Derive S-Parameter Matrices for Interconnected Multi-Port Networks of Arbitrary Size

Question

Question(s)

Is the math below correct? If not, then what should be changed?

If you know of an authoritative reference for this derivation, then please post it in an answer, or in the comments.

Background

After lots of searching to find a good reference with a generalized solution to chain one S-parameter matrix's port onto another S-parameter matrix's port of another arbitrary size, I resorted (reluctantly) to ChatGPT, which provided a surprisingly good formulation.

The "interesting conversation" with ChatGPT concluded here.

(nb, I know that using AI to answer a question is not allowed here, but I am hoping that it is acceptable to verify an AI response, especially when they response is quite complicated mathematically and I want to make sure the result is correct before I use it in real life.)

Overview

The objective is to determine the new S-parameter matrix \$S'^Q\$ after connecting a \$P\$-port network to a \$Q\$-port network. This involves understanding how signals are reflected and transmitted between ports, considering both direct connections and indirect interactions through the attached network.

Definitions

• \$S^Q_{ij}\$: The elements of the original \$Q\$-port S-parameter matrix, where \$i, j \in \{1, 2, \ldots, Q\}\$.

• \$S^P_{ij}\$: The elements of the \$P\$-port S-parameter matrix, represented as:

  $$ S^P = \begin{pmatrix} S^P_{11} & S^P_{12} & \cdots & S^P_{1P} \\ S^P_{21} & S^P_{22} & \cdots & S^P_{2P} \\ \vdots & \vdots & \ddots & \vdots \\ S^P_{P1} & S^P_{P2} & \cdots & S^P_{PP} \end{pmatrix} $$

• Connection: Port \$N\$ of the \$Q\$-port network is connected to port \$M\$ of the \$P\$-port network.

• \$S'^Q_{ij}\$: The elements of the new \$Q\$-port S-parameter matrix after the connection.

Generalized Transformation Equations

The new S-parameters \$S'^Q_{ij}\$ are determined by considering how the connection affects each interaction in the network:

1. For ports \$i\$ and \$j\$, where \$i, j \neq N\$:

   These interactions are modified by the connection through port \$N\$:

   $$ S'^Q_{ij} = S^Q_{ij} + \sum_{k=1}^{P} \frac{S^Q_{iN} S^Q_{Nj} S^P_{kM}}{1 - S^Q_{NN} S^P_{MM}} $$

   • \$S^P_{kM}\$: Represents the S-parameter between the \$k\$-th port of the \$P\$-port network and port \$M\$.

   • Summation over \$k\$: Accounts for contributions from all ports in the \$P\$-port network.

2. For \$(i = N\$ and \$j \neq N)\$ or \$(i \neq N\$ and \$j = N)\$:

   These terms capture direct interactions involving the connected port \$N\$:

   • Transmission from \$i\$ to \$N\$:

     $$ S'^Q_{iN} = \sum_{k=1}^{P} \frac{S^Q_{iN} (1 + S^P_{kk}) - S^Q_{iN} S^Q_{NN} S^P_{kM}}{1 - S^Q_{NN} S^P_{MM}} $$

     This equation is used when \$i \neq N\$ and describes how signals from port \$i\$ are transmitted to port \$N\$ through the \$P\$-port network.

   • Transmission from \$N\$ to \$j\$:

     $$ S'^Q_{Nj} = \sum_{k=1}^{P} \frac{S^Q_{Nj} (1 + S^P_{kk}) - S^Q_{NN} S^Q_{Nj} S^P_{kM}}{1 - S^Q_{NN} S^P_{MM}} $$

     This equation is used when \$j \neq N\$ and describes how signals from port \$N\$ are transmitted to port \$j\$ through the \$P\$-port network.

3. Reflection at port \$N\$:

   The reflection at port \$N\$ considers the connection to port \$M\$:

   $$ S'^Q_{NN} = \sum_{k=1}^{P} \frac{S^Q_{NN} + S^P_{kk} - S^Q_{NN} S^P_{MM} + S^Q_{NN} S^P_{kM} S^P_{kk}}{1 - S^Q_{NN} S^P_{MM}} $$

   This reflects the change in reflection characteristics due to the connection.

Resultant S-parameter Matrix \$S'^Q\$

The complete modified matrix \$S'^Q\$ is expressed as:

$$ S'^Q = \begin{pmatrix} S'^Q_{11} & S'^Q_{12} & \cdots & S'^Q_{1Q} \\ S'^Q_{21} & S'^Q_{22} & \cdots & S'^Q_{2Q} \\ \vdots & \vdots & \ddots & \vdots \\ S'^Q_{Q1} & S'^Q_{Q2} & \cdots & S'^Q_{QQ} \end{pmatrix} $$

Summary

• Generalization: This formulation is applicable for connecting any \$P\$-port network to a single port of a \$Q\$-port network (\$Q \geq P\$), focusing on the interaction between port \$N\$ and port \$M\$.

• Key Concept: The summation over \$k\$ accounts for the effect of each port in the \$P\$-port network, ensuring comprehensive modeling of the interaction.

• Practical Application: This method is essential for analyzing complex network interconnections, such as those found in RF and microwave systems, where multiple networks are interconnected and interact through specific ports.

By following this framework, engineers can accurately model and predict the behavior of interconnected multi-port networks, capturing all possible interactions in the analysis.

Answer 1

The book “Linear Multiport Synthesis”, by Robert Newcomb, has the formula.

An N port network with its final M ports connected to an M port load…. Partition the N port network into 4 parts:

The upper left corner of its matrix is a square matrix N-M rows and columns. Call that s11.

The lower right corner is a square matrix of M rows and M columns. Call that s22.

The upper right part of the matrix call s12. The lower left of the matrix call s21.

Then the s matrix of the unterminated N-M ports is called Sin.

Sin=s11+s12 * sload * inv(I-s22*sload) * s21

This works with any number of ports. In particular, with a two port network terminated in a one port network, all the s’s are scalars, the matrix multiplication is just multiplying numbers, and the inverse is division.

But, it also works with a 8 port terminated with a 5 port, for example, resulting in a three port Sin.