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Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 11
Chapter 4Chapter 4
Circuit TheoremsCircuit Theorems
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 22
Theorems
Linearity
Superposition
Source transformation
Substitution
Thevenin’s Norton’s
Maximum Power Transfer
Tellegen
Reciprocity
Source Transfer
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 33
Linearity PropertyLinearity Property
Linearity is the property of an element Linearity is the property of an element describing a linear relationship between describing a linear relationship between cause and effect.cause and effect.
A linear circuit is one whose output is A linear circuit is one whose output is linearly ( or directly proportional) to its linearly ( or directly proportional) to its input.input.
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 44
Example 4.2
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 55
Superposition(1)Superposition(1)
The superposition principle states that The superposition principle states that voltage across (or current through) an voltage across (or current through) an element in a linear circuit is the algebraic element in a linear circuit is the algebraic sum of the voltages across (or currents sum of the voltages across (or currents through) that element due to each through) that element due to each independent source acting alone.independent source acting alone.
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 66
Steps to Apply Superposition Principle:Steps to Apply Superposition Principle:1.1. Turn off all independent source except one Turn off all independent source except one
source. Find the output(voltage or current) due source. Find the output(voltage or current) due to that active source using nodal or mesh to that active source using nodal or mesh analysis.analysis.
2.2. Repeat step 1 for each of the other Repeat step 1 for each of the other independent sources.independent sources.
3.3. Find the total contribution by adding Find the total contribution by adding algebraically all the contributions due to the algebraically all the contributions due to the independent sources.independent sources.
Superposition(2)Superposition(2)
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 77
j
R1V
e +
-L N
i
j
+
-V1
R1
i1
L N
i2
-L N
+
V2
eR1
21;21 iiiVVV
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 88
Fig. 4.6 For Example 4.3
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 99
Substitution TheoremSubstitution Theorem
20V
+6 I2
4V
4
-
8
I3
V3
I1
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1010
Substitution TheoremSubstitution Theorem
20V8V
-
I3
V3
6
I1+
8I2
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1111
Substitution TheoremSubstitution Theorem
V3
6+
-20V
I1
1A
8
I3
I2
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1212
Substitution TheoremSubstitution Theorem
20V
+6 I2
4V
4
-
8
I3
V3
I1
V3
6+
-20V
I1
1A
8
I3
I2
20V8V
-
I3
V3
6
I1+
8I2
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1313
Substitution Theorem Substitution Theorem
If the voltage across and current through a If the voltage across and current through a branch of a dc bilateral network are branch of a dc bilateral network are known, this branch can be replaced by any known, this branch can be replaced by any combination of elements that will maintain combination of elements that will maintain the same voltage across and current the same voltage across and current through the chosen branch.through the chosen branch.
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1414
Substitution TheoremSubstitution Theorem
N1N N2Vs+
-
Is
Vs
NN1 OR NIs
N1
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1515
Thevenin’s TheoremThevenin’s Theorem
A linear two-terminal circuit can be replaceA linear two-terminal circuit can be replaced by an equivalent circuit consisting of a vd by an equivalent circuit consisting of a voltage source Voltage source Vthth in series with a resistor R in series with a resistor R
th (accompanied voltage source), where Vth (accompanied voltage source), where Vtt
hh is the open-circuit voltage at the terminal is the open-circuit voltage at the terminal
s and Rs and Rthth is the input or equivalent resistan is the input or equivalent resistan
ce at the terminals when the independent ce at the terminals when the independent source are turned off.source are turned off.
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1616
(a) original circuit, (b) the Thevenin equivalent circuit
d
c
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1717
LN LOADV+
-
I
Voc
-
+
LNIs
-
LNo RoI+
+
LN
I
V I
-
+
V=Voc-RoI
Simple Proof by figures
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1818
Thevenin’s TheoremThevenin’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.
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 1919
• 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
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2020
Case 2 If the network has dependent sourcCase 2 If the network has dependent sources, there are two methods to get Rth:es, there are two methods to get Rth:
2. Calculate the open-circuit voltage V2. Calculate the open-circuit voltage Vococ and s and short-circuit current Ihort-circuit current Iscsc at the terminal of the at the terminal of the original circuit, then Rth=Voriginal circuit, then Rth=Vococ/I/Iscsc
VocCircuit
-
+OriginalIsc
Circuit
Original
Rth=Voc/Isc
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2121
Examples
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2222
Norton’s TheoremNorton’s Theorem
A linear two-terminal circuit can be A linear two-terminal circuit can be replaced by an equivalent circuit replaced by an equivalent circuit consisting of a current source Iconsisting of a current source INN in in parallel with a resistor Rparallel with a resistor RNN, where I, where INN is is the short-circuit current through the the short-circuit current through the terminals and Rterminals and RNN is the input or is the input or equivalent resistance at the terminals equivalent resistance at the terminals when the independent sources are when the independent sources are turned off.turned off.
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2323
(a) Original circuit, (b) Norton equivalent circuit
d(c)
N
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2424
Examples
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2525
Maximum Power TransferMaximum Power Transfer
LN V+
-
I
RL
a
b
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2626
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2727
If the load RIf the load RLL is invariable, and R is invariable, and RThTh is varia is variable, then what should Rble, then what should RThTh be to make R be to make RLL g get maximum power?et maximum power?
Maximum Power TransferMaximum Power Transfer(several questions)(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?
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2828
Examples
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 2929
Tellegen TheoremTellegen Theorem If there are b branches in a lumped circIf there are b branches in a lumped circ
uit, and the voltage uuit, and the voltage ukk, current i, current ikk of eac of each branch apply passive sign conventioh branch apply passive sign convention, then we have n, then we have
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3030
Inference of Tellegen TheoremInference of Tellegen Theorem If two lumped circuits and have the If two lumped circuits and have the
same topological graph with b branches, same topological graph with b branches, and the voltage, current of each branch and the voltage, current of each branch apply passive sign convention, then we apply passive sign convention, then we have not onlyhave not only
N̂N
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3131
Example
I1V1
I2
V2NR2
+
-
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3232
I1V1
I2
V2NR2
+
-
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3333
Reciprocity TheoremReciprocity Theorem
R3
R1
I2Vs
R2
4V2
3
6
R3
R1
I2Vs
R2
4V3
6 2
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3434
Case 1 Case 1 The current in any branch of a network, due to The current in any branch of a network, due to a single voltage source E anywhere else in the a single voltage source E anywhere else in the network, will equal the current through the branch in network, will equal the current through the branch in which the source was originally located if the source which the source was originally located if the source is placed in the branch in which the current I was is placed in the branch in which the current I was originally measured.originally measured.
Reciprocity TheoremReciprocity Theorem(only applicable to reciprocity networks)(only applicable to reciprocity networks)
N I2Vs
N Vs'I1'
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3535
Reciprocity TheoremReciprocity Theorem(only applicable to reciprocity networks)(only applicable to reciprocity networks)
V2N +
-
Is
Is'+ N-V1'
Case 2
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3636
Reciprocity TheoremReciprocity Theorem(only applicable to reciprocity networks)(only applicable to reciprocity networks)
Case 3
V2N
Vs
+
-
Is'NI1'
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3737
ExampleExampleAll resistors are 1 , find out i.
i
+ E
--
+ E --
i
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3838
Source Transfer PropertySource Transfer Property• Voltage source transfer
VsR1
R4R2
R5R3
R1
R2
R3
R4Vs
Vs R5
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 3939
Source TransferSource Transfer• Current source transfer
CR2 R3
R4
Is
B
Is
AR1
Basic Theory of Circuits, SJTUBasic Theory of Circuits, SJTU 4040
SummarySummary
Linearity PropertyLinearity Property SuperpositionSuperposition Source TransformaSource Transforma
tiontion Substitution TheorSubstitution Theor
emem Thevenin’s TheoreThevenin’s Theore
mm Norton’s TheoremNorton’s Theorem
Maximum Power TraMaximum Power Transfernsfer
Tellegen TheoremTellegen Theorem Inference of TellegeInference of Tellege
n Theoremn Theorem Reciprocity TheoreReciprocity Theore
mm Source TransferSource Transfer
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