superposition theorm

8/3/2019 Superposition Theorm

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INC 112 Basic Circuit Analysis

Lecture 4


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Special Techniques

Superposition Theorem

Source Transformation

Thevenins Theorem

Nortons Theorem


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Linear elements vs. linear circuits

Linear element: passive element that has a

linear voltage-current relationship, i.e.

v(t)=R*i(t)

Linear dependent source: source whose output

is proportional only to the first power, i.e.,

v1 = 0.6i1-14v2

Linear circuit: composed of independentsources, linear dependent sources, and linear

elements


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One of the properties of linearity

Definition:

voltage across or current thru anyresistor or source may be calculated by

adding all the individual voltage or currents

caused by the separate independent

source acting alone


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Implementation

Other independent

voltage sources give

zero voltage, then act

like short circuits. Other independent

current sources give

zero current, then actlike open circuits.


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Example

I1 = 1A

I2 = 2A

I total = 1+2 = 3A


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Example

I1 = 1A

I2 = 0A

I total = 1+0 = 1A


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Example

Find voltage Vx


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V

VxV

333.9

42)7/12(6

)7/12(42

)4||3(6

)4||3()42(

!

v

!v

!


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V

VxV

333.3

1042

210

4)3||6(

)3||6()10(

!

v

!v

!


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V

VxVxVxVV

6333.3333.9

)10()42(

!!

!


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Example 5.1

use superposition to find ix


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Example 5.1 (cont.)

= 0.2A'

xi = 0.8A

''

xi

= 1.0Axi


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Superposition and dependent source

one cannot use superposition to

dependent source!!!


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Example 5.3


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KVL:

2

021210

'

'''

!

!

x

xxx

i

iii


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Supermesh:

6.0

02)3(12

''

''''''

!

!

x

xxx

i

iii


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A

iiixxx

4.1)6.0(2

'''

!

!

!


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Special Techniques



Thevenins Theorem

Nortons Theorem


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Practical voltage source

LssLiRvv !

s

sLsc

soc

R

vi

vv

!

!


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Practical current source

p

L

sLR

vii !

sLsc

spLoc

ii

iRv

!

!


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Equivalent practical sources

THNTH

NTH

RIV

RR

v!

!

Head of current = +


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Equivalent practical sources (cont.)

The two practical sources are equivalent

only with respect to what transpires at the

load terminals; they are not equivalent

internally!

Note: goal is to have either all current sources or all

voltage sources in the circuitmaking nodal/mesh

analysis easier


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Example

Usesource transformation to findIx


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AIX 3

1

6

2

321

2!!

!


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Special Techniques



Thevenins Theorem

Nortons Theorem


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Thevenins Theorem

Equivalent circuit


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Thevenins Equivalent Circuit

Composed of VTH and

RTHconnected in series

VTH = Voc (open-circuit voltage)

RTH= equivalent R (total R) in the circuit


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A statement of Thevenins Theorem

Disconnect a load network from the circuit

by substituting an open circuit, and find

Voc across the load, which is equivalent to

VTH.

Open load and evaluate RTH by zeroing

out every independent source (open

current source, short voltage source).


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Example

Find Thevenins equivalent circuit

and the current passing thruRL

given RL = 1


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0V

10V

0V 0V

6V 6V

VVTH 61032

3!v

!

Find VTH


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Short voltage source

RTH

;!

v!

!

2.13

232

3210

23||210THR

Find RTH


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Thevenins equivalent circuit

The current thru RL = 1is A423.012.13

6 !


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Example



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Find VTH

0V

5V

0V 0V

3V 3V

VVTH 331

!v!


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Open circuit

current source

RTH

;!

!

15

2310THR

Find RTH


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Example: Bridge circuit



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Find VTH

0V

10V

8V 2V

VTH = 8-2 = 6V


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Find RTH

RTH


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KKK

KKKKRTH

4.28.06.1

1||48||2

!!

!


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HW#3

Read Superposition and source

transformation

Ex

ercise 5.13, 5.24

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