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Some questions on Application Note 1287-11

Question asked by rajsodhi Employee on Jan 8, 2007
Latest reply on Mar 2, 2007 by rajsodhi
Various network analyzer calibration folks have suggested to look at the application note "Specifying Calibration Standards and Kits for Agilent Vector Network Analyzers", Application Note 1287-11.  I found this very helpful.  Thank you.  I did have some questions however.

Basically, I was able to figure out that each of the calibration kit elements are modeled as a potentially lossy transmission line followed by the standard itself.  So a measurement looking into this transmission line terminated by the standard itself would be modeled by the signal flow graph shown on page 6.  I cannot understand why figure 4 corresponds to the model of a transmission line, which is why I'm writing today.  I can see that in the middle of the transmission line model, there is the S-parameter block of a transmission line whose characteristic impedance equals Zo.  This would be the S-parameters [0, exp(-g*l) ; exp(-g*l) 0].  ("g" represents the lowercase Greek letter "gamma") But then there are two more sets of S-parameters represented by the cascade parameters Txi and Txo.  These have S-parameters looking like [G1 1+G2 ; 1+G1 G2].  Later, in equation 1.1, it states that G2=-G1 .  So this means that the S-parameters for XI are [G1 1-G1 ; 1+G1 -G1].  For the sake of argument, let's assume that Zc gives rise to G1 of 0.1.  Then it would seem that the transmission coefficient is greater than one, which doesn't make any sense.  Could someone point to how this signal flow graph representation of the transmission line is derived?

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I tried to derive the scattering parameters for a general transmission line on my own, figuring I could later convert to the chain scattering parameters to incorporate the effect of the offset transmission line. 

According to Pozar, the ABCD matrix of a lossless transmission line is as follows.

[      cos(t), i*zc*sin(t)]
[ i/zc*sin(t),      cos(t)]


Converting to S-parameters, we get:

              [              2     2                           ]
              [-I sin(t) (-zc  + zo )              2           ]
              [----------------------             ----         ]
              [       zo zc %1                     %1          ]
              [                                                ]
              [                                        2     2 ]
              [          2               -I sin(t) (-zc  + zo )]
              [         ----             ----------------------]
              [          %1                     zo zc %1       ]

                                     zc sin(t) I   sin(t) zo I
                    %1 := 2 cos(t) + ----------- + -----------
                                         zo            zc


According to Matthaei, Young and Jones, the ABCD matrix of any transmission line (lossless or otherwise) is as follows.

abcd =

[    cosh(gl), zc*sinh(gl)]
[ sinh(gl)/zc,    cosh(gl)]


Converting to S-parameters as we have done before, we obtain...


             [               2     2                            ]
             [  sinh(gl) (-zc  + zo )              2            ]
             [- ---------------------             ----          ]
             [        zo zc %1                     %1           ]
             [                                                  ]
             [                                          2     2 ]
             [          2                  sinh(gl) (-zc  + zo )]
             [         ----              - ---------------------]
             [          %1                       zo zc %1       ]

                                      zc sinh(gl)   sinh(gl) zo
                   %1 := 2 cosh(gl) + ----------- + -----------
                                          zo            zc


We may now convert to T-parameters.  Is there anything wrong with this approach?  Am I missing something here?  Is it safe to say that the two approaches are the same?  Since I could not reconcile equation 1.4 with the above development, I began to lose confidence, but perhaps the network analyzer calibration gurus at Agilent have dealt with this...

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The last issue involves taking a calibration kit provided by some vendor, and implementing the correction model.  By looking at the user defined calibration kit (coaxial), we may figure out the offset loss, offset delay, offset characteristic impedance, and the models for the "open" and "short" in terms of their inductance and capacitance.  The application note says we can use the following equations.
al = Offset_Loss*Offset_Delay/2/Offset_Zo*sqrt(f/1e9) ;
bl = 2*pi*f*Offset_Delay + al ;
Zc = Offset_Zo + (1 - j)*(Offset_Loss/4/pi/f)*sqrt(f/1e9) ;

My question here is on how to calculate gamma.  In equation 1.1, gamma is defined as gamma= alpha + j*beta.  What is provided in the above equations is alpha*length and beta*length.  Is it simply a matter of taking the equation for al and bl, and constructing gamma*length as gl = al+j*bl?  Or, could it be that since alpha*length shows up in the equation for bl, it has already been accounted for...   if you have a reference on this, that would be most appreciated as well.

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Thanks so much for taking the time to read all of my questions. 
I hope I'll be able to resolve most of them, with your help.

Yours

Raj Sodhi  

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