multiconductor termination network

[rickcwalker]

I'm trying to solve a multiconductor transmission line problem for a practical application.

I've written a classic FDM simulator on a square grid. I'm using a simple relaxation equation to solve for the E field for an arbitrary set of conductors, and then I do a contour integral around each conductor to get the charge on each conductor. I do this setting each conductor to 1.0 volt and all the others to 0.0. I've compared my results to the literature and I think I correctly have an approximate Maxwell capacitance matrix [C0] for vacuum and another [CC] with included dielectrics.

The literature says that [L]=MU*E0*inv[C0] and that [Z] is given by sqrt(MU*E0*inv[C0]*inv[CC]).

I'm putting the upper diagonal terms for L and C into the Ngspice CPL transmission line model. I was hoping to terminate the line in a network that would absorb all modes to eliminate ringing.

If I simply take the [Z] matrix at face value and create a network with all the Rij terms it doesn't properly terminate the line.

How is the [Z] matrix to be interpreted? What is the equation that it is the solution to?

I've got textbooks by Visscher, Paul, and Dworsky, but none of them go beyond generating the [L] and [C] matrices to a practical application of terminating the line for all modes. Any hints are appreciated!

kind regards,
--
Rick Walker

 

[PlanarMetamaterials]

Welcome, Rick.

It sounds like you're on the right track. However, saying that the MTL is terminated in [Zc] is not the same as implementing a network of impedances, each of which has the same value as the entry of [Zc]. This is because the [Zc] corresponds to the total impedance seen between each pair of terminals, whereas of course in application we know that a network of impedances will have many elements in parallel, which each affect the effective impedance seen by each terminal.

I believe Clayton Paul's text has the correct solution. I also wrote an algorithm for this, which takes in [Zc] and returns [ZNetwork], which are the physical values between terminals that you are looking for; see below. (Note: I wrote this c++ function using the Eigen linalg library quite a few years ago, and it was never exhaustively tested. However, I recall being reasonably happy with it).

MatrixXcld TerminationMatrix(const MatrixXcld& Zc)
{
  MatrixXcld Yct = Zc.inverse();
  MatrixXcld YNetwork = -Yct;
  for(unsigned int row = 0; row < YNetwork.rows(); row++)
  {
    complex<long double> ColumnSum = complex<long double>(0,0);
    for(unsigned int col = 0; col < YNetwork.cols(); col++) ColumnSum+= Yct(row,col);
    YNetwork(row,row) = ColumnSum;
  }

  MatrixXcld ZNetwork = YNetwork.cwiseInverse();

  return ZNetwork;
}

I'm also curious about why you are using the "the upper diagonal terms for L and C"? I don't know much about Ngspice, can you elaborate on this format?

 

[rickcwalker]

Hi Planar,

Ngspice has a poorly documented model called CPL. The manual says it takes the upper diagonal terms of the R,L,C,G maxwell matrices and produces an N-port model of the line.

My tool takes an ascii configuration file like this:

# coupler3.geo
# three conductor tranmission line
workspace 200 60
fill_diel 0 0 200 60 3.3
fill_diel 20 30 180 36 4.8
fill_rect 85 29 115 30 // bottom plate
fill_rect 85 36 95 37 // first conductor
fill_rect 105 36 115 37 // second conductor
boundary_fill

and then produces an NGSPICE include file like:

* use like this
* P1 in1 in2 in3 0 out1 out2 out3 0 coupler3

.model coupler3 cpl
+r= 1.00 0.00 0.00
+ 1.00 0.00
+ 1.00
+l= 307.32n 209.96n 209.96n
+ 437.46n 173.19n
+ 437.46n
+g= 0.00 0.00 0.00
+ 0.00 0.00
+ 0.00
+c= 273.14p -93.91p -93.91p
+ 152.94p -15.48p
+ 152.94p
+length=1.0 // in meters

.subckt coupler3_term in1 in2 in3
r11 in1 0 44.6984
r12 in1 in2 8.0905
r13 in1 in3 8.09053
r22 in2 0 -303.918
r23 in2 in3 -69.3351
r33 in3 0 -303.921
.ends term_coupler3

You can see that the specification of R,L,G,C elements are only the upper (N^2+N)/2 elements of the symmetrical matrices. In particular, you can see that the C matrix is in Maxwell form because of the negative off diagonal elements.

The essence of my question is properly generating the "coupler3_term" network. Currently my algorithm from "General equations for the characteristic impedance matrix and termination network of multiconductor transmission lines", Knockaert et al. is giving negative resistor values. Of course, if I am putting the transmission line parameters in the wrong order for SPICE all bets are off. There are a few example networks in the Ngspice source code that make me think I'm doing that part of the problem correctly.

Thank you for your pointers!
--
Rick

 

[PlanarMetamaterials]

Ah, yes, the Knockaert et al. paper may have been the algorithm source I was thinking of, it seems my terminology is similar. I might suggest starting with the three-conductor example provided in the paper to validate the NGSpice.

Good Luck!