S-parameter and ABCD matrix

abcd matrix

Hi all,

Suppose I have a two-port non-symmetric passive network whose S-par is Sx=[S11,S12;S21,S22]=[S11,S12;S12,S22].
Let's say the ABCD matrix of this network is Ax=[A,B;C,D].

If I rotate the device 180 degree, meaning port2 becomes port1 and vice versa, then the S-parameter would be Sy=[S22,S12;S12,S11]. Hence, the ABCD matrix would then be Ay=[D,B;C,A].

Am I missing something here?
In the "Microwave Engineering" by David M. Pozar book, it is written that Ay=inverse[Ax].

regards,
wlcsp

abcd parameters

Assuming that the system is lossless,
A x inv A = I

abcd s parameters

If you flip the terminals of the asymmetric, the ABCD matrix becomes [D B C A]. This is consistent with the S - parameters becoming [S22 S12 S12 S11].

I don't think the Ay=inv(Ax) is true. Consider a potential divider. If I take two potential dividers and flip the ports of one of them and cascade the two I would get a T-network. If Ay=inv(Ax), then the overall ABCD matrix of the T-network would be 'I', which is certainly not correct.

abcd matrix microwave

take an amplifier as an examle, according to the meaning of S parameter. Sx=[S11,S12;S21,S22]
you can also take an inversed amplier as a nomal amplifier ,port2 becomes to port1 and port1 becomes port2，so change 2 to 1 and change 1 to 2 in S parameter matrix will do， i think Sy=[S22,S21;S12,S11] is right，to get ABCD matrix ,you can use function s2abcd in matlab.and from S to ABCD,Formula is also availble in many books.

matlab s2abcd

I fully agree with spminn and molidong.
This question is actually related to TRL calibration technique described in the famous book of David Pozar "Microwave Engineering". It is written (in terms of ABCD matrix):
[Measurement]=[Error][DUT]inv[Error]. I think, this is incorrect.

Let's say
[Error]=[A,B;C,D], then
[Measurement]=[A,B;C,D][DUT][D,B;C,A]

Related to the same topic, given equations 4.74a, 4.74b, 4.75, 4.76a, and 4.76b, Pozar derived the equations 4.77a, 4.77b and 4.78 (please see the attachment).
I believe 4.77a, 4.77b and 4.78 are all incorrect.

Could someone try to solve this set of equations?

regards,
wlcsp

Re: matlab s2abcd

Hi wlcsp
I have the same question about this chapter of TRL calibration from "David Pozer Microwave Engineering".
I think the relationship of calibration matrix as below which you metion is incorrect.
[Measurement]=[Error][DUT]inv[Error].
I guess the author want to express the formula of TRL calibration structure is below. But I don't know why he can say [error2]=inv[error1]
[Measurement]=[Error1][DUT][Error2].
In the book, the error1 and error2 matrix can be derived by [S11 S12 S21 S22] , or as you say that assuming error1 is equal to [A B C D] then we flip error1 and get error2 is equal to [D B C A].
so If we can know the parameter of S11, S12 and S22 etc. We can get error1 and error2 ABCD matrix model.
In the book, the author uses "signal flow method" to derive the relationship equation 4.74a~4.79b. I don't see something wrong.
So I think the 4.80~4.83 is correct. But the question is
"Is [error2] equal to inv[error1]"
:|
Maybe the 3rd edition of Microwave Engineering corrects this issue.

wlcsp said:

I fully agree with spminn and molidong.
This question is actually related to TRL calibration technique described in the famous book of David Pozar "Microwave Engineering". It is written (in terms of ABCD matrix):
[Measurement]=[Error][DUT]inv[Error]. I think, this is incorrect.

Let's say
[Error]=[A,B;C,D], then
[Measurement]=[A,B;C,D][DUT][D,B;C,A]

Related to the same topic, given equations 4.74a, 4.74b, 4.75, 4.76a, and 4.76b, Pozar derived the equations 4.77a, 4.77b and 4.78 (please see the attachment).
I believe 4.77a, 4.77b and 4.78 are all incorrect.

Could someone try to solve this set of equations?

regards,
wlcsp

Click to expand...

Re: matlab s2abcd

macgyfu said:

Hi wlcsp
I have the same question about this chapter of TRL calibration from "David Pozer Microwave Engineering".
I think the relationship of calibration matrix as below which you metion is incorrect.
[Measurement]=[Error][DUT]inv[Error].
I guess the author want to express the formula of TRL calibration structure is below. But I don't know why he can say [error2]=inv[error1]
[Measurement]=[Error1][DUT][Error2].
In the book, the error1 and error2 matrix can be derived by [S11 S12 S21 S22] , or as you say that assuming error1 is equal to [A B C D] then we flip error1 and get error2 is equal to [D B C A].
so If we can know the parameter of S11, S12 and S22 etc. We can get error1 and error2 ABCD matrix model.
In the book, the author uses "signal flow method" to derive the relationship equation 4.74a~4.79b. I don't see something wrong.
So I think the 4.80~4.83 is correct. But the question is
"Is [error2] equal to inv[error1]"
:|
Maybe the 3rd edition of Microwave Engineering corrects this issue.

Click to expand...

i'm currently looking for information about TRL calibration as well.
refer to my first thread.
https://www.edaboard.com/threads/252670/

pozar is the only book i found that describe how to do the calculations.

the 3rd and 4th edition did not correct the mistake.
The TRL chapter is basically copy and paste job in 3rd and 4th edition.

---------- Post added at 16:32 ---------- Previous post was at 16:26 ----------

wlcsp said:

I fully agree with spminn and molidong.
This question is actually related to TRL calibration technique described in the famous book of David Pozar "Microwave Engineering". It is written (in terms of ABCD matrix):
[Measurement]=[Error][DUT]inv[Error]. I think, this is incorrect.

Let's say
[Error]=[A,B;C,D], then
[Measurement]=[A,B;C,D][DUT][D,B;C,A]

Related to the same topic, given equations 4.74a, 4.74b, 4.75, 4.76a, and 4.76b, Pozar derived the equations 4.77a, 4.77b and 4.78 (please see the attachment).
I believe 4.77a, 4.77b and 4.78 are all incorrect.

Could someone try to solve this set of equations?

regards,
wlcsp

Click to expand...

you are right about the mistake made by pozar.
after correcting the mistake, i tried to simplified the equation, but it was really difficult, after long hours, i ended up with a power 5 equation of e^-gamma*L. which i believe is impossible to solve.

an easier method is to use equation 4.80, insert it into the corrected equation 4.77a, this will lead to a cubic equation.
which i solved using matlab, but it doesn give good answers, because it will have more than 1 answer, and the correct one is the one that cause both real and imaginary of gamma to be positive, but some of them do not satisfied this. meaning non of the answer can cause gamma to have both positive real and imaginary terms.