Network Parameter Representations

Basic Network Parameter Representations 

Once a microwave structure has been represented as an N-port network, the next step is to mathematically describe the relationship between signals entering and leaving its ports.

Several parameter systems have been developed for this purpose. Each representation uses a different set of variables and offers advantages for specific applications.

The most common network representations are:

• Impedance Parameters (Z-Parameters)
• Admittance Parameters (Y-Parameters)
• Scattering Parameters (S-Parameters)

Although all three describe the same physical network, microwave engineers primarily use S-parameters because they are easier to measure accurately at high frequencies.

Impedance Parameters (Z-Parameters)

Impedance parameters relate port voltages to port currents.

For an N-port network:

\[ [V] = [Z][I] \]

where:

\[ [V] = \begin{bmatrix} V_1\\ V_2\\ \vdots\\ V_N \end{bmatrix} \]

\[ [I] = \begin{bmatrix} I_1\\ I_2\\ \vdots\\ I_N \end{bmatrix} \]

and:

\[ [Z] = \begin{bmatrix} Z_{11} & Z_{12} & \cdots\\ Z_{21} & Z_{22} & \cdots\\ \vdots & \vdots & \ddots \end{bmatrix} \]

For a two-port network:

\[ \begin{bmatrix} V_1\\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12}\\ Z_{21} & Z_{22} \end{bmatrix} \begin{bmatrix} I_1\\ I_2 \end{bmatrix} \]

Physical Meaning of Z-Parameters

The individual impedance parameters are defined as:

\[ Z_{11} = \left. \frac{V_1}{I_1} \right|_{I_2=0} \]

\[ Z_{22} = \left. \frac{V_2}{I_2} \right|_{I_1=0} \]

\[ Z_{12} = \left. \frac{V_1}{I_2} \right|_{I_1=0} \]

\[ Z_{21} = \left. \frac{V_2}{I_1} \right|_{I_2=0} \]

Determining these parameters requires open-circuit conditions:

\[ I=0 \]

at one or more ports.

At microwave frequencies, creating a true open circuit is difficult because stray capacitance and radiation effects distort measurements.

For this reason, Z-parameters are mainly used in theoretical analysis and lower-frequency RF design.

Admittance Parameters (Y-Parameters)

Admittance parameters are the dual of impedance parameters and relate currents directly to voltages.

The general matrix equation is:

\[ [I] = [Y][V] \]

where:

\[ [Y] = \begin{bmatrix} Y_{11} & Y_{12}\\ Y_{21} & Y_{22} \end{bmatrix} \]

For a two-port network:

\[ \begin{bmatrix} I_1\\ I_2 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12}\\ Y_{21} & Y_{22} \end{bmatrix} \begin{bmatrix} V_1\\ V_2 \end{bmatrix} \]

Physical Meaning of Y-Parameters

\[ Y_{11} = \left. \frac{I_1}{V_1} \right|_{V_2=0} \]

\[ Y_{22} = \left. \frac{I_2}{V_2} \right|_{V_1=0} \]

\[ Y_{12} = \left. \frac{I_1}{V_2} \right|_{V_1=0} \]

\[ Y_{21} = \left. \frac{I_2}{V_1} \right|_{V_2=0} \]

These parameters require short-circuit conditions:

\[ V=0 \]

at selected ports.

At microwave frequencies, true shorts are difficult to realize because conductor inductance, contact resistance, and parasitic effects introduce significant measurement errors.

Consequently, Y-parameters are rarely used for practical microwave measurements.

Why \(Z\) and \(Y\) Parameters Are Limited to Lumped Circuits and Fail for Distributed Systems

While \(Z\) and \(Y\) parameters are mathematically elegant for low-frequency lumped circuits, they become impractical and inaccurate when applied to high-frequency distributed microwave circuits.

1. Difficulty of Realizing True Open and Short Circuits at Microwave Frequencies

• Ideal Opens Do Not Exist: To measure \(Z\)-parameters, one port must be left completely open \((I = 0)\). However, at microwave frequencies, an open-ended wire or trace acts as an antenna, radiating energy into space and creating parasitic capacitance. The current at the terminal is not zero \((I \neq 0)\).

• Ideal Shorts Do Not Exist: To measure \(Y\)-parameters, a port must be shorted to ground \((V = 0)\). At high frequencies, any physical ground wire or connection introduces lead inductance. This non-zero inductive reactance

  \[ X_L = \omega L \]

  prevents the voltage from remaining strictly zero \((V \neq 0)\).

2. Total Voltage and Current Are Not Uniquely Defined

In distributed systems such as waveguides, electromagnetic energy propagates as spatial field vectors \(\vec{E}\) and \(\vec{H}\) rather than distinct currents flowing through wires.

Because physical dimensions are comparable to the wavelength \((\text{Size} \sim \lambda)\), total voltage depends on the path of integration:

\[ V = -\int_{C} \vec{E} \cdot d\vec{l} \]

For modes other than pure TEM, such as TE and TM modes in rectangular waveguides, voltage and current cannot be uniquely defined at a port interface. As a result, definitions such as

\[ Z = \frac{V}{I} \]

lose their physical meaning.

3. High-Frequency Measurement Limitations and Device Instability

• Instrument Probes: Physical probes used to measure total voltage or current introduce stray capacitance, inductance, and loading effects that alter the circuit being measured.

• Active Device Instability: Microwave transistors and active devices often become unstable or self-oscillate when subjected to open-circuit or short-circuit terminations required for \(Z\) and \(Y\) parameter measurements.

Why S-Parameters Are Preferred

Because ideal open and short circuits are difficult to realize at microwave frequencies, engineers use scattering parameters instead of impedance or admittance parameters.

Unlike \(Z\) and \(Y\) parameters, \(S\)-parameters are measured using matched loads, making them practical for high-frequency circuits, transmission lines, waveguides, filters, amplifiers, and antenna systems.

This is why modern microwave engineering relies almost exclusively on vector network analyzer (VNA) measurements based on scattering parameters.