Basic Theory of Circuits, SJTU 1 Chapter 4 Circuit Theorems

Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 11

Chapter 4Chapter 4

Circuit TheoremsCircuit Theorems


Theorems

Linearity

Superposition

Source transformation

Substitution

Thevenin’s Norton’s

Maximum Power Transfer

Tellegen

Reciprocity

Source Transfer


Linearity PropertyLinearity Property

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

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


Example 4.2


Superposition(1)Superposition(1)

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


Steps to Apply Superposition Principle:Steps to Apply Superposition Principle:1.1. Turn off all independent source except one Turn off all independent source except one

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

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

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

Superposition(2)Superposition(2)


j

R1V

e +

-L N

i

j

+

-V1

R1

i1

L N

i2

-L N

+

V2

eR1

21;21 iiiVVV


Fig. 4.6 For Example 4.3


Substitution TheoremSubstitution Theorem

20V

+6 I2

4V

4

-

8

I3

V3

I1



20V8V

-

I3

V3

6

I1+

8I2



V3

6+

-20V

I1

1A

8

I3

I2



20V

+6 I2

4V

4

-

8

I3

V3

I1

V3

6+

-20V

I1

1A

8

I3

I2

20V8V

-

I3

V3

6

I1+

8I2


Substitution Theorem Substitution Theorem

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



N1N N2Vs+

-

Is

Vs

NN1 OR NIs

N1


Thevenin’s TheoremThevenin’s Theorem

A linear two-terminal circuit can be replaceA linear two-terminal circuit can be replaced by an equivalent circuit consisting of a vd by an equivalent circuit consisting of a voltage source Voltage source Vthth in series with a resistor R in series with a resistor R

th (accompanied voltage source), where Vth (accompanied voltage source), where Vtt

hh is the open-circuit voltage at the terminal is the open-circuit voltage at the terminal

s and Rs and Rthth is the input or equivalent resistan is the input or equivalent resistan

ce at the terminals when the independent ce at the terminals when the independent source are turned off.source are turned off.


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

d

c


LN LOADV+

-

I

Voc

-

+

LNIs

-

LNo RoI+

+

LN

I

V I

-

+

V=Voc-RoI

Simple Proof by figures


Thevenin’s TheoremThevenin’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. 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


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

2. Calculate the open-circuit voltage V2. Calculate the open-circuit voltage Vococ and s and short-circuit current Ihort-circuit current Iscsc at the terminal of the at the terminal of the original circuit, then Rth=Voriginal circuit, then Rth=Vococ/I/Iscsc

VocCircuit

-

+OriginalIsc

Circuit

Original

Rth=Voc/Isc


Examples


Norton’s TheoremNorton’s Theorem

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


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

d(c)

N


Examples


Maximum Power TransferMaximum Power Transfer

LN V+

-

I

RL

a

b



If the load RIf the load RLL is invariable, and R is invariable, and RThTh is varia is variable, then what should Rble, then what should RThTh be to make R be to make RLL g get maximum power?et maximum power?

Maximum Power TransferMaximum Power Transfer(several questions)(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?


Examples


Tellegen TheoremTellegen Theorem If there are b branches in a lumped circIf there are b branches in a lumped circ

uit, and the voltage uuit, and the voltage ukk, current i, current ikk of eac of each branch apply passive sign conventioh branch apply passive sign convention, then we have n, then we have


Inference of Tellegen TheoremInference of Tellegen Theorem If two lumped circuits and have the If two lumped circuits and have the

same topological graph with b branches, same topological graph with b branches, and the voltage, current of each branch and the voltage, current of each branch apply passive sign convention, then we apply passive sign convention, then we have not onlyhave not only

N̂N


Example

I1V1

I2

V2NR2

+

-


I1V1

I2

V2NR2

+

-


Reciprocity TheoremReciprocity Theorem

R3

R1

I2Vs

R2

4V2

3

6

R3

R1

I2Vs

R2

4V3

6 2


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

Reciprocity TheoremReciprocity Theorem(only applicable to reciprocity networks)(only applicable to reciprocity networks)

N I2Vs

N Vs'I1'



V2N +

-

Is

Is'+ N-V1'

Case 2



Case 3

V2N

Vs

+

-

Is'NI1'


ExampleExampleAll resistors are 1 , find out i.

i

+ E

--

+ E --

i


Source Transfer PropertySource Transfer Property• Voltage source transfer

VsR1

R4R2

R5R3

R1

R2

R3

R4Vs

Vs R5


Source TransferSource Transfer• Current source transfer

CR2 R3

R4

Is

B

Is

AR1


SummarySummary

Linearity PropertyLinearity Property SuperpositionSuperposition Source TransformaSource Transforma

tiontion Substitution TheorSubstitution Theor

emem Thevenin’s TheoreThevenin’s Theore

mm Norton’s TheoremNorton’s Theorem

Maximum Power TraMaximum Power Transfernsfer

Tellegen TheoremTellegen Theorem Inference of TellegeInference of Tellege

n Theoremn Theorem Reciprocity TheoreReciprocity Theore

mm Source TransferSource Transfer

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Basic Theory of Circuits, SJTU 1 Chapter 4 Circuit Theorems