SEE 1023 Circuit Theory

Concept of Equivalence

Concept of Equivalence

Circuit A and circuit B are equivalent if they have the same I-V characteristics at their terminals.

CircuitA

CircuitB

a

b

b

a

Examples

1

2

55 5

5

55

5/3

15

Examples

3

4

20V

15V

10V

10A 5A 15A

-15 V

-20A

Examples

5

6

10V

5

5A 1010

50V

2A 5

Thevenin’s Theorem

?

Thevenin’s Theorem

Thevenin found that any complex circuit can always be replaced by a simple circuit having a voltage source in series with a resistor.

VTH

RTH

Thevenin’s Theorem

VTH = ?

RTH = ?

VTH

RTH a a

b b

How to find:

Thevenin’s Theorem

If the terminals ab of the circuit B is opened, what is Vab?

VTH

RTH a a

b bCircuit A Circuit B

If the terminals ab of the circuit A is opened, what is the value of Vab?

24 6

2

3

20V

Thevenin’s Theorem

If the independent source of circuit B is killed, what is the equivalent resistance at the terminals ab.

VTH

RTH a a

b bCircuit A Circuit B

24 6

2

3

20V

If all independent sources of circuit A are killed, what is the value of equivalent resistance at the terminals ab.

Thevenin Equivalent Circuit

1. To find VTH:

ComplexCircuit

a

b

DeadcomplexCircuit b

a

Rin

2. To find RTH :

ComplexCircuit

a

b

+

-Vab

Terminals ab are opened and find Vab.

VTH =Vab

All independent sources are killed and find Rin. RTH = Rin

Thevenin Equivalent Resistance

DeadcomplexCircuit b

a

Rin

1 All independent sources are killed and find Rin. RTH = Rin

Rin is the input resistance at terminals ab.

Rin can be obtained by circuit resistance reduction.Warning !!!

This technique is limited to all resistance circuit.

Thevenin Equivalent Resistance

DeadcomplexCircuit b

a

2 All independent sources are killed.

1A current source is injected at terminals ab. Then find Vab.

This technique is known as current injection.

1ARTH = Vab

Thevenin Equivalent Resistance

DeadcomplexCircuit b

a

3 All independent sources are killed.

1V voltage source is applied at terminals ab. Then find I.

This technique is known as voltage application.

1VRTH = 1/I

+-I

Thevenin Equivalent Resistance

4 Let the circuit as it is (live).

Short-circuited terminals ab. Then find ISC.

This technique is known as short-circuit.

ComplexCircuit

b

a

ISC

SC

THTH I

VR

Thevenin’s Theorem

a

b

24 6

2

3

20V

1. Find Thevenin equivalent circuit.

Thevenin’s Theorem

1. To Find VTH

14.00V

0V

R4

2

R6

3

V1

20V

I1

2A

R5

6

0

I2

0AdcR3

4

0A current source

equals open-circuit.

VTH = 14 V

Thevenin’s Theorem

2. To Find RTH

We use 1A current source

injection.

RTH = 6

R3

4

R4

2

R5

6

R6

3

I2

1A

6.000V

V1

0

I1

0

00V

Independent sources

are set to zero.

8

8

4

8

6

8

2. Find Thevenin equivalent circuit

60V

a

b

Thevenin’s Theorem: Example

8

8

8

6

8 60V

6A

a b

3. Find Thevenin equivalent circuit.

Thevenin’s Theorem: Example

8

8 6

8 60V

6Aa b

4. Find Thevenin equivalent circuit.

4

Thevenin’s Theorem: Example

5. Find Thevenin equivalent circuit.

Thevenin’s Theorem: Example

1. To Find VTH

Thevenin’s Theorem: Example

I10

R4

2x

I2

5A

R3

4

20.00V

R2

6

0V

v

0

R1

2

E1

2*V(x)

+-

VTH = 20 V

0A current source

equals open-circuit.

1. To Find RTH

Thevenin’s Theorem: Example

RTH = 6

6.000V

x

E1

2*V(x)

+-

0V

I11A

R4

2

R3

4

v

I2

0

R2

6

R1

2

0

Independent sources

are set to zero.

We use 1A current source

Injection technique.

Thevenin’s Theorem: 1883

Thevenin states that any linear two-terminal circuit could be replaced by a simple circuit having a voltage source, VTH, in series with a resistor, RTH.

VTH

RTH

Thevenin’s Theorem: 1883

Léon Charles Thévenin (1857- 1926)

A French Engineer

Norton’s Theorem: 1926

Norton states that any linear two-terminal circuit could be replaced by a simple circuit having a current source, IN, in parallel with a resistor, RN.

IN RN

a

b

a

b

Norton’s Theorem: 1926

Edward Lawry Norton (1898 - 1983) An US Engineer

Norton Equivalent Circuit (NEC)

1. To find IN:

ComplexCircuit

a

b

2. To find RN :

ComplexCircuit

a

bTerminals ab are shorted

and find ISC.

IN =ISC

RTH = RN. So, RN is to be found exactly the same way as RTH.

ISC

Thevenin’s Theorem

a

b

24 6

2

3

20V

1. Find Norton equivalent circuit (NEC).

Norton’s Theorem

2. Find Norton equivalent circuit (NEC).

24 V 3 A

4

12

a

b

IN = 9 ARN = 3

Answer:

3. Find Norton equivalent circuit (NEC).

Norton’s Theorem: Example

1. To Find Norton curren IN.

Norton’s Theorem: PSpice

R4

2

v

R3

4

R2

60Vdc

3.333A

E1

2*V(x)

+-

R1

2

0

I2

5

x

0V voltage source is used to provide short circuit and to sense IN.

IN = 3.333A

2. To Find Norton resistance RN

Norton’s Theorem: PSpice

RN = 6

6.000V

x

E1

2*V(x)

+-

0V

I11A

R4

2

R3

4

v

I2

0

R2

6

R1

2

0

Independent sources

are set to zero.

We use 1A current source

Injection technique.

Thevenin-Norton Transformation

RTHVTH

RTH

TH

TH

R

V

IN RN NNRI

RN

The NEC is simply the source transformation of the TEC.

Using Thevenin’s theorem to analyze circuits

Find Ia using Thevenin’s theorem

2Vx

8 4

6

8

Ia

60V 6A

+-

-Vx

+

6A60V

Find Va using Thevenin’s theorem

2Vx

8 4

6

8

+-

-Vx

+

6A60V

+-Va

Using Thevenin’s theorem to analyze circuits

Find Ia using Norton’s theorem

2Vx

8 4

6

8

Ia

60V 6A

+-

-Vx

+

6A60V

Using Norton’s theorem to analyze circuits

Find Va using Thevenin’s theorem

2Vx

8 4

6

8

+-

-Vx

+

6A60V

+-Va

Using Thevenin’s theorem to analyze circuits

1. Define the terminals a and b

2. Remove the 8 resistor that connected between the terminals a and b.

(we define this resistor as a load)

3. Find Thevenin equivalent circuit (TEC) for the circuit without the load.

4. Find Ia,Va,or Pa from TEC.

Thevenin’s Theorem to find VIP: Steps

1. Define the terminals a and b

2. Remove the 8 resistor that connected between the terminals a and b.

(we define this resistor as a load)

3. Find Norton equivalent circuit (NEC) for the circuit without the load.

4. Find Ia,Va, or Pa from NEC.

Norton Theorem to find VIP: Steps

Maximum Power Transfer Theorem

VTH

RTH

RL

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?

Maximum Power Transfer Theorem

VTH

RTH

RL

The power absorbed by the load RL:

LTHL

THLL R

RR

VRiP

2

2

i

2

2

THL

LTH

RR

RV

Maximum Power Transfer Theorem

VTh

RTh

RL

The power absorbed by the load RL:

LThL

ThLL R

RR

VRiP

2

2

i

2

2

ThL

LTH

RR

RV

Maximum Power Transfer Theorem

L

L

dR

dPTo find the value of RL for which power,

PRL, is maximum, set to 0:

LTh

LThL2

LTh

4LTh

LThL2

LTh2Th

02

02

RR

RRRRR

RR

RRRRRV

dR

dP

L

RL

Th

2

max, 4R

VP ThRL

Maximum Power Transfer Theorem

ThL RR

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

The maximum power is given by

Th

2

max, 4R

VP ThL

Maximum Power Transfer Theorem• The relationship between PL and RL can be illustrated by the graph shown below.

PL

PL,max Th

2

max, 4R

VP ThL

Example 1

• Find RL to achieve maximum power at RL

• Calculate maximum power at RL

• Find the % of power from the source is

delivered to RL

360 V30

150 RL

Answer: VTh = 300 V, RTh = 25 , Pmax = 900 W, % = 35.71

Example 2

Ro is adjusted for maximum power, find Ro

and the maximum power.

6V 4

6

Ro

+-+

-

Vx

2Vx

Answer: VTh = 12 V, RTh =12 , Pmax = 3 W