Unit 04: Two-Port Networks

Fundamentals of Circuit Analysis (2026) · Chapter 8

Two-Port Networks Z / Y / T / H Parameters Network Connections

Knowledge Structure Overview

Two-Port Networks
└─ 8. Two-Port Networks
    ├─ 8.1 Basic Concepts of Two-Port Networks
    ├─ 8.2 Parameter Equations of Two-Port Networks
    │   ├─ 8.2.1 Impedance Equations and Z Parameters
    │   ├─ 8.2.2 Admittance Equations and Y Parameters
    │   ├─ 8.2.3 Transmission Equations and T Parameters
    │   └─ 8.2.4 Hybrid Equations and H Parameters
    └─ 8.3 Circuit Analysis with Two-Port Networks
        └─ 8.3.1 Two-Port Network Connections

Chapter 8

8.1 Basic Concepts of Two-Port Networks

Definition: A network with two ports (each port consists of a pair of terminals)
Port Condition: In each pair of terminals forming a port, the current entering one terminal equals the current leaving the other terminal
Difference from Four-Terminal Networks: The four terminal currents of a four-terminal network do not necessarily satisfy the port condition
Port Variables:
- Input port (port 1): voltage $\dot{U}_1$, current $\dot{I}_1$
- Output port (port 2): voltage $\dot{U}_2$, current $\dot{I}_2$
Reference Direction: Current is conventionally defined as entering the positive-polarity terminal of each port
Linear Two-Port Network: Contains no independent sources internally; all elements are linear
Symmetric Two-Port Network: The electrical characteristics of both ports are identical
Reciprocal Two-Port Network: Satisfies the reciprocity theorem, $z_{12} = z_{21}$ (or $y_{12} = y_{21}$)

Chapter 8

8.2 Four Parameter Equations at a Glance

Z Parameters

Impedance Equations

Uses currents as independent variables, expresses voltages. Measured under open-circuit conditions.

$\dot{U} = \mathbf{Z}\dot{I}$

Reciprocal: $z_{12}=z_{21}$ Symmetric: $z_{11}=z_{22}$

Y Parameters

Admittance Equations

Uses voltages as independent variables, expresses currents. Measured under short-circuit conditions. $\mathbf{Y}=\mathbf{Z}^{-1}$

$\dot{I} = \mathbf{Y}\dot{U}$

Reciprocal: $y_{12}=y_{21}$ Symmetric: $y_{11}=y_{22}$

T Parameters

Transmission Equations

Describes signal transmission from input to output; suitable for cascade analysis.

$\begin{bmatrix}\dot{U}_1\\\dot{I}_1\end{bmatrix} = \mathbf{T}\begin{bmatrix}\dot{U}_2\\-\dot{I}_2\end{bmatrix}$

Reciprocal: $AD-BC=1$ Symmetric: $A=D$

H Parameters

Hybrid Equations

Mixed use of voltage and current as independent variables. Commonly used in transistor equivalent circuits.

$\begin{bmatrix}\dot{U}_1\\\dot{I}_2\end{bmatrix} = \mathbf{H}\begin{bmatrix}\dot{I}_1\\\dot{U}_2\end{bmatrix}$

Reciprocal: $h_{12}=-h_{21}$ Symmetric: $h_{11}h_{22}-h_{12}h_{21}=1$

Z Parameters Impedance Equations in Detail

$$\dot{U}_1 = z_{11}\dot{I}_1 + z_{12}\dot{I}_2 \qquad \dot{U}_2 = z_{21}\dot{I}_1 + z_{22}\dot{I}_2$$ $$\begin{bmatrix}\dot{U}_1\\\dot{U}_2\end{bmatrix} = \mathbf{Z}\begin{bmatrix}\dot{I}_1\\\dot{I}_2\end{bmatrix}$$

Physical Meaning of Parameters (open-circuit parameters):

$z_{11} = \dfrac{\dot{U}_1}{\dot{I}_1}\bigg|_{\dot{I}_2=0}$ — Input impedance at port 1 with port 2 open-circuited
$z_{12} = \dfrac{\dot{U}_1}{\dot{I}_2}\bigg|_{\dot{I}_1=0}$ — Transfer impedance with port 1 open-circuited
$z_{21} = \dfrac{\dot{U}_2}{\dot{I}_1}\bigg|_{\dot{I}_2=0}$ — Transfer impedance with port 2 open-circuited
$z_{22} = \dfrac{\dot{U}_2}{\dot{I}_2}\bigg|_{\dot{I}_1=0}$ — Input impedance at port 2 with port 1 open-circuited

Measurement Method: Open-circuit port 1 or port 2 alternately, and measure voltage and current.

Reciprocal condition: $z_{12} = z_{21}$ Symmetric condition: $z_{11} = z_{22}$

Y Parameters Admittance Equations in Detail

$$\dot{I}_1 = y_{11}\dot{U}_1 + y_{12}\dot{U}_2 \qquad \dot{I}_2 = y_{21}\dot{U}_1 + y_{22}\dot{U}_2$$ $$\begin{bmatrix}\dot{I}_1\\\dot{I}_2\end{bmatrix} = \mathbf{Y}\begin{bmatrix}\dot{U}_1\\\dot{U}_2\end{bmatrix}$$

Physical Meaning of Parameters (short-circuit parameters):

$y_{11} = \dfrac{\dot{I}_1}{\dot{U}_1}\bigg|_{\dot{U}_2=0}$ — Input admittance at port 1 with port 2 short-circuited
$y_{12} = \dfrac{\dot{I}_1}{\dot{U}_2}\bigg|_{\dot{U}_1=0}$ — Transfer admittance with port 1 short-circuited
$y_{21} = \dfrac{\dot{I}_2}{\dot{U}_1}\bigg|_{\dot{U}_2=0}$ — Transfer admittance with port 2 short-circuited
$y_{22} = \dfrac{\dot{I}_2}{\dot{U}_2}\bigg|_{\dot{U}_1=0}$ — Input admittance at port 2 with port 1 short-circuited

Relation to Z parameters: $\mathbf{Y} = \mathbf{Z}^{-1}$ Reciprocal: $y_{12}=y_{21}$ Symmetric: $y_{11}=y_{22}$

T Parameters Transmission Equations in Detail

$$\dot{U}_1 = A\dot{U}_2 - B\dot{I}_2 \qquad \dot{I}_1 = C\dot{U}_2 - D\dot{I}_2$$ $$\begin{bmatrix}\dot{U}_1\\\dot{I}_1\end{bmatrix} = \mathbf{T}\begin{bmatrix}\dot{U}_2\\-\dot{I}_2\end{bmatrix}$$

Physical Meaning of Parameters:

$A = \dfrac{\dot{U}_1}{\dot{U}_2}\bigg|_{\dot{I}_2=0}$ — Open-circuit voltage transfer ratio
$B = -\dfrac{\dot{U}_1}{\dot{I}_2}\bigg|_{\dot{U}_2=0}$ — Short-circuit transfer impedance
$C = \dfrac{\dot{I}_1}{\dot{U}_2}\bigg|_{\dot{I}_2=0}$ — Open-circuit transfer admittance
$D = -\dfrac{\dot{I}_1}{\dot{I}_2}\bigg|_{\dot{U}_2=0}$ — Short-circuit current transfer ratio

Characteristic: T parameters are suitable for describing signal transmission from input to output.

Reciprocal condition: $AD - BC = 1$ Symmetric condition: $A = D$

H Parameters Hybrid Equations in Detail

$$\dot{U}_1 = h_{11}\dot{I}_1 + h_{12}\dot{U}_2 \qquad \dot{I}_2 = h_{21}\dot{I}_1 + h_{22}\dot{U}_2$$ $$\begin{bmatrix}\dot{U}_1\\\dot{I}_2\end{bmatrix} = \mathbf{H}\begin{bmatrix}\dot{I}_1\\\dot{U}_2\end{bmatrix}$$

Physical Meaning of Parameters:

$h_{11} = \dfrac{\dot{U}_1}{\dot{I}_1}\bigg|_{\dot{U}_2=0}$ — Input impedance with port 2 short-circuited
$h_{12} = \dfrac{\dot{U}_1}{\dot{U}_2}\bigg|_{\dot{I}_1=0}$ — Reverse voltage transfer ratio with port 1 open-circuited
$h_{21} = \dfrac{\dot{I}_2}{\dot{I}_1}\bigg|_{\dot{U}_2=0}$ — Forward current transfer ratio with port 2 short-circuited
$h_{22} = \dfrac{\dot{I}_2}{\dot{U}_2}\bigg|_{\dot{I}_1=0}$ — Output admittance with port 1 open-circuited

Characteristic: H parameters are widely used in transistor equivalent circuits.

Reciprocal condition: $h_{12} = -h_{21}$ Symmetric condition: $h_{11}h_{22} - h_{12}h_{21} = 1$

Summary

Four-Parameter Comparison

Z Parameters

Impedance equations

Measurement: Open-circuit

Reciprocal: $z_{12}=z_{21}$

Symmetric: $z_{11}=z_{22}$

Y Parameters

Admittance equations

Measurement: Short-circuit

Reciprocal: $y_{12}=y_{21}$

Symmetric: $y_{11}=y_{22}$

T Parameters

Transmission equations

Measurement: Open/Short-circuit

Reciprocal: $AD-BC=1$

Symmetric: $A=D$

H Parameters

Hybrid equations

Measurement: Hybrid

Reciprocal: $h_{12}=-h_{21}$

Symmetric: $det(\mathbf{H})=1$

Chapter 8

8.3 Circuit Analysis with Two-Port Networks

Two-Port Network Connections

Cascade

Output of the first network connects to the input of the second

$\mathbf{T} = \mathbf{T}_1 \cdot \mathbf{T}_2 \cdots \mathbf{T}_n$

Parallel-Parallel

Corresponding terminals connected in parallel

$\mathbf{Y} = \mathbf{Y}_1 + \mathbf{Y}_2$

Series-Series

Both input and output connected in series

$\mathbf{Z} = \mathbf{Z}_1 + \mathbf{Z}_2$

Series-Parallel

Input in series, output in parallel

$\mathbf{H} = \mathbf{H}_1 + \mathbf{H}_2$

Parallel-Series

Input in parallel, output in series

$\mathbf{G} = \mathbf{G}_1 + \mathbf{G}_2$

Solution Steps

Choose the appropriate
parameter type

Calculate the two-port
network parameters

Combine parameter equations
with external circuit equations

Solve for voltages
and currents