Hybrid Parameters in Two Port Networks

In this tutorial, you will learn the basics of hybrid and inverse hybrid parameters in two-port networks, including their matrix representation, short-circuit and open-circuit parameters, equivalent circuits, conditions for reciprocity and symmetry, and the differences between hybrid and inverse hybrid parameters.

Contents:

1. Matrix Representation of Hybrid Parameters
2. Short-Circuit Hybrid Parameters in Two-port Network
3. Open Circuit Hybrid Parameters in Two-port Network
4. Features of Hybrid Parameters in Two-port Network
5. Steps to Obtain Hybrid Parameters in Two-port Network
6. Equivalent Circuit for Hybrid Parameters
7. Inverse Hybrid Parameters in Two-Port Networks
8. Open Circuit Inverse Hybrid Parameters in Two-Port Networks
9. Short Circuit Inverse Hybrid Parameters in Two-Port Networks
10. Steps to Obtain the Inverse Hybrid Parameters in Two-Port Networks
11. Equivalent Circuits for Inverse Hybrid Parameters
12. Hybrid Parameters Vs Inverse Hybrid Parameters
13. Conditions for Reciprocity and Symmetry for Hybrid and Inverse Hybrid Parameters

Matrix Representation of Hybrid Parameters

• There are two ports in the network which are port 1 and port 2. The voltage and current in port 1 are V₁ and I₁ respectively and the voltage and current in port 2 are V₂ and I₂ respectively.
• The voltage at port 1, V₁ and current at port 2, I₂ are dependent variables while the voltage at port 2, V₂, and current at port 1, I₁ are independent variables. The direction of the current is such that it enters the network.
• The parameters h₁₁, h₁₂, h₂₁, and h₂₂ are the network functions and are called hybrid parameters.
• The mathematical expression for hybrid parameters is given by these two equations.
  \(V_1=h_{11} I_1+h_{12} V_2\)
  \(I_2=h_{21} I_1+h_{22} V_2\)
• The equations obtained can be represented in matrix form.
  \(\begin{bmatrix}
  V_1 \\
  I_2
  \end{bmatrix}
  =
  \begin{bmatrix}
  h_{11} & h_{12} \\
  h_{21} & h_{22}
  \end{bmatrix}
  \begin{bmatrix}
  I_1 \\
  V_2
  \end{bmatrix}\)
  Where, \(\begin{bmatrix}
  h_{11} & h_{12} \\
  h_{21} & h_{22}
  \end{bmatrix}\) are the hybrid parameters

Short-Circuit Hybrid Parameters in Two-port Network

• The two ports in the circuit are port 1 and port 2 and the current direction is taken such that it enters the network.
• The short circuit hybrid parameters in a two-port network are h₁₁ and h₂₁. The voltage across port 2 is zero when port 2 is short-circuited.
• The short circuit input impedance at port 1 when port 2 is short-circuited is h₁₁.
  \(h_{11}=\frac{V_1}{I_1} \text{ when } V_2=0\)
  Where,
  V₁ is the voltage at port 1
  I₁ is the current at port 1
  h₁₁ is the short circuit input impedance
• The short-circuit forward current gain at port 1 when port 2 is short-circuited is h₂₁.
  \(h_{21}=\frac{I_2}{I_1} \text{ when } V_2=0\)
  Where,
  I₁ is the current at port 1
  I₂ is the current at port 2
  h₂₁ is the short-circuit forward current gain
• The parameter h₁₁ is represented in ohms whereas h₂₁ is a dimensionless quantity.

Open Circuit Hybrid Parameters in Two-port Network

• Consider a two-port network with two ports, port 1 and port 2 where the current direction is such that it enters the network.
• The open-circuit hybrid parameters are h₂₂ and h₁₂ respectively. When port 1 is open-circuited, the current through port 1 is zero.
• The open-circuit output impedance at port 2 when port 1 is open-circuited is h22.
  \(h_{22}=\frac{I_2}{V_2} \text{ when } I_1=0\)
  Where,
  I₂ is the current at port 2
  V₂ is the voltage at port 2
  h₂₂ is the open-circuit output impedance
• The open-circuit reverse voltage gain at port 2 when port 1 is open-circuited is h12.
  \(h_{12}=\frac{V_1}{V_2} \text{ when } I_1=0\)
  Where,
  V₁ is the voltage at port 1
  V₂ is the voltage at port 2
  h₁₂ is the open circuit reverse voltage gain
• h₂₂ is represented in Siemens whereas h₁₂ is a dimensionless quantity.

Features of Hybrid Parameters in Two-port Network

The following points give the features of hybrid parameters in a two-port network.

• Hybrid parameters are used in the analysis of transistor circuits to a greater extent.
• These parameters help in establishing a hybrid relationship where the voltage of one port and the current of the other port are taken as the independent variables.
• Hybrid parameters are also called h parameters.
• They are called the hybrid parameters as dimensionally they represent one impedance, one admittance, a voltage gain, and a current gain.
• A transformer does not have Z and Y parameters but they have hybrid parameters.

Steps to Obtain Hybrid Parameters in Two-port Network

• Let a two-port network have currents I₁ and I₂ at port 1 and port 2 respectively and voltages V₁ and V₂ at port 1 and port 2 respectively.
• The equations for h parameters of the network can be given by the following expressions:
  \(V_1=h_{11} I_1+h_{12} V_2\)
  \(I_2=h_{21} I_1+h_{22} V_2\)
• Substitute voltage in port 2 as zero i.e., short-circuit port 2. The short circuit input impedance and the short circuit forward current gain can be obtained.
  \(V_2=0\)
  \(h_{11}=\frac{V_1}{I_1}\)
  \(h_{21}=\frac{I_2}{I_1} \)
• Substitute current in port 1 as zero i.e., open circuit port 1. The open-circuit reverse voltage gain and open circuit output admittance are obtained.
  \(I_1=0\)
  \(h_{12}=\frac{V_1}{V_2} \)
  \(h_{22}=\frac{I_2}{V_2} \)

Equivalent Circuit for Hybrid Parameters

The given figure shows the equivalent circuit for hybrid parameters.

• The mathematical representation of the hybrid parameters is given as follows:
  \(V_1=h_{11} I_1+h_{12} V_2\)
  \(I_2=h_{21} I_1+h_{22} V_2\)
• The equivalent circuit must satisfy the mathematical representation for hybrid parameters. Thus, these equations are used as the basis to model the circuit.
• The two ports in the circuit are represented by 11’ and 22’.
• The input and output currents are I₁ and I₂ respectively while the voltages across 11’ and 22’ are V₁ and V₂ respectively.
• The short circuit input impedance (h₁₁) and open circuit reverse voltage gain along with voltage in port 2 (h₁₂V₂) are placed in series in port 1.
• The open-circuit output admittance (h₂₂) and short circuit forward current gain along with the current in port 1 (h₂₁I₁) are placed in parallel in port 2.

Inverse Hybrid Parameters in Two-Port Networks

Inverse hybrid parameters, also called g-parameters, are another set of network parameters, where the dependent and independent variables are swapped compared to hybrid parameters.

• The voltages in port 1 and port 2 are V₁ and V₂ respectively and the currents in port 1 and port 2 are I₁ and I₂ respectively. The direction of the current is such that it enters the network.
• The voltage at port 2, V₂ and current at port 1, I₁ are dependent variables while the voltage at port 1, V₁, and current at port 2, I₂ are independent variables.
• The parameters g₁₁, g₁₂, g₂₁, and g₂₂ are the network functions and are called inverse hybrid parameters.
• The mathematical expression for inverse hybrid parameters is given by these two equations.
  \(I_1=g_{11} V_1+g_{12} I_2\)
  \(V_2=g_{21} V_1+g_{22} I_2\)
• The equations obtained can be represented in matrix form.
  \(\begin{bmatrix}
  I_1 \\
  V_2
  \end{bmatrix}
  =
  \begin{bmatrix}
  g_{11} & g_{12} \\
  g_{21} & g_{22}
  \end{bmatrix}
  \begin{bmatrix}
  V_1 \\
  I_2
  \end{bmatrix}\)
  Where, \( \begin{bmatrix}
  g_{11} & g_{12} \\
  g_{21} & g_{22}
  \end{bmatrix}\) are the inverse hybrid parameters

Open Circuit Inverse Hybrid Parameters in Two-Port Networks

• Consider a two-port network with port 1 and port 2 where the current direction is such that it enters the network.
• The open-circuit inverse hybrid parameters are g₂₁ and g₁₁ respectively. When port 2 is open-circuited, the current through port 2 is zero.
• The open-circuit input admittance at port 1 when port 2 is open-circuited is g11.
  \(g_{11}=\frac{I_1}{V_1} \text{ when } I_2=0\)
  Where,
  I₁ is the current at port 1
  V₁ is the voltage at port 1
  g₁₁ is the open-circuit input admittance
• The open-circuit forward voltage gain at port 1 when port 2 is open-circuited is g₂₁.
  \(g_{21}=\frac{V_2}{V_1} \text{ when } I_2=0\)
  Where,
  V₁ is the voltage at port 1
  V₂ is the voltage at port 2
  g₂₁ is the open-circuit forward voltage gain
• g₁₁ is represented in Siemens whereas g₂₁ is a dimensionless quantity.

Short Circuit Inverse Hybrid Parameters in Two-Port Networks

• Let the two ports in the circuit be port 1 and port 2. The direction of the current is taken such that it enters the network.
• The short circuit inverse hybrid parameters in a two-port network are g₁₂ and g₂₂. The voltage across port 1 is zero when port 1 is short-circuited.
• The short-circuit output impedance at port 2 when port 1 is short-circuited is g₂₂.
  \(g_{22}=\frac{V_2}{I_2} \text{ when }V_1=0\)
  Where,
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  V₂ is the voltage at port 2
  I₂ is the current at port 2
  g₂₂ is the short circuit output impedance
• The short-circuit reverse current gain at port 2 when port 1 is short-circuited is g₁₂.
  \(g_{12}=\frac{I_1}{I_2} \text{ when } V_1=0\)
  Where,
  I₁ is the current at port 1
  I₂ is the current at port 2
  g₁₂ is the short-circuit reverse current gain
• g₂₂ is represented in ohms whereas g₁₂ is a dimensionless quantity.

Steps to Obtain the Inverse Hybrid Parameters in Two-Port Networks

• A two-port network has currents I₁ and I₂ at port 1 and port 2 respectively and voltages V₁ and V₂ at port 1 and port 2 respectively.
• The equations for inverse hybrid parameters or g parameters of the network can be given by the following expressions:
  \(I_1=g_{11} V_1+g_{12} I_2\)
  \(V_2=g_{21} V_1+g_{22} I_2\)
• Substitute voltage in port 1 as zero i.e., short-circuit port 1. The short circuit output impedance and the short circuit reverse current gain can be obtained.
  \(V_1=0\)
  \(g_{22}=\frac{V_2}{I_2} \)
  \(g_{12}=\frac{I_1}{I_2} \)
• Substitute current in port 2 as zero i.e., open circuit port 2. The open-circuit forward voltage gain and open circuit input admittance are obtained.
  \(I_2=0\)
  \(g_{11}=\frac{I_1}{V_1} \)
  \(g_{21}=\frac{V_2}{V_1} \)

Equivalent Circuits for Inverse Hybrid Parameters

The given figure shows the equivalent circuit for inverse hybrid parameters.

• The inverse hybrid parameters in the mathematical form are given as follows:
  \(I_1=g_{11} V_1+g_{12} I_2\)
  \(V_2=g_{21} V_1+g_{22} I_2\)
• The equations given above are the basis to model the circuit. The equivalent circuit must satisfy the mathematical representation for inverse hybrid parameters.
• The two ports in the equivalent circuit for inverse hybrid parameters are represented by 11’ and 22’.
• The voltages across 11’ and 22’ are V₁ and V₂ respectively while the input and output currents are I₁ and I₂ respectively.
• The open-circuit input admittance (g₁₁) and short circuit reverse current gain along with the current in port 2 (g₁₂I₂) are placed in parallel in port 1.
• The short circuit output impedance (g₂₂) and open circuit forward voltage gain along with voltage in port 1 (g₂₁V₁) are placed in series in port 2.

Hybrid Parameters Vs Inverse Hybrid Parameters

The following table gives the difference between hybrid parameters and inverse hybrid parameters in a two-port network.

Conditions for Reciprocity and Symmetry for Hybrid and Inverse Hybrid Parameters

Condition for reciprocity

• The short circuit forward current gain is equal to the negative of the open-circuit reverse voltage gain for hybrid parameters.
  \(h_{12}=-h_{21}\)
• The open-circuit forward voltage gain is equal to the negative of short circuit reverse current gain for hybrid parameters.
  \(g_{12}=-g_{21}\)
• If a network follows the above conditions, the network is reciprocal.

Condition for symmetry

• To satisfy the condition for symmetry for hybrid parameters, the determinant of the hybrid parameter matrix, ‘h’ is equal to 1.
  \(h_{11} h_{22}-h_{12} h_{21}=1\)
• To satisfy the condition for symmetry for inverse hybrid parameters, the determinant of the inverse hybrid parameter matrix, ‘g’ must be equal to 1.
  \(g_{11}g_{22}-g_{12} g_{21}=1\)
• In a symmetrical network, with one port open, the ratio of voltage of current at one port will be the same as the ratio of voltage of current at another port.