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DEE1213
ELECTRICAL TECHNOLOGY B
Lecture #2
Advanced Network Theorems (Part 2)
1
Subject Learning Outcome (SLO)
• This Lecture partially contributing to the
fulfillment of the following SLO:
– Use circuit theory to analyze the three phase
power system.
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Lecture Learning Outcomes
Upon completion of this lecture, you will be able to:
• Apply Thevenin’s and Norton’s Theorems to solve complex
circuitry problems
• Understand the Maximum Power Transfer Theorem
3
Review
• Determine the current through the 2 Ω
resistor of the network below. (1 A)
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Electrical Technology B
Thevenin’s Theorem
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THEVENIN’S THEOREM
• Consider the following:
• For purposes of discussion, at this point, we
consider that both networks are composed of
resistors and independent voltage and current
sources
8
Network
1
Network
2
•
•
A
B
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THEVENIN’S THEOREM
• Suppose Network 2 is detached from Network 1
and we focus temporarily only on Network 1.
• Network 1 can be as complicated in structure as
one can imagine. Maybe 45 meshes, 387
resistors, 91 voltage sources and 39 current
sources.
9
Network
1
•
•
A
B
Thevenin Equivalent Voltage, VTH
• Now place a voltmeter across terminals A-B and read
the voltage. We call this the open-circuit voltage.
• No matter how complicated Network 1 is, we read
one voltage. It is either positive at A, (with respect
to B) or negative at A.
• We call this voltage Vos and we also call it VTHEVENIN =
VTH
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Network
1
•
•
A
B
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Deactivate Independent Sources
• We now deactivate all sources of Network 1.
• To deactivate a voltage source, we remove the
source and replace it with a short circuit.
• To deactivate a current source, we remove the
source (Open).
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Example
12
+
_
+
+
_ _
A
B
V1
I2
V2
I1
V3
R1
R2
R3
R4
R1
R2
R3
R4
A
B
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Thevenin Equivalent Resistance, RTH
• The total resistance across the terminals A and
B is called RTHEVENIN and shorten this to RTH.
• For the circuit shown above,
RTH = R4 // [ R3 + (R1 // R2) ]
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R1
R2
R3
R4
A
B
Thevenin Equivalent Circuit
• We had obtained the VTH and RTH. Now, we can replace
Network 1 with the following network.
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VTH
RTH
A
B
+
_
A
B
Network
2
VTH
RTH
+
_
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Example 1
• Find VX by first finding VTH and RTH to the left
of A-B.
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12 4
6 2 VX
30 V +
_
+
_
A
B
Example 1: VTH
• First remove everything to the right of A-B and
determine VTH.
16
12 4
6
30 V +
_
A
B
(30)(6)
10
6 12
AB
V V
Notice that there is no current flowing in the 4 resistor (A-B) is
open. Thus there can be no voltage across the resistor.
12 4
6 2 VX
30 V +
_
+
_
A
B
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Example 1: RTH
• We now deactivate the sources to the left of
A-B and find the resistance seen looking in
these terminals.
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12 4
6
A
B
RTH
RTH = 12||6 + 4 = 8
12 4
6 2 VX
30 V +
_
+
_
A
B
Example 1: Thevenin Equivalent Circuit
• After having found the Thevenin circuit, we
connect this to the load in order to find VX.
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8
10 V
VTH
RTH
2 VX
+
_
+
_
A
B
10 2
2
2 8
( )( )
X
V V
12 4
6 2 VX
30 V +
_
+
_
A
B
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Example 2
• Determine the current flows through the 17 Ω
resistor using Thevenin’s Theorem.
19
+
_
20 V
5
20
10
17
1.5 A
A
B
• We first find VTH with the 17 resistor removed.
• Next we find RTH by looking into terminals A-B with the
sources deactivated.
Example 2: VTH
20
+
_
20 V
5
20
10
1.5 A
A
B
20(20)
(1.5)(10)
(20 5)
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OS AB TH
TH
V V V
V V
+
_
20 V
5
20
10
17
1.5 A
A
B
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Example 2: RTH
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5
20
10
A
B
5(20)
10 14
(5 20)
TH
R
+
_
20 V
5
20
10
17
1.5 A
A
B
Example 2: Thevenin Equivalent Circuit
22
+
_
20 V
5
20
10
17
1.5 A
A
B
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31 V
VTH
RTH
17 VAB
+
_
+
_
A
B
17
AB
V V
1A
17
V
17
17
AB
AB
AB
I
V
I
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Electrical Technology B
Norton’s Theorem
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Norton’s Theorem
• Assume that the network enclosed below is
composed of independent sources and resistors.
• Norton’s Theorem states that this network can be
replaced by a current source shunted by a resistance
R.
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Network I R
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Norton’s Theorem: RN
• In the Norton circuit, the current source is the short
circuit current of the network, that is, the current
obtained by shorting the output of the network. The
resistance is the resistance seen looking into the
network with all sources deactivated. This is the
same as RTH.
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ISS RN = RTH
Example
• Find the Norton equivalent circuit to the left
of terminals A-B for the network shown below.
Connect the Norton equivalent circuit to the
load and find the current in the 50 resistor.
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+
_
20
60
40
50
10 A
50 V
A
B
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Example: ISS
27
+
_
20
60
40
10 A
50 V
ISS
+
_
20
60
40
50
10 A
50 V
A
B
10.7
SS
I A
Replaced by
“SHORT”
Example: RN
28
+
_
20
60
40
50
10 A
50 V
A
B
55
N
R
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Example: Norton Equivalent Circuit
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10.7 A 55 50
+
_
20
60
40
50
10 A
50 V
A
B
𝐴 = 𝜋𝑟2
𝑰𝟓𝟎𝛀 = [55/(55+50)]10.7 = 5.6 A
Electrical Technology B
Maximum Power Transfer Theorem
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Maximum Power Transfer Theorem
• The maximum power transfer theorem states
the following:
– A load will receive maximum power from a linear
bilateral dc network when its total resistive value
is exactly equal to the Thévenin resistance (RTH) of
the network as “seen” by the load.
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Maximum Power Transfer Theorem
• For the Thévenin equivalent circuit, maximum
power will be delivered to the load when
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Applications
• One of the most common applications of the
maximum power transfer theorem is to
speaker systems.
• An audio amplifier (amplifier with a frequency
range matching the typical range of the
human ear) with an output impedance of 8 Ω.
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Applications
