Post on 18-Dec-2015
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TECHNIQUES OF DC CIRCUIT ANALYSIS:TECHNIQUES OF DC CIRCUIT ANALYSIS:Superposition PrincipleSuperposition Principle
Thevenin’s TheoremThevenin’s Theorem
Norton’s TheoremNorton’s Theorem
Source TransformationSource Transformation
Maximum Power TransferMaximum Power Transfer
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• Applies only for LINEAR CIRCUIT
Circuit containing linear elements, linear dependent and independent sources
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 KILLED
• voltage sources are shorted
• current sources are opened
Dependent sources CANNOT be killed !
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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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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
Case 1Case 1 Network with NO dependent sources
• Kill all the independent sources
• Find the equivalent R looking between the terminals
Linear circuit – independent sources killed
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
Case 2Case 2 Network with dependent sources
• Kill 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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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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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