ABCD parameters matrix of unsymmetric coupled lines

[PlanarMetamaterials]

After starting to write up this process, I realized what you want may be more simple than what we've been discussing.

I've been trying to help you determine quantities used in the equations from post #1, but if you're willing to do a numerical simulation, you can effectively bypass that derivation and compute the ABCD matrix directly.

Since you have the [Z] and [Y] parameters of the MTL, you can convert either of them (or average both, for accuracy) to your ABCD matrix directly:

\[ \left[Y\right] = \left[ \begin{array}{cc} \left[Y_A\right] & \left[Y_B\right] \\ \left[Y_C\right] & \left[Y_D\right] \\ \end{array} \right] \ \ \ \ ,\ \ \ \ \left[ABCD\right] = \left[ \begin{array}{cc} -\left[Y_C\right]^{-1} \left[Y_D\right] & -\left[Y_C\right]^{-1} \\ \left[Y_B\right] - \left(\left[Y_A\right] \left[Y_C\right]^{-1} \left[Y_D\right]\right) & -\left[Y_A\right] \left[Y_C\right]^{-1} \\ \end{array} \right] \]

\[ \left[Z\right] = \left[ \begin{array}{cc} \left[Z_A\right] & \left[Z_B\right] \\ \left[Z_C\right] & \left[Z_D\right] \\ \end{array} \right] \ \ \ \ ,\ \ \ \ \left[ABCD\right] = \left[ \begin{array}{cc} \left[Z_A\right] \left[Z_C\right]^{-1} & \left( \left[Z_A\right] \left[Z_C\right]^{-1} \left[Z_D\right] \right) - \left[Z_B\right] \\ \left[Z_C\right]^{-1} & \left[Z_C\right]^{-1} \left[Z_D\right]^{-1} \\ \end{array} \right] \]

Would this solve your problem, or do you still want to go through the analytical process using the equation from post #1?

 

[promach]

do you still want to go through the analytical process using the equation from post #1?

YES, the whole purpose of the thread is to prove analytically the ABCD parameters of asymmetric coupled transmission lines
I wish to proceed with both analytical proof and numerical simulation.

For numerical proof, how do I actualy make use of the csv file output (in post #19) which consists of so much info ?

 

[PlanarMetamaterials]

YES, the whole purpose of the thread is to prove analytically the ABCD parameters of asymmetric coupled transmission lines
I wish to proceed with both analytical proof and numerical simulation.

Alright; for the analytical derivation, I think you have everything you need in post #4; however there are some issues that I've noticed:

1) It seemed strange that the authors of the paper you cited did not include geometrical information about the lines (i.e., [Ti] and [Tv]), and moreover described the modes as odd and even, which is typically only done in the case of symmetric lines. So, I went and read the paper, where I noticed that their derivation is for symmetric coupled lines. As such, the equations you provided on post #1 will not work for asymmetric lines. (Updated: Note that the authors refer to "Unsymmetric Networks" for their Fig. 9, but they are just referring to the terminations of the lines, not the fact that the lines themselves are asymmetric!)

2) The authors chose to use some non-standard mode definitions in their equations 1 and 2 -- specifically, there are some factors of 1/2 used where they typically shouldn't be -- this will result in their derivations having erroneous factors of 2 you won't find elsewhere. It's these mode definitions which depend on line geometry, and are only of this form for symmetric lines.

3) My guesses for the meaning of the author's variables \( Y_{oe} \), \( Y_{oo} \) and \( Z_{oe} \), and \( Z_{oo} \) in post #4 were incorrect: they define these in the paper as well, as the characteristic admittances and impedances of the even and odd modes. I should have written:

\[
\left[Z_{cM} \right] = \left[ \begin{array}{cc} Z_{0e} & 0 \\ 0 & Z_{0o} \\ \end{array} \right]
\left[Z_{cM} \right]^{-1} = \left[ \begin{array}{cc} Y_{0e} & 0 \\ 0 & Y_{0o} \\ \end{array} \right]
\]

For numerical proof, how do I actualy make use of the csv file output (in post #19) which consists of so much info ?

The files contain the 4x4 complex matrices of [Y] and [Z]. They also tell you which entry corresponds to which matrix entry in the header line.

 

[promach]

1) It seemed strange that the authors of the paper you cited did not include geometrical information about the lines (i.e., [Ti] and [Tv]), and moreover described the modes as odd and even, which is typically only done in the case of symmetric lines. So, I went and read the paper, where I noticed that their derivation is for symmetric coupled lines. As such, the equations you provided on post #1 will not work for asymmetric lines.

Wait, why are odd mode and even mode ONLY for symmetric coupled lines ?

From what I understand, asymmetric coupled lines differ only at the termination impedance condition.
I am just a beginner, please correct me if wrong.

 

[PlanarMetamaterials]

Wait, why are odd mode and even mode ONLY for symmetric coupled lines ?

Well, there will still be two supported modes, and you may call them "even" and "odd" if you wish, but the matrices that define the modes ([Ti] and [Tv]) are dependent on geometry, and so will lack the symmetry you see for symmetric lines (i.e., the values I gave in post #4 will be different for asymmetric lines).

 

[promach]

the matrices that define the modes ([Ti] and [Tv]) are dependent on geometry, and so will lack the symmetry you see for symmetric lines (i.e., the values I gave in post #4 will be different for asymmetric lines).

It seems to me that the theory inside MTL book by Paul Clayton is targeted only for asymmetric coupled transmission lines. Or how do I modify equation (7.82) for asymmetric coupled transmission lines ?

By the way, have a look at Asymmetric Coupled Transmission Lines in an Inhomogeneous Medium

 

[PlanarMetamaterials]

It seems to me that the theory inside MTL book by Paul Clayton is targeted only for asymmetric coupled transmission lines.

The book covers the general theory, as well as many specific cases, including both symmetric and asymmetric lines.

Have a look at Asymmetric Coupled Transmission Lines in an Inhomogeneous Medium

Yes, you can see from the derivation in that work that things become a little bit more messy when you're dealing with asymmetric lines. Usually, when you have two asymmetric lines, the two modes are referred to as the c-mode and the pi-mode, as they are in that paper. Was there something in particular you were pointing out?

 

[promach]

Why the matrices that define the modes ([Ti] and [Tv]) are dependent on geometry ?

And how do I modify equation (7.82) in MTL book for asymmetric coupled transmission lines ?

 

[PlanarMetamaterials]

Why the matrices that define the modes ([Ti] and [Tv]) are dependent on geometry ?

Most parameters are geometry-dependent; those parameters in particular describe the relative distribution of voltages and currents of each mode, which depend on the MTL's mutual inductances and capacitances, which of course depends on geometry.

And how do I modify equation (7.82) in MTL book for asymmetric coupled transmission lines ?

This is what I was desribing previously. There are unfortunately no good closed-form expression for an asymmetric microstrip case. I determine these for asymmetric cases through simulation, such as the HFSS setup I gave in post #18.

 

[promach]

For post #21 , how do I interpret the following csv output from HFSS ?

It is bit messy at first look ...

"Freq [GHz]",
"Zt(Conductor1_T1,Conductor1_T1) []","Zt(Conductor2_T1,Conductor1_T1) []","Zt(Conductor3_T1,Conductor1_T1) []",
"Zt(Conductor1_T2,Conductor1_T1) []","Zt(Conductor2_T2,Conductor1_T1) []","Zt(Conductor3_T2,Conductor1_T1) []",
"Zt(Conductor1_T1,Conductor2_T1) []","Zt(Conductor2_T1,Conductor2_T1) []","Zt(Conductor3_T1,Conductor2_T1) []",
"Zt(Conductor1_T2,Conductor2_T1) []","Zt(Conductor2_T2,Conductor2_T1) []","Zt(Conductor3_T2,Conductor2_T1) []",
"Zt(Conductor1_T1,Conductor3_T1) []","Zt(Conductor2_T1,Conductor3_T1) []","Zt(Conductor3_T1,Conductor3_T1) []",
"Zt(Conductor1_T2,Conductor3_T1) []","Zt(Conductor2_T2,Conductor3_T1) []","Zt(Conductor3_T2,Conductor3_T1) []",
"Zt(Conductor1_T1,Conductor1_T2) []","Zt(Conductor2_T1,Conductor1_T2) []","Zt(Conductor3_T1,Conductor1_T2) []",
"Zt(Conductor1_T2,Conductor1_T2) []","Zt(Conductor2_T2,Conductor1_T2) []","Zt(Conductor3_T2,Conductor1_T2) []",
"Zt(Conductor1_T1,Conductor2_T2) []","Zt(Conductor2_T1,Conductor2_T2) []","Zt(Conductor3_T1,Conductor2_T2) []",
"Zt(Conductor1_T2,Conductor2_T2) []","Zt(Conductor2_T2,Conductor2_T2) []","Zt(Conductor3_T2,Conductor2_T2) []",
"Zt(Conductor1_T1,Conductor3_T2) []","Zt(Conductor2_T1,Conductor3_T2) []","Zt(Conductor3_T1,Conductor3_T2) []",
"Zt(Conductor1_T2,Conductor3_T2) []","Zt(Conductor2_T2,Conductor3_T2) []","Zt(Conductor3_T2,Conductor3_T2) []"

1,1.85318297710533 - 1617.8252076386i,0.458236490803877 - 275.363358662524i,0.149987206929787 - 63.0248099805453i,
1.85033796940584 - 1618.70307443157i,0.457682486701148 - 275.59930477006i,0.149770825587866 - 63.113123075575i,
0.458236827310535 - 275.363358661025i,3.21025291749103 - 3057.64703847593i,0.458018863699714 - 275.396441806971i,
0.457779195805881 - 275.601968700337i,3.20313275153284 - 3059.13210583941i,0.457606610909862 - 275.631011423439i,
0.149987359192181 - 63.0248099828175i,0.45801867404188 - 275.396441798628i,1.85339384619673 - 1617.93614845377i,
0.149775902194139 - 63.1170489024334i,0.457497459679077 - 275.632151191042i,1.85067847919482 - 1618.80996415221i,
1.85033796941932 - 1618.70307443175i,0.457778859155545 - 275.601968693563i,0.149775749808132 - 63.1170489014407i,
1.85307557781961 - 1617.8296604526i,0.458045585731561 - 275.36514335357i,0.150011648927944 - 63.025493694943i,
0.457682823321486 - 275.599304776068i,3.20313275149206 - 3059.13210583844i,0.457497649271842 - 275.632151207055i,
0.458045922495418 - 275.365143367856i,3.2099425176227 - 3057.64607921038i,0.457904936317074 - 275.393915650728i,
0.149770977989986 - 63.1131230769746i,0.457606421284723 - 275.631011406787i,1.85067847919815 - 1618.80996415245i,
0.150011801453644 - 63.0254936950657i,0.457904746757289 - 275.393915626404i,1.85354836954978 - 1617.93252449156i

 

[PlanarMetamaterials]

For post #21 , how do I interpret the following csv output from HFSS ?

It is bit messy at first look ...

I need to apologize for a small error: I misstated earlier in post 23 these data represented 4x4 matrices; I was thinking of your case with 2 non-reference conductors per port. The matrix is actually 6x6 since there are three non-reference conductor per port in my file (and two ports).

The information in the header specifies this: Conductor[n] is the nth conductor, and T[m] is the mth port. So for example, the entry labelled "Zt(Conductor3_T2,Conductor2_T1)" represents the impedance between the 3rd non-reference conductor on Port 2, and the 2nd non-reference conductor on Port 1.

Of course, the [Z] matrix is part of the equation \( \vec{V} = \left[Z\right] \vec{I} \), where the vectors \( \vec{V} \) and \( \vec{I} \) are in the same order as the matrix specifies: that is:

\[ \vec{V} = \left[ \begin{array}{c} V_1^1 \\ V_2^1 \\ V_3^1 &\\ V_1^2 \\ V_2^2 \\ V_3^2 \end{array} \right] \]
\[ \vec{I} = \left[ \begin{array}{c} I_1^1 \\ I_2^1 \\ I_3^1 &\\ I_1^2 \\ I_2^2 \\ I_3^2 \end{array} \right] \]

In which the subscripts are the conductor index and the superscripts are the port numbers. One could then re-arrange this system to get the ABCD matrices of post #21 (keeping in mind the presumed orientations and signs of the currents).

 

[promach]

So for example, the entry labelled "Conductor3_T2" is the 3rd non-reference conductor on Port 2.

Would you be able to point out in the following screenshot, which is "Conductor3_T2" ?

--- Updated Aug 24, 2020 ---

in post #23, why are the characteristic admittances and impedances of the even and odd modes placed at the diagonal position of the matrix ?

 

[PlanarMetamaterials]

Would you be able to point out in the following screenshot, which is "Conductor3_T2" ?

The terminals are labelled under "Excitations" in the menu on the left.

in post #23, why are the characteristic admittances and impedances of the even and odd modes placed at the diagonal position of the matrix ?

Modal-domain data represents the independent waves travelling in an MTL; the absence of off-diagonal elements indicates that there is no coupling between them. (As a matter of note; this isn't always possible: see Faria's work: "On the modal decomposition of n‐coupled transmission lines" ; although these are generally considered edge cases which you won't come across in real life).

 

[promach]

there are some factors of 1/2 used where they typically shouldn't be -- this will result in their derivations having erroneous factors of 2 you won't find elsewhere.

How to derive equations (1) and (2) ?

Why did you mention in post #23, that the coefficient of 1/2 is redundant ?

 

[PlanarMetamaterials]

The matrices in those equations are just modal-domain ABCD matrices.

The voltage and current quantities are not derived in that work; they are defined -- which is, unfortunately, why the definitions are incorrect. You may compute these quantities using [Ti] and [Tv], and equations 7.8 and 7.10 from Paul's text.

For example, using the matrices I gave in post #4 (you'll have to invert them), and equations 7.8, yields:

\[ V_e = \frac{1}{2}\left(V_1 + V_2 \right) \]
\[ I_e = \left(I_1 + I_2 \right) \]

\[ V_o = \left(V_1 - V_2 \right) \]
\[ I_o = \frac{1}{2}\left(I_1 - I_2 \right) \]

which you'll notice has some factors of 2 different from the work you're looking at.

 

[promach]

According to posts #21 and #30 , could I infer the following ?

Note: I am assuming that the Z-matrix is arranged in 6-by-6 format.


Z_A = "Zt(Conductor1_T1,Conductor1_T1) []" <-- top left corner

Z_B = "Zt(Conductor3_T2,Conductor1_T1) []" <-- top right corner

Z_C = "Zt(Conductor1_T1,Conductor3_T2) []" <-- bottom left corner

Z_D = "Zt(Conductor3_T2,Conductor3_T2) []" <-- bottom right corner

 

[PlanarMetamaterials]

According to posts #21 and #30 , could I infer the following ?

Note: I am assuming that the Z-matrix is arranged in 6-by-6 format.


Z_A = "Zt(Conductor1_T1,Conductor1_T1) []" <-- top left corner

Z_B = "Zt(Conductor3_T2,Conductor1_T1) []" <-- top right corner

Z_C = "Zt(Conductor1_T1,Conductor3_T2) []" <-- bottom left corner

Z_D = "Zt(Conductor3_T2,Conductor3_T2) []" <-- bottom right corner

The matrix is symmetric, but I think normally "Zt(Conductor3_T2,Conductor1_T1) []" would be expressed at the bottom-left corner and "Zt(Conductor1_T1,Conductor3_T2) []" would be expressed the top-right corner.

The sub-matrices are literally just quarter of the original matrix, so each is 3x3, such that

\[ Z_A = \left[ \begin{array}{ccc} Zt(Conductor1\_T1,Conductor1\_T1) & Zt(Conductor1\_T1,Conductor2\_T1) & Zt(Conductor1\_T1,Conductor3\_T1) \\ Zt(Conductor2\_T1,Conductor1\_T1) & Zt(Conductor2\_T1,Conductor2\_T1) & Zt(Conductor2\_T1,Conductor3\_T1) \\ Zt(Conductor3\_T1,Conductor1\_T1) & Zt(Conductor3\_T1,Conductor2\_T1) & Zt(Conductor3\_T1,Conductor3\_T1) \\ \end{array} \right] \]

etc.

 

[promach]

from post#35 , can I infer that equations from post #1 are wrong ?

 

[PlanarMetamaterials]

That's a good question. It appears that due to the authors' choice of definitions, the errors won't have any physical ramifications. For example, if you were computing the modal impedance or admittance properties (\(Z_{0e} \) or \(Z_{0o} \), etc.) from terminal-domain data, then you would indeed find that the values would fluctuate based on your choice of [Ti] and [Tv].

But in that particular work, the authors don't seem to describe from where they obtained the values of these modal-domain properties, or what the actual values were. Most likely, they calculated them using the same (incorrect) [Ti] and [Tv] matrices -- from terminal-domain data -- such that the errors would literally cancel themselves out when re-computing the terminal-domain ABCD matrices (as appears to be the case, due to the presence of experimental confirmation of the theory).

So strictly, I would say that yes, the equations of Fig. 9 are incorrect -- but only if you are using the correct values of \(Z_{0e} \), \(Z_{0o} \), \(Y_{0e} \) and \(Y_{0o} \).

[Disclaimer: many experts in this field are of the option that the correct values of modal impedance and admittance cannot be computed. If you wish to avoid this maelstrom of discussion, just use a simulator to determine their values. The definitions of [Ti] and [Tv] I gave in post #4 will also work fine for you .]

 

[promach]

Do you have simplified proof of MTL equations 3.27a and 3.27b for the Paul Clayton's MTL book ?