Z Parameters

In the analysis of circuits that have both an input port and an output port, it becomes necessary to describe the relationship between voltages and currents at these two ports without knowing the internal circuit topology. The Z parameters (also called open-circuit impedance parameters) provide exactly this: a complete description of a linear two-port network using four impedance values measured under open-circuit conditions at either port.

Z parameters are foundational in microwave engineering, transmission line analysis, transistor modeling, and filter design. GATE questions frequently test Z parameter definitions, their measurement conditions, the Z matrix for standard circuits like T-networks, and interconversion between parameter sets.

Core Concept: Z Parameter Definitions

A two-port network is any circuit with two pairs of accessible terminals called ports. At each port, a voltage and a current exist. The Z parameter model expresses both port voltages as linear combinations of both port currents. The governing equations are V₁ = Z₁₁I₁ + Z₁₂I₂ and V₂ = Z₂₁I₁ + Z₂₂I₂. These can be written compactly as [V] = [Z][I].

Each Z parameter is defined as a ratio of a port voltage to a port current under the condition that the other port is open-circuited (current forced to zero). Specifically: Z₁₁ = V₁/I₁ when I₂ = 0 (input impedance with output open), Z₂₁ = V₂/I₁ when I₂ = 0 (forward transfer impedance), Z₁₂ = V₁/I₂ when I₁ = 0 (reverse transfer impedance), and Z₂₂ = V₂/I₂ when I₁ = 0 (output impedance with input open).

For a reciprocal network (one containing only passive bilateral elements like R, L, C), Z₁₂ = Z₂₁. This reciprocity condition simplifies analysis significantly. For a symmetrical network (identical from both ports), additionally Z₁₁ = Z₂₂. Both conditions hold for a symmetric passive T-network.

Mathematical Expression: Z Matrix for T-Network

The most commonly tested structure is the T-network (also called a T-circuit or Y-circuit in ladder network terminology). It consists of series impedance ZA in the input arm, series impedance ZB in the output arm, and shunt impedance ZC connected from the junction to the common ground. The Z parameters for this T-network are: Z₁₁ = ZA + ZC, Z₁₂ = Z₂₁ = ZC, and Z₂₂ = ZB + ZC.

This result is physically intuitive. Z₁₁ is the total impedance seen at port 1 with port 2 open, which is ZA in series with ZC. Z₂₂ is ZB in series with ZC similarly. The transfer parameters Z₁₂ and Z₂₁ both equal ZC because the shunt element carries the shared current that creates coupled voltage, confirming reciprocity.

A network is said to have no Z parameters if the Z matrix does not exist. This happens when Z₁₁ and Z₂₂ are infinite, or more practically when the network equations cannot be expressed in the Z form. For example, an ideal series-connected two-port (like two ports in series) always has a Z matrix, but a pure shunt admittance two-port has Z parameters that exist but the Z matrix has specific structure.

Practical Understanding

Z parameters are most naturally measured at low frequencies using an impedance analyzer. At port 1, with port 2 left open, you apply a voltage V₁, measure current I₁, and compute Z₁₁ = V₁/I₁. You also measure V₂ across the open port 2 to get Z₂₁ = V₂/I₁. Then repeat with port 1 open and excitation at port 2.

In transistor modeling at low frequencies, the Z parameter model relates the device terminal voltages to terminal currents. A BJT in common-emitter configuration can be characterized by Z parameters measured from its base-emitter and collector-emitter ports. However, at high frequencies the S parameters (scattering parameters) are preferred because open circuits are difficult to achieve precisely at microwave frequencies.

Given:
T-network: ZA = j10 Ω (inductor), ZB = j10 Ω (inductor), ZC = -j20 Ω (capacitor)
at frequency f = 1 kHz

Why this formula applies:
For a T-network, Z parameters follow: Z11=ZA+ZC, Z12=Z21=ZC, Z22=ZB+ZC.

Formula:
Z₁₁ = ZA + ZC
Z₁₂ = Z₂₁ = ZC
Z₂₂ = ZB + ZC

Substitution:
Z₁₁ = j10 + (-j20) = -j10 Ω
Z₁₂ = Z₂₁ = -j20 Ω
Z₂₂ = j10 + (-j20) = -j10 Ω

Z matrix:
| -j10   -j20 |
| -j20   -j10 |

Check reciprocity: Z₁₂ = Z₂₁ = -j20 Ω ✓ (passive network)
Check symmetry: Z₁₁ = Z₂₂ = -j10 Ω ✓ (ZA = ZB)

Final Answer:
Z₁₁ = Z₂₂ = -j10 Ω, Z₁₂ = Z₂₁ = -j20 Ω
Network is both reciprocal and symmetrical.

Exam Tip: Z parameters are measured with one port OPEN. Y parameters are measured with one port SHORT. Remember: Z = Open, Y = Short. For a T-network Z₁₂ = ZC (shunt element). For reciprocal networks Z₁₂ = Z₂₁. GATE often gives a T or pi network and asks for Z or Y parameters directly.

Key Mechanism Points

• Z parameters are defined under open-circuit port conditions. Z₁₁ and Z₂₁ are found by opening port 2 (I₂ = 0); Z₁₂ and Z₂₂ by opening port 1 (I₁ = 0).
• For a T-network: Z₁₁ = ZA + ZC, Z₂₂ = ZB + ZC, Z₁₂ = Z₂₁ = ZC. The shunt element ZC is directly the transfer impedance.
• Reciprocity condition Z₁₂ = Z₂₁ holds for all passive bilateral networks containing only R, L, C elements (no controlled sources).
• Symmetry condition Z₁₁ = Z₂₂ holds when the network looks identical from both ports, as in a symmetric T.
• Z parameters may not exist for certain two-port configurations, such as an ideal gyrator or circuits that cannot be described by the open-circuit voltage-current relationship.

Quick Revision

• Z parameters: V₁ = Z₁₁I₁ + Z₁₂I₂ and V₂ = Z₂₁I₁ + Z₂₂I₂. Matrix form [V] = [Z][I].
• Measurement: Z₁₁ = V₁/I₁|I₂=0, Z₂₁ = V₂/I₁|I₂=0, Z₁₂ = V₁/I₂|I₁=0, Z₂₂ = V₂/I₂|I₁=0.
• T-network: Z₁₁ = ZA+ZC, Z₂₂ = ZB+ZC, Z₁₂ = Z₂₁ = ZC.
• Reciprocal network: Z₁₂ = Z₂₁. Symmetrical network: additionally Z₁₁ = Z₂₂.
• Exam trap: Z parameters use OPEN circuit, not short circuit. Short circuit is used for Y parameters.
• Units of all Z parameters are ohms (Ω). They are impedance quantities by definition.
• Z matrix of series-connected two-ports: add the individual Z matrices element by element.