L4 Thevinin Norton Superposition

1

L4: Basic Circuit Analysis

S Sundar Kumar Iyer

Dept. of Electrical Engineering

IIT Kanpur

Acknowledgements to Baquer Mazhari, A.R. Harish, Adrish Banerjee

07.01.2013

2

Recap L3

• Equivalent Circuit – Resistance; series, parallel, …

• Node and Mesh Analysis

Take home message

• Basic results of resistance combination

• Series-parallel analysis not enough

3

v1 v2

v3

Node 1 and node 2 are merged together into a super node. KCL is applied to the super node

Sum of currents leaving a super node is zero

1 3 2 31 2

2 1 3 4

0v v v vv v

R R R R

− −+ + + =

Super Node

4

Mesh-3

Super mesh

Super Mesh

5

Today Let Us Look at …

• Thévenin Equivalent

• Norton Equivalent

• Superposition

The above a useful tools for circuit analysis

6

Any circuitAny circuit

Thévenin Equivalent Circuits

7

We can apply Thevenin’s theorem to any part of the circuit

Vt1

Rt1

Vt2

Rt2

8

Thévenin Equivalent Circuits

What is Vt ?

t o cV v=

Any circuit

Any circuit

+

Voc

-

+

Voc

-

9

Thévenin Equivalent Circuits

What is Rt ?

scisc

i

Any circuitAny circuit

t oct

sc sc

V vR

i i= =

10

Summary

t o cV v=

+

Voc

-

o ct

sc

vR

i=

sci

11

Examples

t o cV v=

+

Voc

-

2

2 1

1 5 5t

RV

R R= × =

+

1

0 .1 5ssc

vi A

R= =

3 3 .3o ct

sc

vR

i= = Ω

12

13

Currents and voltages in the circuit are only due to Independent Sources

14

Finding the Thévenin Resistance Directly

Suppose we make all independent sources zero in the circuit

Circuit with no Independent sources

tR

15

1.Turn off independent sources in the original network:

-A voltage source becomes a short circuit

-A current source becomes an open circuit

2. Compute the resistance between the terminals

Finding the Thévenin Resistance Directly

Circuit with no Independent sources

tR

16

0 V

0 A

17

5 2 04

5 2 0eq

R×

= = Ω+

18

Find Thévenin resistance for each of the circuits shown below

19

Circuit with dependent Sources

?t

R =

VZ

IZ

Zt

Z

VR

I=

20

Norton’s equivalent

sci

sci

n scI i=

Any circuit Any circuit

How do we find IN ?

21

Norton’s equivalent

+

Voc

-

+

Voc

-

o c n tv I R= ×o c

t

sc

vR

i=

Any circuit Any circuit

How do we find RN ?

22

Norton’s equivalent: Summary

n scI i=o c

t

sc

vR

i=

sci

+

Voc

-

23

24

25

Examples

26

tn

t

VI

R=

t n tV I R= ×

Source Transformation

27

Example

28

Use source transformation to solve for the indicated currents

29

Linearity and Superposition

IRV =

kVkIR =

Increasing the current by a constant k

RIV 11 =

Response to two excitations:

RIV 22 =

( )212121 VVRIRIRIIV +=+=+=

30

The superposition principle states that the total response is the sum of the responses to each of the independent sources acting individually.

Superposition Principle

31

Example

Circuit with only voltage source active. Current source is open circuited.

Circuit with only current source active. Voltage source is open circuited.

1 2Ti i i= +1 2Tv v v= +

32

Example

1 2Ti i i= +1 2Tv v v= +

33

10V is discarded by short circuit

Example

34

Maximum Power Transfer for dc circuits

RL

R

VS

What value of RL will give rise to maximum load power ?

S

L

VI

R R=

+

2 2

2( )

LL L S

L

RP I R V

R R= = ×

+

0L

L

P

R

∂=

∂

LR R=2

m ax4

SL

L

VP

R=

I

35

RL5V

1KΩ 1 6 .2 5L LR K P m W= ⇒ =

1 0 2L LR K P m W= ⇒ =

0 .2 3 .4 7L LR K P m W= ⇒ =

Maximum power is delivered to the load when RL = R

36

General Case

RL

Resistors and Sources RLVt

Rt

Maximum power is delivered to the load when RL = Rt

37

Using Thevenin’s theorem, find the equivalent circuit to the left of the terminals in the circuit shown below. Hence find i.

6o cv V= 3tR = Ω

1 .5i A=