Abstract:
The invention related to a method and circuit that is used to compensate for S-parameters of a passive circuit which do not satisfy passivity. The method includes the following steps: (1) getting S-parameters which do not satisfy passivity, these S-parameters being composed of an S-parameter matrix S; (2) computing matrix [S×S′], wherein matrix S′ is a complex conjugate transposed matrix of the S-parameter matrix S; (3) computing the eigenvalues of the matrix [S×S′], and choosing an eigenvalue Ψ whose real part real(Ψ) is the biggest; (4) computing a compensating value ξ, the compensating value ξ being equal to real(Ψ) 1/2 ×(1+ε), wherein the ε is a very small positive number; and (5) dividing each of the S-parameters by the compensating value ξ to get the compensated S-parameters.

Description:
BACKGROUND 
     1. Technical Field 
     The present invention relates to a method and apparatus for the compensation of S-parameters of a passive circuit, and more particularly to a method to compensate S-parameters to satisfy passivity. 
     2. Description of Related Art 
     S-parameters are transmission and reflection coefficients for a circuit computed from measurements of voltage waves traveling toward and away from a port or ports of the circuit. Further, the S-parameters are related to frequency. In general, S-parameters are expressed either in terms of a magnitude and phase or in an equivalent form as a complex number having a real part and an imaginary part. Referring to  FIGS. 1 and 2 , a passive circuit  10  includes a port  11  and a port  12 . A set of four S-parameters, namely S 11 , S 12 , S 21 , and S 22  each represented by a complex number, provide a complete characterization of the performance of the two ports  11 , and  12  of the passive circuit  10  at a single frequency. These S-parameters form an S-parameter matrix. Because the circuit  10  is a passive circuit, the S-parameter matrix should satisfy passivity: real (eigenvalue[E−S×S′])≧0. It means that the real part of the eigenvalue of the matrix [E−S×S′] is not smaller than 0. The matrix S′ is a complex conjugate transposed matrix of the S-parameters matrix. However, in many instances, the measured S-parameters do not satisfy passivity for many different reasons, such as directivity and crosstalk related to signal leakage, source and load impedance mismatches related to reflections, and so on. 
     What is needed, therefore, is a method to compensate for the S-parameters of a passive circuit to satisfy passivity. 
     SUMMARY 
     A method is used to compensate for S-parameters of a passive circuit which do not satisfy passivity. The method includes the following steps: (1) getting S-parameters which do not satisfy passivity, these S-parameters being composed of an S-parameter matrix S; (2) computing matrix [S×S′], wherein matrix S′ is a complex conjugate transposed matrix of the S-parameter matrix S; (3) computing the eigenvalues of the matrix [S×S′], and choosing an eigenvalue Ψ whose real part real(Ψ) is the biggest; (4) computing a compensating value ξ, the compensating value ξ being equal to real(Ψ) 1/2 ×(1+ε), wherein the ε is a very small positive number; and (5) dividing each of the S-parameters by the compensating value ξ to get the compensated S-parameters. 
     Other advantages and novel features of the present invention will become more apparent from the following detailed description of preferred embodiment when taken in conjunction with the accompanying drawings, in which: 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a passive circuit connected to a test system; 
         FIG. 2  is a diagram of a relationship between an input and an output of the passive circuit; 
         FIG. 3  is a flow diagram of the test system measuring the S-parameters of a circuit; 
         FIG. 4  is a flow diagram of the compensating steps; 
         FIG. 5  is a block diagram of a compensating part of the test system of  FIG. 1 ; and 
         FIG. 6  is a block diagram of the test system of  FIG. 1  measuring S-parameters of a signal trace. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring to  FIG. 1 , a test system  20  for measuring S-parameters of the ports  11  and  12  of a passive circuit  10  is shown. The test system  20  includes a measuring part  22  and a compensating part  23 . 
       FIG. 3  shows a flow chart of the test system  20  measuring S-parameters. Firstly, in step  201 , the measuring part  22  measures S-parameters of the passive circuit  10  at a given frequency. Then in step  202  the test system  20  then computes if the measured S-parameters satisfy passivity. If the S-parameters satisfy passivity, then in step  205  the test system  20  outputs the original S-parameters. If the S-parameters do not satisfy passivity, then in step  204  the compensating part  23  adjusts the S-parameters to satisfy passivity, then in step  205  the test system  20  outputs the compensated S-parameters. 
     Referring to  FIGS. 4 and 5 ,  FIG. 4  shows the detailed steps of the step  204  of  FIG. 3 .  FIG. 5  shows modules of the compensating part  23 . The compensating part  23  includes a matrix computing module  31 , an eigenvalue computing module  32 , a compensating value computing module  33 , and a compensating module  34 . The eigenvalue computing module  32  is connected to the matrix computing module  31 , the compensating value computing module  33  is connected to the eigenvalue computing module  32 , and the compensating module  34  is connected to the compensating value computing module  33 . 
     The detailed steps of compensating the S-parameters includes the following steps:
         Step  301 , the compensating part  23  gets the S-parameters that do not satisfy passivity.   Step  302 , the matrix computing module  31  computes the matrix [S×S′]. The eigenvalue computing module  32  computes the eigenvalues of the matrix [S×S′]. Because the S-parameters do not satisfy passivity, there is an eigenvalue λ of the matrix [E−S×S′] whose real part is the smallest, and less than 0, that is real(λ)&lt;0. According to the formula of eigenvalue[E−S×S′]=1−eigenvalue[S×S′], there is an eigenvalue Ψ of the matrix [S×S′] whose real part is the biggest, and greater than 1, for real(Ψ)=1−real(λ)&gt;1.   Step  303 , the compensating value computing module  33  then computes a compensating value ξ according to the compensating value formula of ξ=real(Ψ) 1/2 ×(1+ε). In the above formula, the ε is a very small positive number.   Step  304 , the compensating module  34  compensates the S-parameters according to the compensating formula of S*=S/ξ to get the compensated S-parameters S*. Regarding the compensated S-parameters S*, the estimating formula used to estimate passivity is: real(eigenvalue[E−S*×S*′])=1−real(eigenvalue[S*×S*′])=1−real(eigenvalue[S/ξ×S′/ξ])=1−real(eigenvalue[S×S′])/[real(Ψ)×(1+ε) 2 ]. Because real(Ψ) is greater than 1, ε is a very small positive number, and the real(eigenvalue[S×S′]) is not bigger than real(Ψ). Therefore, real(eigenvalue[E−S*×S*′])/real(Ψ)×(1+ε) 2  is smaller than 1, and real(eigenvalue[E−S*×S*′])=1−real(eigenvalue[S×S′])/[real(Ψ)×(1+ε) 2 ]&gt;0. Thus, the compensated S-parameter matrix S* satisfies passivity.       

     Referring to  FIGS. 1 to 5 , described below is an embodiment of the compensating method used in a two port passive circuit. The two ports are the same kind of ports. The measuring part  22  measures S-parameters of the two port passive circuit at a given frequency (step  201 ), and forms an S-parameter matrix S=[(S 11 , S 12 ), (S 21 , S 22 )]. Because the two ports are the same, S 11  is equal to S 22 , and S 12  is equal to S 21 . Therefore, the S-parameter matrix S is equal to [(S 11 , S 12 ), (S 12 , S 11 )]. In the above equation, S 11  is (R 11 +I 11 i), and S 12  is (R 12 +I 12 i). If in step  202  it is found that the S-parameter matrix S does not satisfy passivity, then step  204  is performed to produce adjusted S-parameters, as detailed below:
         Step  301 , the compensating part  23  gets the S-parameter matrix S=[(S 11 , S 12 ), (S 12 , S 11 )].   Step  302 , the matrix computing module  31  computes the matrix [S×S′]=[(R 11   2 +R 12   2 +I 11   2 +I 12   2 , 2×R 11 ×R 12 +2×I 11 ×I 12 ),(2×R 11 ×R 12 +2×I 11 ×I 12 , R 11   2 +R 12   2 +I 11   2 +I 12   2 )]. The eigenvalue computing module  32  computes the eigenvalues of the matrix [S×S′], and the eigenvalues are Ψ 1 =(I 11 +I 12 ) 2 +(R 11 +R 12 ) 2 , and Ψ 2 =(I 11 −I 12 ) 2 +(R 11 −R 12 ) 2 . Then, the two eigenvalues are compared to get an eigenvalue Ψ max  from the two eigenvalues whose real part is the biggest real part of all the eigenvalues.   Step  303 , the compensating value computing module  33  computes a compensating value ξ according to the compensating value formula of ξ=real(Ψ max ) 1/2 ×(1+ε). In the above formula, ε is a very small positive number.   Step  304 , the compensating module  34  compensates the S-parameter matrix S according to the compensating module S*=S/ξ to get the compensated S-parameter matrix S* which satisfies passivity.       

     A realistic example is described below using the compensating method to compensate S-parameters. Referring to  FIG. 6 , a signal trace  60  is laid on a printed circuit board  50 . The procedure is as follows:
         Step  201 , the test system  20  measures S-parameters of the signal trace  60  at a frequency of 4.6372 GHZ, and forms an S-parameter matrix:   [(−0.2608352337196621+0.3476273125912422i,   0.7203853298827190+0.5405228611018837i),   (0.7203853298827190+0.5405228611018837i,   −0.2608352337196621+0.3476273125912422i)].   Step  202 , in the above described S-parameter matrix, eigenvalues of the matrix [E−S×S′] are 2.978085395288764×10 −6  and −2.487071395607110×10 −6 . One of the eigenvalues is smaller than 0, so the S-parameters of the signal trace  60  do not satisfy passivity.   Step  204 , then, the compensating part  23  compensates by adjusting the S-parameters of the signal trace  60 , as described below:   Step  301 , the compensating part  23  gets the S-parameter matrix S=   [(−0.2608352337196621+0.3476273125912422i,   0.7203853298827190+0.5405228611018837i),   (0.7203853298827190+0.5405228611018837i,   −0.2608352337196621+0.3476273125912422i)].   Step  302 , the matrix computing module  31  computes the matrix [S×S′]. The eigenvalue computing module  32  computes the eigenvalues of the matrix [S×S′]. Then, compare the eigenvalues, and get an eigenvalue Ψ max =1.000002487071396 whose real part is the biggest.   Step  303 , the compensating value computing module  33  then computes a compensating value ξ according to the compensating value formula of ξ=real(Ψ max ) 1/2 ×(1+ε)=1.000001243535925. In the above equation, the ε is set to 1×10 −12 .   Step  304 , the compensating module  34  compensates the S-parameter matrix S according to the compensating module S*=S/ξ to get the compensated S-parameter matrix S*=   [(−0.2608349093620819+0.3476268803047282, 0.7203844340587956+0.5405221889431238i), (0.7203844340587956+0.5405221 889431238I, −0.2608349093620819+0.3476268803047282)].       

     For the above matrix S*, the eigenvalues of the matrix [S*×S*′] are 5.465145198668697×10 −6 , and 2.000122290161193×10 −12 . Both of the two eigenvalues are bigger than 0, so the matrix S* satisfies passivity. 
     It is to be understood, however, that even though numerous characteristics and advantages of the present invention have been set forth in the foregoing description, together with details of the structure and function of the invention, the disclosure is illustrative only, and changes may be made in detail, especially in matters of shape, size, and arrangement of parts within the principles of the invention to the full extent indicated by the broad general meaning of the terms in which the appended claims are expressed.