Z Parameters (Impedance Parameters) of a Two-Port Network

Z Parameters (Impedance Parameters) of a Two-Port Network

Theory

Z-parameters (also called Impedance parameters) are used to represent a two-port network in terms of voltages and currents. They express port voltages as functions of port currents.

V₁ = Z₁₁·I₁ + Z₁₂·I₂
V₂ = Z₂₁·I₁ + Z₂₂·I₂

Matrix form:

| V₁ | | Z₁₁ Z₁₂ | | I₁ |
| V₂ | = | Z₂₁ Z₂₂ | | I₂ |

Where:
Z₁₁ → Input impedance (output open)
Z₂₂ → Output impedance (input open)
Z₁₂, Z₂₁ → Transfer impedances
Unit: Ohms (Ω)

How to Calculate Parameters

Condition 1: I₂ = 0 (Output Open Circuit)
Z₁₁ = V₁ / I₁
Z₂₁ = V₂ / I₁

Condition 2: I₁ = 0 (Input Open Circuit)
Z₂₂ = V₂ / I₂
Z₁₂ = V₁ / I₂

Example: Ladder Network

Given a ladder network with series and shunt resistors, find Z-parameters using open-circuit conditions.

Step 1: Find Z₁₁ and Z₂₁ (I₂ = 0)

• Output open → no current in last branch
• Equivalent reduction gives:

Z₁₁ = 9.37 Ω
Z₂₁ = 5.26 Ω

Step 2: Find Z₂₂ and Z₁₂ (I₁ = 0)

• Input open → no current in input branch
• Equivalent reduction gives:

Z₂₂ = 17.47 Ω
Z₁₂ = 5.26 Ω

Final Z Matrix

| 9.37 5.26 |
| 5.26 17.47 | Ω

Shortcut Using ABCD Parameters

If ABCD parameters are known:

Z₁₁ = A / C
Z₂₁ = 1 / C
Z₂₂ = D / C
Z₁₂ = (AD − BC) / C

Advantage: Faster and less error-prone for ladder networks.

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