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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

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Recap L3

• Equivalent Circuit – Resistance; series, parallel, …

• Node and Mesh Analysis

Take home message

• Basic results of resistance combination

• Series-parallel analysis not enough

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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

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Mesh-3

Super mesh

Super Mesh

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Today Let Us Look at …

• Thévenin Equivalent

• Norton Equivalent

• Superposition

The above a useful tools for circuit analysis

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Any circuitAny circuit

Thévenin Equivalent Circuits

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

Vt1

Rt1

Vt2

Rt2

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Thévenin Equivalent Circuits

What is Vt ?

t o cV v=

Any circuit

Any circuit

+

Voc

-

+

Voc

-

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Thévenin Equivalent Circuits

What is Rt ?

scisc

i

Any circuitAny circuit

t oct

sc sc

V vR

i i= =

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Summary

t o cV v=

+

Voc

-

o ct

sc

vR

i=

sci

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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

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

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Finding the Thévenin Resistance Directly

Suppose we make all independent sources zero in the circuit

Circuit with no Independent sources

tR

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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

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0 V

0 A

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5 2 04

5 2 0eq

R×

= = Ω+

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Find Thévenin resistance for each of the circuits shown below

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Circuit with dependent Sources

?t

R =

VZ

IZ

Zt

Z

VR

I=

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Norton’s equivalent

sci

sci

n scI i=

Any circuit Any circuit

How do we find IN ?

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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 ?

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Norton’s equivalent: Summary

n scI i=o c

t

sc

vR

i=

sci

+

Voc

-

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Examples

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tn

t

VI

R=

t n tV I R= ×

Source Transformation

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Example

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Use source transformation to solve for the indicated currents

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Linearity and Superposition

IRV =

kVkIR =

Increasing the current by a constant k

RIV 11 =

Response to two excitations:

RIV 22 =

( )212121 VVRIRIRIIV +=+=+=

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

Superposition Principle

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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= +

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Example

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

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10V is discarded by short circuit

Example

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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

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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

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General Case

RL

Resistors and Sources RLVt

Rt

Maximum power is delivered to the load when RL = Rt

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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=