THÉVENIN & NORTON EQUIVALENT CIRCUITS

Thévenin Equivalent Circuit

• Any linear 2-terminal (1-port) and active network can be

replaced by an equivalent circuit consisting of an independent

voltage source in series with a resistor.

network

of

sources

and

resistors

≡

–+

VTh

RTh

RL

iL+

vL

–

a

b

RL

iL+

vL

–

a

b

Thévenin equivalent circuitActual circuit

Norton Equivalent Circuit

• Any linear 2-terminal (1-port) network of independent voltage

sources, independent current sources, and linear resistors can be

replaced by an equivalent circuit consisting of an independent current

source in parallel with a resistor without affecting the operation of the

rest of the circuit.

Norton equivalent circuit

network

of

sources

and

resistors

≡RL

iL+

vL

–

a

b

a

RL

iL+

vL

–

iN

b

RN

Determination of Thévenin/Norton Equivalent

Calculate the open-circuit voltage, voc

Calculate the short-circuit current, isc

Note that isc is in the direction of the

open-circuit voltage drop across the

terminals a,b !

sc

oc

Th

ocTh

i

vR

vV

network

of

sources

and

resistors

a

b

+

voc

–

network

of

sources

and

resistors

a

b

isc

Alternative Method of Calculating RTH

For a network containing only independent sources

and linear resistors:

1. Set all independent sources to zero

voltage source short circuit

current source open circuit

2. Find equivalent resistance Req between the terminals by inspection

Or, set all independent sources to zero

1. Apply a test voltage source VTEST

2. Calculate ITEST

TEST

TEST

ThI

VR

network of

independent

sources and

resistors, with

each source

set to zero

Req

network of

independent

sources and

resistors, with

each source

set to zero

ITEST

–+

VTEST

Finding IN

IN ≡ isc = VTh/RTh

Analogous to calculation of Thevenin Equivalent Circuit:

• Find open-circuit voltage and short-circuit current

• Or, find short-circuit current and Norton (Thevenin) resistance

Finding IN and RN

• We can derive the Norton equivalent circuit from a Thévenin equivalent

circuit simply by making a source transformation:

RLRN

iL

iN

+

vL

–

a

b

–+

RL

iL+

vL

–

vTh

RTh

sc

Th

Th

N

sc

oc

ThN ; i

R

vi

i

vRR

a

b

Answer: VTh= 10V, RTH = 1W

Find Thevenin’s and Norton equivalent circuits between the terminals of Ro.

Exercise

Find Thevenin’s equivalent circuit between the terminals of Ro.

Exercise

Open Circuit to find VTH

Short Circuit to find IN

Maximum Power Transfer

Objectives

• To find the resistance that absorb maximum power from an

active circuit.

• To calculate the maximum power that can be transferred from

a circuit to a resistance.

13

• Suppose that a general network N

has a resistive load as shown.

LL

L

dPSolve 0 to find where P is max imum

dR

Now we might consider two questions:

• For what value of RL is maximum power delivered to RL?

• What is the maximum power that can be delivered to RL?

To answer these questions

1) Replace N by a Thevenin Equivalent Circuit

2) Determine a general expression for power to RL

3)

independent

sources,

dependent sources,

and resistors

Network N

RL

Maximum Power Transfer Theorem

A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit.

L

2

LTh

ThL

2L R

RR

VRip

LTh

LThL2

LTh

4LTh

LThL2

LTh2Th

02

02

RR

RRRRR

RR

RRRRRV

dR

dp

L

–+

VTh

RTh

RL

iL+

vL

–

Thévenin equivalent circuit

To find the value of RL for which p is

maximum, set to 0:

Power absorbed by load resistor:

LdR

dp

Maximum Power Transfer Theorem

• The relationship between PL and RL can be illustrated by the graph

shown below.

Maximum Power Transfer Theorem

Example

• Find Ro for maximum power and Pmax.

Example

Solution

Solution