Question

How do you find the impedance seen at the input of a two-port network?

Answer 1

To find the impedance seen at the input of a two-port network, you can use the concept of a scattering matrix (S-matrix) or T-parameters (T-matrix). Both approaches are commonly used in microwave engineering and network analysis.

Here, I'll explain how to find the input impedance using the S-parameters, which is one of the most popular methods:

Understand S-parameters:
The S-parameters describe the relationship between the incident and reflected waves at the ports of a two-port network. For a two-port network, the S-matrix is represented as follows:

[b1, b2] = [S11, S12; S21, S22] * [a1, a2]

where a1 and a2 are the incident waves at Port 1 and Port 2, respectively, and b1 and b2 are the reflected waves at those ports.

S11 and S22 are the reflection coefficients at Port 1 and Port 2 when only one port is excited, while S12 and S21 are the transmission coefficients from one port to the other when the other port is terminated with a matched impedance.

Find the S-parameters of the two-port network:
You can measure or calculate the S-parameters of the two-port network experimentally or through simulation tools like network analyzers or electromagnetic simulation software.

Calculate the input impedance:
To find the input impedance (Zin) seen at the input (Port 1) of the two-port network when Port 2 is terminated with a matched impedance (Z0), follow these steps:

Set a2 to Z0 (matched impedance) and a1 to zero (open circuit or terminated with a high impedance).

The equation for the S-matrix becomes:
[b1, b2] = [S11, S12; S21, S22] * [0, Z0]

Since there is no incident wave at Port 1 (a1 = 0), b1 must be zero (no reflection at Port 1). Therefore, the equation simplifies to:
0 = S11 * 0 + S12 * Z0

Solving for S12, we get:
S12 = 0

Now, the input impedance (Zin) is given by:
Zin = Z0 * (1 + S11) / (1 - S11)

Substitute the value of S12 (0) into the equation, and you get:
Zin = Z0 * (1 + S11) / (1 - S11)

Calculate the magnitude and phase of Zin based on your S-parameters.

That's it! You now have the input impedance (Zin) seen at the input of the two-port network when Port 2 is terminated with a matched impedance (Z0).