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Chapter 4

Circuit Theorems

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Linearity Property

• Linearity is the property of an element describing a linear relationship between cause and effect.

• A linear circuit is one whose output is linearly ( or directly proportional) to its input.

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Fig. 4.4 For Example 4.2

.1,5

;3,15

AIothenAIsif

AIothenAIsif

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Superposition(1)

• The superposition principle states that voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.

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• Steps to Apply Superposition Principle:1. Turn off all independent source except one source. Find

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

2. Repeat step 1 for each of the other independent sources.

3. Find the total contribution by adding algebraically all the contributions due to the independent sources.

Superposition(2)

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j

R1V

e +

-L N

i

j

+

-V1

R1

i1

L N

i2

-L N

+

V2

eR1

21;21 iiiVVV

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Fig. 4.6 For Example 4.3

21 vvv

Vv

VvVv

10

82;21

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Source Transformation(1)

• A source transformation is the process of replacing a voltage source Vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa. Vs=isR or is=Vs/R

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Source Transformation(2)

• It also applies to dependent sources:

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Fig. 4.17 for Example, find out Vo

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So, we get vo=3.2V

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7

2A

6V2A

I

Example: find out I (use source transformation )

AI 5.0

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Substitution Theorem

20V

+6 I2

4V

4

-

8

I3

V3

I1

20V8V

-

I3

V3

6

I1+

8I2

V3

6+

-20V

I1

1A

8

I3

I2

I1=2A, I2=1A, I3=1A, V3=8V

I1=2A, I2=1A, I3=1A, V3=8V

I1=2A, I2=1A, I3=1A, V3=8V

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Substitution Theorem

• If the voltage across and current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.

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Substitution Theorem

N1N N2Vs+

-

Is

Vs

NN1 OR NIs

N1

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Thevenin’s Theorem

• A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source Vth in series with a resistor Rth, where Vth is the open-circuit voltage at the terminals and Rth is the input or equivalent resistance at the terminals when the independent source are turned off.

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(a) original circuit, (b) the Thevenin equivalent circuit

d

c

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LN LOADV+

-

I

Voc

-

+

LNIs

-

LNo RoI+

+

LN

I

V I

-

+

V=Voc-RoI

Simple Proof by figures

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Thevenin’s Theorem

Consider 2 cases in finding Rth:

• Case 1 If the network has no dependent sources, just turn off all independent sources, calculate the equivalent resistance of those resistors left.

• Case 2 If the network has dependent sources, there are two methods to get Rth:

1.

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Thevenin’s Theorem• Case 2 If the network has dependent sources, there are

two methods to get Rth:

1. Turn off all the independent sources, apply a voltage source v0 (or current source i0) at terminals a and b and determine the resulting current i0 (or resulting voltage v0), then Rth= v0/ i0

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• Case 2 If the network has dependent sources, there are two methods to get Rth:

2. Calculate the open-circuit voltage Voc and short-circuit current Isc at the terminal of the original circuit, then Rth=Voc/Isc

Thevenin’s Theorem

VocCircuit

-

+OriginalIsc

Circuit

Original

Rth=Voc/Isc

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Examples

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Norton’s Theorem

• A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN is the short-circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the independent sources are turned off.

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(a) Original circuit, (b) Norton equivalent circuit

d(c)

N

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Examples

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Maximum Power Transfer

LN V+

-

I

RL

a

b

Replacing the original network by its Thevenin equivalent, then the power delivered to the load is

LLTh

ThL R

RR

VRip 22 )(

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Power delivered to the load as a function of RL

Th

ThThL

LThTh

L

R

VpandRRyieldsso

RLRTh

RRT

dR

dp

4

0

2

32

We can confirm that is the maximum power by showing that

02

2

LdR

pd

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• If the load RL is invariable, and RTh is variable, then what should RTh be to make RL get maximum power?

Maximum Power Transfer(several questions)

• If using Norton equivalent to replace the original circuit, under what condition does the maximum transfer occur?

• Is it true that the efficiency of the power transfer is always 50% when the maximum power transfer occurs?

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Examples

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Tellegen Theorem

• If there are b branches in a lumped circuit, and the voltage uk, current ik of each branch apply passive sign convention, then we have

b

kkkiu

1

0

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Inference of Tellegen Theorem

• If two lumped circuits and have the same topological graph with b branches, and the voltage, current of each

branch apply passive sign convention, then we have not only

N̂N

0ˆ0ˆ

0ˆˆ0

11

11

k

b

kk

b

kkk

k

b

kk

b

kkk

iuiualsobut

iuiu

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Example

.,3

,10,4;2,2

,6,2.

21

1221

12

thenVoutfindAIget

canWeVVRWhenVVAIgetcanWe

VVRWhenonlyresistorsincludingnetworkaisN

I1V1

I2

V2NR2

+

-

b

kkk

b

kkk IVIVIVIVIVIV

TheoremTellegenthetoAccording

32211

32211 0;0

b

k

b

kkkkk

kkkkkkkk

IVIV

IVIIRIRIIVand

3 3

VV

VV

IVIVIVIV

42

2)2(10

42)3(6

2

22

22112211

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Reciprocity Theorem

R3

R1

I2Vs

R2

4V 2 3

6 R3

R1

I2Vs

R2

4V3 6 2

AI3

12 AI

3

12

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• Case 1 The current in any branch of a network, due to a single voltage source E anywhere else in the network, will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current I was originally measured.

Reciprocity Theorem(only applicable to reciprocity networks)

N I2Vs

N Vs'I1'Vs

I

Vs

Iexistsactually

IIthenVsVsif

2

'

'1:

2'1'

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Reciprocity Theorem(only applicable to reciprocity networks)

V2N +

-

Is

Is'+ N-V1'

Case 2

Is

V

Is

Vexistsactually

VVthenIsIsif

2

'

'1:

2'1'

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Reciprocity Theorem(only applicable to reciprocity networks)

Case 3

V2N

Vs

+

-

Is'NI1'

Vs

V

Is

Iexistsactually

VIthenIsVsif

2

'

'1:

2'1'

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example

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Source Transfer• Voltage source transfer

VsR1

R4R2

R5R3

R1

R2

R3

R4Vs

Vs R5

An isolate voltage source can then be transferred to a voltage source in series with a resistor.

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Source Transfer• Current source transfer

B

R1 R4

R2

Is

C R3

A

CR2 R3

R4

Is

B

Is

AR1

Examples

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Summary

• Linearity Property• Superposition• Source Transformation• Substitution Theorem• Thevenin’s Theorem• Norton’s Theorem

• Maximum Power Transfer

• Tellegen Theorem• Inference of Tellegen

Theorem• Reciprocity Theorem• Source Transfer