Two-Port Network

Two-Port Network

Two-Port Network is a four terminals electrical network or circuit with two pairs of terminals. These terminals are connected with an external circuit.

In Two-Port Network, one port is known as input port and the second port is known as an output port. It is used to the mathematical analysis of the electrical circuit to isolate a portion of the large network.

One-Port Network

One-port Network consists of only two terminals where the current enters from one terminal and leaves from the second terminal. i.e. resistor, inductor, and capacitor. The below figure shows the representation of the One-port Network.

One-Port Network
One-Port Network

Here one pair of terminals 1 and 1’ is available. In the above figure, the current enters from the terminal-1 and leaves from the terminal-1’.

Two-Port Network

In a two-port network, there are two pairs of terminals. The current enters from the one terminal and leaves from the second terminal. There are two sources of voltages. The below figure shows a graphical representation of the two-port network.

Two-Port Network
Two-Port Network

As shown in the above figure, there are four variables; V1, V2, I1, and I2. From four variables, any two variables are independent, and the remaining two variables are the dependent variable. Hence, it forms six pairs of equations.

In these equations, the coefficients of independent variables are called Parameters. These six parameters for two-port network that are listed below.

  • Z Parameter (Impedance Parameter)
  • Y Parameter (Admittance Parameter)
  • T Parameter (Transmission Parameter) (ABCD Parameter)
  • T’ Parameter (Inverse ABCD Parameter)
  • H Parameter (Hybrid Parameter)
  • H’ Parameter (Inverse Hybrid Parameter)

Let’s find equations for all parameters.

Z Parameter

In the Z parameter, V1, V2 variables are dependent variables, and I1, I2 are the independent variables. The coefficients of the independent variable (I1 and I2) are known as Z Parameter.

In this parameter, the voltages are a function of currents.

    \[ V_1 = f(I_1,I_2) \]

    \[ V_2 = f(I_1, I_2) \]

And the equation for Z parameter is,

    \[ V_1 = Z_1_1 I_1 + Z_1_2 I_2 \]

    \[ V_2 = Z_2_1 I_1 + Z_2_2 I_2 \]

(1)   \begin{gather*} \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix}  \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} \end{gather*}

Where, Z11, Z12, Z21, and Z22 are known as the Z parameter.

From the above equations, if we consider the port-2 is an open circuit. Hence, the current pass through the port-2 is zero. So, I2 is equal to zero. Put this value in the above equation.  And we get the value of Z11 and Z21.

    \[ Z_{11} = \left. \frac{V_1}{I_1} \right\vert_{I_2 = 0} \]

    \[ Z_{21} = \left. \frac{V_2}{I_1} \right\vert_{I_2 = 0} \]

Similarly, if we consider the port-1 is open. So, I1 is zero. Put this value in equations of the Z parameter and we get the value of Z12 and Z22.

    \[ Z_{12} = \left. \frac{V_1}{I_2} \right\vert_{I_1 = 0} \]

    \[ Z_{22} = \left. \frac{V_2}{I_2} \right\vert_{I_1 = 0} \]

As we can see, the value of the Z parameter is a ratio of voltage and current. Hence, this parameter is also known as the Impedance parameter. Hence, the unit of all Z parameter is ohm.

Y Parameter

In the Y parameter, I1, I2 variables are dependent variables, and V1, V2 are the independent variables. The coefficients of the independent variable (V1 and V2) are known as Y Parameter.

In this parameter, the currents are a function of voltages.

    \[ I_1 = f(V_1, V_2) \]

    \[ I_2 = f(V_1, V_2) \]

The equations of Y parameter is;

    \[ I_1 = Y_1_1 V_1 + Y_1_2 V_2 \]

    \[ I_2 = Y_2_1 V_1 + Y_2_2 V_2 \]

(2)   \begin{gather*} \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{gather*}

Where, Y11, Y12, Y21, and Y22 are known as the Y parameter.

From the above equations, if we consider the port-2 is short. So, V2 is zero. Put this value in equations of the Y parameter and we get the value of Y11 and Y21.

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

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

Similarly, if we consider the port-1 is short. So, V1 is zero. Put this value in equations of the Y parameter and we get the value of Y12 and Y22.

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

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

As we can see in the value of the Y parameter, it is a ratio of current and voltage that gives an admittance. Hence, this parameter is also known as the admittance parameter. The unit of all Y parameter is mho.

H Parameter

In the H parameter, V1, I2 variables are dependent variables, and V2, I1 are the independent variable. The coefficients of the independent variable (V2 and I1) are known as H Parameter.

    \[ V_1 = f(I_1, V_2) \]

    \[ V_2 = f(I_1, V_2) \]

The equations of H parameter is;

    \[ V_1 = h_{11} I_1 + h_{12} V_2 \]

    \[ I_2 = h_{21} I_1 + h_{22} V_2 \]

(3)   \begin{gather*} \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} \end{gather*}

Where, h11, h12, h21, and h22 are known as the H parameter.

From the above equations, if we consider the port-2 is short. Hence, V2 is zero. Put this value in equations of the H parameter and we get the value of h12 and h22.

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

    \[ h_{21} = \left. \frac{I_2}{I_1} \right\vert_{V_2 = 0} \]

Similarly, if we consider the port-1 is open. So, I1 is zero. Put this value in equations of the H parameter and we get the value of h12 and h22.

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

    \[ h_{22} = \left. \frac{I_2}{V_2} \right\vert_{I_1 = 0} \]

This parameter is also known as Hybrid Parameter. Here, the unit of h11 is ohm, h22 is mho, and h12 & h21 are unitless.

H’ Parameter

In H’ parameter, I1, V2 variables are dependent variables, and V1, I2 are independent variables. The coefficients of the independent variable (V1 and I2) are known as H’ Parameter.

    \[ I_1 = f(V_1,I_2) \]

    \[ V_2 = f(V_1,I_2) \]

The equations of H’ parameter is;

    \[ I_1 = h'_{11} V_1 + h_{12}' I_2 \]

    \[ V_2 = h'_{21} V_1 + h_{22}' I_2 \]

(4)   \begin{gather*} \begin{bmatrix} I_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} h'_{11} & h'_{12} \\ h'_{21} & h'_{22} \end{bmatrix}  \begin{bmatrix} V_1 \\ I_2 \end{bmatrix} \end{gather*}

Where, h’11, h’12, h’21, and h’22 are known as the H’ parameter.

From the above equations, if we consider the port-2 is open. Hence, I2 is zero. Put this value in equations of the H’ parameter and we get the value of h’11 and h’21.

    \[ h'_{11} = \left. \frac{I_1}{V_1} \right\vert_{I_2 = 0} \]

    \[ h'_{21} = \left. \frac{V_2}{V_1} \right\vert_{I_2 = 0} \]

Similarly, if we consider the port-1 is short. So, V1 is zero. Put this value in equations of the H’ parameter and we get the value of h’12 and h’22.

    \[ h'_{12} = \left. \frac{I_1}{I_2} \right\vert_{V_1 = 0} \]

    \[ h'_{22} = \left. \frac{V_2}{I_2} \right\vert_{V_1 = 0} \]

In H’ Parameter, the variables are inverse compared to the H Parameter. Therefore, this parameter is also known as Inverse Hybrid Parameter. Here, the unit of h11 is mho, h22 is ohm, and h12 & h21 are unitless.

T Parameter

In T parameter, V1, I1 variables are dependent variables, and V2, I2 are independent variables. The coefficients of the independent variable (V2 and -I2) are known as T Parameter.

Here, the input part is expressed in terms of the output part.

    \[ V_1 = f(V_2, -I_2) \]

    \[ I_1 = f(V_2, -I_2) \]

The equations of T parameter is;

    \[ V_1 = AV_2 - BI_2 \]

    \[ I_1 = CV_2 - DI_2 \]

(5)   \begin{gather*} \begin{bmatrix} V_1 \\ I_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix}  \begin{bmatrix} V_2 \\ -I_2 \end{bmatrix} \end{gather*}

Where, A, B, C, and D are known as the T parameter.

From the above equations, if we consider the port-2 is open. Hence, I2 is zero. Put this value in equations of the T parameter and we get the value of A and C.

    \[ A = \left. \frac{V_1}{V_2} \right\vert_{I_2 = 0} \]

    \[ C = \left. \frac{I_1}{V_2} \right\vert_{I_2 = 0} \]

Similarly, if we consider the port-2 is short. So, V2 is zero. Put this value in equations of the T parameter and we get the value of B and D.

    \[ B = - \left. \frac{V_1}{I_2} \right\vert_{V_2 = 0} \]

    \[ D = - \left. \frac{I_1}{I_2} \right\vert_{V_2 = 0} \]

T parameter is called a Transmission Parameter or ABCD parameter. Here, unit of B is ohm, C is mho and A & D are unitless.

T’ Parameter

In T’ parameter, V2, I2 variables are dependent variables, and V1, I1 are independent variables. The coefficients of the independent variable (V1 and -I1) are known as T’ Parameter.

Here, the variable of the output port is expressed in terms of the variable of the input port.

    \[ V_2 = f(V_1,I_1) \]

    \[ I_2 = f(V_1,I_1) \]

The equations of T’ parameter is;

    \[ V_2 = A'V_1 - B'I_1 \]

    \[ I_2 = C'V_1 - D'I_1 \]

(6)   \begin{gather*} \begin{bmatrix} V_2 \\ I_2 \end{bmatrix} = \begin{bmatrix} A' & B' \\ C' & D' \end{bmatrix}  \begin{bmatrix} V_1 \\ -I_1 \end{bmatrix} \end{gather*}

Where, A’, B’, C’, and D’ are known as the T’ parameter.

From the above equations, if we consider the port-1 is open. Hence, I1 is zero. Put this value in equations of the T’ parameter and we get the value of A’ and C’.

    \[ A' = \left. \frac{V_2}{V_1} \right\vert_{I_1 = 0} \]

    \[ C' = \left. \frac{I_2}{V_1} \right\vert_{I_1 = 0} \]

Similarly, if we consider the port-1 is short. So, V1 is zero. Put this value in equations of the T’ parameter and we get the value of B’ and D’.

    \[ B' = - \left. \frac{V_2}{I_1} \right\vert_{V_1 = 0} \]

    \[ D' = - \left. \frac{I_2}{I_1} \right\vert_{V_1 = 0} \]

In T’ Parameter, the variables are inverse than the T Parameter. Therefore, this parameter is also known as Inverse Transmission Parameter or Inverse ABCD Parameter.

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