ECE%201131%20-%20Electric%20Circuits

1
ECE 1131 - Electric Circuits

• Farah Diyana binti Abdul Rahman

Circuit Theorems
2
Circuit Theorems Overview

• Linearity
• Superposition
• Source Transformation
• Thévenin and Norton Equivalents
• Maximum Power Transfer

3
Introduction

• In this chapter we consider five circuit
  theorems
• A source transformation allows us to replace
  source and series resistor by a current source
  and parallel resistor and vice versa. Doing so
  does not change the element current or voltage of
  any other element of the circuit.
• Superposition says that the response of a linear
  circuit to several inputs working together us
  equal to the sum of the responses to each of the
  inputs working separately.
• Thevenin's theorem allows us to replace part of
  circuit by a voltage source and series resistor.
  Doing so does not change the element current or
  voltage of any other element of the circuit.
• Norton's theorem allows us to replace part of
  circuit by a current source and parallel
  resistor. Doing so does not change the element
  current or voltage of any other element of the
  circuit.
• The maximum power transfer theorem describes the
  condition under which one circuit transfers as
  much power as possible to another circuit.

4
Linearity Defined

• Given a function f(x)?
• y f(x)?
• y1 f(x1)?
• y2 f(x2)?
• the function f(x) is linear if and only if
• f(a1x1 a2x2) a1y1 a2y2
• for any two inputs x1 and x2 and any constants a1
  and a2

5
Example 1 Linearity Ohms Law

• Is Ohms law linear?
• v f(i)?
• iR
• v1 i1R
• v2 i2R
• f(a1i1 a2i2) (a1i1 a2i2 )R
• a1(i1R) a2 (i2 R)?
• a1v1 a2v2

6
Example 2 Linearity Ohms Law

• Is the power dissipated by a resistor a linear
  function of the current?
• p f(i)?
• i2R
• p1 i12R
• p2 i22R
• f(a1i1 a2i2) (a1i1 a2i2 ) 2R
• a12 i12R 2a1a2i1i2R a2
  2i22 R
• ? a1p1 a2p2

7
Linear Circuits

• A linear circuit is one whose output is linearly
  related (or directly proportional) to its input
• In this class we will only consider circuits in
  which the voltage and currents are linearly
  related to the independent sources
• For circuits, the inputs are represented by
  independent sources
• The current through and voltage across each
  circuit element is linearly proportional to the
  independent source amplitude
• Will focus on how to apply this principle

8
Example 3 Linearity Circuit Analysis

• Solve for vo as a function of Vs

9
Example 3 Linearity Circuit Analysis (cont.)?

• Is vo a linear function of Vs ?
• If we had solved the circuit for Vs 10V, could
  we find vo for Vs 20V without having to
  reanalyze the circuit?

10
Example 4 Linearity Circuit Analysis

• Solve for vo as a function of Vs and Is

11
Example 4 Linearity Circuit Analysis (cont.)?

• If Is 0, then vo is a linear function of Vs
• If Vs 0, then vo is a linear function of Is
• This holds true in general
• When used for circuit analysis, this is called
  superposition

12
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 current through) that element
  due to each independent source acting alone
• To apply this principle for analysis, we follow
  these steps
• Turn off all independent sources except one. Find
  the output (voltage or current) due to that
  source.
• Repeat Step 1 for each independent source.
• Add the contribution of each source to find the
  total output.

13
Example 5 Superposition

• Solve for vo using superposition
• First, find the contribution due to the 10V
  source.
• This means we must turn off the current source.
• How do you turn off a current source?
• What is the equivalent of turning off a current
  source.

14
Example 5 Superposition (cont.)?

• Solve for vo due to the 10V source
• Second, find the contribution due to the 2mA
  source.
• This means we must turn off the voltage source.
• How do you turn off a voltage source?
• What is the equivalent of turning off a voltage
  source.

15
Example 5 Superposition (cont.)?
Solve for vo due to the 2mA source
16
Example 5 Superposition (cont.)?

• Finally, solve for vo by adding the contributions
  due to both sources.
• What if the 10V source had been a 20V source. Is
  there an easy way to find vo in this case?

17
Example 6 Superposition

• Use the principle of superposition to find vo.
• We will find the contribution due to the 35V
  source first
• So we must first turn off the current source

18
Example 6 Superposition (cont.)?
Solve for vo with the 7mA source turned off
19
Example 6 Superposition (cont.)?
Solve for vo with the 35V source turned off
20
Superposition Review

• The superposition theorem states that a circuit
  can be analyzed with only one source of power at
  a time, the corresponding component voltages and
  currents algebraically added to find out what
  theyll do with all power sources in effect
• To negate all but one power source for analysis,
  replace any source of voltage with a wire
  replace any current source with an open (break)
  circuit

21
Superposition Final Remarks

• Superposition is based on circuit linearity
• Must analyze as many circuits as there are
  independent sources
• Dependent sources are never turned off
• As with the examples, is usually more work than
  combining resistors, the node voltage analysis,
  or mesh current analysis
• Is an important idea
• If you want to consider a range of values for an
  independent source, is sometimes easier than
  these methods
• Although multiple circuits must be analyzed, each
  simpler than the original because all but one of
  the independent sources is turned off
• Will be necessary when we discuss sinusoidal
  circuit analysis

22
Example 7
23
Example 7 (cont.)?
24
(No Transcript)
25
Example 7 (cont.)?
26
Example 7 (cont.)?
27
Example 8
28
Example 8 (cont.)?
29
Example 8 (cont.)?
30
Example 8 (cont.)?
31
Source Transformation Introduction

• Recall that we discussed how to combine networks
  of resistors to simplify circuit analysis
• Series combinations
• Parallel combinations
• Delta ? Wye Transformations
• We can also apply this idea to certain
  combinations of sources and resistors
• Why transforms circuits?
• Combine resistors values together
• Many circuits can be completely simplified into a
  circuit with a single resistor and a single
  source.

32
Source Transformation Concept

• Source Transformation The replacement of a
  voltage source in series with a resistor by a
  current source in parallel with a resistor or
  vice versa
• The two circuits are equivalent if they have the
  same current-voltage relationship at their
  terminals

33
Source Transformation Proof

• A two-terminal circuit element is defined by its
  voltage-current relationship
• Relationship can be found by applying a voltage
  source to the element and finding the
  relationship to current
• Equivalently, can apply a current source and find
  relationship to voltage

34
Source Transformation Proof (cont.)?
35
Source Transformation Dependent Sources

• Also works with dependent sources
• Arrow of the current source must point towards
  the positive terminal of the voltage source
• Does not work if R 0

36
Voltage Sources Resistor Series Equivalent

• Recall Voltage sources in series add
• Recall Resistors in series add
• Mixture of both in series also has an equivalent
• Equivalent voltage source Sum of the voltages
• Equivalent resistance Sum of the resistors
• Proof possible by KVL

37
Current Sources Resistors Parallel Equivalent

• Recall Current sources in parallel add
• Recall The conductance of resistors in parallel
  adds
• Mixture of both in parallel also has an
  equivalent
• Equivalent current source Sum of the currents
• Equivalent resistance Parallel combination
• Proof possible by KCL

38
Example 9 Source Transformation
Use source transformation to determine the
current and power in the 8 ? resistor
39
Example 9 Source Transformation (cont.)?
40
Example 9 Source Transformation (cont.)?
41
Example 10 Source Transformation
Use source transformation to find io in the
circuit
42
Example 10 Source Transformation (cont.)?
43
Example 11 Source Transformation
44
Thévenin's Theorem

• Thévenin's Theorem A linear two-terminal circuit
  is electrically equivalent to a voltage source in
  series with a resistor
• This applies to any two terminals in a circuit
• This is a surprising result
• Proof in textbook We will focus on how to apply
• Better model of physical power supplies like
  batteries

45
Norton's Theorem

• Norton's Theorem A linear two-terminal circuit
  is electrically equivalent to a current source in
  parallel with a resistor
• The Norton equivalent can be obtained by a source
  transformation of the Thévenin's equivalent and
  vice versa
• This implies RTh RN and VTh RThIN
• In lectures, I will denote RN and RTh as simply
  Req

46
Finding Thévenin Norton Equivalents

• Recall Two terminal circuits are only equivalent
  if they have the same voltage-current
  relationship
• This means regardless of what is connected to the
  terminals, all three devices must behave the same
• Consider
• Open-Circuit Voltage
• Short-Circuit Current
• This is sufficient, but there are two other
  methods

47
Finding Thévenin Norton Equivalents Resistance

• If we set all of the independent sources equal to
  zero in all three circuits, they should all have
  the same resistance
• With the dependent sources removed, it should be
  relatively easy to find the internal resistance
  of the circuit
• If the circuit has dependent sources, this can be
  tricky

48
Thévenin Norton Equivalents Resistance (cont.)?

• If the circuit has dependent sources, we need to
  find the voltage-current relationship for the
  circuit
• Easiest to hook up a voltage source (or current
  source) and calculate the current (or voltage)?
• The source can have any value
• Then Req V/I
• If the circuit has dependent sources, Req may be
  negative

49
Thévenin Norton Equivalents Summary

• To find Thevenin or Norton equivalent of a
  two-terminal circuit, must do two or three tasks
• Find the open-circuit voltage Voc
• Find the short-circuit current Isc
• Find the internal resistance Ri
• Then you can find the equivalent values by the
  following equations

50
Methods of Finding a Thevenin Equivalent Circuit

• If the circuit contains
• 1. Resistors and independent sources
• Connect an open circuit between terminals a and
  b. Find voc vab, the voltage across the open
  circuit.
• Deactivate the independent sources. Find Rth by
  circuit resistance reduction.

51
Methods of Finding a Thevenin Equivalent Circuit

• 2. Resistors and independent dependent sources
  or
• Resistors and independent sources
• Connect an open circuit between terminals a and
  b. Find voc vab, the voltage across the open
  circuit.
• Connect a short circuit between terminals a and
  b. Find isc, the current directed from a to b in
  the short circuit.
• Set all independent sources to zero, then connect
  1-A current source from terminal b to terminal a.
  Determine vab.
• Then Rth vab/1 or Rth voc/isc
• 3. Resistors and dependent sources (no
  independent sources)?
• Note that voc 0.
• connect 1-A current source from terminal b to
  terminal a. Determine vab.
• Then Rth vab/1

52
Thevenin's Theorem
The two most important source transformations,
Thevenin's Source, and Norton's Source,
Let's say that the source is a collection of
voltage sources, current sources and resistances,
while the load is a collection of resistances
only. Both the source and the load can be
arbitrarily complex, but we can conceptually say
that the source is directly equivalent to a
single voltage source and resistance (figure (a)
below).

Figure b
Figure a
We can determine the value of the resistance Rs
and the voltage source, vs by attaching an
independent source to the output of the circuit,
as in figure (b) above.
53
Thevenin's Theorem (cont.)?
In this case we are using a current source, but a
voltage source could also be used. By varying i
and measuring v, both vs and Rs can be found
using the following equation We can easily see
from this that if the current source is set to
zero (equivalent to an open circuit), then v is
equal to the voltage source, vs. This is also
called the open-circuit voltage, voc. This is an
important concept, because it allows us to model
what is inside a unknown (linear) circuit, just
by knowing what is coming out of the circuit.
This concept is known as Thévenin's Theorem after
French telegraph engineer, and the circuit
consisting of the voltage source and resistance
is called the Thévenin Equivalent Circuit.
54
Norton's Theorem
Recall from above that the output voltage, v, of
a Thévenin equivalent circuit can be expressed
as Now, let's rearrange it for the output
current, i This is equivalent to a KCL
description of the following circuit. We can call
the constant term vs/Rs the source current,
is. When the above circuit is disconnected
from the external load, the current from the
source all flows through the resistor, producing
the requisite voltage across the terminals, voc.
Also, if we were to short the two terminals of
our circuit, the current would all flow through
the wire, and none of it would flow through the
resistor (current divider rule). In this way, the
circuit would produce the short-circuit current
isc (which is exactly the same as the source
current is).
55
(No Transcript)
56
Open Circuit Voltage

• If the current flowing from a source is zero,
  then the source is connected to an open circuit
• The voltage at the source terminals with i(t)
  equal to zero is called the open circuit voltage
• voc(t)?

57
Short Circuit Current

• If the voltage across the source terminals is
  zero, then the source is connected to a short
  circuit
• The current that flows when v(t) equals zero is
  called the short circuit current
• isc(t)?

58
(No Transcript)
59
(No Transcript)
60
Open and Short Circuits
We can also observe the following The value of
the Thévenin voltage source is the open-circuit
voltage The value of the Norton current source
is the short-circuit current We can say that,
generally,
61
Example 12
62
Example 12 (cont.)?
63
Example 13
64
(No Transcript)
65
Example 14
66
(No Transcript)
67
Thevenin and Norton Equivalent Circuits
68
Thevenin and Norton Equivalent Circuits
69
Thevenin and Norton Equivalent Circuits
70
Thevenin and Norton Equivalent Circuits
71
Thevenin and Norton Equivalent Circuits
72
Thevenin and Norton Equivalent Circuits
73
Thevenin and Norton Equivalent Circuits
74
Thevenin and Norton Equivalent Circuits
75
Thevenin and Norton Equivalent Circuits
76
Norton's Equivalent Circuit

• Norton equivalent circuit is related to the
  Thevenin equivalent circuit by a source
  transformation
• If the circuit contains
• Resistors and independent sources
• Connect a short circuit between terminal a and b.
  Find isc, the current directed from a to b in the
  short circuit
• Deactivate the independent sources. Find Rn Rth
  by circuit deduction

77
Norton's Equivalent Circuit (cont.)?

• Resistors and independent and dependent sources
  or Resistors and independent sources
• Connect an open circuit between terminals a and
  b. Find voc vab, or the voltage across the open
  circuit
• Connect short circuit between terminal a and b.
  Find isc, the current directed from a to b in the
  short circuit
• Set all independent sources to zero, then connect
  a 1-A current source from terminal b to terminal
  a. Determine vab. Then Rn Rth vab/1
• Resistors and dependent sources (no independent
  sources)?
• Note that isc 0
• Connect a 1-A current source from terminal b to
  terminal a. Determine vab. Then Rn Rth vab/1

78
Norton's example
79
Norton's example (cont.)?
80
Norton's bonus question
81
Maximum Power Transfer

• What load resistance RL will maximize the power
  absorbed by the resistor?

The maximum power transfer theorem states that
the maximum power delivered to a load by a source
is attained when the load resistance, RL, is
equal to the Thevenin resistance, Rth, of the
source
82
Maximum Power Transfer Derivation
Goal Find the value of RL that maximizes the
power absorbed by RL This can only be
true if RL Req
83
Maximum Power Transfer Summary

• Finding the load resistance that maximizes power
  transfer is usually a top-step process
• Find the Thevenin or Norton equivalent
• Find the load resistance RL

84
Max. Power Transfer example
85
Max. Power Transfer example (cont.)?
86
Max. Power Transfer example (cont.)?
87
Max. Power Transfer example (cont.)?