Download Topic 4 Network Theorem

Faculty of Engineering and
Technical Studies
CIRCUIT THEORY
TUTORIAL 2 –
Topic 4: Network Theorem
Wei Wen Shyang
EBEC3103 Circuit Theory
Jan 2005
1
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Objectives
At the end of this topic, you should be able to:
 apply the superposition theorem for circuit analysis
 apply Thevenin’s theorem to simplify the circuit for
analysis
 apply Norton’s theorem to simplify the circuit for analysis
 understand maximum power transfer and perform circuit
conversion
2
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Superposition Theorem
• The Superposition theorem states that if a linear system
is driven by more than one independent power source,
the total response is the sum of the individual
responses. The following example will show the step of
finding branches current using superpostion theorem
3
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Refer to the Figure 4.1, determine the branches current using superposition
theorem.
120 V
6
2
i1
i3
i2
3
i4
4
12 A
Figure 4.1
Solution
• The application of the superposition theorem is shown in Figure
4.1, where it is used to calculate the branch current. We begin by
calculating the branch current caused by the voltage source of 120
V. By substituting the ideal current with open circuit, we
deactivate the current source, as shown in Figure 4.2.
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
6
120 V
i'1
2
v1
i'2
3
i'3
i'4
4
Figure 4.2
• To calculate the branch current, the node voltage across the
3Ω resistor must be known. Therefore
v 1  120 v 1
v1
=0


6
3 24
where v1 = 30 V
The equations for the current in each branch,
5
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
120  30
= 15 A
i'1 =
6
30
i'2 =
= 10 A
3
30
i'3 = i'4 =
=5A
6
In order to calculate the current cause by the current source, we
deactivate the ideal voltage source with a short circuit, as shown
2
6
i 1"
i 2"
3
i 3"
i4"
4
12 A
6
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
• To determine the branch current, solve the node voltages across the
3Ω dan 4Ω resistors as shown in Figure 4.4.
2
6
+
v3
+
3
v4
-
4
12 A
-
The two node voltages are
v3 v3 v3  v4
 
3 6
2
=0
v4  v3 v4

 12 = 0
2
4
7
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
• By solving these equations, we obtain
• v3 = -12 V
• v4 = -24 V
Now we can find the branches current,
8
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
To find the actual current of the circuit, add the currents due
to both the current and voltage source,
9
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Thevenin’s Theorem
• Thevenin and Norton equivalent circuits are techniques use
to simplify circuits. Figure 4.5 shows the Thevenin
equivalent circuit which represents any circuit from multiple
sources (dependent and independent) and resistors.
Rangkaian
berintangan yang
mengandungi
punca-punca
bergantung dan tak
bergantung
RTh
a
a
VTh
b
Figure 4.5 (a) General circuit.
b
(b) Thevenin equivalent circuit
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Example
Refer to the Figure 4.6, find the Thevenin equivalent circuit.
4
5
+
25 V
20 
+
v1
3A
-
a
vab
-
b
Solution
• In order to find the Thevenin equivalent circuit for the circuit shown in
Figure 4.6, calculate the open circuit voltage, vab. Note that when the a, b
terminals are open, there is no current flow to 4Ω resistor. Therefore, the
voltage vab is the same as the voltage across the 3A current source,
labeled v1.
• To find the voltage v1, solve the equations for the singular node voltage.
By choosing the bottom right node as the reference node,
11
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
v 1  25 v 1

3  0
5
20
• By solving the equation, v1 = 32 V. Therefore, the Thevenin voltage Vth for
the circuit is 32 V.
• The next step is to short circuit the terminals and find the short circuit
current for the circuit shown in Figure 4.7. Note that the current is in the
same direction as the falling voltage at the terminal.
4 a
5
+
+
25 V
20 
v2
3A
Figure 4.7
vab
isc
b
12
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Current isc can be found if v2 is known. By using the bottom
right node as the reference node, the equationfor v2 becomes
By solving the above equation, v2 = 16 V. Therefore, the short circuit
current isc is
v 2  25 v 2
v

3 2  0
5
20
4
The Thevenin resistance RTh is
Figure 4.8 shows the Thevenin equivalent circuit for the Figure 4.6.
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Figure 4.8
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Norton’s Theorem
• The Norton equivalent circuit contains an independent
current source which is parallel to the Norton
equivalent resistance. It can be derived from the
Thevenin equivalent circuit by using source
transformation. Therefore, the Norton current is
equivalent to the short circuit current at the terminal
being studied, and Norton resistance is equivalent to
Thevenin resistance.
15
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Example 4.3
Derive the Thevenin and Norton equivalent circuits of Figure 4.6.
4
5
25 V
a
3A
20 
b
Solution
Step 1: Source transformation (The 25V voltage source is converted to a 5 A
current source.)
4
5A
5
20 
a
3A
b
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Step 2: Combination of parallel source and parallel resistance
4
8A
a
4
b
Step 3: Source transformation (combined serial resistance to produce the
Thevenin equivalent circuit.)
8
a
32 V
b
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
17
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Step 4: Source transformation (To produce the Norton equivalent circuit. The
current source is 4A (I = V/R = 32 V/8 ))
a
4A
8
b
Figure 4.9 Steps in deriving Thevenin and Norton equivalent circuits.
18
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Maximum Power Transfer
• Maximum power transfer can be illustrated by Figure
4.10. Assume that a resistance network contains
independent and dependent sources, and terminals a
and b to which the resistance RL is connected. Then
determine the value of RL that allows the delivery of
maximum power to the load resistor.
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Resistance network
which contains
dependent and
independent
sources
Figure 4.10
20
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
• Maximum power transfer happens when the load resistance
RL is equal to the Thevenin equivalent resistance, RTh. To
find the maximum power delivered to RL,
2
2
pmax =
VTh R L
2R L 
2
=
VTh
4R L
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Circuit Transformation
• The configuration of circuit connection can be changed to
make the calculation easier. There are TWO type of
transformations which are Delta () to star connection ()
and vice versa.
a
Ra
R1
R2
Rc
c
R3
Rb
b
Figure 4.12 Delta and Star Circuit Connection
22
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
• Delta () to star (Y) transformation:
R1 R2
Ra 
R 1  R2  R3
R2 R3
Rb 
R 1  R2  R3
R1 R3
Rc 
R 1  R2  R3
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
• Star (Y) to Delta () transformation:
Ra Rb  RbRc  RcRa
R1 
Rb
Ra Rb  RbRc  RcRa
R2 
Rc
Ra Rb  RbRc  RcRa
R3 
Ra
24
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Faculty of Engineering and
Technical Studies
Thank You
25
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia