Chapter 04 Network Theorems

8/8/2019 Chapter 04 Network Theorems

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2010 Ch. 4 Network Theorems 1

Topics to be DiscussedTopics to be Discussed

Superposition Theorem. Thevenins Theorem.

Nortons Theorem.

Maximum Power Transfer Theorem. Maximum Power Transfer Theorem for AC

Circuits.

Millmans Theorem.

Reciprocity Theorem.

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Network Theorems

Some special techniques, known as network

theorems and network reduction methods, have

been developed.

These drastically reduce the labour needed tosolve a network.

These also provide simple conclusions and good

insight into the problems.

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SuperpositionSuperposition

PrinciplePrinciple

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Superposition Theorem

The response (current orvoltage) in a linearnetwork

at any point due to multiple sources (current and/or

emf) (including linear dependent sources),

can be calculated by summing the effects of eachsource considered separately,

all other sources turned OFF or made

inoperative.

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Turning off the sources

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2010 Ch. 4 Network Theorems 6Next


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Linear Dependent Source

It is a source whose output current or voltage isproportional only to the first power of some

current or voltage variable in the network or to the

sum of such quantities.

Examples :

linear.notis6.06.0but,

linear,is166.0

21

2

1

21

viv

oriv

viv

s

s

s

!

!

!

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Application

Problem : Consider two 1-V batteries in

series with a 1- resistor. Let us apply the

principle of superposition, and find the

power delivered by both the batteries.

Solutions : Power delivered by only onesource working at a time is

P1

= 1 W

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Therefore, the power delivered by both thesources,

P= 2P1

= 2 W

The above answer is obviously wrong,

because it is a wrong application of

the superposition theorem.

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Example 1

Find the current Iin the network given,using the superposition theorem.

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Solution :

A0.375!

!

v!

4.0

15.0

3.01.0

3.05.01I

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A0.1

A0.2

!!@!

v

!

21

3

2

I

3.01.0

1080

II

I

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Example 2

Using superposition theorem, find current ix in thenetwork given.

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Solution :

A05.015050

101 !

!i

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A3015050

150402 !v!i

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A3015050

501203 !

v!i

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A0.05!

!

!

303005.0

321 iiiix

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BenchmarkExample 3

Find voltage v across 3- resistor by applyingthe principle of superposition.

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Solution :

Using current divider,

A

3

2

)32(1

14 !

v!i

V2.0)(3A)(2/34

!v!v!@ Riv

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Using current-divider, the voltage v5 across 3-

5

15 A (3 ) 2.5V1 (2 3)

v ! v v ; ! -

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By voltage divider,

V3.0321

366 !v!v

4 5 6 2.0 2.5 3.0v v v v@ ! ! ! 2.5 V

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Find current i2 across R2 resistor by applying the

principle of superposition. Where R1=R2=R3=1-

and S=10 , b= 5 , = .

Example 4


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Thevenins Theorem

It was first proposed by a French telegraph

engineer, M.L. Thevenin in 1883.

There also exists an earlier statement of the

theorem credited to Helmholtz.

Hence it is also known as Helmholtz-Thevenin

Theorem.

It is useful when we wish to find the responseonly in a single resistance in a big network.

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Thevenins Theorem

Any two terminals AB of a network

composed of linear passive and active

elements may by replaced by a simple

equivalent circuit consisting of

1. an equivalent voltage source Voc,and

2. an equivalent resistance th in series.

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The voltage Voc is equal to the potential

difference between the two terminals AB causedby the active network with no externalresistance connected to these terminals.

The series resistance Rth is the equivalentresistance looking back into the network at theterminals AB with all the sources within the

network made inactive, ordead.

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Illustrative Example 3

Using Thevenins theorem, find the current inresistorR2 of 2 .

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Solution :

1. Designate the resistor R2

as load.

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2. Pull out the load resistor and enclose the remaining

network within a dotted box.

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3. Temporarily remove the load resistor R2, leaving the

terminals A and B open .

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4. Find the open-circuit voltage across the terminals A-

B,

11.2!v!

!!

!

12.47

A;4.25

21

14

728

ABV

I

5. This is called Thevenin voltage, VTh = VAB = 11.2 .

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6. Turn OFF all the sources in the circuit

Find the resistance between terminals A and B. This is

the Thevenin resistance, RTh. Thus,

1 41 || 4

1 4ThR

v! ; ; ! !

0.8

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7. The circuit within the dotted box is replaced by the

Thevenins equivalent, consisting of a voltage source of

VTh in series with a resistor RTh,

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8. The load resistor R2 is again connected to Theveninsequivalent forming a single-loop circuit.

The current I2 through this resistor is easily calculated,

Th2

Th 2

11.2

0.8 2

VI

R R! ! !

4 A

Important Comment

Theequivalent circuitreplacesthe circuit withinthe

box only

forthe

effects

externaltothe

box.

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Example 4

Using Thevenins Theorem, find the current in the

ammeterA of resistance 1.5 connected in an

unbalanced Wheatstone bridge shown.

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Solution :

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V6!vv!

!!@

!

!

!

!

65.1475.0

A5.162

12

andA7

5.0412

12

2

1

BDADABocVVVV

I

I

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Ans. -1 A

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BenchmarkExample 5Again consider ourbenchmark example to determine

voltage across 3- resistor by applying Thevenins

theorem.

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Solution :

We treat the 3- resistor as load.

Thevenin voltage VTh is the open-circuit voltage(withRL removed).

We use source transformation.

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V5!v!@ 15ThV

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To compute RTh, we turn off all the sources in the

circuit within box and get the circuit

Thus, RTh = 3 .

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V2.5!

v!33

35

LV

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Nortons Theorem

It is dual of Thevenins Theorem.

A two terminal network containing linear

passive and active elements can be replaced

by an equivalent circuit of a constant-current source in parallel with a resistance.

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The value of the constant-current source is

the short-circuit current developed whenthe terminals of the original network are

short circuited.

The parallel resistance is the resistance

looking back into the original network with

all the sources within the network madeinactive (as in Thevenins Theorem).

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Example 6

Obtain the Nortons equivalent circuit with respect tothe terminals AB for the network shown, and hence

determine the value of the current that would flow

through a load resistor of 5 if it were connected

across terminals AB.

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Solution : When terminals A-B are shorted

1 2

10 5

5 10 I I I @ ! ! ! 2.5

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Turning OFF the sources,

3

10!

v!@

105

105NR

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A1!

v!

!

5)3/10(

)3/10(

5.2LN

N

NL RR

R

II

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Power Transferred to the Load

Consider the circuit :

Source Load

r

ERLp

(Variable)

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p

RL0

pmax

RL= r

Maximumpoweristransferredwhen

RL = r.

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Proof

? A 0

02)()(

zero.toequalnumeratorpute,maximizingor

)(

1)(21)(4

2

2

2

!!

vvv!@

!

L

LLL

L

LLL

L

L

L

RrRrRrR

rR

rRRrRE

dR

dp

RrR

Ep

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Maximum Power Transfer Theorem

Maximum power is drawn form a sourcewhen the Load Resistance is equal to the

Source Internal Resistance.

When maximum power transfer condition issatisfied, we say that the load is matched

with the source.

Under maximum power transfer condition,

the efficiency of the source is only 50 %.

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What is the maximum power that a sourceof emfE and internal resistance r can

ever deliver ?

Ans.

r

E

4

2

Available Power

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Prove that under maximum power transfer

condition, the efficiency of the source is only

50 %.

%50

%100)(2

2

!

v!| rRI

RI

P

P

L

L

in

o

L

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Example 7The open-circuit voltage of a standard car-battery is 12.6

V, and the short-circuit current is approximately 300 A.

What is the available power from the battery ?

Solution : The output impedance of the battery,

oco

sc

12.60.042

300

VR

I! ! ! ;

Therefore, the available power22 2

ocThavl

Th o

(12.6)

4 4 4 0.042

VVP

R R! ! ! !

v

945 W

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Millmans Theorem

A number of parallel voltage sources V1, V2, V3 ,Vnwith internal resistances R1, R2, R3, Rn,

respectively can be replaced by a single voltage

source Vin series with equivalent resistance R.

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n

nn

GGGG

GVGVGVGVV

!

...

...

321

332211

1 2 3

1 1

...n

RG G G G G

! !

and

Equivalent Circuit

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Reciprocity Theorem

In a linear bilateral network, if a voltage source Vin a branch A produces a current Iin any other

branch B, then the same voltage source Vacting

in the branch B would produce the same current I

in branch.

The ratio V/Iis known as the transfer

resistance.

Letusverifythereciprocitytheorem byconsideringanexample.

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Example 8

In the network shown, find the current in branch B dueto the voltage source of 36 V in branch A.

Now transfer the voltage source to branch B and find

the current in branch A.

Is the reciprocity theorem established ?

0Also, determine the transfer resistance from branch A

to branch B.

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Solution : The equivalent resistance for the voltage

source,;!!! 94324)]13(||12[2eqR

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The current supplied by the voltage source = 36/9 =4 A.

Using current divider, the current Iin branch B,

A3!

v!

412

124I

Now, transferring the voltage source to branch B,

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ReviewReview

Superposition Theorem.

Thevenins Theorem.

Nortons Theorem.

Maximum Power Transfer Theorem.

Maximum Power Transfer Theorem for AC Circuits.

Millmans Theorem.

Reciprocity Theorem.

Chapter 04 Network Theorems

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