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In class we've been introduced to two port networks, and I was wondering: Is there a physical significance to a singular matrix in a two port network, or is it simply where the mathematical model breaks down?

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It doesn't mean that the model breaks down, but it is usually a symptom that the model is idealized.

Consider the simplified common-emitter AC model for a BJT:

schematic

simulate this circuit – Schematic created using CircuitLab

It is clear that:

\begin{align*} I_2 &= h_{fe} \cdot I_1 \\ V_1 &= h_{ie} \cdot I_1 \end{align*}

which can be rewritten in matrix form as:

\$ \begin{bmatrix} V_1 \\ I_2 \end{bmatrix} = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix} \cdot \begin{bmatrix} I_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} h_{ie} & 0 \\ h_{fe} & 0 \end{bmatrix} \cdot \begin{bmatrix} I_1 \\ V_2 \end{bmatrix} \$

from which you can see the h-parameter matrix is singular, having the second column made up of zeroes, but still the model is valid, although really idealized, since it neglects the output conductance \$h_{oe}\$ and the \$h_{re}\$ coefficient.

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