# Patent

## Title
Method for modeling varactor by direct extraction of parameters

## Abstract

A method for modeling a varactor with a MOS structure, and transforms an s-parameter obtained by the measurement using measurement equipment into a y-parameter and a z-parameter and then directly extracts parameters required for the modeling by means of equations in accordance with embodiments. The modeling can be made reflecting the parameters of the varactor varied according to frequency so that the more accurate RF modeling of a passive device can be made and the accurate modeling of the varactor can be made through the direct extraction method of the parameters so that the parameters of the varactor having a physical meaning can be extracted without using an expensive CAD tool, the time required for optimizing the parameters can be reduced, a computing time can be shorten, an initial condition dependence generated in performing the optimization is not required, and a pattern for a test is not required.

## Background
The present application claims priority under 35 U.S.C. 119 to Korean Patent Application No. 10-2006-0137299 (filed on Dec. 29, 2006), which is hereby incorporated by reference in its entirety.
BACKGROUND
The process of manufacturing a semiconductor integrated circuit (IC) device may be very complicated and expensive. Therefore, prior to actually manufacturing the IC device, the IC device may be subjected to a simulation process verifying that it is operated according to a specified design. The most representative program used for an operation simulation of a circuit is a simulation program with integrated circuit emphasis (SPICE). The SPICE simulates a circuit model by using equations mathematically defining the operation of several circuit devices included in the circuit design and then provides the results as verification information. In order to simulate the circuit, various devices should be modeled. For this, a Berkeley short-channel (BSIM) insulated-gate field effect transistor (IGFET) has been the most widely used.
A varactor may be implemented having a MOS structure in a semiconductor integrated circuit device. Such a MOS varactor may have a high Q index and may occupy a small area when compared to a pn junction-type varactor. The MOS varactor may be used as an LC tank of a voltage controlled oscillator (VCO) for generating different VCO oscillating frequencies. The MOS varactor is one of important RF passive devices such as an analog filter, a switch capacitor circuit, a data converter, an RF circuit, and the like that may be applied to various fields. Since the characteristics of the MOS varactor may be varied according to the operating frequency, a physical geometry of the device, and a bias condition, an RF modeling is important.
However, in the modeling of the varactor, since the real component of impedance may be determined by the serial connection of R, L, and C, it may have a value that is not varied according to frequency. However, the real component in the actual measured value of impedance may vary according to frequency. The existing modeling method cannot explain this change. Accordingly, since it may be difficult to accurately model the device, the operation prediction of the circuit may not be accurate.
Moreover, optimally modeling the varactor using a CAD tool may be time-consuming and incurs a great deal of expense.
SUMMARY
Embodiments relate to a method for modeling a varactor, and more particularly to an RF modeling method of a varactor through a direct extraction method of parameters for explaining the parameters of the varactor varied according to frequency. Embodiments relate to an accurate RF modeling method of a varactor. Embodiments relate to an RF modeling method capable of explaining the change of parameters in a varactor according frequency. Embodiments relate to reducing the time required to extract parameters of a varactor having a physical meaning and optimize the parameter without using an expensive CAD tool.



DRAWINGS
Example FIG. 1 illustrates a varactor, in accordance with embodiments.
Example FIG. 2 illustrates an equivalent circuit diagram of a varactor, in accordance with embodiments.
Example FIG. 3A illustrates a real component Re(ZA) of port 1 gate impedance ZA obtained through a parameter direction extraction method, in accordance with embodiments.
Example FIG. 3B illustrates an imaginary component Im(ZA) of port 1 gate impedance ZA obtained through a parameter direction extraction, in accordance with embodiments.
Example FIG. 4A illustrates −1/Im(ZA) of port 1 gate impedance ZA obtained through a parameter direction extraction method, in accordance with embodiments.
Example FIG. 4B illustrates −1/Im(ZA)/ω of port 1 gate impedance ZA obtained through a parameter direction extraction method, in accordance with embodiments.
Example FIGS. 5A and 5B illustrates the parameters in question for obtaining Lgate through a parameter direction extraction method in accordance with embodiments.
Example FIGS. 6A to 6G illustrate an s-parameter plot from a comparison of an s-parameter obtained through a parameter direction extraction method in accordance with embodiments and an s-parameter obtained through a measurement using measurement equipment.



DESCRIPTION
As illustrated in example FIGS. 1 and 2, MOS varactor 10 can be formed of a structure (i.e., a branch) including a pair of MOS capacitors 20 formed on and/or over a semiconductor substrate such as p-type semiconductor substrate 12. Each MOS capacitor can include gate insulating film 22 formed on and/or over semiconductor substrate 12 and gate electrode 24 formed on and/or over gate insulating film 22. A pair of spacers 25 can be formed on sidewalls of gate insulating film 22 and gate electrode 24. Source 26 and drain 28 can be formed under sidewalls 25 in an n+ area of semiconductor substrate 12.
P-type substrate 12 can be connected to a ground through p+ region 14. Gate electrode 24 may be connected to common gate electrode (Gate) which is Port 1. Source 26 and drain 28 can be electrically connected to bulk terminal (Bulk). The bulk terminal in MOS varactor 10 having a 2-port network structure can be connected to ground through the P-type substrate 12.
As illustrated in example FIG. 2, in an equivalent circuit diagram, Lgate can serve as an overall inductance of Port 1 via and a gate. Rgate can serve as a unit cell via/contact resistance of Port 1 and the gate. Cpar can serve as a parasitic capacitance of the varactor while Cgate can serve as a variable capacity of a MOS-type variable capacitance. Rsd can serve as a unit cell via/contact resistance of Port 2 and a bulk. Lsd can serve as an overall inductance of Port 2 via and the bulk terminal. Dnwpsub can serve as a diode provided between an N-well and P-type substrate 12. Rsub can serve as a resistor for P-type substrate 12 and Csub can serve as a capacitor of P-type substrate 12.
Cgate can be represented by equation 1 below, wherein Cgmin is capacitance in the largest reverse bias, dCg is capacity coefficient, Vg is potential difference across Cgate, and dVg and Vgnorm are voltage coefficient.

Cgate=Cgmin+dCg×[1+tan h[(Vg−dVg)/Vgnorm]]  (1)

In accordance with embodiments, the following fact and devises can be provided in a method for modeling the varactor using a direct extraction method based on the fact.
First, unlike an inductor that requires the accurate modeling of the substrate in order to make Q index large since the loss due to the substrate is large, the varactor in accordance with embodiments has a sufficiently large Q index and the influence of the characteristics of the substrate on the characteristics of the varactor is relatively small.
Secondly, for the varactor in accordance with embodiments, Rgate can serve as a resistance component by way of a polysilicon regardless of a gate bias. Third, Lgate can have an inductance value through the gate connection. Fourth, Cox can serve as a capacitor by way of a gate insulating film (oxide film). Fifth, Cgate can serve as a capacitor made by way of a depletion layer, whereby its value can be very small in a depletion state. However, when the accumulation state by the increase of voltage, its value can be large. Sixth, Rsd can be represented by channel resistance from the source to the depletion layer. Lastly, the real component of Port 1 gate impedance ZA and Port 2 bulk impedance ZB can be high in frequency dependence.
In order to obtain Port 1 impedance ZA, Port 2 impedance ZB and substrate impedance ZC in the measured s-parameter, when transforming the measured s-parameter into a y-parameter and a z-parameter and then using the following equations 2 to 8 devised in accordance with embodiments, the values of Rgate, Cvar, Lgate, Rsd, and Lsd can be directly extracted without separately using an expensive CAD tool. Herein, the parameter Cvar can be formed of Cgate and Cpar as illustrated in example FIG. 2. Parasitic capacitance Cpar of the varactor can be associated with gate leakage current, whereby if this value is small enough to be disregarded, Cvar and Cgate can be the same.




  
    
      
        
          Re
          ⁡
          
            (
            
              Z
              A
            
            )
          
        
        =
        
          
            Re
            ⁡
            
              (
              
                
                  Z
                  11
                
                -
                
                  Z
                  12
                
              
              )
            
          
          =
          
            R
            gate
          
        
      
    
    
      
        (
        2
        )
      
    
  
  
    
      
        
          Im
          ⁡
          
            (
            
              Z
              A
            
            )
          
        
        =
        
          
            Im
            ⁡
            
              (
              
                
                  Z
                  11
                
                -
                
                  Z
                  12
                
              
              )
            
          
          =
          
            
              ω
              ⁢
              
                
              
              ⁢
              
                L
                gate
              
            
            -
            
              1
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  C
                  var
                
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  
  
    
      
        
          1
          
            Im
            ⁡
            
              (
              
                Z
                A
              
              )
            
          
        
        =
        
          
            1
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  L
                  gate
                
              
              -
              
                1
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  
                    C
                    var
                  
                
              
            
          
          =
          
            
              
                
                  -
                  ω
                
                ⁢
                
                  
                
                ⁢
                
                  C
                  var
                
              
              
                1
                -
                
                  
                    ω
                    2
                  
                  ⁢
                  
                    L
                    gate
                  
                  ⁢
                  
                    C
                    var
                  
                
              
            
            ≈
            
              
                -
                ω
              
              ⁢
              
                
              
              ⁢
              
                C
                var
              
            
          
        
      
    
    
      
        (
        4
        )
      
    
  
  
    
      
        
          Im
          ⁡
          
            (
            
              Z
              A
            
            )
          
        
        =
        
          
            Im
            ⁡
            
              (
              
                
                  Z
                  11
                
                -
                
                  Z
                  12
                
              
              )
            
          
          =
          
            
              ω
              ⁢
              
                
              
              ⁢
              
                L
                gate
              
            
            -
            
              1
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  C
                  var
                
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  
  
    
      
        
          Im
          (
          
            Z
            A
          
          )
        
        +
        
          
            1
            
              ω
              ⁢
              
                
              
              ⁢
              
                C
                var
              
            
          
          ⁢
          ω
          ⁢
          
            
          
          ⁢
          L
        
      
    
    
      
        (
        6
        )
      
    
  
  
    
      
        
          Re
          ⁡
          
            (
            
              Z
              B
            
            )
          
        
        =
        
          
            Re
            ⁡
            
              (
              
                
                  Z
                  22
                
                -
                
                  Z
                  12
                
              
              )
            
          
          =
          
            R
            sd
          
        
      
    
    
      
        (
        7
        )
      
    
  
  
    
      
        
          Im
          ⁡
          
            (
            
              Z
              B
            
            )
          
        
        =
        
          
            Im
            ⁡
            
              (
              
                
                  Z
                  22
                
                -
                
                  Z
                  12
                
              
              )
            
          
          =
          
            j
            ⁢
            
              
            
            ⁢
            ω
            ⁢
            
              
            
            ⁢
            
              L
              sd
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




In the above equations 2 to 8, Z11, Z12, Z21, and Z22 can be values obtained by transforming the s-parameter obtained by measurement equipment such as a vector network analyzer, where ω can be a constant and Cj can be junction capacitance between the n+ area and the p-type substrate.
The Rgate of parameters directly extracted using the above equations 2 to 8 is illustrated in example FIGS. 3A and 3B, the Cvar is illustrated in example FIGS. 4A and 4B, and Lgate is illustrated in FIGS. 5A and 5B.
The real component and imaginary component of substrate impedance Zc can be represented by the following equations:




  
    
      
        
          Re
          ⁡
          
            (
            
              Z
              C
            
            )
          
        
        =
        
          
            Re
            ⁡
            
              (
              
                Z
                12
              
              )
            
          
          =
          
            
              R
              sub
            
            
              1
              +
              
                
                  ω
                  2
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  C
                  sub
                  2
                
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          
            -
            1
          
          
            Im
            ⁡
            
              (
              
                Z
                C
              
              )
            
          
        
        =
        
          
            
              ω
              ⁢
              
                
              
              ⁢
              
                C
                j
              
            
            +
            
              
                ω
                3
              
              ⁢
              
                R
                sub
                2
              
              ⁢
              
                C
                sub
                2
              
              ⁢
              
                C
                j
              
            
          
          
            1
            +
            
              
                ω
                2
              
              ⁢
              
                R
                sub
                2
              
              ⁢
              
                
                  C
                  sub
                
                ⁡
                
                  (
                  
                    
                      C
                      sub
                    
                    +
                    
                      C
                      j
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




In a low frequency area of equation 10, Rsub can be obtained and thus, Csub can be obtained using equation 11.
The varactor illustrated in example FIG. 1 can be manufactured using a 0.13 μm CMOS process, whereby the varactor can be structured in a comb structure in order to reduce serial parasitic resistance. In the manufactured varactor in accordance with embodiments, a line width of metal can be 2.5 μm, its length can be 2 μm, a branch can be 1, and Nf can be 32. In such a varactor and which is in a wafer state, the s-parameter can be measured from 100 MHz to 10.1 GHz using an E8361A vector network analyzer and a cascade RF probe. Since the parasitic component is included in the wafer state, a two-step de-embedding can be performed to correct this.
As illustrated in example FIGS. 6A to 6G, the comparison result of the s-parameter measured on the varactor manufactured in accordance with embodiments and the parameters directly extracted through equations 2 to 11. Parameters S(4,4), S(4,3), S(3,3), S(3,1) are parameters directly extracted through the equations provided in accordance with embodiments and parameters S(2,2), S(2,1), S(1,2), S(1,1) are values measured by way of the equipment under the conditions described above.
As illustrated in the s-parameter plot of example FIGS. 6A to 6G, the parameters directly extracted in accordance with embodiments can be conformed to the measured parameters.
In accordance with embodiments, the modeling can be made reflecting the parameters of the varactor varied according to frequency so that a more accurate RF modeling of a passive device can be made. Also, an accurate modeling of the varactor can be made through the direct extraction method of the parameters so that the parameters of the varactor having a physical meaning can be extracted without using an expensive CAD tool and the time required for optimizing the parameters can be reduced. Finally, since embodiments can include the direct extraction method of the parameters, a computing time can be shortened, an initial condition dependency generated in performing the optimization is not required, and a pattern for a test is not required.
Although embodiments have been described herein, it should be understood that numerous other modifications and embodiments can be devised by those skilled in the art that will fall within the spirit and scope of the principles of this disclosure. More particularly, various variations and modifications are possible in the component parts and/or arrangements of the subject combination arrangement within the scope of the disclosure, the drawings and the appended claims. In addition to variations and modifications in the component parts and/or arrangements, alternative uses will also be apparent to those skilled in the art.

## Claims

1. A method comprising:
using a processor to perform steps of:
providing MOS varactor including a gate, a source, and a drain formed over a p-type semiconductor substrate,
calculating a first port gate impedance ZA, a second port bulk impedance ZB and a substrate impedance ZC in a measured s-parameter using a first model equation


Re(ZA)=Re(Z11−Z12)=Rgate 



and a second model equation




  
    
      Im
      ⁡
      
        (
        
          Z
          A
        
        )
      
    
    =
    
      
        Im
        ⁡
        
          (
          
            
              Z
              11
            
            -
            
              Z
              12
            
          
          )
        
      
      =
      
        
          ω
          ⁢
          
            
          
          ⁢
          
            L
            gate
          
        
        -
        
          1
          
            ω
            ⁢
            
              
            
            ⁢
            
              C
              var
            
          
        
      
    
  
  ,



where Lgate represents the overall inductance of the first port via and the gate, Rgate represents a unit cell via/contact resistance of the first port and the gate, and Cvar is formed of Cgate and Cpar, where Cgate represents a variable capacity of a MOS-type variable capacitance and Cpar represents a parasitic capacitance of the varactor, parameters Z11 and Z12 are values obtained by transforming the measured s-parameter, and ω is a constant.

2. The method of claim 1, wherein Cvar is directly extracted from equations:




  
    
      
        
          
            1
            
              Im
              ⁡
              
                (
                
                  Z
                  A
                
                )
              
            
          
          =
          
          ⁢
          
            
              1
              
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  
                    L
                    gate
                  
                
                -
                
                  1
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    
                      C
                      var
                    
                  
                
              
            
            =
            
              
                
                  
                    -
                    ω
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    C
                    var
                  
                
                
                  1
                  -
                  
                    
                      ω
                      2
                    
                    ⁢
                    
                      L
                      gate
                    
                    ⁢
                    
                      C
                      var
                    
                  
                
              
              ≈
              
                
                  -
                  ω
                
                ⁢
                
                  
                
                ⁢
                
                  C
                  var
                
              
            
          
        
        ,
      
    
  
  
    
      
        
          
            Im
            (
            
              Z
              A
            
            )
          
          =
          
          ⁢
          
            
              Im
              ⁡
              
                (
                
                  
                    Z
                    11
                  
                  -
                  
                    Z
                    12
                  
                
                )
              
            
            =
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  L
                  gate
                
              
              -
              
                1
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  
                    C
                    var
                  
                
              
            
          
        
        ,
        
          
        
        ⁢
        and
      
    
  
  
    
      
        
          
            Im
            ⁡
            
              (
              
                Z
                A
              
              )
            
          
          +
          
            1
            
              ω
              ⁢
              
                
              
              ⁢
              
                C
                var
              
            
          
        
        =
        
        ⁢
        
          ω
          ⁢
          
            
          
          ⁢
          
            L
            .

3. The method of claim 2, wherein the parameters Rsub and Csub are extracted from equations:




  
    
      
        
          Re
          ⁡
          
            (
            
              Z
              C
            
            )
          
        
        =
        
        ⁢
        
          
            Re
            ⁡
            
              (
              
                Z
                12
              
              )
            
          
          =
          
            
              
                R
                sub
              
              
                1
                +
                
                  
                    ω
                    2
                  
                  ⁢
                  
                    R
                    sub
                    2
                  
                  ⁢
                  
                    C
                    sub
                    2
                  
                
              
            
            ⁢
            
              
            
            ⁢
            and
          
        
      
    
  
  
    
      
        
          
            
              -
              1
            
            
              Im
              ⁡
              
                (
                
                  Z
                  C
                
                )
              
            
          
          =
          
          ⁢
          
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  C
                  j
                
              
              +
              
                
                  ω
                  3
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  C
                  sub
                  2
                
                ⁢
                
                  C
                  j
                
              
            
            
              1
              +
              
                
                  ω
                  2
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  
                    C
                    sub
                  
                  ⁡
                  
                    (
                    
                      
                        C
                        sub
                      
                      +
                      
                        C
                        j
                      
                    
                    )
                  
                
              
            
          
        
        ,
      
    
  



where Cj is a junction capacitance between a n-type region in the semiconductor substrate area and the p-type semiconductor substrate.

4. The method of claim 1, wherein parameters Rsd, Lsd, and Lgate are directly extracted from equations:


Re(ZB)=Re(Z22−Z12)=Rsd and Im(ZB)Im(Z22−Z12)=jωLsd,

where Rsd represents a unit cell via/contact resistance of the second port and a bulk terminal, Lsd represents an overall inductance of the second port via and the bulk terminal, Lgate represents an overall inductance of the first port via and the gate and Z21 and Z22 are values obtained by transforming the measured s-parameter.

5. The method of claim 2, wherein parameters Rsd, Lsd, and Lgate are directly extracted from equations:


Re(ZB)Re(Z22−Z12)=Rsd and Im(ZB)=Im(Z22−Z12)=jωLsd.

6. The method of claim 5, wherein parameters Rsub and Csub are directly extracted from equations:




  
    
      
        
          Re
          ⁡
          
            (
            
              Z
              C
            
            )
          
        
        =
        
        ⁢
        
          
            Re
            ⁡
            
              (
              
                Z
                12
              
              )
            
          
          =
          
            
              
                R
                sub
              
              
                1
                +
                
                  
                    ω
                    2
                  
                  ⁢
                  
                    R
                    sub
                    2
                  
                  ⁢
                  
                    C
                    sub
                    2
                  
                
              
            
            ⁢
            
              
            
            ⁢
            and
          
        
      
    
  
  
    
      
        
          
            
              -
              1
            
            
              Im
              ⁡
              
                (
                
                  Z
                  C
                
                )
              
            
          
          =
          
          ⁢
          
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  C
                  j
                
              
              +
              
                
                  ω
                  3
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  C
                  sub
                  2
                
                ⁢
                
                  C
                  j
                
              
            
            
              1
              +
              
                
                  ω
                  2
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  
                    C
                    sub
                  
                  ⁡
                  
                    (
                    
                      
                        C
                        sub
                      
                      +
                      
                        C
                        j
                      
                    
                    )
                  
                
              
            
          
        
        ,
      
    
  



where Cj is a junction capacitance between a n+-type region in the semiconductor substrate area and the p-type semiconductor substrate.

7. The method of claim 4, wherein the parameters Rsub and Csub are directly extracted from equations:




  
    
      
        
          Re
          ⁡
          
            (
            
              Z
              C
            
            )
          
        
        =
        
        ⁢
        
          
            Re
            ⁡
            
              (
              
                Z
                12
              
              )
            
          
          =
          
            
              
                R
                sub
              
              
                1
                +
                
                  
                    ω
                    2
                  
                  ⁢
                  
                    R
                    sub
                    2
                  
                  ⁢
                  
                    C
                    sub
                    2
                  
                
              
            
            ⁢
            
              
            
            ⁢
            and
          
        
      
    
  
  
    
      
        
          
            
              -
              1
            
            
              Im
              ⁡
              
                (
                
                  Z
                  C
                
                )
              
            
          
          =
          
          ⁢
          
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  C
                  j
                
              
              +
              
                
                  ω
                  3
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  C
                  sub
                  2
                
                ⁢
                
                  C
                  j
                
              
            
            
              1
              +
              
                
                  ω
                  2
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  
                    C
                    sub
                  
                  ⁡
                  
                    (
                    
                      
                        C
                        sub
                      
                      +
                      
                        C
                        j
                      
                    
                    )
                  
                
              
            
          
        
        ,
      
    
  



where Cj is a junction capacitance between a n+-type region in the semiconductor substrate area and the p-type semiconductor substrate.

8. The method of claim 1, wherein the parameters Rsub and Csub are extracted from equations:




  
    
      
        
          Re
          ⁡
          
            (
            
              Z
              C
            
            )
          
        
        =
        
        ⁢
        
          
            Re
            ⁡
            
              (
              
                Z
                12
              
              )
            
          
          =
          
            
              
                R
                sub
              
              
                1
                +
                
                  
                    ω
                    2
                  
                  ⁢
                  
                    R
                    sub
                    2
                  
                  ⁢
                  
                    C
                    sub
                    2
                  
                
              
            
            ⁢
            
              
            
            ⁢
            and
          
        
      
    
  
  
    
      
        
          
            
              -
              1
            
            
              Im
              ⁡
              
                (
                
                  Z
                  C
                
                )
              
            
          
          =
          
          ⁢
          
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  C
                  j
                
              
              +
              
                
                  ω
                  3
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  C
                  sub
                  2
                
                ⁢
                
                  C
                  j
                
              
            
            
              1
              +
              
                
                  ω
                  2
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  
                    C
                    sub
                  
                  ⁡
                  
                    (
                    
                      
                        C
                        sub
                      
                      +
                      
                        C
                        j
                      
                    
                    )
                  
                
              
            
          
        
        ,
      
    
  



where Cj is a junction capacitance between a n+-type region in the semiconductor substrate area and the p-type semiconductor substrate.

9. The method of claim 1, wherein Rgate comprises a resistance component by way of a polysilicon regardless of a gate bias.

10. The method of claim 1, wherein Lgate has an inductance value through a gate connection.

11. The method of claim 1, wherein Cgate comprises a capacitor made by way of a depletion layer, whereby its value can be very small in a depletion state.

12. The method of claim 1, wherein Rsd comprises a channel resistance from the source to the depletion layer.

13. The method of claim 1, wherein ZA and ZB are high in frequency dependence.

14. A method, using a processor, for modeling a MOS varactor including a gate, a source, and a drain formed over a p-type semiconductor substrate, the varactor having parameters including a first port gate impedance ZA and a second port bulk impedance ZB, a substrate impedance ZC, a parameter Rgate representing a unit cell via/contact resistance of the first port and the gate that are directly extracted from the following model equations;



  
    
      
        Re
        ⁢
        
          
        
        ⁢
        
          (
          
            Z
            A
          
          )
        
      
      =
      
        
          Re
          ⁢
          
            
          
          ⁢
          
            (
            
              
                Z
                11
              
              -
              
                Z
                12
              
            
            )
          
        
        =
        
          R
          gate
        
      
    
    ,
    
      

    
    ⁢
    
      
        Im
        ⁢
        
          
        
        ⁢
        
          (
          
            Z
            A
          
          )
        
      
      =
      
        
          Im
          ⁢
          
            
          
          ⁢
          
            (
            
              
                Z
                11
              
              -
              
                Z
                12
              
            
            )
          
        
        =
        
          
            ω
            ⁢
            
              
            
            ⁢
            
              L
              gate
            
          
          -
          
            1
            
              ω
              ⁢
              
                
              
              ⁢
              
                C
                var
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        Re
        ⁢
        
          
        
        ⁢
        
          (
          
            Z
            B
          
          )
        
      
      =
      
        
          Re
          ⁢
          
            
          
          ⁢
          
            (
            
              
                Z
                22
              
              -
              
                Z
                12
              
            
            )
          
        
        =
        
          R
          sd
        
      
    
    ,
    
      

    
    ⁢
    
      
        Im
        ⁢
        
          
        
        ⁢
        
          (
          
            Z
            B
          
          )
        
      
      =
      
        
          Im
          ⁢
          
            
          
          ⁢
          
            (
            
              
                Z
                22
              
              -
              
                Z
                12
              
            
            )
          
        
        =
        
          j
          ⁢
          
            
          
          ⁢
          ω
          ⁢
          
            
          
          ⁢
          
            L
            sd
          
        
      
    
    ,
    
      

    
    ⁢
    
      
        Re
        ⁢
        
          
        
        ⁢
        
          (
          
            Z
            C
          
          )
        
      
      =
      
        
          Re
          ⁢
          
            
          
          ⁢
          
            (
            
              Z
              12
            
            )
          
        
        =
        
          
            R
            sub
          
          
            1
            +
            
              
                ω
                2
              
              ⁢
              
                
              
              ⁢
              
                R
                sub
                2
              
              ⁢
              
                
              
              ⁢
              
                C
                sub
                2
              
            
          
        
      
    
    ,
    and
  




  
    
      
        
          -
          1
        
        
          Im
          ⁢
          
            
          
          ⁢
          
            (
            
              Z
              C
            
            )
          
        
      
      =
      
        
          
            ω
            ⁢
            
              
            
            ⁢
            
              C
              j
            
          
          +
          
            
              ω
              3
            
            ⁢
            
              
            
            ⁢
            
              R
              sub
              2
            
            ⁢
            
              
            
            ⁢
            
              C
              sub
              2
            
            ⁢
            
              
            
            ⁢
            
              C
              j
            
          
        
        
          1
          +
          
            
              ω
              2
            
            ⁢
            
              
            
            ⁢
            
              R
              sub
              2
            
            ⁢
            
              
            
            ⁢
            
              C
              sub
            
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  C
                  sub
                
                +
                
                  C
                  j
                
              
              )
            
          
        
      
    
    ,
  



where Lgate represents the overall inductance of the first port via and the gate, Rgate represents the unit cell via/contact resistance of the first port and the gate, Cpar represents a parasitic capacitance of the varactor, Cgate represents a variable capacity of variable capacitance, Rsd represents a unit cell via/contact resistance of the second port and a bulk terminal, Lsd represents an overall inductance of the second port via and the bulk terminal, Rsub represents a resistance of the p-type semiconductor substrate, and Csub represents a capacitance of the p-type semiconductor substrate.

15. The method of claim 14, wherein the parameter Cvar is directly extracted from the following model equations:




  
    
      
        
          
            1
            
              Im
              ⁡
              
                (
                
                  Z
                  A
                
                )
              
            
          
          =
          
          ⁢
          
            
              1
              
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  
                    L
                    gate
                  
                
                -
                
                  1
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    
                      C
                      var
                    
                  
                
              
            
            =
            
              
                
                  
                    -
                    ω
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    C
                    var
                  
                
                
                  1
                  -
                  
                    
                      ω
                      2
                    
                    ⁢
                    
                      L
                      gate
                    
                    ⁢
                    
                      C
                      var
                    
                  
                
              
              ≈
              
                
                  -
                  ω
                
                ⁢
                
                  
                
                ⁢
                
                  C
                  var
                
              
            
          
        
        ,
      
    
  
  
    
      
        
          
            Im
            ⁡
            
              (
              
                Z
                A
              
              )
            
          
          =
          
          ⁢
          
            
              
                I
                ⁢
                m
              
              ⁡
              
                (
                
                  
                    Z
                    11
                  
                  -
                  
                    Z
                    12
                  
                
                )
              
            
            =
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  L
                  gate
                
              
              -
              
                1
                
                  ω
                  ⁢
                  
                    
                  
                  ⁢
                  
                    C
                    var
                  
                
              
            
          
        
        ,
        and
      
    
  
  
    
      
        
          
            Im
            ⁡
            
              (
              
                Z
                A
              
              )
            
          
          +
          
            1
            
              ω
              ⁢
              
                
              
              ⁢
              
                C
                var
              
            
          
        
        =
        
        ⁢
        
          ω
          ⁢
          
            
          
          ⁢
          
            L
            .

16. The method of claim 15, wherein the parameters Rsd, Lsd, and Lgate are directly extracted from the following model equations:


Re(ZB)=Re(Z22−Z12)=Rsd, and



Im(ZB)=Im(Z22−Z12)=jωLsd.

17. The method of claim 15, wherein the parameters Rsub and Csub are extracted from the following model equations:




  
    
      
        
          
            Re
            ⁡
            
              (
              
                Z
                C
              
              )
            
          
          =
          
          ⁢
          
            
              Re
              ⁡
              
                (
                
                  Z
                  12
                
                )
              
            
            =
            
              
                R
                sub
              
              
                1
                +
                
                  
                    ω
                    2
                  
                  ⁢
                  
                    R
                    sub
                    2
                  
                  ⁢
                  
                    C
                    sub
                    2
                  
                
              
            
          
        
        ,
        and
      
    
  
  
    
      
        
          
            -
            1
          
          
            Im
            ⁡
            
              (
              
                Z
                C
              
              )
            
          
        
        =
        
        ⁢
        
          
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  C
                  j
                
              
              +
              
                
                  ω
                  3
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  C
                  sub
                  2
                
                ⁢
                
                  C
                  j
                
              
            
            
              1
              +
              
                
                  ω
                  2
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  
                    C
                    sub
                  
                  ⁡
                  
                    (
                    
                      
                        C
                        sub
                      
                      +
                      
                        C
                        j
                      
                    
                    )
                  
                
              
            
          
          .

18. The method of claim 14, wherein the parameters Rsub and Csub are directly extracted from the following model equations:




  
    
      
        
          
            Re
            ⁡
            
              (
              
                Z
                C
              
              )
            
          
          =
          
          ⁢
          
            
              Re
              ⁡
              
                (
                
                  Z
                  12
                
                )
              
            
            =
            
              
                R
                sub
              
              
                1
                +
                
                  
                    ω
                    2
                  
                  ⁢
                  
                    R
                    sub
                    2
                  
                  ⁢
                  
                    C
                    sub
                    2
                  
                
              
            
          
        
        ,
        and
      
    
  
  
    
      
        
          
            -
            1
          
          
            Im
            ⁡
            
              (
              
                Z
                C
              
              )
            
          
        
        =
        
        ⁢
        
          
            
              
                ω
                ⁢
                
                  
                
                ⁢
                
                  C
                  j
                
              
              +
              
                
                  ω
                  3
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  C
                  sub
                  2
                
                ⁢
                
                  C
                  j
                
              
            
            
              1
              +
              
                
                  ω
                  2
                
                ⁢
                
                  R
                  sub
                  2
                
                ⁢
                
                  
                    C
                    sub
                  
                  ⁡
                  
                    (
                    
                      
                        C
                        sub
                      
                      +
                      
                        C
                        j
                      
                    
                    )
                  
                
              
            
          
          .