EQUIVALENT CIRCUITS

Ho Kyung Kim, Ph.D.hokyung@pusan.ac.kr

School of Mechanical EngineeringPusan National University

Basic Experiment and Design of Electronics

Outline

• Superposition theorem

• Thévenin’s and Norton’s theorem

• Maximum power transfer theorem

• Wheatstone bridge

2

• In any linear circuit containing multiple independent sources, the current or voltage at any point in the network may be calculated as the algebraic sum of the individual contributions of each source acting alone.

Superposition theorem

212121

BBBBBB iiR

vR

vR

vvi

021 Rivv BB

– in order to set a voltage source equal to zero, replace it with a short circuit

– in order to set a current source equal to zero, replace it with an open circuit

• When viewed from the load, any network composed of ideal voltage and current sources, and of linear resistors, may be represented by an equivalent circuit consisting of an ideal voltage source vT in series with an equivalent resistance RT.

Thévenin's theorem

① find the equivalent resistance presented by the circuit at its terminal② compute the Thévenin voltage

• When viewed from the load, any network composed of ideal voltage and current sources, and of linear resistors, may be represented by an equivalent circuit consisting of an ideal current source iN in parallel with an equivalent resistance RN.

Norton's theorem

① find the equivalent resistance presented by the circuit at its terminal② compute the Norton current

Determination of equivalent resistance

① remove the load② zero all independent voltage and current sources③ compute the total resistance between load terminals

Note that the computed resistance is equivalent to that which would be encountered by a current source connected to the circuit in place of the load.

321 || RRRRT

Computing the Thévenin voltage

① remove the load, leaving the load terminals open-circuited② define the open-circuit voltage vOC across the open load terminals③ solve for vOC

④ Thévenin voltage, vT = vOC

• The equivalent (Thévenin) source voltage is equal to the open-circuit voltage at the load terminals (with the load removed).

vS

R1 R3

R2 RL

iL+

vOC

-

+vOC-

+ 0 V -

21

22 RR

Rvvv SROC

L

S

LT

TL RRRR

RRR

v

RRv

i

321

21

2

)||(

Computing the Norton current

① replace the load with a short circuit② define the short-circuit current iSC to be the Norton equivalent current③ solve for iSC

④ Norton current, iN = iSC

• The Norton equivalent current is equal to the short-circuit current that would flow if the load were replaced by a short circuit.

321 Rv

Rv

RvvS

213231

32

RRRRRRRR

vv S

213231

2

3 RRRRRRR

vRvii SSCN

• Then, how much power can be transferred to the load from the source under the most ideal conditions?

Maximum power transfer theorem

• The Thévenin and Norton models imply that some of the power generated by the source will necessarily be dissipated by the internal circuits within the source.

Thévenin equivalent

LLL RiP 2

LT

TL RR

vi

LLT

TL R

RRv

P 2

2

)(

0

L

L

RP

0)(

)(2)(4

222

LT

LTLTLTT

L

L

RRRRRvRRv

RP

Power absorbed by the load, RL;

Load current, iL;

Then, we have;

Maximum PL can be obtained when

RL = RT

• a resistive circuit• widely used as measurement circuit

Wheatstone bridge

The source voltage divides between each resistor pair according to the voltage divider rule;

21

2

RRRvv Sad

x

xSbd RR

Rvv

3

x

xSbdadab RR

RRR

Rvvvv

321

2

and

Self-assigned HW

• Rizzoni (5th ed.), Ch. 3– 51, 52, 53, 54, 56, 58, 59, 60, 73, 75

14