Introduction
When analyzing complex electrical networks, we often treat a section of the circuit as a "black box" with two pairs of terminals — an input port and an output port. This is the two-port network model, and it's one of the most powerful tools in network analysis.
The Z-parameter (also called open-circuit impedance parameter) is one of several ways to characterize a two-port network. It relates the port voltages to the port currents using a 2×2 impedance matrix. In this article, we'll derive the Z-parameters, understand their physical meaning, learn how to find them experimentally, and solve a numerical example.
Table of Contents
What is a Two-Port Network?
A two-port network is any linear circuit with two pairs of terminals:
- Port 1 (Input) — terminals where current I1 enters and voltage V1 is measured
- Port 2 (Output) — terminals where current I2 enters and voltage V2 is measured
The convention is: current entering a port is positive. The network can contain resistors, capacitors, inductors, and dependent sources — but no independent sources inside.
Examples of two-port networks: filters, amplifiers, transformers, transmission lines, and attenuators.
Z-Parameter Definition
The Z-parameters express voltages as functions of currents:
V2 = Z21 · I1 + Z22 · I2
In matrix form:
| V1 | | Z11 Z12 | | I1 |
| V2 | | Z21 Z22 | | I2 |
Each Z-parameter has the unit of ohms (Ω) — hence the name "impedance parameters."
Finding Z-Parameters (Open Circuit Method)
Z-parameters are found by open-circuiting one port at a time and measuring the voltage-to-current ratios. This is why they're called "open-circuit parameters."
Case 1: Open-Circuit Port 2 (I2 = 0)
When the output port is open-circuited, no current flows through it (I2 = 0). The equations reduce to:
Z21 = V2 / I1 |I₂=0
Case 2: Open-Circuit Port 1 (I1 = 0)
When the input port is open-circuited, no current flows through it (I1 = 0). The equations reduce to:
Z22 = V2 / I2 |I₁=0
Physical Meaning of Each Parameter
Equivalent Circuit of Z-Parameter
The Z-parameter equations can be represented by an equivalent circuit using two voltage-controlled voltage sources:
The equivalent circuit consists of:
- Port 1: impedance Z11 in series with a dependent voltage source Z12·I2
- Port 2: impedance Z22 in series with a dependent voltage source Z21·I1
This circuit satisfies both Z-parameter equations simultaneously and is useful for simulation and analysis.
Symmetry and Reciprocity Conditions
Two important properties of two-port networks can be determined from Z-parameters:
Key fact: All networks made of only passive elements (R, L, C) without dependent sources are reciprocal (Z12 = Z21). Networks with transistors or dependent sources are generally non-reciprocal.
Solved Example
Find the Z-parameters of the following T-network with Za = 3Ω, Zb = 5Ω, Zc = 4Ω (series-series configuration):
Solution
Case 1: I2 = 0 (Port 2 open)
With Port 2 open, current I1 flows through Za and Zc in series (no current through Zb):
Z21 = V2/I1 = Zc = 4 Ω (voltage across Zc only)
Case 2: I1 = 0 (Port 1 open)
With Port 1 open, current I2 flows through Zb and Zc in series:
Z12 = V1/I2 = Zc = 4 Ω (voltage across Zc only)
Z-matrix:
| 4 9 | Ω
Verification: Z12 = Z21 = 4 Ω → network is reciprocal ✓ (expected, since it's a passive network). Z11 ≠ Z22 → network is NOT symmetrical (because Za ≠ Zb).
When Z-Parameters Don't Exist
Z-parameters don't exist for networks where the port current cannot be treated as an independent variable. Specifically:
- Networks with a series connection at the input or output that forces a voltage regardless of current (ideal voltage source across a port)
- Ideal series-series feedback amplifiers in certain configurations
In such cases, other parameter sets (Y-parameters, h-parameters, or ABCD parameters) may be more appropriate.
FAQs
Why are Z-parameters called "open-circuit" parameters?
Because each parameter is measured by open-circuiting one port (making the current at that port zero). The open-circuit condition is the defining measurement setup for Z-parameters.
What is the difference between Z-parameters and Y-parameters?
Z-parameters express voltages as functions of currents ([V] = [Z][I]), while Y-parameters express currents as functions of voltages ([I] = [Y][V]). They are matrix inverses of each other: [Y] = [Z]−1.
How do you find Z-parameters for a T-network?
For a T-network with series arms Za, Zb and shunt arm Zc: Z11 = Za + Zc, Z22 = Zb + Zc, Z12 = Z21 = Zc.
Can Z-parameters be complex numbers?
Yes. For AC circuits with reactive elements (L and C), Z-parameters are complex impedances with real (resistive) and imaginary (reactive) parts. They are frequency-dependent.
What does it mean if Z12 = 0?
It means there is no reverse coupling — a current at Port 2 produces no voltage at Port 1. This is characteristic of unilateral devices like ideal transistor amplifiers.
Conclusion
Z-parameters provide a complete characterization of a linear two-port network by relating port voltages to port currents through a 2×2 impedance matrix. They are found experimentally by open-circuiting each port in turn. The conditions Z11 = Z22 (symmetry) and Z12 = Z21 (reciprocity) reveal important network properties at a glance.
For network analysis problems, Z-parameters are especially convenient for series-connected two-port networks, where the overall Z-matrix is simply the sum of individual Z-matrices.