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# Network Theory - Two-Port Networks

In general, it is easy to analyze any electrical network, if it is represented with an equivalent model, which gives the relation between input and output variables. For this, we can use **two port network** representations. As the name suggests, two port networks contain two ports. Among which, one port is used as an input port and the other port is used as an output port. The first and second ports are called as port1 and port2 respectively.

**One port network** is a two terminal electrical network in which, current enters through one terminal and leaves through another terminal. Resistors, inductors and capacitors are the examples of one port network because each one has two terminals. One port network representation is shown in the following figure.

Here, the pair of terminals, 1 & 1’ represents a port. In this case, we are having only one port since it is a one port network.

Similarly, **two port network** is a pair of two terminal electrical network in which, current enters through one terminal and leaves through another terminal of each port. Two port network representation is shown in the following figure.

Here, one pair of terminals, 1 & 1’ represents one port, which is called as **port1** and the other pair of terminals, 2 & 2’ represents another port, which is called as **port2**.

There are **four variables** V_{1}, V_{2}, I_{1} and I_{2} in a two port network as shown in the figure. Out of which, we can choose two variables as independent and another two variables as dependent. So, we will get six possible pairs of equations. These equations represent the dependent variables in terms of independent variables. The coefficients of independent variables are called as **parameters**. So, each pair of equations will give a set of four parameters.

## Two Port Network Parameters

The parameters of a two port network are called as **two port network parameters** or simply, two port parameters. Following are the types of two port network parameters.

- Z parameters
- Y parameters
- T parameters
- T’ parameters
- h-parameters
- g-parameters

Now, let us discuss about these two port network parameters one by one.

### Z parameters

We will get the following set of two equations by considering the variables V_{1} & V_{2} as dependent and I_{1} & I_{2} as independent. The coefficients of independent variables, I_{1} and I_{2} are called as **Z parameters**.

$$V_1 = Z_{11} I_1 + Z_{12} I_2$$

$$V_2 = Z_{21} I_1 + Z_{22} I_2$$

The **Z parameters** are

$$Z_{11} = \frac{V_1}{I_1}, \: when \: I_2 = 0$$

$$Z_{12} = \frac{V_1}{I_2}, \: when \: I_1 = 0$$

$$Z_{21} = \frac{V_2}{I_1}, \: when \: I_2 = 0$$

$$Z_{22} = \frac{V_2}{I_2}, \: when \: I_1 = 0$$

Z parameters are called as **impedance parameters** because these are simply the ratios of voltages and currents. Units of Z parameters are Ohm (Ω).

We can calculate two Z parameters, Z_{11} and Z_{21}, by doing open circuit of port2. Similarly, we can calculate the other two Z parameters, Z_{12} and Z_{22} by doing open circuit of port1. Hence, the Z parameters are also called as **open-circuit impedance parameters**.

### Y parameters

We will get the following set of two equations by considering the variables I_{1} & I_{2} as dependent and V_{1} & V_{2} as independent. The coefficients of independent variables, V_{1} and V_{2} are called as **Y parameters**.

$$I_1 = Y_{11} V_1 + Y_{12} V_2$$

$$I_2 = Y_{21} V_1 + Y_{22} V_2$$

The **Y parameters** are

$$Y_{11} = \frac{I_1}{V_1}, \: when \: V_2 = 0$$

$$Y_{12} = \frac{I_1}{V_2}, \: when \: V_1 = 0$$

$$Y_{21} = \frac{I_2}{V_1}, \: when \: V_2 = 0$$

$$Y_{22} = \frac{I_2}{V_2}, \: when \: V_1 = 0$$

Y parameters are called as **admittance parameters** because these are simply, the ratios of currents and voltages. Units of Y parameters are mho.

We can calculate two Y parameters, Y_{11} and Y_{21} by doing short circuit of port2. Similarly, we can calculate the other two Y parameters, Y_{12} and Y_{22} by doing short circuit of port1. Hence, the Y parameters are also called as **short-circuit admittance parameters**.

### T parameters

We will get the following set of two equations by considering the variables V_{1} & I_{1} as dependent and V_{2} & I_{2} as independent. The coefficients of V_{2} and -I_{2} are called as **T parameters**.

$$V_1 = A V_2 - B I_2$$

$$I_1 = C V_2 - D I_2$$

The **T parameters** are

$$A = \frac{V_1}{V_2}, \: when \: I_2 = 0$$

$$B = -\frac{V_1}{I_2}, \: when \: V_2 = 0$$

$$C = \frac{I_1}{V_2}, \: when \: I_2 = 0$$

$$D = -\frac{I_1}{I_2}, \: when \: V_2 = 0$$

T parameters are called as transmission parameters or **ABCD parameters**. The parameters, A and D do not have any units, since those are dimension less. The units of parameters, B and C are ohm and mho respectively.

We can calculate two parameters, A and C by doing open circuit of port2. Similarly, we can calculate the other two parameters, B and D by doing short circuit of port2.

### T ’ parameters

We will get the following set of two equations by considering the variables V_{2} & I_{2} as dependent and V_{1} & I_{1} as independent. The coefficients of V_{1} and -I_{1} are called as **T’ parameters**.

$$V_2 = A' V_1 - B' I_1$$

$$I_2 = C' V_1 - D' I_1$$

The **T’ parameters** are

$$A' = \frac{V_2}{V_1}, \: when\: I_1 = 0$$

$$B' = -\frac{V_2}{I_1}, \: when\: V_1 = 0$$

$$C' = \frac{I_2}{V_1}, \: when\: I_1 = 0$$

$$D' = -\frac{I_2}{I_1}, \: when \: V_1 = 0$$

T’ parameters are called as inverse transmission parameters or **A’B’C’D’ parameters**. The parameters A’ and D’ do not have any units, since those are dimension less. The units of parameters, B’ and C’, are Ohm and Mho respectively.

We can calculate two parameters, A’ and C’, by doing an open circuit of port1. Similarly, we can calculate the other two parameters, B’ and D’, by doing a short circuit of port1.

### h-parameters

We will get the following set of two equations by considering the variables V_{1} & I_{2} as dependent and I_{1} & V_{2} as independent. The coefficients of independent variables, I_{1} and V_{2}, are called as **h-parameters**.

$$V_1 = h_{11} I_1 + h_{12} V_2$$

$$I_2 = h_{21} I_1 + h_{22} V_2$$

The h-parameters are

$$h_{11} = \frac{V_1}{I_1},\: when\: V_2 = 0$$

$$h_{12} = \frac{V_1}{V_2},\: when\: I_1 = 0$$

$$h_{21} = \frac{I_2}{I_1},\: when\: V_2 = 0$$

$$h_{22} = \frac{I_2}{V_2},\: when\: I_1 = 0$$

h-parameters are called as **hybrid parameters**. The parameters, h_{12} and h_{21}, do not have any units, since those are dimension-less. The units of parameters, h_{11} and h_{22}, are Ohm and Mho respectively.

We can calculate two parameters, h_{11} and h_{21} by doing short circuit of port2. Similarly, we can calculate the other two parameters, h_{12} and h_{22} by doing open circuit of port1.

The h-parameters or hybrid parameters are useful in transistor modelling circuits (networks).

### g-parameters

We will get the following set of two equations by considering the variables I_{1} & V_{2} as dependent and V_{1} & I_{2} as independent. The coefficients of independent variables, V_{1} and I_{2} are called as **g-parameters**.

$$I_1 = g_{11} V_1 + g_{12} I_2$$

$$V_2 = g_{21} V_1 + g_{22} I_2$$

The **g-parameters** are

$$g_{11} = \frac{I_1}{V_1},\: when\: I_2 = 0$$

$$g_{12} = \frac{I_1}{I_2},\: when\: V_1 = 0$$

$$g_{21} = \frac{V_2}{V_1},\: when\: I_2 = 0$$

$$g_{22} = \frac{V_2}{I_2},\: when \: V_1 = 0$$

g-parameters are called as **inverse hybrid parameters**. The parameters, g_{12} and g_{21} do not have any units, since those are dimension less. The units of parameters, g_{11} and g_{22} are mho and ohm respectively.

We can calculate two parameters, g_{11} and g_{21} by doing open circuit of port2. Similarly, we can calculate the other two parameters, g_{12} and g_{22} by doing short circuit of port1.