SJTU 1
Chapter 4
Circuit Theorems
SJTU 2
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.
SJTU 3
Fig. 4.4 For Example 4.2
.1,5
;3,15
AIothenAIsif
AIothenAIsif
SJTU 4
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.
SJTU 5
• 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.
2. Repeat step 1 for each of the other independent sources.
3. Find the total contribution by adding algebraically all the contributions due to the independent sources.
Superposition(2)
SJTU 6
j
R1V
e +
-L N
i
j
+
-V1
R1
i1
L N
i2
-L N
+
V2
eR1
21;21 iiiVVV
SJTU 7
Fig. 4.6 For Example 4.3
21 vvv
Vv
VvVv
10
82;21
SJTU 8
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
SJTU 9
Source Transformation(2)
• It also applies to dependent sources:
SJTU 10
Fig. 4.17 for Example, find out Vo
SJTU 11
So, we get vo=3.2V
SJTU 12
7
2A
6V2A
I
Example: find out I (use source transformation )
AI 5.0
SJTU 13
Substitution Theorem
20V
+6 I2
4V
4
-
8
I3
V3
I1
20V8V
-
I3
V3
6
I1+
8I2
V3
6+
-20V
I1
1A
8
I3
I2
I1=2A, I2=1A, I3=1A, V3=8V
I1=2A, I2=1A, I3=1A, V3=8V
I1=2A, I2=1A, I3=1A, V3=8V
SJTU 14
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.
SJTU 15
Substitution Theorem
N1N N2Vs+
-
Is
Vs
NN1 OR NIs
N1
SJTU 16
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.
SJTU 17
(a) original circuit, (b) the Thevenin equivalent circuit
d
c
SJTU 18
LN LOADV+
-
I
Voc
-
+
LNIs
-
LNo RoI+
+
LN
I
V I
-
+
V=Voc-RoI
Simple Proof by figures
SJTU 19
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.
SJTU 20
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
SJTU 21
• Case 2 If the network has dependent sources, there are two methods to get Rth:
2. Calculate the open-circuit voltage Voc and short-circuit current Isc at the terminal of the original circuit, then Rth=Voc/Isc
Thevenin’s Theorem
VocCircuit
-
+OriginalIsc
Circuit
Original
Rth=Voc/Isc
SJTU 22
Examples
SJTU 23
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.
SJTU 24
(a) Original circuit, (b) Norton equivalent circuit
d(c)
N
SJTU 25
Examples
SJTU 26
Maximum Power Transfer
LN V+
-
I
RL
a
b
Replacing the original network by its Thevenin equivalent, then the power delivered to the load is
LLTh
ThL R
RR
VRip 22 )(
SJTU 27
Power delivered to the load as a function of RL
Th
ThThL
LThTh
L
R
VpandRRyieldsso
RLRTh
RRT
dR
dp
4
0
2
32
We can confirm that is the maximum power by showing that
02
2
LdR
pd
SJTU 28
• If the load RL is invariable, and RTh is variable, then what should RTh be to make RL get maximum power?
Maximum Power Transfer(several questions)
• 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?
SJTU 29
Examples
SJTU 30
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
kkkiu
1
0
SJTU 31
Inference of Tellegen Theorem
• If two lumped circuits and have the same topological graph with b branches, and the voltage, current of each
branch apply passive sign convention, then we have not only
N̂N
0ˆ0ˆ
0ˆˆ0
11
11
k
b
kk
b
kkk
k
b
kk
b
kkk
iuiualsobut
iuiu
SJTU 32
Example
.,3
,10,4;2,2
,6,2.
21
1221
12
thenVoutfindAIget
canWeVVRWhenVVAIgetcanWe
VVRWhenonlyresistorsincludingnetworkaisN
I1V1
I2
V2NR2
+
-
b
kkk
b
kkk IVIVIVIVIVIV
TheoremTellegenthetoAccording
32211
32211 0;0
b
k
b
kkkkk
kkkkkkkk
IVIV
IVIIRIRIIVand
3 3
VV
VV
IVIVIVIV
42
2)2(10
42)3(6
2
22
22112211
SJTU 33
Reciprocity Theorem
R3
R1
I2Vs
R2
4V 2 3
6 R3
R1
I2Vs
R2
4V3 6 2
AI3
12 AI
3
12
SJTU 34
• 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.
Reciprocity Theorem(only applicable to reciprocity networks)
N I2Vs
N Vs'I1'Vs
I
Vs
Iexistsactually
IIthenVsVsif
2
'
'1:
2'1'
SJTU 35
Reciprocity Theorem(only applicable to reciprocity networks)
V2N +
-
Is
Is'+ N-V1'
Case 2
Is
V
Is
Vexistsactually
VVthenIsIsif
2
'
'1:
2'1'
SJTU 36
Reciprocity Theorem(only applicable to reciprocity networks)
Case 3
V2N
Vs
+
-
Is'NI1'
Vs
V
Is
Iexistsactually
VIthenIsVsif
2
'
'1:
2'1'
SJTU 37
example
SJTU 38
Source Transfer• Voltage source transfer
VsR1
R4R2
R5R3
R1
R2
R3
R4Vs
Vs R5
An isolate voltage source can then be transferred to a voltage source in series with a resistor.
SJTU 39
Source Transfer• Current source transfer
B
R1 R4
R2
Is
C R3
A
CR2 R3
R4
Is
B
Is
AR1
Examples
SJTU 40
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