Thevenin and Norton Theorems

7/29/2019 Thevenin and Norton Theorems

1/12

Thevenin Theorem in Sinusoidal Steady Analysis

Aim: To obtain a simple equivalent circuit for a 1-port circuit thatconsists of linear, time-invariant resistors, capacitors, inductorsand independent sources.

Thevenin Equivalent:

+

_v

i

1-port

circuit

+

_V

I

+_

ZTH

VTH

ZTh Thevenin impedance

Equivalent impedance between terminalswhen sources are set to zero.

VTh Open circuit voltage

THThVIZV

The voltage of the port when the port isleft as open circuit.

Thevenin Theorem insinusoidal steadyanalysis: A 1-port circuitthat consists of linearresistor, capacitor,

inductors andindependent sources hasa Thevenin equivalentcircuit in sinusoidalsteady state if the port

voltage phasor can beuniquely determined for agiven port currentphasor, in other words, ifthe 1-port is current-controlled.


2/12

Norton Theorem in Sinusoidal Steady Analysis

Aim: To obtain a simple equivalent circuit for a 1-port circuit thatconsists of linear, time-invariant resistors, capacitors, inductorsand independent sources.

Norton Equivalent:

+

_v

i

1-port

circuit

Norton Theorem insinusoidal steadyanalysis: A 1-port circuitthat consists of linearresistor, capacitor,

inductors andindependent sources hasa Norton equivalentcircuit in sinusoidalsteady state if the port

current phasor can beuniquely determined for agiven port voltage phasor,in other words, if the 1-port is voltage-controlled.

NNIVGI

+

_V

I

YNIN

GN Norton conductance

IN Short circuit current

Equivalent conductance betweenterminals when sources are set to zero.

The current through the port when the

port is short-circuited.


3/12

How to obtain Thevenin equivalent circuit?

+

_

V

I

1-portcircuit

Connect a sinusoidal current source to the port.

I*V*

+

_ Solve the circuit using sinusoidal steady analysis

and obtain a relation between phasors I* and V*.

Use I=I* and V=-V* to obtain a relationbetween I and V.

+

_V

I

1-portcircuit

Set the values of independent sources tozero.

Calculate the equivalent impedance ZTh = V / I. Assume that I=0 (open-circuit the port) and

calculate Vth=V taking into account allindependent sources .

There exist two methods for this!


4/12

How to obtain Norton equivalent circuit?

Connect a sinusoidal voltage source to the port. Solve the circuit using sinusoidal steady analysis

and obtain a relation between phasors I* and V*.

Use I=-I* and V=V* to obtain a relationbetween I and V.

+

_V

I

1-portcircuit

Set the values of independent sources tozero.

Calculate the equivalent admitance GN = I / V. Assume that V=0 (short-circuit the port) and

calculate IN=I taking into account allindependent sources .

There exist two methods for this!

+

_

V

I

1-portcircuit

I*

V*

+

_

+-


5/12

Thevenin Equivalent: THTH VIZV

If 1-port is not current-controlled there is no Thevenin eq..

Norton Equivalent:NNIVYI

If 1-port is not voltage-controlled there is no Norton eq..

0THZTH

TH

THZ

VVZ

I 1 ,0THZ No Norton equivalent!

,0NY No Thevenin equivalent!

NY

NI

0NYN

N

NY

II

YV

1

THV

THZ

Interchange between Thevenin and Norton

From Thevenin to Norton:

From Norton to Thevenin:


6/12

Example 1: Find the Thevenin equivalent circuit for the following circuit!

Example 2: Find the Norton equivalent circuit for the following circuit! Check your answercomparing it to Example 1.


7/12

Circuit(Network) Functions in Sinusoidal SteadyAnalysis

+

_E1

IS

linear,time-

independentelements

0

0

)()(0

000

s

d

TI

I

VV

jwNjwM

IAA

)(jwT

How does Vdk affect Is ? Depending on w!

Ssk

d IwT

wTcofactorwV

k )(det

)()( ,

Assume that

there is only one

source.

0


8/12

+

_E1

IS

)(

)()(

jwQ

jwP

I

wV

S

dk

)(jwP are polynomials

in (jw) with real coefficients.)(jwQ

m

m

n

n

S

d

jwbjwbjwb

jwajwajwa

I

wVk

)(...)()(

)(...)()()(2

21

2

21

This depends on thecircuit but not on thevalue of Is .

linear,time-

independentelements

Circuit(Network) Functions in Sinusoidal SteadyAnalysis

How does Vdk affect Is ? Depending on w!


9/12

One can define many other circuit functions:

)(

)(

wI

wV

s

dk Impedance Function

)(

)(1

wI

wV

s

Input Impedance Function

)(

)(

1 jwV

jwVkd Voltage Transfer Function

)()(

jwIjwI

s

k Current Transfer Function


10/12

Symmetries of Circuit Functions

Lemma: Let be a polynomial in complex variable s withreal coefficients.

1)

2) (z is called as a root of n(z).)

jwssn ),(

)()( snsn

0)(0)( znzn

Proof: 011

1 ...)( nsnsnsnsnk

k

k

k

Rnnnn

kk

011 ,,...,

1) ...)( 0111 nsnsnsnsnk

k

k

k

01

1

1... nsnsnsn

k

k

k

k

01

1

1... nsnsnsn

k

k

k

k

)(sn

2) 00)(0)( znzn

0)(0)()()( znznznznFrom (1)


11/12

Circuit function: )(

)(

)( jwd

jwn

jwH

)(

)()(jwHj

ejwHjwH

Symmetry Property: The magnitude of any circuit function is an evenfunction of w and its phase is an odd function of w.

Proof:

)()()(

jwdjwnjwH

)()()(

jwdjwnjwH

)()(

jwdjwn

jwjw and from Lemma)(

)()(

jwd

jwnjwH

)( jwH

)()()()( jwHjwHjwHjwHw

Since the phase of is .z z )()( jwHjwH

Theorem: For a circuit in sinusoidal steady state, any circuit function is

well-defined and is the ratio of two polynomials in (jw) with real

coefficients if det(T(jw)) is nonzero.


12/12

+_ Vs (t) 1-portcircuit

so VjwHV )(

Result:In order to find the behaviour of the circuit for the frequency w, one

should find and .sVjwH ,)( sVjwH

),(

sVj

ss eVV

oVj

oo eVV

sV

s

jwH

o eVejwHV

)()(

sVjwH

so

eVjwHV

)(

)(

so VjwHV )( so VjwHV )(

sso VjwHwtVjwHtv )(cos)(

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Thevenin and Norton Theorems