Circuit Theorems

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Circuit Theorems

• Dr. Mustafa Kemal Uyguroglu

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Circuit Theorems Overview

• Introduction
• Linearity
• Superpositions
• Source Transformation
• Thévenin and Norton Equivalents
• Maximum Power Transfer

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INTRODUCTION
A large complex circuits
Simplify circuit analysis
Circuit Theorems
?Thevenins theorem ? Norton theorem ?Circuit
linearity ? Superposition ?source
transformation ? max. power transfer
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Linearity Property

• A linear element or circuit satisfies the
  properties of
• Additivity requires that the response to a sum
  of inputs is the sum of the responses to each
  input applied separately.
• If v1 i1R and v2 i2R
• then applying (i1 i2)
• v (i1 i2) R i1R i2R v1 v2

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Linearity Property

•  
• Homogeneity    
• If you multiply the input (i.e. current) by some
  constant K, then the output response (voltage) is
  scaled by the same constant.
• If v1 i1R then K v1 K i1R

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Linearity Property

• A linear circuit is one whose output is linearly
  related (or directly proportional) to its input.

Suppose vs 10 V gives i 2 A. According to the
linearity principle, vs 5 V will give i 1 A.
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Linearity Property - Example
Solve for v0 and i0 as a function of Vs
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Linearity Property Example (continued)
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Linearity Property - Example
Ladder Circuit
3 A
5 A
1 A
3 V -
6 V -
2 A
2 A
5 V -
8V -
14 V -
This shows that assuming I0 1 A gives Is 5 A
the actual source current of 15 A will give I0
3 A as the actual value.
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Superposition

• The superposition principle states that the
  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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Steps to apply superposition principle

• Turn off all independent sources except one
  source. Find the output (voltage or current) due
  to that active source using nodal or mesh
  analysis.
• Turn off voltages sources short voltage
  sources make it equal to zero voltage
• Turn off current sources open current sources
  make it equal to zero current
• 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.
• Dependent sources are left intact.

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Superposition - Problem
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2mA Source Contribution
I0 -4/3 mA
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4mA Source Contribution
I0 0
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12V Source Contribution
I0 -4 mA
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Final Result
I0 -4/3 mA I0 0 I0 -4 mA I0 I0
I0 I0 -16/3 mA
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Example

• find v using superposition

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one independent source at a time, dependent
source remains
KCL i i1 i2 Ohm's law i v1 / 1
v1 KVL 5 i (1 1) i2(2) KVL 5 i(1 1)
i1(2) 2v1 10 i(4) (i1i2)(2) 2v1 10
v1(4) v1(2) 2v1 v1 10/8 V

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Consider the other independent source
KCL i i1 i2 KVL i(1 1) i2(2) 5
0i2(2) 5 i1(2) 2v2Ohm's law i(1)
v2v2(2) i2(2) 5 0 gt i2 -(52v2)/2i2(2)
5 i1(2) 2v2-2v2 (i - i2)(2) 2v2-2v2
v2 (52v2)/2(2) 2v2-4v2 2v2 5
2v2-8v2 5 gt v2 - 5/8 V from
superposition v -5/8 10/8 v 5/8 V
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Source Transformation

• 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

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Source Transformation
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Source Transformation
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Source Transformation

• Equivalent sources can be used to simplify the
  analysis of some circuits.
• A voltage source in series with a resistor is
  transformed into a current source in parallel
  with a resistor.
• A current source in parallel with a resistor is
  transformed into a voltage source in series with
  a resistor.

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Example 4.6

• Use source transformation to find vo in the
  circuit in Fig 4.17.

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Example 4.6
Fig 4.18
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Example 4.6

• we use current division in Fig.4.18(c) to get
• and

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Example 4.7

• Find vx in Fig.4.20 using source transformation

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Example 4.7

• Applying KVL around the loop in Fig 4.21(b) gives
• 
  (4.7.1)
• Appling KVL to the loop containing only the 3V
  voltage source, the resistor, and vx yields
  
• 
  (4.7.2)

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Example 4.7

• Substituting this into Eq.(4.7.1), we obtain
• Alternatively
• thus

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Thevenins Theorem

• Any circuit with sources (dependent and/or
  independent) and resistors can be replaced by an
  equivalent circuit containing a single voltage
  source and a single resistor.
• Thevenins theorem implies that we can replace
  arbitrarily complicated networks with simple
  networks for purposes of analysis.

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Implications

• We use Thevenins theorem to justify the concept
  of input and output resistance for amplifier
  circuits.
• We model transducers as equivalent sources and
  resistances.
• We model stereo speakers as an equivalent
  resistance.

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Independent Sources (Thevenin)
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No Independent Sources
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Introduction

• Any Thevenin equivalent circuit is in turn
  equivalent to a current source in parallel with a
  resistor source transformation.
• A current source in parallel with a resistor is
  called a Norton equivalent circuit.
• Finding a Norton equivalent circuit requires
  essentially the same process as finding a
  Thevenin equivalent circuit.

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Computing Thevenin Equivalent

• Basic steps to determining Thevenin equivalent
  are
• Find voc
• Find RTh

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Thevenin/Norton Analysis

• 1. Pick a good breaking point in the circuit
  (cannot split a dependent source and its control
  variable).
• 2. Thevenin Compute the open circuit voltage,
  VOC.
• Norton Compute the short circuit current,
  ISC.
• For case 3(b) both VOC0 and ISC0 so skip step
  2

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Thevenin/Norton Analysis

• 3. Compute the Thevenin equivalent resistance,
  RTh
• (a) If there are only independent sources, then
  short circuit all the voltage sources and open
  circuit the current sources (just like
  superposition).
• (b) If there are only dependent sources,
  then must use a test voltage or current source in
  order to calculate
• RTh VTest/Itest
• (c) If there are both independent and
  dependent sources, then compute RTh from VOC/ISC.

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Thevenin/Norton Analysis

• 4. Thevenin Replace circuit with VOC in series
  with RTh
• Norton Replace circuit with ISC in parallel
  with RTh
• Note for 3(b) the equivalent network is merely
  RTh , that is, no voltage (or current) source.
• Only steps 2 4 differ from Thevenin Norton!