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Chapter 4
Circuit Theorems
Basic Theory of Circuits, SJTU
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Linearity
Substitution
Superposition
Maximum
Power Transfer
Tellegen
Theorems
Source
transformation
Reciprocity
Thevenin’s
Source
Transfer
Norton’s
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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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Example 4.2
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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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V  V 1  V 2; i  i1  i 2
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Fig. 4.6 For Example 4.3
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Substitution Theorem
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Substitution Theorem
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Substitution Theorem
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Substitution Theorem
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Substitution Theorem

If the voltage across and current through a
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
OR
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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 (accompanied voltage
source), 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
+
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.
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• 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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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

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
a
RL
b
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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
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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
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Example
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Reciprocity Theorem
2
6
3
2
6
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.
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Reciprocity Theorem
(only applicable to reciprocity networks)
Case 2
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Reciprocity Theorem
(only applicable to reciprocity networks)
Case 3
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Example
All resistors are 1 , find out i.
+ E --
i
+ E -i
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Source Transfer Property
• Voltage source transfer
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Source Transfer
• Current source transfer
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Summary
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Linearity Property
Superposition
Source
Transformation
Substitution
Theorem
Thevenin’s
Theorem
Norton’s Theorem

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


Maximum Power
Transfer
Tellegen Theorem
Inference of
Tellegen Theorem
Reciprocity
Theorem
Source Transfer
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