# CHAPTER 6 TWO PORT NETWORKS.

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CHAPTER 6 TWO PORT NETWORKS

OBJECTIVES To understand about two – port networks and its functions.
To understand the different between z-parameter, y-parameter, ABCD- parameter and terminated two port networks. To investigate and analysis the behavior of two – port networks.

SUB - TOPICS 6-1 Z – PARAMETER 6-2 Y – PARAMETER 6-3 ABCD – PARAMETER
6-4 TERMINATED 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. (Fig. a) 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. (Fig. b)

One port or two terminal circuit
Two port or four terminal circuit It is an electrical network with two separate ports for input and output. No independent sources.

6-1 Z – PARAMETER Z – parameter also called as impedance parameter and the units is ohm (Ω) The “black box” is replace with Z-parameter is as shown below. + V1 - I1 I2 V2 Z11 Z21 Z12 Z22 where the z terms are called the impedance parameters, or simply z parameters, and have units of ohms.

z11 = Open-circuit input impedance
z21 = Open-circuit transfer impedance from port 1 to port 2 z12 = 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 When I2 = 0(open circuit port 2). Redraw the circuit. Ia I1 +
40Ω 240Ω 120Ω + V1 _ V2 I1 Ia Ib

When 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 10I2 I2 I1

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

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

6-2 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 where the y terms are called the admittance parameters, or simply y parameters, and they have units of Siemens.

y21 = Short-circuit transfer admittance from port 1 to port 2 y12 = Short-circuit transfer admittance from port 2 to port 1 y22 = Short-circuit output admittance

Example 3 Find the Y – parameter of the circuit shown below. 5Ω 15Ω
20Ω + V1 _ V2 I1 I2

Solution V2 = 0 20Ω + V1 _ I1 I2 Ia

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

Example 4 Find the Y – parameters of the circuit shown. V1 V2 -j20Ω
+ _ V1 V2 -j20Ω 10Ω j4Ω 10I2 I2 I1

Solution i) V2 = 0 (short – circuit port 2). Redraw the circuit. V1
+ _ V1 10Ω j4Ω 10I2 I2 I1

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

6-3 T (ABCD) PARAMETER T – parameter or also 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. + V1 - I1 I2 V2 A11 C21 B12 D22

where the T terms are called the transmission parameters, or simply T or ABCD parameters, and each parameter has different units. B= negative short-circuit transfer impedance () A=open-circuit voltage ratio C= open-circuit transfer admittance (S) D=negative short-circuit current ratio

Example 5 Find the ABCD – parameter of the circuit shown below. 2Ω 10Ω
+ V2 _ I1 I2 V1

Solution i) I2 = 0, 10Ω + V2 _ I1 V1

ii) V2 = 0, 10Ω I1 I2 + V1 _ I1 + I2 In matrix form;

6-4 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 Terminated two-port parameter can be implement to z-parameter, Y-parameter and ABCD-parameter.

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 6  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 Vs Z

Solution Z-parameter equation becomes; KVL at the output;
Subs. (3) into (2); Subs. (4) into (1); Hence, the input impedance; 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 7 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 = -10I1 (1) (2) (3)

But from Figure (b), we know that V1 = 50 – 10I1 and I2 =0
(3) Into (1) -10I1 = 4V2 – 20I2 I1 = -0.4V2 + 2I2 (4) Into (2) -0.4V2 + 2I2 = 0.1V2 – 2I2 0.5V2 = 4I2 (4) (5) From (5); ZTH = V2/I2 = 8Ω (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