Z-Parameters
Understanding Z-Parameters
Z-parameters describe a network in terms of impedance. Each parameter has a clear physical meaning: Z11 is the input impedance with the output open-circuited, Z21 is the forward transfer impedance, and so on. They are particularly useful in circuit analysis because they relate directly to the physical impedance values of components.
Z-Parameter Matrix
- Z11 (input impedance): V1/I1 with I2 = 0 (output open). The impedance seen looking into port 1 with port 2 open-circuited.
- Z12 (reverse transfer impedance): V1/I2 with I1 = 0. Describes how current at port 2 affects voltage at port 1.
- Z21 (forward transfer impedance): V2/I1 with I2 = 0. Describes how current at port 1 creates voltage at port 2.
- Z22 (output impedance): V2/I2 with I1 = 0. The impedance seen looking into port 2 with port 1 open-circuited.
Conversion from S-Parameters
At microwave frequencies, Z-parameters are not measured directly because creating stable open-circuit conditions is impractical (stray capacitance and radiation prevent true open circuits). Instead, S-parameters are measured with a VNA and mathematically converted to Z-parameters using well-known formulas.
When to Use Z-Parameters
- Series-connected networks: Z-parameters of series-connected two-ports simply add.
- Component modeling: Z-parameters directly give impedance values used in circuit simulation.
- Low-frequency circuits: Below a few hundred MHz, Z-parameters can be measured directly and provide intuitive impedance values.
V1 = Z11 × I1 + Z12 × I2
V2 = Z21 × I1 + Z22 × I2
Z-matrix:
[V] = [Z] × [I]
Conversion from S-parameters:
Z = Z0 × (I + S) × (I - S)^(-1)
where I = identity matrix, Z0 = reference impedance
For reciprocal networks: Z12 = Z21
For symmetric networks: Z11 = Z22
Network Parameter Comparison
| Parameter | Termination | Best For | Cascade Rule |
|---|---|---|---|
| Z-parameters | Open circuit | Series networks | Addition |
| Y-parameters | Short circuit | Parallel networks | Addition |
| S-parameters | Matched load | Microwave measurement | Signal flow graph |
| ABCD matrix | N/A | Cascaded two-ports | Multiplication |
| h-parameters | Mixed OC/SC | Transistor models | N/A |
Frequently Asked Questions
What are Z-parameters?
Z-parameters describe a network as a matrix of impedance values relating port voltages to port currents under open-circuit conditions. Z11 is input impedance, Z21 is forward transfer impedance, Z12 is reverse transfer impedance, and Z22 is output impedance.
Why are S-parameters used instead of Z-parameters at microwave frequencies?
At microwave frequencies, open and short circuit terminations are impractical because they cause radiation and parasitic effects. S-parameters use matched terminations (50 ohms), which are easy to implement at any frequency. S-parameters can be mathematically converted to Z-parameters when needed.
How do you convert S-parameters to Z-parameters?
Use the matrix formula Z = Z0(I+S)(I-S)^-1, where I is the identity matrix and Z0 is the reference impedance (typically 50 ohms). For single elements, Z11 = Z0(1+S11)(1-S11)^-1 for a one-port device. All modern VNAs and simulation tools perform this conversion automatically.