Circuit Theorems

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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
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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
• 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.

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Fig. 4.6 For Example 4.3
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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. VsisR or
  isVs/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 vo3.2V
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Example find out I (use source transformation )
I
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Substitution Theorem
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I12A, I21A, I31A, V38V
I12A, I21A, I31A, V38V
I12A, I21A, I31A, V38V
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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
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Thevenins 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
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Simple Proof by figures

VVoc-RoI
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Thevenins 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

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Thevenins Theorem

• Case 2 If the network has dependent sources,
  there are two methods to get Rth
• 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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Thevenins Theorem

• 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 RthVoc/Isc

RthVoc/Isc
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Examples
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Nortons 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
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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
a
b
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Power delivered to the load as a function of RL
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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 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

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Example
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Reciprocity Theorem
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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
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Source Transfer

• Voltage source transfer

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

Examples
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Summary

• Maximum Power Transfer
• Tellegen Theorem
• Inference of Tellegen Theorem
• Reciprocity Theorem
• Source Transfer

• Linearity Property
• Superposition
• Source Transformation
• Substitution Theorem
• Thevenins Theorem
• Nortons Theorem