3, John Wiley &amp; Sons, Inc., Toronto, CA, 1994 and D. DeZutter, et al., “Skin Effect Modeling Based on a Differential Surface Admittance Operator,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 8, pp. 2526-2538, August. 2005. The full-periphery EFIE solver yields dc resistance (independent from d as expected) of 8.929 mΩ/m, which agrees with the dc resistance value in C. R. Paul,  Analysis of Mulliconduclor Transmission Lines , Ch. 3, John Wiley &amp; Sons, Inc., Toronto, CA, 1994. The dc inductance was computed to be 600.6 nH/m (599.5 nH/rn in C. R. Paul,  Analysis of Mulliconductor Transmission Lines , Ch. 3, John Wiley &amp; Sons, Inc., Toronto, CA, 1994) for d=2 mm and 412.6 nH/m (413 nH/m in D. DeZutter, et al., “Skin Effect Modeling Based on a Differential Surface Admittance Operator,”  IEEE Trans. Microwave Theory Tech ., vol. 53, no. 8, pp. 2526-2538, August. 2005) for d=0.5 mm. 
         [0068]    Attention is now drawn to full-peripheral impedance for skin effect approximation in a three-dimensional electric field integral equation. 
         [0069]    Surface Electric Field Integral Equation And Conductor Loss Model 
         [0070]    Let us consider a generic planar interconnect structure, depicted in  FIG. 4 , occupying volume V in the medium of L homogeneous dielectric layers, which are infinitely extending in the xy-plane and exhibit inhomogeneity along the z-axis. The interconnect is made up of a very good conducting material where conduction current is dominant and displacement current is negligible. The layers are characterized by complex permittivity {circumflex over (∈)} l =∈ l +σ l /(jω∈ l ) and permeability μ l  within intervals h l &lt;z&lt;h l+1  where l=1, . . . , L and i=√{square root over (− 1 )}. In the above and throughout the following derivations the time-harmonic field variation is assumed and suppressed for brevity. The electric field in a multilayered medium is related to volumetric electric current density j through the vector potential A as 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       ( 
                       r 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         E 
                         inc 
                       
                        
                       
                         ( 
                         r 
                         ) 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           ω 
                           + 
                           
                             
                               1 
                               
                                 ω 
                                  
                                 
                                   ε 
                                   ^ 
                                 
                                  
                                 μ 
                               
                             
                              
                             
                               
                                 ∇ 
                                 ∇ 
                               
                               · 
                             
                           
                         
                         ) 
                       
                        
                       
                         
                           A 
                            
                           
                             ( 
                             r 
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 where 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     A 
                      
                     
                       ( 
                       r 
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       V 
                     
                      
                     
                       
                         
                           
                             
                               G 
                               _ 
                             
                             A 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 r 
                                 ′ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           j 
                            
                           
                             ( 
                             
                               r 
                               ′ 
                             
                             ) 
                           
                         
                       
                        
                       
                         
                            
                           
                             v 
                             ′ 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0071]    In above noted equation (14),  G   A  is the layered medium Greens function in standard form 
         [0000]          G     A   ={circumflex over (x)}{circumflex over (x)}G   A,xx   +ŷŷG   A,xx   +{circumflex over (z)}{circumflex over (z)}G   A,zz   +{circumflex over (z)}{circumflex over (x)}G   A,zx   +{circumflex over (z)}ŷG   A,zx .  (15)
 
         [0072]    The electric field integral equation (EFIE) with respect to current j flowing in the conductor due to given excitation E inc  follows from Ohm&#39;s law E(r)=σ −1 j(r) enforced inside the conductor 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         σ 
                         
                           - 
                           1 
                         
                       
                        
                       
                         j 
                          
                         
                           ( 
                           r 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           E 
                           inc 
                         
                          
                         
                           ( 
                           r 
                           ) 
                         
                       
                       - 
                       
                         
                           ( 
                           
                             ω 
                             + 
                             
                               
                                 1 
                                 
                                   ω 
                                    
                                   
                                     ε 
                                     ^ 
                                   
                                    
                                   μ 
                                 
                               
                                
                               
                                 
                                   ∇ 
                                   ∇ 
                                 
                                 · 
                               
                             
                           
                           ) 
                         
                          
                         
                           
                             ∫ 
                             V 
                           
                            
                           
                             
                               
                                 
                                   
                                     G 
                                     _ 
                                   
                                   A 
                                 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     , 
                                     
                                       r 
                                       ′ 
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     ′ 
                                   
                                   ) 
                                 
                               
                             
                              
                             
                                
                               
                                 v 
                                 ′ 
                               
                             
                           
                         
                       
                     
                   
                   , 
                   
                     r 
                     ∈ 
                     
                       V 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0073]    Numerical solution of the volumetric integral equation (14) may be time consuming. Therefore, we use the surface impedance approximation to reduce the volumetric EFIE to the surface integral equation. Considering the rectangular cross-section of the conductors with local coordinate system introduced as shown in  FIG. 4  we approximate dependence of the volumetric current density j across the thickness of the conductor according to the skin-effect 
         [0000]    
       
         
           
             
               
                 
                   
                     j 
                      
                     
                       ( 
                       r 
                       ) 
                     
                   
                   ≅ 
                   
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           
                             k 
                             σ 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   J 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       t 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         k 
                                         σ 
                                       
                                        
                                       
                                         ( 
                                         
                                           T 
                                           - 
                                           
                                             z 
                                              
                                             
                                               ( 
                                               r 
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   J 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       b 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     
                                       z 
                                        
                                       
                                         ( 
                                         r 
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             T 
                           
                         
                       
                     
                     + 
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           
                             k 
                             σ 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   J 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       l 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         k 
                                         σ 
                                       
                                        
                                       
                                         ( 
                                         
                                           x 
                                            
                                           
                                             ( 
                                             r 
                                             ) 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   J 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       r 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         k 
                                         σ 
                                       
                                        
                                       
                                         ( 
                                         
                                           X 
                                           - 
                                           
                                             x 
                                              
                                             
                                               ( 
                                               r 
                                               ) 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             W 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0074]    where J(r t ), J(r b ), J(r l ), J(r r ) are the surface current densities the projection of radius-vector r the top, bottom, left, and right walls of the conductor, respectively, k σ =√{square root over (ωμ 0/ 2)}( 1 −i) is the wavenumber of conductor. In equation (17) the conductor&#39;s thickness and its width at location r are denoted as T and W, respectively. On substitution of formula (17) into the integral equation we obtain 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       V 
                     
                      
                     
                       
                         
                           
                             
                               G 
                               _ 
                             
                             A 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 r 
                                 ′ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           j 
                            
                           
                             ( 
                             
                               r 
                               ′ 
                             
                             ) 
                           
                         
                       
                        
                       
                          
                         
                           v 
                           ′ 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       
                         ∫ 
                         
                           S 
                           t 
                         
                       
                        
                       
                         
                           ∫ 
                           
                             
                               z 
                               ′ 
                             
                             = 
                             0 
                           
                           T 
                         
                          
                         
                           
                             
                               
                                 
                                   G 
                                   _ 
                                 
                                 A 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   , 
                                   
                                     
                                       r 
                                       ′ 
                                     
                                     - 
                                     
                                       
                                         
                                           n 
                                           ^ 
                                         
                                          
                                         
                                           ( 
                                           
                                             r 
                                             ′ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         z 
                                         ′ 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               
                                 
                                   J 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       ′ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     
                                       z 
                                       ′ 
                                     
                                   
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   σ 
                                 
                               
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     T 
                                   
                                 
                               
                             
                           
                            
                           
                              
                             
                               z 
                               ′ 
                             
                           
                            
                           
                              
                             
                               s 
                               ′ 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           S 
                           b 
                         
                       
                        
                       
                         
                           ∫ 
                           
                             
                               z 
                               ′ 
                             
                             = 
                             0 
                           
                           T 
                         
                          
                         
                           
                             
                               
                                 
                                   G 
                                   _ 
                                 
                                 A 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   , 
                                   
                                     
                                       r 
                                       ′ 
                                     
                                     + 
                                     
                                       
                                         
                                           n 
                                           ^ 
                                         
                                          
                                         
                                           ( 
                                           
                                             r 
                                             ′ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         z 
                                         ′ 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               
                                 
                                   J 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       ′ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     
                                       z 
                                       ′ 
                                     
                                   
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   σ 
                                 
                               
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     T 
                                   
                                 
                               
                             
                           
                            
                           
                              
                             
                               z 
                               ′ 
                             
                           
                            
                           
                              
                             
                               s 
                               ′ 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           S 
                           l 
                         
                       
                        
                       
                         
                           ∫ 
                           
                             
                               x 
                               ′ 
                             
                             = 
                             0 
                           
                           W 
                         
                          
                         
                           
                             
                               
                                 
                                   G 
                                   _ 
                                 
                                 A 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   , 
                                   
                                     
                                       r 
                                       ′ 
                                     
                                     + 
                                     
                                       
                                         
                                           n 
                                           ^ 
                                         
                                          
                                         
                                           ( 
                                           
                                             r 
                                             ′ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         x 
                                         ′ 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               
                                 
                                   J 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       ′ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     
                                       x 
                                       ′ 
                                     
                                   
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   σ 
                                 
                               
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     W 
                                   
                                 
                               
                             
                           
                            
                           
                              
                             
                               x 
                               ′ 
                             
                           
                            
                           
                              
                             
                               s 
                               ′ 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           S 
                           r 
                         
                       
                        
                       
                         
                           ∫ 
                           
                             
                               x 
                               ′ 
                             
                             = 
                             0 
                           
                           W 
                         
                          
                         
                           
                             
                               
                                 
                                   G 
                                   _ 
                                 
                                 A 
                               
                                
                               
                                 ( 
                                 
                                   r 
                                   , 
                                   
                                     
                                       r 
                                       ′ 
                                     
                                     - 
                                     
                                       
                                         
                                           n 
                                           ^ 
                                         
                                          
                                         
                                           ( 
                                           
                                             r 
                                             ′ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         x 
                                         ′ 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             · 
                             
                               
                                 
                                   J 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       ′ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     
                                       x 
                                       ′ 
                                     
                                   
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   σ 
                                 
                               
                               
                                 1 
                                 - 
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       σ 
                                     
                                      
                                     W 
                                   
                                 
                               
                             
                           
                            
                           
                              
                             
                               x 
                               ′ 
                             
                           
                            
                           
                             
                                
                               
                                 s 
                                 ′ 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0075]    where the source radius-vector r′ resides on the conductor surface S, i.e. r′∈S, S being the union of the top, bottom, left, and right conductor walls S t , S b , S l , and S r  respectively. Assuming small thickness of conductor we can ignore the variation of the Green&#39;s function across the conductor cross-section  G   A ( r ,r′±{circumflex over (n)}(r′)z′)≅  G   A (r,r′) and  G   A (r,r′±{circumflex over (n)}(r′)x′)≅G A (r,r′) in equation (18). Such approximation allows us to evaluate analytically the integrals over conductor thickness T and width W in equation (18) and combine the four integral terms over the top, bottom, left, and right surfaces, yielding 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       V 
                     
                      
                     
                       
                         
                           
                             
                               G 
                               _ 
                             
                             A 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 r 
                                 ′ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           j 
                            
                           
                             ( 
                             
                               r 
                               ′ 
                             
                             ) 
                           
                         
                       
                        
                       
                          
                         
                           v 
                           ′ 
                         
                       
                     
                   
                   ≅ 
                   
                     
                       ∫ 
                       S 
                     
                      
                     
                       
                         
                           
                             
                               G 
                               _ 
                             
                             A 
                           
                            
                           
                             ( 
                             
                               r 
                               , 
                               
                                 r 
                                 ′ 
                               
                             
                             ) 
                           
                         
                         · 
                         
                           J 
                            
                           
                             ( 
                             
                               r 
                               ′ 
                             
                             ) 
                           
                         
                       
                        
                       
                         
                            
                           
                             s 
                             ′ 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0076]    Thus, instead of the volume integral equation (16) we obtain the surface integral equation with respect to the unknown surface current densities 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                       s 
                     
                      
                     
                       { 
                       
                         J 
                          
                         
                           ( 
                           
                             r 
                             ′ 
                           
                           ) 
                         
                       
                       } 
                     
                   
                   = 
                   
                     
                       
                         n 
                         ^ 
                       
                        
                       
                         ( 
                         r 
                         ) 
                       
                     
                     × 
                     
                         
                       
                         
                           [ 
                           
                             
                               
                                 E 
                                 inc 
                               
                                
                               
                                 ( 
                                 r 
                                 ) 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   ω 
                                   + 
                                   
                                     
                                       1 
                                       
                                         ω 
                                          
                                         
                                           ε 
                                           ^ 
                                         
                                          
                                         μ 
                                       
                                     
                                      
                                     
                                       
                                         ∇ 
                                         ∇ 
                                       
                                       · 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 
                                   ∫ 
                                   S 
                                 
                                  
                                 
                                   
                                     
                                       
                                         
                                           G 
                                           _ 
                                         
                                         A 
                                       
                                        
                                       
                                         ( 
                                         
                                           r 
                                           , 
                                           
                                             r 
                                             ′ 
                                           
                                         
                                         ) 
                                       
                                     
                                     · 
                                     
                                       J 
                                        
                                       
                                         ( 
                                         
                                           r 
                                           ′ 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                      
                                     
                                       s 
                                       ′ 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         r 
                         , 
                         
                           
                             r 
                             ′ 
                           
                           ∈ 
                           
                             S 
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0077]    In equation (20) Z s {J(r′)} denotes the surface impedance operator relating tangential electric field on the surface of conductor to the values of the surface current densities J(r′) at the corresponding points on the top, bottom, left, and right sides of the conductor according to E(r)=σ −1 j(r) and the relationship in equation (17) between the volumetric and surface current densities 
         [0000]    
       
         
           
             
               
                 
                     
                   
                     
                       
                         
                           
                             
                               
                                 n 
                                 ^ 
                               
                                
                               
                                 ( 
                                 r 
                                 ) 
                               
                             
                             × 
                             
                               E 
                                
                               
                                 ( 
                                 r 
                                 ) 
                               
                             
                           
                           = 
                             
                            
                           
                             
                               Z 
                               s 
                             
                              
                             
                               { 
                               
                                 J 
                                  
                                 
                                   ( 
                                   
                                     r 
                                     ′ 
                                   
                                   ) 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               [ 
                               
                                 
                                   E 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       ∈ 
                                       
                                         S 
                                         t 
                                       
                                     
                                     ) 
                                   
                                 
                                 , 
                                 
                                   E 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       ∈ 
                                       
                                         S 
                                         b 
                                       
                                     
                                     ) 
                                   
                                 
                                 , 
                                 
                                   E 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       ∈ 
                                       
                                         S 
                                         l 
                                       
                                     
                                     ) 
                                   
                                 
                                 , 
                                 
                                   E 
                                    
                                   
                                     ( 
                                     
                                       r 
                                       ∈ 
                                       
                                         S 
                                         r 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             
                               t 
                               p 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   σ 
                                 
                               
                               σ 
                             
                             × 
                           
                         
                       
                     
                     
                       
                         
                             
                            
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       1 
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             T 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             σ 
                                           
                                            
                                           T 
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             T 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             σ 
                                           
                                            
                                           x 
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             W 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               k 
                                               σ 
                                             
                                              
                                             
                                               ( 
                                               
                                                 W 
                                                 - 
                                                 x 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             W 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             σ 
                                           
                                            
                                           T 
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             T 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       1 
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             T 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             σ 
                                           
                                            
                                           x 
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             W 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               k 
                                               σ 
                                             
                                              
                                             
                                               ( 
                                               
                                                 W 
                                                 - 
                                                 x 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             W 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               k 
                                               σ 
                                             
                                              
                                             
                                               ( 
                                               
                                                 T 
                                                 - 
                                                 z 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             T 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             σ 
                                           
                                            
                                           z 
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             T 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       1 
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             W 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             σ 
                                           
                                            
                                           W 
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             W 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               k 
                                               σ 
                                             
                                              
                                             
                                               ( 
                                               
                                                 T 
                                                 - 
                                                 z 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             T 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             σ 
                                           
                                            
                                           z 
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             T 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                          
                                         
                                           
                                             - 
                                              
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             k 
                                             σ 
                                           
                                            
                                           W 
                                         
                                       
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             W 
                                           
                                         
                                       
                                     
                                   
                                   
                                     
                                       1 
                                       
                                         1 
                                         - 
                                         
                                            
                                           
                                             
                                               - 
                                                
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               k 
                                               σ 
                                             
                                              
                                             W 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             · 
                           
                         
                       
                     
                     
                       
                         
                             
                            
                           
                             [ 
                             
                               
                                 
                                   
                                     J 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             ′ 
                                           
                                            
                                           
                                             ( 
                                             r 
                                             ) 
                                           
                                         
                                         ∈ 
                                         
                                           S 
                                           t 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     J 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             ′ 
                                           
                                            
                                           
                                             ( 
                                             r 
                                             ) 
                                           
                                         
                                         ∈ 
                                         
                                           S 
                                           b 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     J 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             ′ 
                                           
                                            
                                           
                                             ( 
                                             r 
                                             ) 
                                           
                                         
                                         ∈ 
                                         
                                           S 
                                           l 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     J 
                                      
                                     
                                       ( 
                                       
                                         
                                           
                                             r 
                                             ′ 
                                           
                                            
                                           
                                             ( 
                                             r 
                                             ) 
                                           
                                         
                                         ∈ 
                                         
                                           S 
                                           r 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0078]    Superscript t p  in formula (21) denotes transposition. 
         [0079]    Method of Moments Discretization of the Surface Impedance Operator 
         [0080]    In order to make surface EFIE (8) amenable to Rao-Wilton-Glission (RWG) solution (as per S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shapes,”  IEEE Trans. Antennas Propag ., Vol. 30, pp. 409-418, May 1982 and K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media, Part I: Theory,”  IEEE Trans. Antennas Propag ., Vol. 38, pp. 335-344, March 1990) the interconnect surface is modeled by a triangulated plane extruded to its corresponding thickness ( FIG. 4 ). The discussion of the full periphery surface impedance operator RWG discretization is detailed below. Considering a single conductor the traditional RWG discretization of the unknown current J in (20) over N basis functions ƒ is equivalently formulated as a discretization over the ramp-functions (half-RWG functions) situated at the top, bottom, left, and right walls of conductor as follows 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       J 
                        
                       
                         ( 
                         r 
                         ) 
                       
                     
                     ≃ 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         
                           J 
                           j 
                         
                          
                         
                           
                             f 
                             j 
                           
                            
                           
                             ( 
                             r 
                             ) 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         
                           w 
                           = 
                           t 
                         
                         , 
                         b 
                         , 
                         l 
                         , 
                         r 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           1 
                         
                         
                           M 
                           w 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           3 
                         
                          
                         
                           
                             I 
                             
                               j 
                               , 
                               m 
                             
                             w 
                           
                            
                           
                             
                               
                                 R 
                                 
                                   j 
                                   , 
                                   m 
                                 
                                 w 
                               
                                
                               
                                 ( 
                                 r 
                                 ) 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0081]    where index W denotes which wall the m th  triangle supporting ramp R j.m   w  is situated on and index j runs over three ramps on each triangle in according with  FIG. 4 . 
         [0082]    The ramp-function discretization equation (22) of the unknown current J under the surface impedance operator Z s  in equation (20) followed by testing of the resultant electric field Z s {J} with ramp-function R i,n   t  in of top-wall triangle S n   t  yields 
         [0000]                  R   i,n   t ( r ), Z   s   {J}             =             R   i,n   t ( r ), Z   tt   s   J ( r   m   t ( r ))+ Z   t,b   s   J ( r   m   b ( r ))+ Z   tl   s ( r )) J ( r   m   l ( r ))+ Z   tr   s   J ( r   m   r ( r ))           (23)
 
         [0083]    where J(r m   t ), J(r m   b ), J(r m   l ), J(r m   r ), are the current values at the corresponding points r m   t , r m   b , r m   l , r m   r , source triangles S m   t , S m   b , S m   l , S m   r  at the top, bottom, left, and right walls, respectively, overlapping with triangle S n   t  as shown in  FIG. 4  and  FIG. 5 . The current values on each source triangle are defined as a superposition of three ramp-functions 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         J 
                          
                         
                           ( 
                           
                             r 
                             m 
                             t 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           3 
                         
                          
                         
                           
                             I 
                             
                               j 
                               , 
                               m 
                             
                             t 
                           
                            
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               t 
                             
                              
                             
                               ( 
                               
                                 r 
                                 m 
                                 t 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         J 
                          
                         
                           ( 
                           
                             
                               r 
                               . 
                             
                             m 
                             b 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           3 
                         
                          
                         
                           
                             I 
                             
                               j 
                               , 
                               m 
                             
                             b 
                           
                            
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               b 
                             
                              
                             
                               ( 
                               
                                 r 
                                 m 
                                 b 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         J 
                          
                         
                           ( 
                           
                             r 
                             m 
                             l 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           3 
                         
                          
                         
                           
                             I 
                             
                               j 
                               , 
                               m 
                             
                             l 
                           
                            
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               l 
                             
                              
                             
                               ( 
                               
                                 r 
                                 m 
                                 l 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         J 
                          
                         
                           ( 
                           
                             r 
                             m 
                             r 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           3 
                         
                          
                         
                           
                             I 
                             
                               j 
                               , 
                               m 
                             
                             r 
                           
                            
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               r 
                             
                              
                             
                               ( 
                               
                                 r 
                                 m 
                                 r 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0084]    and the corresponding values of the surface impedance operator are 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                       tt 
                       s 
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                        
                       
                         1 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 σ 
                               
                                
                               T 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
                       Z 
                       tb 
                       s 
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                        
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             T 
                           
                         
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 σ 
                               
                                
                               T 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       Z 
                       tl 
                       s 
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                        
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             x 
                           
                         
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 k 
                                 σ 
                               
                                
                               W 
                             
                           
                         
                       
                     
                   
                   , 
                   
                     
                       
                         Z 
                         tr 
                         s 
                       
                        
                       
                         ( 
                         r 
                         ) 
                       
                     
                     = 
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                         
                         σ 
                       
                        
                       
                         
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   k 
                                   σ 
                                 
                                  
                                 
                                   ( 
                                   
                                     W 
                                     - 
                                     x 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             1 
                             - 
                             
                                
                               
                                 
                                   - 
                                    
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   k 
                                   σ 
                                 
                                  
                                 W 
                               
                             
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0085]    In equation (24) I j,m   w , w=t, b, l, r represent the sought unknown coefficients in method of moments. From equations (23) through (25) we notice that each, ith ramp R i,n   t  on nth top wall, the observation triangle S n   t  has non-zero inner products          R i,n   t , R j,m   w            with each of three ramps R j,m   w , j=1, 2, 3, on overlapping triangles S m   w , w=t, b, l, r thus, yielding the following 12 nonzero entries in each row of the discretized surface impedance operator 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                       
                         j 
                         , 
                         m 
                       
                       t 
                     
                      
                     
                       Z 
                       tt 
                       s 
                     
                      
                     
                       〈 
                       
                         
                           
                             R 
                             
                               i 
                               , 
                               n 
                             
                             t 
                           
                            
                           
                             ( 
                             r 
                             ) 
                           
                         
                         , 
                         
                           
                             R 
                             
                               j 
                               , 
                               m 
                             
                             t 
                           
                            
                           
                             ( 
                             
                               
                                 r 
                                 m 
                                 t 
                               
                                
                               
                                 ( 
                                 r 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       〉 
                     
                   
                   = 
                   
                     
                       I 
                       
                         j 
                         , 
                         m 
                       
                       t 
                     
                      
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           k 
                           σ 
                         
                       
                       σ 
                     
                      
                     
                       1 
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             T 
                           
                         
                       
                     
                      
                     
                       
                         ∫ 
                         
                           S 
                           n 
                           t 
                         
                       
                        
                       
                         
                           
                             
                               R 
                               
                                 i 
                                 , 
                                 n 
                               
                               t 
                             
                              
                             
                               ( 
                               r 
                               ) 
                             
                           
                           · 
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               t 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   m 
                                   t 
                                 
                                  
                                 
                                   ( 
                                   r 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           S 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       I 
                       
                         j 
                         , 
                         m 
                       
                       b 
                     
                      
                     
                       Z 
                       tb 
                       s 
                     
                      
                     
                       〈 
                       
                         
                           
                             R 
                             
                               i 
                               , 
                               n 
                             
                             t 
                           
                            
                           
                             ( 
                             r 
                             ) 
                           
                         
                         , 
                         
                           
                             R 
                             
                               j 
                               , 
                               m 
                             
                             b 
                           
                            
                           
                             ( 
                             
                               
                                 r 
                                 m 
                                 b 
                               
                                
                               
                                 ( 
                                 r 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                       〉 
                     
                   
                   = 
                   
                     
                       I 
                       
                         j 
                         , 
                         m 
                       
                       b 
                     
                      
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           k 
                           σ 
                         
                       
                       σ 
                     
                      
                     
                       
                          
                         
                           
                             - 
                              
                           
                            
                           
                               
                           
                            
                           
                             k 
                             σ 
                           
                            
                           T 
                         
                       
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             T 
                           
                         
                       
                     
                      
                     
                       
                         ∫ 
                         
                           S 
                           n 
                           t 
                         
                       
                        
                       
                         
                           
                             
                               R 
                               
                                 i 
                                 , 
                                 n 
                               
                               t 
                             
                              
                             
                               ( 
                               r 
                               ) 
                             
                           
                           · 
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               b 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   m 
                                   b 
                                 
                                  
                                 
                                   ( 
                                   r 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           S 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       I 
                       
                         j 
                         , 
                         m 
                       
                       l 
                     
                      
                     
                       〈 
                       
                         
                           
                             R 
                             
                               i 
                               , 
                               n 
                             
                             t 
                           
                            
                           
                             ( 
                             r 
                             ) 
                           
                         
                         , 
                         
                           
                             
                               Z 
                               tl 
                               s 
                             
                              
                             
                               ( 
                               r 
                               ) 
                             
                           
                            
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               l 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   m 
                                   l 
                                 
                                  
                                 
                                   ( 
                                   r 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                       〉 
                     
                   
                   = 
                   
                     
                       I 
                       
                         j 
                         , 
                         m 
                       
                       l 
                     
                      
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           k 
                           σ 
                         
                       
                       σ 
                     
                      
                     
                       1 
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             W 
                           
                         
                       
                     
                      
                     
                       
                         ∫ 
                         
                           S 
                           n 
                           t 
                         
                       
                        
                       
                         
                           
                             
                               R 
                               
                                 i 
                                 , 
                                 n 
                               
                               t 
                             
                              
                             
                               ( 
                               r 
                               ) 
                             
                           
                           · 
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               l 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   m 
                                   l 
                                 
                                  
                                 
                                   ( 
                                   r 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             x 
                           
                         
                          
                         
                            
                           S 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       I 
                       
                         j 
                         , 
                         m 
                       
                       r 
                     
                      
                     
                       〈 
                       
                         
                           
                             R 
                             
                               i 
                               , 
                               n 
                             
                             t 
                           
                            
                           
                             ( 
                             r 
                             ) 
                           
                         
                         , 
                         
                           
                             
                               Z 
                               tr 
                               s 
                             
                              
                             
                               ( 
                               r 
                               ) 
                             
                           
                            
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               r 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   m 
                                   r 
                                 
                                  
                                 
                                   ( 
                                   r 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                       〉 
                     
                   
                   = 
                   
                     
                       I 
                       
                         j 
                         , 
                         m 
                       
                       r 
                     
                      
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           k 
                           σ 
                         
                       
                       σ 
                     
                      
                     
                       1 
                       
                         1 
                         - 
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               k 
                               σ 
                             
                              
                             W 
                           
                         
                       
                     
                      
                     
                       
                         ∫ 
                         
                           S 
                           n 
                           t 
                         
                       
                        
                       
                         
                           
                             
                               R 
                               
                                 i 
                                 , 
                                 n 
                               
                               t 
                             
                              
                             
                               ( 
                               r 
                               ) 
                             
                           
                           · 
                           
                             
                               R 
                               
                                 j 
                                 , 
                                 m 
                               
                               r 
                             
                              
                             
                               ( 
                               
                                 
                                   r 
                                   m 
                                   r 
                                 
                                  
                                 
                                   ( 
                                   r 
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             
                               
                                 k 
                                 σ 
                               
                                
                               
                                 ( 
                                 
                                   W 
                                   - 
                                   x 
                                 
                                 ) 
                               
                             
                           
                         
                          
                         
                            
                           S 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0086]    The integrals in equations (26) through (29) can be evaluated numerically to arbitrary precision. 
         [0087]    Since various modifications can be made in my invention as herein above described, and many apparently widely different embodiments of same made within the spirit and scope of the claims without department from such spirit and scope, it is intended that all matter contained in the accompanying specification shall be interpreted as illustrative only and not in a limiting sense.