Question
Is the math below correct?
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:
For ports \$i\$ and \$j\$, where \$i, j \neq N\$:
These interactions are modified by the connection through port \$N\$:
$$
S'^Q_{ij} = S^Q_{ij} + \sum_{k=1}^{P} \frac{S^Q_{iN} S^Q_{Nj} S^P_{kM}}{1 - S^Q_{NN} S^P_{MM}}
$$
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.
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.
