Calculating input impedance of 3 port network

I am trying to derive a expression for calculating the input impedance of a 3 port network to use as direct calculating code and avoid SPICE/simulator solving of the same.

I am able to solve the input impedance of a 2 port system with a load, $Z_{load}$, connected to port 2. The impedance looking into port 1 would be (by solving basic 2 port theory):

$$Z_{in} = Z_{11} - \frac{Z_{21}Z_{12}}{Z_{22}+Z_{load}}$$

If I have a 3 port network with 2 ports connected to different loads, $Z_{load1}$ and $Z_{load2}$, how can I generate an expression for $Z_{in}$ looking in from port 1 starting from only the Z-matrix, $Z_{load1}$ and $Z_{load2}$?

Answer

The 3-port can be described in 3 equations, using

$$\left(\begin{matrix} V_1 \\ V_2 \\ V_3 \end{matrix}\right) = \left(\begin{matrix} Z_{11} & Z_{12} & Z_{13} \\ Z_{21} & Z_{22} & Z_{23} \\ Z_{31} & Z_{32} & Z_{33} \end{matrix}\right)\left(\begin{matrix} I_1 \\ I_2 \\ I_3\end{matrix}\right)$$

I will now load ports 1 and 2, while using port 3 as the input. Adding two loads adds two new equations

$$\begin{align} V_1 &= -Z_{L1}\cdot I_1 \\ V_2 &= -Z_{L2}\cdot I_2 \end{align}$$

This can be inserted into the matrix notation:

$$\left(\begin{matrix} Z_{11} & Z_{12} & Z_{13} \\ Z_{21} & Z_{22} & Z_{23} \\ Z_{31} & Z_{32} & Z_{33} \end{matrix}\right)\left(\begin{matrix} I_1 \\ I_2 \\ I_3\end{matrix}\right) = \left(\begin{matrix} -Z_{L1}\cdot I_1 \\ -Z_{L2}\cdot I_2 \\ V_3 \end{matrix}\right)$$

Which is the same as

$$\left(\begin{matrix} Z_{11} + Z_{L1} & Z_{12} & Z_{13} \\ Z_{21} & Z_{22} + Z_{L2} & Z_{23} \\ Z_{31} & Z_{32} & Z_{33} \end{matrix}\right)\left(\begin{matrix} I_1 \\ I_2 \\ I_3\end{matrix}\right) = \left(\begin{matrix} 0 \\ 0 \\ V_3 \end{matrix}\right)$$

Since we want to solve for $Z_{in} = \frac{V_3}{I_3}$, we can use Cramer's rule to find

$$I_3 = \frac{\left|\begin{matrix} Z_{11} + Z_{L1} & Z_{12} & 0 \\ Z_{21} & Z_{22} + Z_{L_2} & 0 \\ Z_{31} & Z_{32} & V_3 \end{matrix}\right|}{\left|\begin{matrix} Z_{11} + Z_{L1} & Z_{12} & Z_{13} \\ Z_{21} & Z_{22} + Z_{L2} & Z_{23} \\ Z_{31} & Z_{32} & Z_{33} \end{matrix}\right|}\ $$

Or also

$$Z_{in} = \frac{V_3}{I_3} = \frac{\left|\begin{matrix} Z_{11} + Z_{L1} & Z_{12} & Z_{13} \\ Z_{21} & Z_{22} + Z_{L2} & Z_{23} \\ Z_{31} & Z_{32} & Z_{33} \end{matrix}\right|}{\left|\begin{matrix} Z_{11} + Z_{L1} & Z_{12} & 0 \\ Z_{21} & Z_{22} + Z_{L_2} & 0 \\ Z_{31} & Z_{32} & 1 \end{matrix}\right|}$$