TWO PORT NETWORKS

SUB - TOPICS Z – PARAMETER Y – PARAMETER T (ABCD) – PARAMETER TERMINATED TWO PORT NETWORKS

OBJECTIVES TO UNDERSTAND ABOUT TWO – PORT NETWORKS AND ITS FUNTIONS. TO UNDERSTAND THE DIFFERENT BETWEEN Z – PARAMETER, Y – PARAMETER, T – PARAMETER AND TERMINATED TWO PORT NETWORKS. TO INVERTIGATE AND ANALYSIS THE BEHAVIOUR OF TWO – PORT NETWORKS.

TWO – PORT NETWORKS A pair of terminals through which a current may enter or leave a network is known as a port. Two terminal devices or elements (such as resistors, capacitors, and inductors) results in one – port network. Most of the circuits we have dealt with so far are two – terminal or one – port circuits.

A two – port network is an electrical network with two separate ports for input and output. It has two terminal pairs acting as access points. The current entering one terminal of a pair leaves the other terminal in the pair.

One – port network Two – port network I + V Linear network - I1 + V2

Two (2) reason why to study two port – network: Such networks are useful in communication, control system, power systems and electronics. Knowing the parameters of a two – port network enables us to treat it as a “black box” when embedded within a larger network.

From the network, we can observe that there are 4 variables that is I1, I2, V1and V2, which two are independent. The various term that relate these voltages and currents are called parameters.

Z – PARAMETER Z – parameter also called as impedance parameter and the units is ohm (Ω) Impedance parameters is commonly used in the synthesis of filters and also useful in the design and analysis of impedance matching networks and power distribution networks. The two – port network may be voltage – driven or current – driven.

Two – port network driven by voltage source. Two – port network driven by current sources. Linear network I1 I2 +  V1 V2 I1 I2 + V1 - Linear network V2

The “black box” is replace with Z-parameter is as shown below. The terminal voltage can be related to the terminal current as: + V1 - I1 I2 V2 Z11 Z21 Z12 Z22 (1) (2)

In matrix form as: The Z-parameter that we want to determine are z11, z12, z21, z22. The value of the parameters can be evaluated by setting: 1. I1= 0 (input port open – circuited) 2. I2= 0 (output port open – circuited)

Thus,

Where; z11 = open – circuit input impedance. z12 = open – circuit transfer impedance from port 1 to port 2. z21 = open – circuit transfer impedance from port 2 to port 1. z22 = open – circuit output impedance.

Example 1 Find the Z – parameter of the circuit below. 40Ω 240Ω 120Ω + V1 _ V2 I1 I2

Solution I2 = 0(open circuit port 2). Redraw the circuit. 40Ω 240Ω 120Ω + V1 _ V2 I1 Ia Ib

ii) I1 = 0 (open circuit port 1). Redraw the circuit. 40Ω 240Ω 120Ω + V1 _ V2 Iy I2 Ix

In matrix form:

Example 2 Find the Z – parameter of the circuit below + _ V1 V2 -j20Ω 10Ω j4Ω 2Ω 10I2 I2 I1

Solution i) I2 = 0 (open circuit port 2). Redraw the circuit. + V1 _ j4Ω 2Ω I1 V2 I2 = 0

ii) I1 = 0 (open circuit port 1). Redraw the circuit. + _ V1 V2 -j20Ω 10Ω 10I2 I2 I1 = 0

Y - PARAMETER Y – parameter also called admittance parameter and the units is siemens (S). The “black box” that we want to replace with the Y-parameter is shown below. + V1 - I1 I2 V2 Y11 Y21 Y12 Y22

The terminal current can be expressed in term of terminal voltage as: In matrix form: (1) (2)

The y-parameter that we want to determine are Y11, Y12, Y21, Y22 The y-parameter that we want to determine are Y11, Y12, Y21, Y22. The values of the parameters can be evaluate by setting: i) V1 = 0 (input port short – circuited). ii) V2 = 0 (output port short – circuited). Thus;

Example 1 Find the Y – parameter of the circuit shown below. 5Ω 15Ω 20Ω + V1 _ V2 I1 I2

Solution V2 = 0 5Ω 20Ω + V1 _ I1 I2 Ia

ii) V1 = 0 In matrix form; 5Ω 15Ω + V2 _ I1 I2 Ix

Example 2 (circuit with dependent source) Find the Y – parameters of the circuit shown. + _ V1 V2 -j20Ω 10Ω j4Ω 2Ω 10I2 I2 I1

Solution i) V2 = 0 (short – circuit port 2). Redraw the circuit. + _ 10Ω j4Ω 2Ω 10I2 I2 I1

ii) V1 = 0 (short – circuit port 1). Redraw the circuit. + _ V2 -j20Ω 10Ω j4Ω 2Ω 10I2 I2 I1

T (ABCD) PARAMETER T – parameter or ABCD – parameter is a another set of parameters relates the variables at the input port to those at the output port. T – parameter also called transmission parameters because this parameter are useful in the analysis of transmission lines because they express sending – end variables (V1 and I1) in terms of the receiving – end variables (V2 and -I2).

The “black box” that we want to replace with T – parameter is as shown below. The equation is: + V1 - I1 I2 V2 A11 C21 B12 D22

In matrix form is: The T – parameter that we want determine are A, B, C and D where A and D are dimensionless, B is in ohm (Ω) and C is in siemens (S). The values can be evaluated by setting i) I2 = 0 (input port open – circuit) ii) V2 = 0 (output port short circuit)

Thus; In term of the transmission parameter, a network is reciprocal if;

Example Find the ABCD – parameter of the circuit shown below. 2Ω 10Ω + V2 _ I1 I2 V1 4Ω

Solution i) I2 = 0, 2Ω 10Ω + V2 _ I1 V1

ii) V2 = 0, 2Ω 10Ω I1 I2 + V1 _ 4Ω I1 + I2

TERMINATED TWO – PORT NETWORKS In typical application of two port network, the circuit is driven at port 1 and loaded at port 2. Figure below shows the typical terminated 2 port model. + V1 - I1 I2 V2  Zg ZL Vg Two – port network

Zg represents the internal impedance of the source and Vg is the internal voltage of the source and ZL is the load impedance. There are a few characteristics of the terminated two-port network and some of them are;

The derivation of any one of the desired expression involves the algebraic manipulation of the two – port equation. The equation are: 1) the two-port parameter equation either Z or Y or ABCD. For example, Z-parameter,

2) KVL at input, 3) KVL at the output, From these equations, all the characteristic can be obtained.

Example 1 For the two-port shown below, obtain the suitable value of Rs such that maximum power is available at the input terminal. The Z-parameter of the two-port network is given as With Rs = 5Ω,what would be the value of + V1 - I1 I2 V2  Rs 4Ω Vs Z

Solution Z-parameter equation becomes; KVL at the output; Subs. (3) into (2)

Subs. (4) into (1) For the circuit to have maximum power,

To find at max. power transfer, voltage drop at Z1 is half of Vs From equations (3), (4), (5) & (6) Overall voltage gain,

Example 2 The ABCD parameter of two – port network shown below are. The output port is connected to a variable load for a maximum power transfer. Find RL and the maximum power transferred.

Solution ABCD parameter equation becomes V1 = 4V2 – 20I2 I1 = 0.1V2 – 2I2 At the input port, V1 = -10I (1) (2) (3)

(3) Into (1) -10I1 = 4V2 – 20I2 I1 = -0.4V2 2I2 (2) = (4) 0.1V2 – 2I2 = -0.4V2 + 2I2 0.5V2 = 4I2 From (5); ZTH = V2/I2 = 8Ω (4) (5) (6)

But from Figure (b), we know that V1 = 50 – 10I1 and I2 =0 Sub. these into (1) and (2) 50 – 10I1 = 4V2 I1 = 0.1V2 (7) (8)

Sub (8) into (7) V2 = 10 Thus, VTH = V2 = 10V RL for maximum power transfer, RL = ZTH = 8Ω The maximum power P = I2RL = (VTH/2RL)2 x RL = V2TH/4RL = 3.125W