Admittance Parameter (Y Parameter)

Admittance Parameter (Y Parameter)

The admittance parameter (also known as Y parameter) is used in many areas of application in electrical engineering like telecommunication, and power electronics. Similar to the Z parameter, it is also used to describe the behavior of a linear electrical network and the small-signal response of non-linear networks.

Y parameters are also known as short-circuited admittance parameters. Because, to calculate the values of these parameters, we need to short each port of the two-port network.

As we know, the admittance is a ratio of current to the voltage. And it is reciprocal of the impedance (Z). The equation of admittance is as;

    \[ [Y] = \frac{[I]}{[V]} \]

    \[ [I] = [Y][V] \]

The above equation is in matrix form. Where [Y] shows an admittance matrix. Consider the below two-port network.

Two-port Network
Two-port Network

    \[ I_1 = f_1(V_1,V_2) \]

    \[ I_2 = f_2(V_1,V_2) \]

(1)   \begin{equation*} \begin{align} I_1 &= Y_{11}V_1 + Y_{12}V_2 \\ I_2 &= Y_{21}V_1 + Y_{22}V_2 \end{align} \end{equation*}

The matrix form of the above equations is;

(2)   \begin{equation*} \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} \end{equation*}

In this equation, the four admittance parameters are represented as Y11, Y12, Y21, and Y22. We will calculate the value of each parameter.

Calculation of Y Parameter

We need to perform two cases, to calculate the values of each admittance parameter. In the first case, we will short the port-1. In this condition, the voltage across the port-1 V1 is zero. And in the second case, we will short the port-2. In this condition, the voltage across the port-2 V2 is zero.

First Case

In his condition, the port-1 is short-circuited. And hence, the voltage of port-1 V1 is zero. This condition is as shown in the below figure.

First case port-1 short
First case port-1 short

Put the value V1=0 in equation-1,

    \[ Y_{12} = \left \frac{I_1}{V_2} \right\vert_{V_1 = 0} \]

    \[ Y_{22} = \left \frac{I_2}{V_2} \right\vert_{V_1 = 0} \]

Second Case

In this case, the port-2 is short-circuited. And hence, the voltage of port-2 V2 is zero. this condition is as shown in the below figure.

Second case port-2 short
Second case port-2 short

    \[ Y_{11} = \left \frac{I_1}{V_1} \right\vert_{V_2 = 0} \]

    \[ Y_{21} = \left \frac{I_2}{V_1} \right\vert_{V_2 = 0} \]

As we can see, the value of the Y parameter is a ratio of Current to the voltage. Therefore, this parameter is also known as the Admittance Parameter. Hence, the unit of all Y parameter is mho.

Reciprocity Conditions for Y Parameter

A network is said to be reciprocal if the port-2 voltage is due to applied current at port-1 is the same as the port-1 voltage when the applied current is the same as port-2.

Generally, the network that consists of passive elements (resistor, inductor, and capacitor) only, that network are reciprocal networks and the network that consists of the active element (transistor or generator), that networks are not a reciprocal network.

Consider the below figure for the first condition,

Reciprocity Condition-1 Y Parameter
Reciprocity Condition-1 Y Parameter

From above figure,

    \[V_1=V_s \quad V_2=0 \quad I_2=-I_2' \]

Put these values in equation-1,

    \[ I_1 = Y_{11}V_s + Y_{12}(0) \]

    \[ -I_2' = Y_{21}V_s + Y_{22}(0) \]

    \[ -I_2' = Y_{21}V_s \]

    \[ \frac{V_s}{I_2'} = \frac{-1}{Y_{21}} \]

Consider the below figure for second condition,

Reciprocity Condition-2 Y Parameter
Reciprocity Condition-2 Y Parameter

    \[ V_2 = V_s \quad V_1 = 0 \quad I_1' = -I_1 \]

Put these values in equation-1,

    \[ -I_1' = Y_{11}(0) + Y_{12}V_s \]

    \[ I_2 = Y_{21}(0) + Y_{22}V_s \]

    \[ -I_1' = Y_{12}V_s \]

    \[ \frac{V_s}{I_1'} = \frac{-1}{Y_{12}} \]

The network is said to be reciprocal if,

    \[ \frac{V_s}{I_2'} = \frac{V_s}{I_1'} \]

    \[ \frac{-1}{Y_{21}} = \frac{-1}{Y_{12}} \]

    \[ Y_{21} = Y_{12} \]

If the two-port network, fulfil this condition, the electrical network is said to a Reciprocal Network.

Symmetry Conditions for Y Parameter

The electrical network is said to be symmetrical, if the input impedance is equal to the output impedance. It is not necessary that, this circuits are physically symmetrical.

Consider the below figure for the first Symmetry condition of Y parameter,

Symmetry Condition-1 Y Parameter
Symmetry Condition-1 Y Parameter

From the above figure,

    \[ V_1 = V_s \quad I_2 = 0 \]

Put these values in the main equations of Z parameter.

    \[ I_1 = Y_{11}V_s + Y_{12}V_2 \]

    \[ 0 = Y_{21}V_s + Y_{22}V_2 \]

    \[ V_2 = - \frac{Y_{21}}{Y_{22}} V_s \]

    \[ I_1 = Y_{11}V_s + Y_{12}\left[-\frac{Y_{21}}{Y_{22}}\right] V_s \]

    \[ I_1 = \left[ Y_{11} - Y_{12} \frac{Y_{21}}{Y_{22}}\right] V_s \]

    \[ I_1 = \left[ \frac{Y_{11}Y_{22} - Y_{12}Y_{21}}{Y_{22}} \right] V_s \]

    \[ \frac{V_s}{I_1} = \frac{Y_{22}}{Y_{11}Y_{22} - Y_{12}Y_{21}} \]

Now, consider the below figure for the second Symmetry condition of Y parameter,

Symmetry Condition-2 Y Parameter
Symmetry Condition-2 Y Parameter

    \[ V_2 = V_s \quad I_1 = 0 \]

Put these values in the main equations of Y parameter.

    \[ 0 = Y_{11}V_1 + Y_{12}V_s \]

    \[ I_2 = Y_{21}V_1 + Y_{22}V_s \]

    \[ V_1 = \frac{-Y_{12}}{Y_{11}} V_s \]

    \[ I_2 = Y_{21} \frac{-Y_{12}}{Y_{11}} V_s + Y_{22}V_s \]

    \[ I_2 = \left[Y_{21} \frac{-Y_{12}}{Y_{11}} + Y_{22} \right] V_s \]

    \[ I_2 = V_s \left[\frac{Y_{11}Y_{22}-Y_{12}Y_{21}}{Y_{11}} \right] \]

    \[ \frac{I_2}{V_s} = \left[\frac{Y_{11}Y_{22}-Y_{12}Y_{21}}{Y_{11}} \right] \]

    \[ \frac{V_s}{I_2} = \left[\frac{Y_{11}}{Y_{11}Y_{22}-Y_{12}Y_{21}} \right] \]

For symmetrical networks, both the port impedances must be equal,

    \[ \frac{V_s}{I_1} = \frac{V_s}{I_2} \]

    \[ \frac{Y_{22}}{Y_{11}Y_{22} - Y_{12}Y_{21}} = \frac{Y_{11}}{Y_{11}Y_{22}-Y_{12}Y_{21}} \]

    \[ Y_{11} = Y_{22} \]

If the two-port network, fulfil this condition, the electrical network is said to a Symmetric Network.

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