Transcript Chapter 4 Circuit Theorems

Chapter 4
Circuit Theorems
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Linearity Property
• Linearity is the property of an element
describing a linear relationship between
cause and effect.
• A linear circuit is one whose output is
linearly ( or directly proportional) to its
input.
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Fig. 4.4 For Example 4.2
if
if
Is  15 A, then Io  3 A;
Is  5 A, then Io  1A.
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Superposition(1)
• The superposition principle states that
voltage across (or current through) an
element in a linear circuit is the algebraic
sum of the voltages across (or currents
through) that element due to each
independent source acting alone.
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Superposition(2)
•
Steps to Apply Superposition Principle:
1.
Turn off all independent source except one source. Find
the output(voltage or current) due to that active source
using nodal or mesh analysis.
Repeat step 1 for each of the other independent sources.
Find the total contribution by adding algebraically all the
contributions due to the independent sources.
2.
3.
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j
i
e
+
L N
R1
V
-
j
i1
L N
+
i2
e
R1
V1
-
L N
+
R1
V2
-
V  V 1  V 2; i  i1  i 2
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Fig. 4.6 For Example 4.3
v  v1  v2
v1  2V ; v 2  8V
 v  10V
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Source Transformation(1)
• A source transformation is the process of replacing
a voltage source Vs in series with a resistor R by a
current source is in parallel with a resistor R, or
vice versa. Vs=isR or is=Vs/R
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Source Transformation(2)
• It also applies to dependent sources:
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Fig. 4.17 for Example, find out Vo
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So, we get vo=3.2V
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Example: find out I (use
source transformation )
2A
I
2A
6V
7
I  0.5 A
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Substitution Theorem
I1
6
I3
+
I2
8
V3
20V
-
I1=2A,
I2=1A,
I3=1A,
V3=8V
I1
4
6
4V
20V
I1
I3
+
I2
8
V3
6
8V
I3
+
I2
8
V3
1A
20V
-
-
I1=2A,
I2=1A,
I3=1A,
V3=8V
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I1=2A,
I2=1A,
I3=1A,
V3=8V
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Substitution Theorem
• If the voltage across and current through
any branch of a dc bilateral network are
known, this branch can be replaced by any
combination of elements that will maintain
the same voltage across and current through
the chosen branch.
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Substitution Theorem
Is
+
Vs
N
N1
N2
-
Vs
N1
Is
OR
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N1
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Thevenin’s Theorem
• A linear two-terminal circuit can be
replaced by an equivalent circuit consisting
of a voltage source Vth in series with a
resistor Rth, where Vth is the open-circuit
voltage at the terminals and Rth is the input
or equivalent resistance at the terminals
when the independent source are turned off.
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(a) original circuit, (b) the Thevenin equivalent circuit
c
d
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Simple Proof by figures
I
+
V
LN
LOAD
LN
-
LN
I
+
V
I
-
+
Voc
LNo
+
+
RoI
Is
-
-
V=Voc-RoI
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Thevenin’s Theorem
Consider 2 cases in finding Rth:
•
Case 1 If the network has no dependent sources, just
turn off all independent sources, calculate the
equivalent resistance of those resistors left.
•
Case 2 If the network has dependent sources, there
are two methods to get Rth:
1.
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Thevenin’s Theorem
•
Case 2 If the network has dependent sources, there are
two methods to get Rth:
1. Turn off all the independent sources, apply a voltage
source v0 (or current source i0) at terminals a and b and
determine the resulting current i0 (or resulting voltage
v0), then Rth= v0/ i0
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Thevenin’s Theorem
•
2.
Case 2 If the network has dependent sources, there are
two methods to get Rth:
Calculate the open-circuit voltage Voc and short-circuit
current Isc at the terminal of the original circuit, then
Rth=Voc/Isc
Original
+
Voc
Circuit
Original
Isc
Circuit
Rth=Voc/Isc
-
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Examples
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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 in parallel with a
resistor RN, where IN is the short-circuit
current through the terminals and RN is the
input or equivalent resistance at the
terminals when the independent sources are
turned off.
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(a) Original circuit, (b) Norton equivalent circuit
N
(c)
d
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Examples
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Maximum Power Transfer
Replacing the original network by its Thevenin equivalent, then the power
delivered to the load is
V
p  i 2 RL  (
LN
I
+
V
Th
RTh  RL
) 2 RL
a
RL
b
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Power delivered to the load as a function of RL
dp
2  RTh  RL  
 TTh 
0
3
dRL
 RTh  RL  
so
yields
RL  RTh
and
VTh2
p
4 RTh
We can confirm that is the maximum
2
power by showing that
d p
0
2
dRL
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Maximum Power Transfer
(several questions)
•
If the load RL is invariable, and RTh is variable,
then what should RTh be to make RL get
maximum power?
• If using Norton equivalent to replace the original
circuit, under what condition does the maximum
transfer occur?
• Is it true that the efficiency of the power transfer
is always 50% when the maximum power transfer
occurs?
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Examples
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Tellegen Theorem
• If there are b branches in a lumped circuit,
and the voltage uk, current ik of each branch
apply passive sign convention, then we have
b
u
k 1
i 0
k k
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Inference of Tellegen Theorem
• If two lumped circuits N and N̂ have the same topological
graph with b branches, and the voltage, current of each
branch apply passive sign convention, then we have not
only
b
u i
k 1
k k
b
 uˆ iˆ
0
k 1
b
but also
 uˆ i
k 1
k k
k k
0
0
b
 u iˆ
k 1
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k k
0
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Example
N is a network including resistors only. When R2  2, V1  6V ,
We can get I1  2 A, V2  2V ; When R2  4, V1  10V , We can
get I1  3 A, find out V2 then.
I1
N
V1
According to the Tellegen Theorem
I2
R2
+
V2
-

b
V I   V I
k 3
k k
k 3
k k
b
k 3
k 3
V1 I1  V2 I 2  Vk I k  0 ; V1I1  V2I 2   VkI k  0
and Vk I k  RI k I k  RI k I k  VkI k
b
b
 V1 I1  V2 I 2  V1I1  V2I 2
V2
2
6  (3)  2   10  (2)  V2 
4
2
 V2  4V
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Reciprocity Theorem
R1
4V
Vs
2
R2
6
R3
3
I2
I2
1
I2  A
3
R1
R2
2
6
R3
3
Vs
4V
1
I2  A
3
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Reciprocity Theorem
(only applicable to reciprocity networks)
• Case 1
The current in any branch of a network, due to a single
voltage source E anywhere else in the network, will equal the
current through the branch in which the source was originally
located if the source is placed in the branch in which the current I
was originally measured.
N
Vs
I1'
N
I2
Vs  Vs' then I1'  I 2
I1' I 2
actually exists :

Vs' Vs
if
V s'
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Reciprocity Theorem
(only applicable to reciprocity networks)
Case 2
Is
N
Is  Is ' then V 1'  V 2
V 1' V 2
actually exists :

Is ' Is
if
+
V2
-
+
N
Is'
V1'
-
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Reciprocity Theorem
(only applicable to reciprocity networks)
Case 3
N
if
V2
-
Vs
I1'
Vs  Is ' then I1'  V 2
I1' V 2
actually exists :

Is ' Vs
+
N
Is'
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example
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Source Transfer
•
Voltage source transfer
R2
R2
R4
R1
Vs
Vs
R4
Vs
R5
R1
R5
R3
R3
An isolate voltage source can then be transferred to a
voltage source in series with a resistor.
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Source Transfer
•
Current source transfer
R2
R3
C
R2
C
R3
Is
Is
Is
B
R1
A
B
R4
R1
A
R4
Examples
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Summary
•
•
•
•
•
•
Linearity Property
Superposition
Source Transformation
Substitution Theorem
Thevenin’s Theorem
Norton’s Theorem
• Maximum Power
Transfer
• Tellegen Theorem
• Inference of Tellegen
Theorem
• Reciprocity Theorem
• Source Transfer
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