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TECHNIQUES OF DC CIRCUIT ANALYSIS:TECHNIQUES OF DC CIRCUIT ANALYSIS:Superposition PrincipleSuperposition Principle

Source TransformationSource Transformation

Thevenin’s TheoremThevenin’s Theorem

Norton’s TheoremNorton’s Theorem

Maximum Power TransferMaximum Power Transfer

SKEE 1023SKEE 1023

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• Applies only for LINEAR CIRCUIT

Circuit containing only linear circuit elements

A LINEAR relationship between voltage and

current

What do we mean by a linear relationship?

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What do we mean by a linear relationship?

When the relationship fulfilled 2 properties:

• Homogeneity (scaling)

• Additivity

f(x) = y f(kx) = ky = kf(x)

f(x) = y f(x1 + x2) = f(x1) + f(x2) = y1 + y2

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Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone.

All other independent sources are deactivated

• voltage sources are shorted

• current sources are opened

Note that dependent sources CANNOT be deactivated !

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Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone.

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Superposition Principle: The voltage across an element ( or the current through an element) of a linear circuit containing more than one independent source, is the algebraic sum the voltage across that element (or the current through that element) due to each independent source acting alone.

• may involve MORE work

• cannot be applied to power calculation – find i or v first (using superposition) before calculating power !

• most suitably used when involved with sources of different properties or types, e.g. different frequencies, mixture of DC and AC, etc.

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Source Transformation: A tool used to simplify circuit; a process of replacing a voltage source in series with a resistor by a current source in parallel with a resistor or vice versa

vs

R

a

b

is R

a

b

If the circuit is equivalent at terminal a-b, their open-circuit and short-circuit characteristics are similar

voc = vs

isc = vs/R

voc = isR

isc = is

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Source Transformation: A tool used to simplify circuit; a process of replacing a voltage source in series with a resistor by a current source in parallel with a resistor or vice versa

vs

R

a

b

is R

a

b

voc = vs

isc = vs/R

voc = isR

isc = is

RivorRv

i sss

s

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Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor

Linear two-terminalcircuit

Load

+

V

+

V

LoadVTh

RTh VVThTh= ?= ?

RRThTh= ?= ?

In 1883, M.L. Thevenin proposed a theorem …….In 1883, M.L. Thevenin proposed a theorem …….

I

I

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Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor

To determine VTh

Linear two-terminalcircuit Load

VTh

RTh

Load

=

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Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor

To determine VTh

Linear two-terminalcircuit Load

VTh

RTh

open circuit voltage = Voc

+

= VThLoad

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Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor

To determine VTh

Linear two-terminalcircuit

VTh

RTh

open circuit voltage = Voc

+

= VTh

open circuit voltage = Voc

+

Load

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Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor

To determine VTh

Linear two-terminalcircuit

VTh

RTh

open circuit voltage = Voc

+

= VTh

open circuit voltage = Voc

+

VVThTh = V = Vococ = Open circuit voltage = Open circuit voltage

= VTh (Since the circuit is equivalent)

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Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor

To determine RTh - Method 1

Short circuit current, isc =

Vth

R th

RTh VTh

i sc

Linear two-terminalcircuit

VTh

RThisc

a

b

a

b

isc

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Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor

To determine RTh – Method 2

Pre-requisite: circuit with NO dependent sources

Deactivate all the independent sources

Linear circuit – independent sources killed

Rin = RThRin = RTh

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Thevenin’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source in series with a resistor

To determine RTh – Method 3

Deactivate all the independent sources - dependent sources stay as they are

• Introduce a voltage (or current) source.

LinearCircuit – ONLY dependent sources killed

+-

vo

io

RTh is calculated as:

o

oTh i

vR

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Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor

43 years later, E.L. Norton proposed a similar theorem. ….

IINN= ?= ?

RRNN= ?= ?

Linear two-terminalcircuit

Load

+

V

I

+

V

I

LoadIN RN

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Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor

To determine IN

Linear circuit

ININ

RN

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Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor

IN= Short circuit current

To determine IN

INRN

Linear circuit

Short circuit current = IN

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Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor

To determine IN

IN= Short circuit currentINRN

Linear circuit

Short circuit current = IN

IINN = I = Iscsc = Short circuit current = Short circuit current

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Norton’s Theorem: A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source in parallel with a resistor

To determine RN

SIMILAR METHOD AS HOW TO OBTAIN RSIMILAR METHOD AS HOW TO OBTAIN RThTh

RRNN = R = RThTh

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Relationship between Norton’s and Thevenin’s equivalents

Linear two-terminalcircuit

b

a

INRN

b

a

VThRTh

b

a

OROR

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Relationship between Norton’s and Thevenin’s equivalents

INRN

b

a

VThRTh

b

a

Since both circuits are equivalent, voc must be the same

N

ThThNNNTh I

VRRRIV

sc

oc

iv

NNoc RIv +

Thoc Vv +

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Maximum Power Transfer

Linear circuitRL What would be the value of RL for

power delivered to it become MAXIMUM?

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Maximum Power Transfer

Linear circuitRL What would be the value of RL for

power delivered to it become MAXIMUM?

VTh

RTh

L

2

LTh

LTh

L R

RRR

V

P L

2

LTh

Th RRR

V

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0 10 20 30 40 50 600.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

RL

p

Maximum Power Transfer

L

2

LTh

LTh

L R

RRR

V

P L

2

LTh

Th RRR

V

Rl=linspace(1,60,500);Vth=10;Rth=12;p=((Vth./(Rl+Rth)).^2).*Rl;

plot(Rl,p,'r');grid;

Maximum power

RL = 12

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Maximum Power Transfer

L

2

LTh

LTh

L R

RRR

V

P L

2

LTh

Th RRR

V

Mathematically, we evaluate RL when 0dRdP

L

L

0)RR(

VR

)RR(V2

dRdP

2LTh

2Th

L3LTh

2Th

L

L

0

)RR(RRR2V

dRdP

3LTh

LThL2Th

L

L

ThL RR

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Using PSpice to verify Norton’s and Thevenin’s Theorems

Find Thevenin equivalent at terminals a-b

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Using PSpice to verify Norton’s and Thevenin’s Theorems

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Using PSpice to verify Norton’s and Thevenin’s Theorems

0

R9

2

- ++-

E2

E

R8

2

R7

6

R6

4

I3

1AacTRAN = 0

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Using PSpice to verify Norton’s and Thevenin’s Theorems

0

R9

2

- ++-

E2

E

R8

2

R7

6

R6

4

I3

1AacTRAN = 0

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Using PSpice to verify Norton’s and Thevenin’s Theorems

0

R9

2

- ++-

E2

E

R8

2

R7

6

R6

4

I3

1AacTRAN = 0

I4

1AacTRAN = 1

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Using PSpice to verify Norton’s and Thevenin’s Theorems

R8

26.000V

1.333V

R6

4

4.000V

R7

6

I3

1AacTRAN = 0

0

R9

2

- ++-

E2

E

I4

1AacTRAN = 1

RTh = 6/1 = 6

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Using PSpice to verify Norton’s and Thevenin’s Theorems

I4

1AacTRAN = 0

R8

2

R7

6

20.00V

R6

4

6.667V

0

- ++-

E2

E

20.00V

I3

1AacTRAN = 5

R9

2

VTh = 20V