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Usual method for transmission (ABCD) matrix construction, from 2 ports EM simulation results, is to first calculate S parameters for each port i.e.:

s11 = port{1}.Vref / port{1}.Vinc;
s12 = port{1}.Vref / port{2}.Vinc;
s21 = port{2}.Vref / port{1}.Vinc;
s22 = port{2}.Vref / port{2}.Vinc;

Then to forge ABCD matrix from S parameters and Zref (used in EM simulation):

A = ((1+s11) * (1-s22) + s12 * s21) / (2*s21);
B = Zref   * ((1+s11) * (1+s22) - s12*s21) / (2*s21);
C = (1 / Zref) * ((1-s11)*(1-s22) - s12*s21) / (2*s21);
D = ((1-s11) * (1+s22) + s12.*s21) / (2*s21);

My question is: Is there direct (one step) way, to calculate ABCD matrix, from EM simulation results, apart from c/p whole S parameter equations into ABCD calc?

JYelton
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Danko Dnevic
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  • What parameters are available from your simulation other than S? If Vref and Vinc, why not just substitute into the ABCD formula quoted? – Tim Williams Aug 14 '23 at 23:44
  • Simulation (openEMS), natively, provides incident/reflected voltages and currents for given frequency and Zo, per port. S parameters are calculated as stated. I'm already substituting in two steps (S from readings, then conversion to ABCD). Now, I'm looking for more elegant way. I tried that, but my math is not good enough. Couldn't find anything more then just definitions A=V1/V2 (I2==0) etc. – Danko Dnevic Aug 15 '23 at 00:02

1 Answers1

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Well... just doing some algebra on it,

Let:
port{1}.Vref be \$V_{R1}\$
port{2}.Vref be \$V_{R2}\$
port{1}.Vinc be \$V_{I1}\$
port{2}.Vinc be \$V_{I2}\$

Then,

\begin{eqnarray} s_{11} = \frac{V_{R1}}{V_{I1}} & \qquad s_{12} = \frac{V_{R1}}{V_{I2}} & \qquad s_{21} = \frac{V_{R2}}{V_{I1}} & \qquad s_{22} = \frac{V_{R2}}{V_{I2}} \\ \end{eqnarray}

substituting,

\begin{eqnarray} A = \frac{ \left(1 + \frac{V_{R1}}{V_{I1}}\right) \left(1 - \frac{V_{R2}}{V_{I2}}\right) + \frac{V_{R1}}{V_{I2}} \frac{V_{R2}}{V_{I1}} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ B = Z_\textrm{ref} \frac{ \left(1 + \frac{V_{R1}}{V_{I1}}\right) \left(1 + \frac{V_{R2}}{V_{I2}}\right) - \frac{V_{R1}}{V_{I2}} \frac{V_{R2}}{V_{I1}} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ C = \frac{1}{Z_\textrm{ref}} \frac{ \left(1 - \frac{V_{R1}}{V_{I1}}\right) \left(1 - \frac{V_{R2}}{V_{I2}}\right) - \frac{V_{R1}}{V_{I2}} \frac{V_{R2}}{V_{I1}} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ D = \frac{ \left(1 - \frac{V_{R1}}{V_{I1}}\right) \left(1 + \frac{V_{R2}}{V_{I2}}\right) + \frac{V_{R1}}{V_{I2}} \frac{V_{R2}}{V_{I1}} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ \end{eqnarray}

distributing \$V_{I1}\$,

\begin{eqnarray} A = \frac{ \left(\frac{V_{I1} + V_{R1}}{V_{I1}}\right) \left(\frac{V_{I2} - V_{R2}}{V_{I2}}\right) + \frac{V_{R1}}{V_{I2}} \frac{V_{R2}}{V_{I1}} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ B = Z_\textrm{ref} \frac{ \left(\frac{V_{I1} + V_{R1}}{V_{I1}}\right) \left(\frac{V_{I2} + V_{R2}}{V_{I2}}\right) - \frac{V_{R1}}{V_{I2}} \frac{V_{R2}}{V_{I1}} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ C = \frac{1}{Z_\textrm{ref}} \frac{ \left(\frac{V_{I1} - V_{R1}}{V_{I1}}\right) \left(\frac{V_{I2} - V_{R2}}{V_{I2}}\right) - \frac{V_{R1}}{V_{I2}} \frac{V_{R2}}{V_{I1}} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ D = \frac{ \left(\frac{V_{I1} - V_{R1}}{V_{I1}}\right) \left(\frac{V_{I2} + V_{R2}}{V_{I2}}\right) + \frac{V_{R1}}{V_{I2}} \frac{V_{R2}}{V_{I1}} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ \end{eqnarray}

gathering \$V_{I1} V_{I2}\$,

\begin{eqnarray} A = \frac{1}{V_{I1} V_{I2}} \frac{ \left(V_{I1} + V_{R1}\right) \left(V_{I2} - V_{R2}\right) + V_{R1} V_{R2} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ B = Z_\textrm{ref} \frac{1}{V_{I1} V_{I2}} \frac{ \left(V_{I1} + V_{R1}\right) \left(V_{I2} + V_{R2}\right) - V_{R1} V_{R2} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ C = \frac{1}{Z_\textrm{ref}} \frac{1}{V_{I1} V_{I2}} \frac{ \left(V_{I1} - V_{R1}\right) \left(V_{I2} - V_{R2}\right) - V_{R1} V_{R2} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ D = \frac{1}{V_{I1} V_{I2}} \frac{ \left(V_{I1} - V_{R1}\right) \left(V_{I2} + V_{R2}\right) + V_{R1} V_{R2} }{ 2 \frac{V_{R2}}{V_{I1}} } \\ \end{eqnarray}

distribute some more,

\begin{eqnarray} A = & \dfrac{ V_{I1} V_{I2} + V_{R1} V_{I2} - V_{I1} V_{R2} - V_{R1} V_{R2} + V_{R1} V_{R2} }{ 2 V_{I2} V_{R2} } \\ B = Z_\textrm{ref} & \dfrac{ V_{I1} V_{I2} + V_{R1} V_{I2} + V_{I1} V_{R2} + V_{R1} V_{R2} - V_{R1} V_{R2} }{ 2 V_{I2} V_{R2} } \\ C = \dfrac{1}{Z_\textrm{ref}} & \dfrac{ V_{I1} V_{I2} - V_{R1} V_{I2} - V_{I1} V_{R2} + V_{R1} V_{R2} - V_{R1} V_{R2} }{ 2 V_{I2} V_{R2} } \\ D = & \dfrac{ V_{I1} V_{I2} - V_{R1} V_{I2} + V_{I1} V_{R2} - V_{R1} V_{R2} + V_{R1} V_{R2} }{ 2 V_{I2} V_{R2} } \\ \end{eqnarray}

notice rightmost terms cancel out,

\begin{eqnarray} A = \frac{ V_{I1} V_{I2} + V_{R1} V_{I2} - V_{I1} V_{R2} }{ 2 V_{I2} V_{R2} } \\ B = Z_\textrm{ref} \frac{ V_{I1} V_{I2} + V_{R1} V_{I2} + V_{I1} V_{R2} }{ 2 V_{I2} V_{R2} } \\ C = \frac{1}{Z_\textrm{ref}} \frac{ V_{I1} V_{I2} - V_{R1} V_{I2} - V_{I1} V_{R2} }{ 2 V_{I2} V_{R2} } \\ D = \frac{ V_{I1} V_{I2} - V_{R1} V_{I2} + V_{I1} V_{R2} }{ 2 V_{I2} V_{R2} } \\ \end{eqnarray}

Doesn't seem to be any more useful reduction to be done here.

Tim Williams
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  • Wow, that is really hard work. Just to remind you, if S parameters equations are inserted, when forging ABCD, that different versions of VR1 and VR2 (lets name them VR1s and VR2s ) are used to get s21 and s12. Reason is switching active port side. When port1 is active, s11 and s21 are obtained (using VR1, VI1 and VR2s). When port2 is active, s22 and s12 are obtained (using VR2, VI2 and VR1s). – Danko Dnevic Aug 15 '23 at 17:30
  • I think everything got carried through, so that should be fine. YMMV; I'm just some guy on the internet, remember. – Tim Williams Aug 15 '23 at 17:37
  • Great then. I'm going to test it. Thank you very much. BTW, I'm sorry, if I sounded arrogantly, about that port flipping issue. That wasn't my intention. – Danko Dnevic Aug 15 '23 at 17:47
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    Oh no problem, it's a valid concern, I understand. If the equations worked as-is, and I didn't drop anything in the math, I don't see why these won't. But that leaves some gaps, and you'll have to test if it checks out, ultimately. – Tim Williams Aug 15 '23 at 17:49