UNIT-3 Network Theorems

8/8/2019 UNIT-3 Network Theorems

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, 16 2010 Ch. 4 Network Theorems 1

Topics to be Discussed Topics to be Discussed

Superposition Theorem.Thevenins Theorem.

Nortons Theorem.

Maximum Power Transfer Theorem.Maximum Power Transfer Theorem for ACCircuits.Millmans Theorem.Reciprocity Theorem.Tellegens Theorem.

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N etwork 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 goodinsight into the problems.

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S uperpositionS uperpositionPrinciplePrinciple

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Turning off the sources

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Linear Dependent Source

I t is a source whose output current or voltage is proportional only to the first power of somecurrent or voltage variable in the network or to thesum of such quantities.Examples :

linear.notis6.06.0 but,

linear,is166.0

21

2

1

21

viv

or iv

viv

s

s

s

!!

!

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A pplication

Problem : Consider two 1-V batteries inseries 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 P 1 = 1 W

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Therefore, the power delivered by both thesources,

P = 2 P 1 = 2 W

The above answer is obviously wrong , because it is a wrong application of

the superposition theorem.

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Ex ample 1

Find the current I in the network given,using the superposition theorem.

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Solution :

A0.375!!v

!4.015.0

3.01.03.05.0

1 I

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Ex ample 2

U sing superposition theorem, find current i x in thenetwork given.

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Solution :

A05.015050

101

!!i

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A3015050

501203 !v!i

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A0.05!

!

!

303005.0321

iiii x

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B enchmark Exa mple 3

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

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Solution :

U sing current divider,

A

3

2

)32(1

14 !v!i

V2.0)(3A )(2/34 !v!v!@ Riv

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U sing current-divider, the voltage v 5 across 3-

515 A (3 ) 2.5 V

1 (2 3)v ! v v ; ! -

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B y voltage divider,

V3.03213

66 !v!v

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

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F ind current i 2 across R 2 resistor by applying theprinciple of superposition. Where R 1=R 2=R 3=1-and S=10 , b= 5 , = .

Exa mple 4


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T hevenins T heorem

I t was first proposed by a French telegraphengineer, M.L. Thevenin in 1883.There also exists an earlier statement of the

theorem credited to Helmholtz.Hence it is also known as Helmholtz-TheveninTheorem.

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

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T hevenins T heoremAny two terminals AB of a network composed of linear passive and activeelements may by replaced by a simpleequivalent circuit consisting of

1. an equivalent voltage source V oc,and2. an equivalent resistance th in series .

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The voltage V oc is equal to the potentialdifference between the two terminals AB caused

by 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 , or dead.

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I llustrative Example 3

U sing Thevenins theorem, find the current inresistor R2 of 2 .

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Solution :1. Designate the resistor R

2as load.

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2. Pull out the load resistor and enclose the remaining

network within a dotted bo x .

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3. Temporarily remove the load resistor R 2, leaving the

terminals A and B open .

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4. F ind the open-circuit voltage across the terminals A-

B,

11.2!v!

!!!

12.47

A;4 .2521

14728

ABV

I

5. T his is called Th evenin voltage , V T h = V AB = 11.2 .

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6. Turn O FF all the sources in the circuit

Find the resistance between terminals A and B . This isthe Th evenin resistance , RTh. Thus,

1 41 || 4

1 4T h R

v! ; ; ! ! 0.8

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7. T he circuit within the dotted bo x is replaced by theTh evenins equivalent , consisting of a voltage source of

V T h in series with a resistor R T h ,

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8. T he load resistor R 2 is again connected to T heveninsequivalent forming a single-loop circuit.

T he current I 2 through this resistor is easily calculated,

Th2

Th 2

11.20.8 2

V I

R R! ! !

4 A

Impor tant C omm ent

The equivalent c ir c uit repla c es the c ir c uit within the box o

nlyfo

r the

e

ff e

cts

external t

othe

box.

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Example 4U sing Thevenins Theorem, find the current in theammeter A of resistance 1.5 connected in anunbalanced Wheatstone bridge shown.

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Solution :

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V6!vv!!!@

!!

!!

65.147 5.0

A5.162

12and

A75.0412

12

2

1

BD AD ABocV V V V

I

I

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Ans. - 1 A

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B enchmark Exa mple 5Again consider our b enc hm ark exa m ple to determinevoltage across 3- resistor by applying Theveninstheorem.

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Solution :We treat the 3- resistor as load.

Thevenin voltage V Th is the open-circuit voltage(with RL 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 thecircuit within box and get the circuit

Thus, RTh = 3 .

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V2.5!v!33

35 LV

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Thevenins Theorem for dependent

sourcesCase- I : When circuit contain both dependent

and independent sources.(i) The open circuit voltage is determined as

usual with the sources activated or alive.(ii) A sort circuited is applied across the terminal

ab and the value of sort circuit current i sc isfound as usual.(iii) Now the thevenins resistance R th = Voc/isc

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Thevenins Theorem for dependent

sourcesCase- II : When circuit contain only dependent

sources.(i) I n this case, V oc = 0.(ii) We connect 1A source to terminal ab and

calculate the value of V ab.

(iii) Now the thevenins resistance R th = Vab/1

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WORKED EX AMPL E 3


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WORKED EX AMPL E 3

Fi nd T hevenins E quivalent circuit across terminal ab.


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Nortons T heorem

I t 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 areshort circuited .

The parallel resistance is the resistancelooking back into the original network with

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

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Ex ample 6

Obtain the Nortons equivalent circuit with respect tothe terminals A B for the network shown, and hencedetermine the value of the current that would flowthrough a load resistor of 5 if it were connected

across terminals A B .

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Solution : When terminals A- B are shorted

1 210 55 10

I I I @ ! ! ! 2.5

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Turning OFF the sources,

310

!v

!@

105105

N R

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A1!v!!

5)3/10()3/10(

5.2L N N

NL R R R

I I

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Nortons Theorem for dependent

sourcesCase- I : When circuit contain both dependent

and independent sources.

(i) The open circuit voltage is determined asusual with the sources activated or alive.

(ii) A sort circuited is applied across the terminal

ab and the value of sort circuit current i sc isfound as usual.(iii) Now the Nortons resistance R N = Voc/isc

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Nortons Theorem for dependent

sourcesCase- II : When circuit contain only dependent

sources.

(i) I n this case, I SC = 0.(ii) We connect 1A source to terminal ab and

calculate the value of V ab.

(iii) Now the thevenins resistance R N = Vab/1

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Fi nd Nortons E quivalent circuit across terminal ab.

WORKED EX AMPL E 3


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Power T ransferred to the Load

Consider the circuit :

S ourc e L o ad

r E R L p(Variable)

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p

R L0

p m a x

R L = r

M ax imum pow e r is t r an sf e rr ed wh en

R L = r .

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Proof

? A 0

02)()(

zero.toequalnumerator pute,maximizingor

)(1)(21)(

4

2

2

2

!!

vvv!@

!

L

L L L

L

L L L

L

L

L

Rr Rr Rr R

r R

r R Rr R E dR

dp

Rr R

E p

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M a x imum Power T ransfer T heorem

Maximum power is drawn form a sourcewhen the Load Resistance is equal to theSource I nternal Resistance.

When maximum power transfer condition issatisfied, we say that the load is ma t c hed with the source.

U nder maximum power transfer condition,the efficiency of the source is only 50 %.

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What is the maximum power that a sourceof emf E and internal resistance r canever deliver ?

Ans .

r

E 4

2

A vaila b le Pow er

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Cli ck


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Prove that under maximum power transfer condition, the efficiency of the source is only50 %.

%50

%100)(22

!

v!| r R I R I

P P

L

L

in

o

L

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Ex ample 7The open-circuit voltage of a standard car-battery is 12.6V, and the short-circuit current is approximately 300 A.What is the available power from the battery ?

S o lu t io n : The output impedance of the battery,

oco

sc

12.60.0 4 2

300

V R

I ! ! ! ;

Therefore, the available power

22 2ocTh

avlTh o

(12.6)4 4 4 0.0 4 2

V V P

R R! ! ! !

v94 5 W

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Cli ck

Cli ck


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M illmans Theorem

A number of parallel voltage sources V 1, V 2, V 3 ,V n with internal resistances R1, R2, R3, Rn ,respectively can be replaced by a single voltagesource V in series with equivalent resistance R.

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n

nn

GGGG

GV GV GV GV V !

...

...

321

332211

1 2 3

1 1... n

RG G G G G

! ! and

E quivalent C ir cuit

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R eciprocity Theorem

I n a linear bilateral network, if a voltage source V in a branch A produces a current I in any other branch B , then the same voltage source V actingin the branch B would produce the same current I

in branch.The ratio V/I is known as the transfer resistance.L et us veri f y the re c ipr oc ity the o re m b y co nsidering an e xa m ple .

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Ex ample 8In the network shown, find the current in branch

Bdueto the voltage source of 36 V in branch A.

Now transfer the voltage source to branch B and findthe current in branch A.

I s the reciprocity theorem established ?

Also, determine the transfer resistance from branch Ato branch B .

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The current supplied by the voltage source = 36/9 = 4 A.U sing current divider, the current I in branch B ,

A3!v!412

124 I

Now, transferring the voltage source to branch B ,

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The equivalent resistance for the voltage source,

;!!! 81431)]42(12[3eq R

The current supplied by the voltage source = 36/8 = 4 .5 A. U sing current divider, the current I in branchA,

A3!v!612

125.4' I

The transfer resistance

12!!!3

36 I V

Rt r

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Ex ample 9I n the network shown in Fig 4 2, Verify Tellegen 's theorem. The

values of components used are presented below.

V1= 15V, V 2= 6V, R 1= 2 , R 2= 3 , R 3= 3 , R 4 = 2

R 5= 5 , R 6= 3 .

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R eview R eview

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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UNIT-3 Network Theorems