1

ECE 3144 Lecture 23

Dr. Rose Q. Hu

Electrical and Computer Engineering Department

Mississippi State University

2

Method of finding Thevenin equivalent Circuit

Number of method

If the circuit contains:

Thevenin

equivalent circuit

1 Resistors and independent sources only (case 1)

a) Connect an open circuit between terminals a and b. Find voc

= vab the voltage across the open circuit

b) Deactivate the independent sources (replace independent voltage sources with short circuits and independent current sources with open circuits). Find RTH by methods introduced in Chapter 2.

2 Resistors and independent and dependent sources (case 2) or case 1

a) Connect an open circuit between terminals a and b. Find voc = vab the voltage across the open terminals.

b) Connect a short circuit between terminals a and b. Find isc, the current directed from a to b in the short circuit.

c) Calculate RTH = voc/isc.

3 Resistors and dependent sources only.

a) Note that voc = 0

b) Connect a 1-A current source from terminal b to terminal a. Determine vab

c) Then RTH = vab/1.

3

Method of finding Norton equivalent Circuit

Number of method

If the circuit contains:

Norton

equivalent circuit

1 Resistors and independent sources only (case 1)

a) Connect a short circuit between terminals a and b. Find isc, the current directed from a to b in the short circuit.

b) Deactivate the independent sources (replace independent voltage sources with short circuits and independent current sources with open circuits). Find RN=RTH by methods introduced in Chapter 2.

2 Resistors and independent and dependent sources (case 2) or case 1

a) Connect an open circuit between terminals a and b. Find voc = vab the voltage across the open terminals.

b) Connect a short circuit between terminals a and b. Find isc, the current directed from a to b in the short circuit.

c) Calculate RN = RTH = voc/isc.

3 Resistors and dependent sources only.

a) Note that isc = 0

b) Connect a 1-A current source from terminal b to terminal a. Determine vab

c) Then RN = RTH = vab/1.

4

Maximum Power Transfer• Many applications of circuits require that the maximum power

available from a source be transferred to a load resistor RL.

• Consider the circuit network A, terminated with load RL. A can represent any circuit network, say power utility systems. Power utility systems are designed to transport the power to the load RL with the greatest efficiency by reducing the losses on the power lines and power sources themselves.

• How to calculate the maximum power efficiency? We know that the general circuit A can be reduced to its Thevenin (Norton) circuit.

A

i

5

Maximum Power Transfer

• For the given general circuit, we wish to what is the power delivered to the load resistor RL.

• Since current i is

• The power delivered to RL is

• Since VTH and RTH are fixed for a given source, the power delivered is a function of load resistor RL. To find RL that maximizes the power, we differentiate the power with respect to RL:

LRip 2

LTH

TH

RR

vi

2

2

)( LTH

LTH

RR

Rvp

0)(

))(2()(4

222

LTH

LTHLTHTHLTH

L RR

RRRvvRR

dR

dp

0)(2)( 2 LTHLLTH RRRRR

LTH RR

=>

=>

6

Maximum Power transfer• Confirm that RTH= RL gives the maximum power transfer instead of the minimum power

transfer. We know that – If RL = 0 => p = i2RL = 0 => p is minimum in this case– if RL = =>p = vi = 0 => p is also minimum in this case.– So RTH=RL gives maximum power transfer.

• The maximum power achieved when RTH = RL

• The maximum power transfer theorem states that the maximum power delivered by a source represented by its Thevenin circuit (Norton circuit) is attained when the load is equal to the Thevenin (Norton) resistance RTH (RN).

• The maximum power delivered is v2TH /4RL for Thevenin equivalent source The power

attained as RL varies is shown as

L

TH

LL

LTH

R

v

RR

Rvp

4)(

2

2

2

max

2max )1(

4

x

x

p

p

TH

L

R

Rx Where

Power vs. load resistance

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

RL / RTH

p / p

max

7

Efficiency of power transfer• The efficiency of power transfer is defined as the ratio of the power delivered to the load,

pout, to the power supplied by the source, pin. Therefore we have the efficiency as

• In the ideal source case, all the power supplied by the source is absorbed by the load =>=1. • For practical sources we have discussed, the maximum power transfer happened when RL =

RTH. • For the maximum case, power supplied by the source pin and the power absorbed by the

load pout are

• Therefore only 50% efficiency can be achieved at maximum power transfer conditions

in

out

p

p

L

TH

LTH

THTHTHin

R

v

RR

vvivp

2)(

2

L

THout

R

vp

4

2

%50max

in

out

p

p

8

Maximum power transfer for Norton Equivalent circuits

• We may also use Norton circuit to represent circuit Aio

A

2

222

)( LN

LNNL

RR

RRiRip

02

2

LdR

pd=> Maximum power occurs when RN = RL = RTH

The maximum power delivered is

4

2

maxLN Ri

p

9

Examples

• Provided in the class.

10

Homework for Lecture 23

• Problems 4.70, 4.71, 4.72, 4.73

• Due March 18