ABCD parameters matrix of unsymmetric coupled lines

promach said:

1. What do you mean by This type of comparison is commonly used to derive relations between the modal and terminal domain representations of various properties. ?

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I mean, comparing modal and terminal-domain equations in order to relate the two. For example, in Paul's text, he compares equations 7.7 (the terminal domain equations) with 7.9 (the modal domain equations) to get the conversion of propagation constants between domains 7.10.

promach said:

2. Why the matrices inside equations (6a) and (6b) only contain entries at the diagonal ?

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Similar to the case of the modal and terminal propagation constants, the matrices of the per-unit-length admittances and impedances are diagonal because we choose them to be that way. Let me explain it another way:

We naturally know the terminal-domain representation of the system. We are trying to find out how to get the properties of the decoupled system (modal domain) from this terminal domain. So we know in advance that we are looking for the decoupled system (which, by definition, contains only diagonal matrices), and what we are trying to solve is the change of bases that will convert between the two representations. The math shows us that these change of bases are the transformation matrices [Tv] and [Ti] (specifically, equations 4 in the paper).

Quoted from post 71 : If we look at the modal-domain wave equations (7.9), we see that [T_V]⁻¹[Z][Y][T_V] is the matrix of diagonal modal propagation constants (squared). Since the matrix is diagonal, the propagation of a single mode's voltage or current does not depend on the voltages or currents of the other modes; i.e., it is decoupled from the others.

How exactly does a diagonal matrix contributes to propagation of a single mode's voltage or current does not depend on the voltages or currents of the other modes ?

One side question, how to use these modal-domain wave equations (7.9) to derive the ABCD parameters of unsymmetric coupled lines which are the equations (2) inside **broken link removed** ?

promach said:

Quoted from post 71 : If we look at the modal-domain wave equations (7.9), we see that [T_V]⁻¹[Z][Y][T_V] is the matrix of diagonal modal propagation constants (squared). Since the matrix is diagonal, the propagation of a single mode's voltage or current does not depend on the voltages or currents of the other modes; i.e., it is decoupled from the others.

How exactly does a diagonal matrix contributes to propagation of a single mode's voltage or current does not depend on the voltages or currents of the other modes ?

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You may observe this mathematically by expanding the terminal-domain wave equation (7.7) and the modal-domain wave equation (7.9).

In the terminal domain,
\[ \frac{d^2}{dz^2} \vec{V}(z) = [Z_T][Y_T]\vec{V}(z) = [\Gamma^2_T]\vec{V}(z)\]
\[ \frac{d^2}{dz^2} V_1(z) = {\Gamma^2_T}_{11}V_1(z) + {\Gamma^2_T}_{12}V_2(z) + {\Gamma^2_T}_{13}V_3(z) \dots\]
\[ \frac{d^2}{dz^2} V_2(z) = {\Gamma^2_T}_{21}V_1(z) + {\Gamma^2_T}_{22}V_2(z) + {\Gamma^2_T}_{23}V_3(z) \dots\]
etc.

while in the modal domain, the propagation constant matrix is diagonal, such that:

\[ \frac{d^2}{dz^2} \vec{V}(z) = [Z_M][Y_M]\vec{V}(z) = [\Gamma^2_M]\vec{V}(z)\]
\[ \frac{d^2}{dz^2} V_1(z) = {\Gamma^2_M}_{11}V_1(z)\]
\[ \frac{d^2}{dz^2} V_2(z) = {\Gamma^2_M}_{22}V_2(z)\]
etc.

promach said:

One side question, how to use these modal-domain wave equations (7.9) to derive the ABCD parameters of unsymmetric coupled lines which are the equations (2) inside **broken link removed** ?

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The wave equations aren't used directly, but one would use them to obtain the transformations from the modal domain to the terminal domain, which are then used in my post #2.

As for the paper you mentioned, it looks like they just assume certain transformation matrices [T_I] and [T_V], without discussing the nature of the modes at all (I would assume this can be found in one or more of the provided references).