Circuit Theorems

Instructor: Chia-Ming TsaiElectronics Engineering

National Chiao Tung UniversityHsinchu, Taiwan, R.O.C.

Contents• Introduction

• Linearity Property

• Superposition

• Source Transformation

• Thevenin’s Theorem

• Norton’s Theorem

• Maximum Power Transfer

• Summary

Introduction

• To develop analysis techniques applicable to linear circuits

• To simplify circuit analysis and help handling the complexity

Linearity Property

• Homogeneity (Scaling) + Additivity

• Homogeneity property:

• Additivity property:

• A linear circuit follows the relationship below

kiRkviRv

212121

2211 and

vvRiRiRiiv

RivRiv

InputConstantOutput

Superposition• Based on linearity property

• Statement– The voltage across (current through) an element

is the algebraic sum of the voltage across (current through) that element due to each independent source acting alone.

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

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

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

1

1

11

M

N

MN

IvIv

VvVv

IIVVv

Example 1

+

=

V 1082 11

vvv

Example 2

+=

"0

'00 iii

Example 3

321 iiii

Source Transformation

i1 i2

R

Vii

R

Vvi

abS

abS

2

1

RivR

vi

ii

SSS

S

or

Let 21

Applicable to Dependent Sources

RivR

vi SS

SS or

i1 i2

Example 1

Example 2

V 5.7

A 5.4

013

01853

x

x

x

v

i

vi

vi

Thevenin’s TheoremA variableelement

Thevenin equivalent circuit

Continued

ThLTh

LLLL

LTh

ThL

VRR

RIRV

RR

VI

A simple voltage divider

=

Proof of Thevenin’s Theorem

+

=

=

All indep.sources on

All indep.sources off

1v+_

+_

2v

2111 )0,...,0;0,...,0;(),...,;,...,;0( vvivIIVVvv MN

Finding VTh and RTh

• If no dependent sources included

Continued• If dependent sources inclu

ded, two methods can be applied to determine RTh

– External voltage source method

– External current source method

Example 1

=RTh

VTh

Example 2

=RTh

voc

1

0iRTh

Norton’s Theorem

Norton equivalent circuit inTh RRRN

Proof of Norton’s Theorem

+

=

=

All indep.sources on

All indep.sources off

1211 ),...,;,...,;0()0,...,0;0,...,0;( iiIIVViVii MN

Example

= RNIN

Maximum Power Transfer

Th

Th

ThL

LTh

LThTh

LTh

LThLLThTh

L

LLTh

ThL

R

Vp

RR

RR

RRV

RR

RRRRRV

dR

dp

RRR

VRip

4

0

2

sfer,power tran maximum achieve To

2

max

3

2

4

22

2

2

A linear circuit

Applications: Source Modeling

sLs

LL v

RR

Rv

s

Ls

LL v

RR

Rv

sLs

LL v

RR

Rv

Voltage source

Current source

sLs

LL v

RR

Rv

sLp

pL i

RR

Ri

0sR

pR