INC 112 Basic Circuit Analysis

Week 6

Other Techniques

Special Techniques

• Superposition Theorem

• Thevenin’s Theorem

• Norton’s Theorem

• Source Transformation

Source Transformation

RLRNINRL

RTH

VTH

THNTH

NTH

RIV

RR

When we do not need to know the current and the voltage inside the box, we can interchange the circuit inside without affecting other parts in the circuit.

Example

Use source transformation to find Ix

1Ω 2Ω

3Ω1A1V Ix

1Ω 2Ω

3Ω1A1V Ix

1Ω

2Ω

3Ω1A Ix1A

1Ω

2Ω

3Ω1A Ix1A

1Ω

2Ω

3Ω Ix2A

1Ω 2Ω

3Ω Ix2V

AI X 3

1

6

2

321

2

I

VVOC

ISC

Maximum Power Theorem

High RL: high voltage, low currentLow RL: high current, low voltage

Maximum Power Theorem

From Thevenin or Norton equivalent circuit, the RL that

gives the maximum power is RL that equals Rth.

RL that gives the maximum power is RL = Rth

Proof

RL

RTH

VTH

2

22

2

)( LTH

LTHL

LTH

TH

L

RR

RVR

RR

VP

RIP

LTH

TH

RR

VI

where

Substitutein theequation

0)(

)(2)(4

222

LTH

LTHLTHTHLTH

L RR

RRRVVRR

dR

dP

LTH

LLTH

LTHLTHTHLTH

LTH

LTHLTHTHLTH

L

RR

RRR

RRRVVRR

RR

RRRVVRR

dR

dP

2)(

)(2)(

0)(

)(2)(

222

4

222

Therefore, RL = Rth is the point that yields maximum power

Find the first derivative. Let it equal zero to find critical points.

Example

3Ω

2Ω

10V

10Ω

RL

2Ω

Find RL that gives the maximum power and find that power.

13.2

6V RL

Thevenin’s equivalent circuit

RL that gives the maximum power is 13.2Ω

Maximum Power is WR

V68.0

2.13

)2/6( 22

Voltage Sources in Real World

3V 3V

0.01Ω

I

V

I

V

Ideal Voltage Source Real Voltage Source

Current Sources in Real World

I

V

I

V

Ideal Current Source Real Current Source

1A 1A 10M

Delta-Wye Conversion (∆-Y)

Ra

Rb

RcR1 R2

R3

1

133221

3

133221

2

133221

R

RRRRRRRc

R

RRRRRRRb

R

RRRRRRRa

RcRbRa

RcRaR

RcRbRa

RbRcR

RcRbRa

RaRbR

3

2

1

Example

4Ω

2Ω

1Ω

5Ω

3Ω

3/2Ω

2Ω

3/8Ω

5Ω

1/2Ω

13/2Ω19/8Ω

1/2Ω

159/71Ω