Alexander Ch 04 Final r 1

8/12/2019 Alexander Ch 04 Final r 1

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Alexander-Sadiku

Fundamentals of Electric CircuitsChapter 4

Circuit Theorems

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.


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Circuit Theorems - Chapter 4

4.1 Motivation

4.2 Linearity Property

4.3 Superposition

4.4 Source Transformation

4.5 Thevenins Theorem

4.6 Nortons Theorem

4.7 Maximum Power Transfer


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If you are given the following circuit, arethere any other alternative(s) to determinethe voltage across 2Wresistor?

What are they? And how?

Can you work it out by inspection?

4.1 Motivation (1)


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4.2 Linearity Property (1)

It is the property of an element describing a linear relationshipbetween cause and effect.

A linear circuit is one whose output is linearly related(ordirectly proportional) to its input.

Homogeneity (scaling) property

v = i R k v = k i R

Additive property

v1= i1R and v2= i2R

v = (i1+ i2) R = v1+ v2


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4.2 Linearity Property (2)

Example 1

By assume Io = 1 A, use linearity to find the actual value of Io in thecircuit shown below.

*Refer to in-class illustration, text book, answer Io = 3A


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4.4 Source Transformation (1)

An equivalent circuit is one whose v-icharacteristics are identical with the

original circuit.

Itis the process of replacing a voltagesource vSin series with a resistor Rby a

current source iSin parallel with a resistorR,or vice versa.


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(a) Independent source transform

(b) Dependent source transform

The arrow of thecurrent source isdirected towardthe positiveterminal of thevoltage source.

The sourcetransformation is

not possible whenR = 0 for voltagesource and R = for current source.

++

++

--

--


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Example 4

Find ioin the circuit shown below using source transformation.

*Refer to in-class illustration, textbook, answer io= 1.78A


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4.3 Superposition Theorem (1)

It states that the voltage across(or currentthrough) an element in a linear circuit is thealgebraic sumof the voltage across (or currents

through) that element due to EACH independentsource acting alone.

The principle of superposition helps us to analyze

a linear circuit with more than one independentsource by calculating the contribution of eachindependent source separately.


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We consider the effects of 8A and 20V oneby one, then add the two effects togetherfor final vo.



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Steps to apply superposition principle

1. Turn off all independent sources except one

source. Find the output (voltage or current)due to that active source using nodal ormesh analysis.

2. Repeat step 1 for each of the other independentsources.

3. Find the total contribution by addingalgebraically all the contributions due to the

independent sources.


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Two things have to be keep in mind:

1. When we say turn off all other independent

sources: Independent voltage sources are replaced

by 0 V (short circuit) and Independent current sources are replaced

by 0 A (open circuit).

2. Dependent sources are leftintact becausethey are controlled by circuit variables.


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Example 2

Use the superposition theorem to findv in the circuit shown below.

3A is discardedby open-circuit

6V is discardedby short-circuit

*Refer to in-class illustration, text book, answer v= 10V


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Example 3

Use superposition to find vxinthe circuit below.

*Refer to in-class illustration, text book, answer Vx = 12.5V

2A is discarded byopen-circuit

20 v1

410 V+

(a)

0.1v14

2 A

(b)

20

0.1v2

v2

10V is discardedby open-circuit

Dependant sourcekeep unchanged


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4.5 Thevenins Theorem (1)

It states that a linear two-terminalcircuit (Fig. a) can be replaced by anequivalent circuit (Fig. b) consistingof a voltage source V

THin series with

a resistor RTH,

where

VTH is the open-circuit voltage at the

terminals.

RTH is the input or equivalent resistance atthe terminals when the independentsources are turned off.


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Example 5

Using Thevenins theorem,find the equivalent circuit to

the left of the terminals inthe circuit shown below.Hence find i.

*Refer to in-class illustration, textbook, answer VTH= 6V, RTH= 3W, i = 1.5A

6

4

(a)

RTh

6

2A

6

4

(b)

6 2A

+

VTh


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Example 6

Find the Thevenin equivalentcircuit of the circuit shown

below to the left of theterminals.

*Refer to in-class illustration, textbook, answer VTH= 5.33V, RTH= 0.44 W

6 V

5 Ix

4

+

(a)

1.5Ix

i1

i2

i1 i2

3

o

+VTh

b

a

1.5Ix 1 V+

30.5I

x

5

(b)

a

b

4

Ix i


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4.6 Nortons Theorem (1)It states that a linear two-terminal circuitcan be replaced by an equivalent circuitof a current source I

Nin parallel with a

resistor RN,

Where

INis the short circuit current through

the terminals.

RNis the input or equivalent resistanceat the terminals when the independentsources are turned off.

The Thevenins and Norton equivalent circuits are

related by a source transformation.


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4.6 Nortons Theorem (2)

Example 7

Find the Norton equivalentcircuit of the circuit shown

below.

*Refer to in-class illustration, textbook, RN= 1W, IN= 10A.

2

(a)

6

2vx

+

+vx

+vx

1V+ix

i

2

(b)

6 10 A

2vx

+

+vx Isc


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4.7 Maximum Power Transfer (1)

2

max

4

T h

L T H

T h

VR R P

R= =

If the entire circuit is replaced byits Thevenin equivalentexcept forthe load, the power delivered tothe load is:

The power transfer profile withdifferent RL

For maximum power dissipated

in RL, Pmax, for a given RTH,and VTH,

L

LTh

Th

L R

RR

VRiP

2

2


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Example 8

Determine the value of RLthat willdraw the maximum power fromthe rest of the circuit shown below.

Calculate the maximum power.

2

4

1 V+

(a)

1

3vx

+

i

v0+ vx

9 V+

io

1 +VTh

+

3vx

2

+ vx 4

(b)

Fig. a

=> To determine RTH

Fig. b

=> To determine VTH

*Refer to in-class illustration, textbook, RL= 4.22W, Pm= 2.901W

4.7 Maximum Power Transfer (2)

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