Short Circuit Admittance (Y) Parameters in Two Port Networks

In this tutorial, you will learn about short-circuit admittance parameters, including their definition, driving point and transfer admittance parameters, and the equivalent circuit for Y parameters. You will also explore conditions for reciprocity and symmetry, the steps to obtain Y parameters, differences between Z and Y parameters, and the π-equivalent circuit for reciprocal networks.

Contents:

1. What is Short-circuit Admittance Parameters?
2. Driving Point Admittance Parameters in a Two-port Network
3. Transfer Admittance Parameters in a Two-port Network
4. Equivalent Circuit for Y Parameters
5. Conditions for Reciprocity and Symmetry for Y Parameters
6. Steps to Obtain Short-Circuit Admittance Parameters
7. Difference Between Z Parameters and Y Parameters
8. π-Equivalent Circuit for Reciprocal Networks

What is Short-circuit Admittance Parameters?

Short-circuit admittance parameters, often denoted as Y parameters, characterize the relationship between currents and voltages in a two-port network when one or both ports are short-circuited. These parameters are particularly useful for analyzing circuits in terms of admittance rather than impedance.

• The voltage and current in the input terminals of a two-port network are V₁ and I₁ respectively and the voltage and current in the output terminals are V₂ and I₂ respectively.
• The currents I₁ and I₂ are dependent variables while the voltages V₁ and V₂ are independent variables. The direction of the current is such that it enters the network.
• The admittances Y₁₁, Y₁₂, Y₂₁, and Y₂₂ are the network functions and are called admittance or Y parameters.
• The current equation for port 1 and port 2 are given by the following expression:
  \(I_1=Y_{11} V_1+Y_{12} V_2\)
  \(I_2=Y_{21} V_1+Y_{22} V_2\)
• The above equations can be represented in matrix form.
  \([I]=[Y][V]\)
  \(\begin{bmatrix}
  I_1 \\
  I_2
  \end{bmatrix}
  =
  \begin{bmatrix}
  Y_{11} & Y_{12} \\
  Y_{21} & Y_{22}
  \end{bmatrix}
  \begin{bmatrix}
  V_1 \\
  V_2
  \end{bmatrix}\)
  Where, [V] is the voltage matrix
  [I] is the current matrix
  [Y] is the admittance matrix

Driving Point Admittance Parameters in a Two-port Network

The driving point admittances are the parameters that describe the relationship between the input voltage and current at a specific port when the other port is short-circuited.

• 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 admittances Y₁₁ and Y₂₂ are the driving point admittance of a two-port network.
• Short-circuit parameters also called the Y parameters and are obtained when the parameters are calculated with each port shorted.
• When port 2 is short-circuited, the voltage across that port is zero. The driving point admittance at port 1 when port 2 is short-circuited is Y11.
  \(Y_{11}=\frac{I_1}{V_1} \text{ when } V_2=0\)
  Where,
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  V₁ is the voltage at the input terminals
  I₁ is the current at the input terminals
  Y₁₁ is the short-circuit input admittance
• When port 1 is short-circuited, the voltage across that port is zero. The driving point admittance at port 2 when port 1 is short-circuited is Y₂₂.
  \(Y_{22}=\frac{I_2}{V_2} \text{ when } V_1=0\)
  Where,
  V₂ is the voltage at the output terminals
  I₂ is the current at the output terminals
  Y₂₂ is the short-circuit output admittance

Transfer Admittance Parameters in a Two-port Network

The transfer admittances represent the relationship between the voltage at one port and the current at another port.

• A two-port network has a voltage and current of V₁ and I₁ respectively in the input terminals and the voltage and current of V₂ and I₂ respectively in the output terminals.
• 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 admittances Y₁₂ and Y₂₁ are the transfer admittances in a two-port network.
• The voltage across port 2 is zero when it is short-circuited. The transfer admittance at port 1 when port 2 is short-circuited is Y₂₁.
  \(Y_{21}=\frac{I_2}{V_1} \text{ when } V_2=0\)
  Where,
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  V₂ is the voltage at the output terminals
  I₁ is the current at the input terminals
  Y₂₁ is the short-circuit forward transfer admittance
• The voltage across port 1 is zero when it is short-circuited. The transfer admittance at port 2 when port 1 is short-circuited is Y₁₂.
  \(Y_{12}=\frac{I_1}{V_2} \text{ when }V_1=0\)
  Where,
  V₁ is the voltage at the input terminals
  I₂ is the current at the output terminals
  Y₁₂ is the short-circuit reverse transfer admittance

Equivalent Circuit for Y Parameters

The given figure shows the equivalent circuit for short-circuit admittance parameters.

• The mathematical representation of the short circuit admittance parameters is given as follows:
  \(I_1=Y_{11} V_1+Y_{12} V_2\)
  \(I_2=Y_{21} V_1+Y_{22} V_2\)
• The equivalent circuit must satisfy the mathematical representation for Y parameters. These equations are thus, used to model the circuit.
• The two ports in the equivalent circuit are represented by 11’ and 22’.
• The input and output currents are I₁ and I₂ respectively while the input and output voltages are V₁ and V₂ respectively.
• The driving point admittance Y₁₁ and Y₂₂ are represented by admittance blocks.
• The transfer admittance Y₁₂ and Y₂₁ are represented by dependent current sources.

Conditions for Reciprocity and Symmetry for Y Parameters

The given points are the condition for reciprocity and symmetry for short-circuit admittance parameters.

Condition for reciprocity

• A reciprocal network has equal transfer admittance. Its short-circuit forward transfer admittance and short-circuit reverse transfer admittance are equal.
  \(Y_{12}=Y_{21}\)
• This is a reciprocal or bilateral network that follows the reciprocity theorem.
• Such networks are linear and have no dependent sources.

Condition for symmetry

• A symmetrical network has equal driving point admittance. Its short-circuit input admittance and short-circuit output admittance are equal.
  \(Y_{11}=Y_{22}\)
• The electrical properties in a symmetrical network are not affected by when the input and output terminals are interchanged.
• A symmetrical network expresses a mirror-like symmetry between port 1 and port 2 along an imaginary axis in the middle.

Steps to Obtain Short-Circuit Admittance Parameters

To determine the Y parameters experimentally:

• 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 parameters are obtained when they are calculated with each port shorted.
• The current equations of the network can be given by the following expressions:
  \(I_1=Y_{11} V_1+Y_{12} V_2\)
  \(I_2=Y_{21} V_1+Y_{22} V_2\)
• Substitute voltage in port 2 as zero i.e short-circuit port 2. The driving point input admittance and the forward transfer admittance at port 1 when port 2 is short-circuited can be calculated by the following expressions:
  \(V_2=0\)
  \(Y_{11}=\frac{I_1}{V_1}\)
  \(Y_{21}=\frac{I_2}{V_1}\)
• Substitute voltage in port 1 as zero i.e short-circuit port 1. The driving point output admittance and reverse transfer admittance at port 2 when port 1 is short-circuited can be calculated by the following expressions:
  \(V_1=0\)
  \(Y_{22}=\frac{I_2}{V_2} \)
  \(Y_{12}=\frac{I_1}{V_2} \)

Difference Between Z Parameters and Y Parameters

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

π-Equivalent Circuit for Reciprocal Networks

The given diagram represents the π equivalent reciprocal circuit.

• A π-equivalent circuit is only valid for reciprocal networks. The values of Y_A, Y_B, and Y_C can be calculated by comparing the equations with the current equations representing the Y parameters.
  \(I_1=Y_{11} V_1+Y_{12} V_2………….(1)\)
  \(I_2=Y_{21} V_1+Y_{22} V_2…………..(2)\)
• Applying nodal analysis in the first node of the circuit, the following equation is obtained.
  \(I_1=Y_B V_1+Y_A (V_1-V_2)\)
  \(I_1=(Y_A+Y_B)V_1-Y_A V_2…………(3)\)
• Applying nodal analysis in the second node of the circuit, the following equation is obtained.
  \(I_2=Y_C V_2+Y_A (V_2-V_1)\)
  \(I_2=(Y_A+Y_C)V_2-Y_A V_2…………(4)\)
• Comparing equations (1) and (3) the values of Y₁₁ and Y₁₂ are obtained.
  \(Y_12 =-Y_A\)
  \(Y_A =-Y_{12}\)
  \(Y_{11}=Y_A+Y_B=Y_B-Y_{12}\)
  \(Y_B=Y_{11}+Y_{12}\)
• Comparing equations (2) and (4) the values of Y₂₂ and Y₂₁ are obtained.
  \(Y_{21} =-Y_A\)
  \(Y_A =-Y_{21}\)
  \(Y_{22}=Y_A+Y_C=Y_C-Y_{21}\)
  \(Y_C=Y_{22}+Y_{21}\)
• Thus, it is seen that the network is reciprocal as the transfer admittances are the same. The different admittances in the circuit are as follows:
  \(Y_A =-Y_{12} =-Y_{21}\)
  \(Y_B=Y_{11}+Y_{12}\)
  \(Y_C=Y_{22}+Y_{21}\)