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TECHNIQUES OF DC CIRCUIT ANALYSIS:
Superposition Principle
Source Transformation
Thevenin’s Theorem
Norton’s Theorem
Maximum Power Transfer
SKEE 1023
1
•
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?
2
When the relationship fulfilled 2 properties:
•
Homogeneity (scaling)
f(x) = y  f(kx) = ky = kf(x)
•
Additivity
f(x) = y  f(x1 + x2) = f(x1) + f(x2) = y1 + y2
What do we mean by a linear relationship?
3
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.
5
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.
6
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
R
a
vs
a
is
b
R
b
voc = isR
voc = vs
If the circuit is equivalent at terminal a-b, their open-circuit and
short-circuiti characteristics
are similar
isc = is
= v /R
sc
s
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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
R
a
a
vs
is
R
b
b
voc = isR
voc = vs
isc = is
isc = vs/R
is 
vs
R
or
v s  i sR
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Thevenin’s
: A linear
two-terminal
circuit can be
In 1883,
M.L. Theorem
Thevenin
proposed
a theorem
…….
replaced by an equivalent circuit consisting of a voltage source in
series with a resistor
I
Linear twoterminal
circuit
+
V
Load

RTh
I
VTh= ?
+
VTh
V

Load
RTh= ?
9
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
RTh
VTh
Load
=
Linear twoterminal
circuit
Load
10
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
RTh
+
VTh
Loadvoltage = Voc
open circuit
= VTh

Linear twoterminal
circuit
Load
11
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
RTh
+
VTh
open circuit voltage = Voc
= VTh

Linear twoterminal
circuit
+
Load
open circuit voltage = Voc

12
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
RTh
+
VTh
open circuit voltage = Voc
= VTh

VTh = Voc = Open circuit voltage
Linear twoterminal
circuit
+
= VTh voltage
(Since=the
open circuit
Voc circuit is equivalent)

13
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
isc
RTh
a
VTh
b
Short circuit current, isc =
RTh 
isc
Linear twoterminal
circuit
Vth
R th

a
VTh
i sc

b
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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 = RTh
Rin = 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
Linear
Circuit –
ONLY
dependent
sources
killed
•
io
vo
+
-
Introduce a voltage (or current) source.
RTh is calculated as:
R Th 
vo
io
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Norton’s Theorem: A linear two-terminal circuit can be replaced
43byyears
later, E.L. Norton proposed a similar theorem. ….
an equivalent circuit consisting of a current source in parallel with
a resistor
I
+
Linear twoterminal
circuit
Load
V

IN = ?
I
IN
RN
+
V

Load
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
IN
RN
IN
Linear
circuit
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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
Linear
circuit
RN
IN= Short circuit current
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
RN
IN= Short circuit current
IN = Isc = Short circuit current
Linear
circuit
Short circuit current
= IN
20
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 RTh
RN = RTh
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Relationship between Norton’s and Thevenin’s equivalents
a
IN
RN
b
Linear twoterminal
circuit
a
OR
b
VTh
RTh
a
b
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Relationship between Norton’s and Thevenin’s equivalents
a
+
IN
v oc  INR N
RN

b
Since both circuits are equivalent,
VT h vocvmust
oc
VT h  INR N  Rbe

R


the
same
N
Th
is c
IN
+
v oc  VTh
VTh
RTh
a

b
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Maximum Power Transfer
Linear circuit
RL
What would be the value of RL for
power delivered to it become
MAXIMUM?
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Maximum Power Transfer
RTh
VTh
Linear circuit
RL
What would be the value of RL for
power delivered to it become
MAXIMUM?
2
  RL  
 VTh 
 
2

 VTh 
 R Th  R L  

 R L
PL 
 
RL
 R Th  R L 
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Maximum Power Transfer
Maximum power
2.4
p
Rl=linspace(1,60,500);
Vth=10;
Rth=12;
p=((Vth./(Rl+Rth)).^2).*Rl;
2.2
2
1.8
1.6
2
  R L 1.4
plot(Rl,p,'r');  V 
1.2
2
Th 

grid;
 VTh 
 R Th  R L  1

 R L
PL 
 
0.8
RL
 R Th  R L 
0.6
0.4
RL = 12 
0
10
20
30
40
50
60
RL
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Maximum Power Transfer
2
  RL  
 VTh 


2



R

R
 VTh 
L 
 Th

 R L
PL 
 
RL
 R Th  R L 
Mathematically, we evaluate RL when
dPL
0
dRL
dPL
 2VTh2
VTh2

R 
0
3 L
2
dRL (R Th  RL )
(R Th  RL )
dPL VTh2  2R L  R Th  R L 

0
3
dRL
(R Th  R L )
 RL  RTh
27
Using PSpice to verify Norton’s and Thevenin’s Theorems
Find Thevenin equivalent at terminals a-b
28
Using PSpice to verify Norton’s and Thevenin’s Theorems
29
Using PSpice to verify Norton’s and Thevenin’s Theorems
+
-
E
+
-
E2
R9
R8
2
2
I3
1Aac
TR AN =
0
R6
R7
4
6
0
30
Using PSpice to verify Norton’s and Thevenin’s Theorems
+
-
E
+
-
E2
R9
R8
2
2
I3
1Aac
TR AN =
0
R6
R7
4
6
0
31
Using PSpice to verify Norton’s and Thevenin’s Theorems
+
-
E
+
-
E2
R9
R8
2
2
I3
1 Aa c
TR AN =
0
I4
R6
R7
4
6
1 Aa c
TR AN =
1
0
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Using PSpice to verify Norton’s and Thevenin’s Theorems
+
-
E
RTh = 6/1 = 6
+
-
1.333V
E2
4.000V
R9
R8
2
2
6.000V
I3
1Aac
TRAN =
0
I4
R6
R7
4
6
1Aac
TRAN =
1
0
33
Using PSpice to verify Norton’s and Thevenin’s Theorems
+
-
E
VTh = 20V
+
-
6.667V
E2
20.00V
R9
R8
2
2
20.00V
I3
1Aac
TRAN =
5
I4
R6
R7
4
6
1Aac
TRAN =
0
0
34