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Circuit Theorems•Introduction•Linearity Property•Superposition•Source Transformation•Thevenin’s Theorem•Norton’s Theorem•Maximum Power Transfer•Summary

Introduction

•To develop analysis techniques applicable tolinear circuits.

•To simplify circuit analysis and help handlethe complexity.

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Linear Circuits•A linear circuit is one whose output is linearly

related (or directly proportional) to its input•Linear circuit consist of• linear elements• linear dependent sources• independent sources

Example 4.1•For the circuit in fig 4.2 find I0 when vs=12V

and vs=24V.

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•KVL

Eqs(4.1.1) and (4.1.3) we get

0412 21 svii03164 21 sx vvii

12ivx becomes)2.1.4(

01610 21 svii

(4.1.1)(4.1.2)

(4.1.3)

2121 60122 iiii

Example 4.1

Eq(4.1.1), we get

When

When

Showing that when the source value is doubled, I0doubles.

76076 22

ss

vivi

A7612

20 iIV12sv

A7624

20 iIV24sv

Example 4.1

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Linearity Property•Homogeneity (Scaling) + Additivity•Homogeneity property:

•Additivity property:

•A linear circuit follows the relationship below:

)()( kvRkiiRv

)(

Applyingand

212121

21

2211

vvRiRiRiiviii

RivRiv

InputConstantOutput

Superposition

),,0;0,,0(...)0,,0,,0;0,,0()0,,0,;0,,0(

)0,,0;,0,,0()0,,0;0,,0,,0()0,,0;0,,0,(

),,;,,(

)0,,0,,,0;0,,0(

)0,,0;0,,0,,,0(

)equations.lineargivelawsOhm'andKCL,KVL(

sourcestindependen:,

),,;,,(

bygiveniscurrent)(orvoltageDCthesystemlinearaFor*

21

21

11

,

,

1,

1,11

M

N

MN

mmIm

nnVn

mn

M

mmmI

N

nnnVMN

IvIvIv

VvVvVv

IIVVv

IAIv

VAVv

IV

IAVAIIVVv

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Superposition•Based on linearity property•Statement–The voltage across (current through) an element is

the algebraic sum of the voltage across (currentthrough) that element due to each independentsource acting alone.

•A turned-off voltage source = a short circuit•A turned-off current source = an open circuit

),...,0;0,...,0(...)0,...,;0,...,0(

)0,...,0;,...,0(...)0,...,0;0,...,(

),...,;,...,(

1

1

11

M

N

MN

IvIv

VvVv

IIVVv

Superposition

•Superposition involves more work but simplercircuits.•Superposition is not applicable to the effect on

power.

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Comments on Superposition

SS

SS

II

VV

CCaSSaCSaCS

CS

aCS

RiivvRivRiv

0:sourceCurrent

0:sourceVoltage

sources,tindependenFor

)()(

.signalcontrol:

signalcontrolled:whereletsources,dependentFor

)()(,resistorsFor

element.circuiteachinpropertydesiredKeep1.:tsrequiremenTwo

212122

11

212122

11

Input 1 LinearCircuit

Input 2 LinearCircuit

00

0

,KVLFor

,KCLForsatisfied.bemustlawscircuitTwo.2

2,1,2,

1,

2,1,2,1,2,2,

1,1,

RTRTRT

RT

leavingleavingenteringenteringleavingentering

leavingentering

VVV

V

IIIIII

II

Comments on Superposition

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Example 1

=

(V)10828141

13

124

6

21

vvv+

Example 2

+=

"0

'00 iii

Keep dependent sources!

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Example 3

321 iiii

Source Transformation

21,arbitraryFor iiVab

•Transformations between voltage source andcurrent source.•Equivalence means identical i-v characteristics.

i1 i2

Vab

+

_Vab

+

_

1

2

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Cont’d

22

11

RV

ii

RVv

i

abS

abS

Rv

iRRR

RRRv

i

RRV

Rv

i

Vii

SS

SS

abS

S

ab

and

011

and0

)11

()(

.arbitraryforLet

21

121

121

21

i1 i2

Vab

+

_Vab

+

_

1

2

Applicable to Dependent Sources

RivRv

i SSS

S or

i1 i2

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Source Transformation•Impossible source Transformation:•Ideal voltage source (R = 0)•Ideal current source (R = )

Example 1

34

43

12/6

12/3

4-2

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Example 2

V5.7A5.4

(2)013

(1)01853

x

x

x

vi

vi

vi

xv25.04

6/2 13

I-V Characteristic•For a linear two-terminal network, its I-V curve (DC)

must be a straight line in the I-V plane.

i

v

i

v

R

VS(VS||R)

R VS

R||IS

IS(IS R)

: Series connection||: Parallel connection

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Equivalent Circuits

i

vd

Slope=1/c

(i)i

+v_

(ii)i

+v_ c

dcv

i

dciv

dciv

-d/c

d

d/c

c

c

•Equivalent circuit:same i-v relation at theterminals

Thevenin’s TheoremA variableelement

Thevenin equivalent circuit

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Cont’d

ThLTh

LLLL

LTh

ThL

VRR

RIRV

RRV

I

= A simple voltage divider.

Proof of Thevenin’s Theorem

iv

R

iRiAv

B

Vvvi

iAvABV

AR

ViRBiA

iAvAiAv

iiivvv

oc

mmi

M

mnnv

N

n

mmi

M

mnnv

N

n

MN

Th

Th0

0

Th

,1,10Th

0Th

ThTh00

,1,10

2121

.0haveweoff,turnedsourcestindependeninternalallWhen*

0When*

where

ion,superpositBy*

,,,,,,sourcestindependenwithcircuitlinearA*

=

RTh

VTh

identical i-vcharacteristic

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Finding VTh and RTh

•If no dependent sources included.–RTh is the equivalent input resistance of the

resistor network .

A resistornetwork

Cont’d•If dependent sources

included, two methodscan be applied todetermine RTh.–External voltage source

method.

–External current sourcemethod.

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Example 1

=RTh

VTh

Example 2

=RTh

voc

1

0iRTh

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2

RTh

VTh RN= IN

Norton’s Theorems

Th

Th

eqTh

ation,transformsourceBy

RV

I

RRR

N

N

Norton’s Theorem

Norton equivalent circuitinTh RRRN

ThThN RVI

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Example

= RNIN

V

IR = 0

R =

•Req = 0–External voltage source (vo = 1V)

may violate the requirement vo= 0V.–Use external current source to find

RTh .

•Req = –External current source (io = 1A)

may violate the requirement io= 0A.–Use external voltage source to find

RTh .

•For others–Both exist.

More Comments

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Maximum Power Transfer

Th

2Th

max

Th

3Th

Th2Th

4Th

Th2

Th2Th

2

2

4

0

2

sfer,power tranmaximumachieveTo*

bygivenistodeliveredpowerThe*

RV

p

RR

RRRR

V

RRRRRRR

VdRdp

RRR

VRip

R

L

L

L

L

LLL

L

LLTh

ThL

L

A linear circuit

Applications: Source Modeling

sLs

LL v

RRR

v

sLs

LL v

RRR

v

sLs

LL v

RRR

v

Voltage source

Current source

sLs

LL v

RRR

v

sLp

pL i

RR

Ri

0sR

pR

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Resistance Measurement

21

3132

32

21

21

0

or

,0When

RRR

RRRRR

vRR

Rvv

RRR

v

i

xx

x

x

The Wheatstone bridge

i0

2

31If

RR

RR

x