[SI-LIST] Re: S-parameter Bisection - summary

  • From: "Yuriy Shlepnev" <shlepnev@xxxxxxxxxxxxx>
  • To: "'Moeller, Merrick'" <mmoeller@xxxxxxxxxxxxxxx>, <si-list@xxxxxxxxxxxxx>
  • Date: Fri, 13 Nov 2009 12:53:24 -0800

Merrick,

You have started very interesting thread and have got a lot of useful
recommendations. I enjoyed reading it and just wanted to summarize a few
things.

The problem of splitting a reciprocal, balanced symmetrical multiport into
two parts cannot be solved in general - it is un-determined inverse problem.
Let me explain. S-matrix of a two-port (one-conductor line segment with
connectors for instance) have 2 independent complex parameters S11=S22 and
S12=S21. If we want to split it in the middle, the S-parameters of each half
are identical, but there are 3 independent parameters to find due to the
un-symmetry of each half - SA11, SA12=SA21 and SA22. If we convert S into T
matrix and SA into TA, then the following matrix equation arise:
T=TA*P*TA^-1*P
where P is the permutation matrix (to interchange b and a) and TA is the
matrix with 3 unknowns. Only 2 elements of T are independent due to symmetry
and reciprocity and 3 elements in TA due to the reciprocity.
As you can see this is not a square root of matrix problem - it becomes such
only if both halves of the multiport are also symmetrical (T=TA*TA in this
case, or two identical line segments without transitions). Otherwise it is a
problem with only 2 independent non-linear equations and 3 unknowns.
One way to solve it is to add more equations - measure S-parameters for
lines with different lengths for instance. The problem is very sensitive to
noise and physical non-identity of the test fixtures.  In addition the
problem is solvable with frequency-dependent uncertainty in the
characteristic or in the normalization impedance of the port in the middle
(you are lucky if it is close to 50). There must be an experiment with a
resistive element to remove the impedance uncertainty (there was a theorem
about that in some very old books on microwave theory).

The alternative approach based on guessed equivalent circuit with a line in
the middle and some circuit modeling the transitions is as good as the line
model that it is based upon. The line model in such approach should be
verified over the frequency band of interest (include frequency-dependent
effects of dielectric, conductor and high-frequency dispersion). Otherwise
the error is just hidden into the transition circuits and propagate further
into the de-embedded circuits.

Best regards,
Yuriy Shlepnev
www.simberian.com 


-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx] On
Behalf Of Moeller, Merrick
Sent: Tuesday, November 10, 2009 10:04 AM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] S-parameter Bisection

 
Is it possible to split a passive S-parameter matrix into two equal
sections? 

 

For example; given s-parameters for a transmission line of a 10" length
can the matrix be split into two 5" lengths. Assume that the passive
s-parameter obeys reciprocity. 

 

The reason I'm asking is that I have a test fixture on each end of my
measurement that I'm trying to de-embed. The insertion loss can be
subtracted very easily, but the return loss and other parameters have a
great deal of error dealing with de-embedding from one side of the DUT
only. I'm hoping to break the fixture into two equal parts without
having a direct point of measurement to do so otherwise. 

 

 

Merrick M. Moeller

 



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  • » [SI-LIST] Re: S-parameter Bisection - summary - Yuriy Shlepnev