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CHAPTER 6 TWO PORT NETWORKS

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**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.

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**SUB - TOPICS 6-1 Z – PARAMETER 6-2 Y – PARAMETER 6-3 ABCD – PARAMETER**

6-4 TERMINATED TWO PORT NETWORKS

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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)

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**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.

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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.

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**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

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**Example 1 Find the Z – parameter of the circuit below. 40Ω 240Ω 120Ω +**

V1 _ V2 I1 I2

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**Solution When I2 = 0(open circuit port 2). Redraw the circuit. Ia I1 +**

40Ω 240Ω 120Ω + V1 _ V2 I1 Ia Ib

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**When I1 = 0 (open circuit port 1). Redraw the circuit.**

40Ω 240Ω 120Ω + V1 _ V2 Iy I2 Ix In matrix form:

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**Example 2 Find the Z – parameter of the circuit below + _ V1 V2 -j20Ω**

10Ω j4Ω 2Ω 10I2 I2 I1

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

j4Ω 2Ω I1 V2 I2 = 0

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**ii) I1 = 0 (open circuit port 1). Redraw the circuit.**

+ _ V1 V2 -j20Ω 10Ω 10I2 I2 I1 = 0

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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.

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**y11 = Short-circuit input admittance**

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

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**Example 3 Find the Y – parameter of the circuit shown below. 5Ω 15Ω**

20Ω + V1 _ V2 I1 I2

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

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

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**Example 4 Find the Y – parameters of the circuit shown. V1 V2 -j20Ω**

+ _ V1 V2 -j20Ω 10Ω j4Ω 2Ω 10I2 I2 I1

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**Solution i) V2 = 0 (short – circuit port 2). Redraw the circuit. V1**

+ _ V1 10Ω j4Ω 2Ω 10I2 I2 I1

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**ii) V1 = 0 (short – circuit port 1). Redraw the circuit.**

+ _ V2 -j20Ω 10Ω j4Ω 2Ω 10I2 I2 I1

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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

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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

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**Example 5 Find the ABCD – parameter of the circuit shown below. 2Ω 10Ω**

+ V2 _ I1 I2 V1 4Ω

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

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ii) V2 = 0, 2Ω 10Ω I1 I2 + V1 _ 4Ω I1 + I2 In matrix form;

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**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.

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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;

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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.

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**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 4Ω Vs Z

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**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,

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**To find at max. power transfer, voltage drop at Z1 is half of**

Vs From equations (3), (4), (5) & (6), Overall voltage gain,

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**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.

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**Solution ABCD parameter equation becomes V1 = 4V2 – 20I2**

I1 = 0.1V2 – 2I2 At the input port, V1 = -10I1 (1) (2) (3)

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**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

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1 TWO PORT NETWORKS CHAPTER 6. 2 OBJECTIVES To understand about two – port networks and its functions. To understand the different between z- parameter,

1 TWO PORT NETWORKS CHAPTER 6. 2 OBJECTIVES To understand about two – port networks and its functions. To understand the different between z- parameter,

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