Linear Circuits - ECE 2040: Circuit Analysisece2040.ece.gatech.edu/videos/Module2Handouts.pdf · 2 Why is this course called linear circuits? What does the linear mean? ... to solve

5/23/2013

1

Linear Circuits

An introduction to electric circuit elements and a study of circuits containing such devices.

Dr. Bonnie H. FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering

School of Electrical and Computer Engineering

Concept Map: Module 2

2

Background Resistive Circuits

Reactive Circuits

Frequency Analysis

Power

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3 4

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Resistive Circuits

Resistive Circuits

Concept Map

3

Background

Reactive Circuits

Frequency Analysis

Power

current, voltage, sources, resistance, circuits

Power

Resistive Circuits

Resistive Circuits Resistors Ohms Law Kirchoffs Laws Series and

parallel resistors

Superposition Solution methods Max Power Configurations Sensors

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Nathan V. ParrishPhD Candidate & Graduate Research AssistantSchool of Electrical and Computer Engineering


Resistivity and Ohms Law

Learn how materials resist the flow of currentLearn about Ohms law a law relating current and voltage through materialsFind resistance of materials from their dimensions and electric properties

Define resistance Calculate conductance from resistance Apply Ohms Law to find currents, voltages,

or resistances Calculate the resistance of a material using

its dimensions and electrical properties

Lesson Objectives

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Ohms Law

6

Resistance and Conductance

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nitrogen

14.007

N7

helium

He4.0026

2

neon

Ne20.180

10

F18.998

9oxygen

O15.999

8carbon

C12.011

6boron

B10.811

5

argon

Ar39.948

18chlorine

Cl35.453

17sulfur

S32.065

16phosphorus

P30.974

15silicon

Si28.086

14aluminium

Al26.982

13

krypton

Kr83.798

36bromine

Br79.904

35selenium

Se78.96

34arsenic

As74.922

33germanium

Ge72.64

32gallium

Ga69.723

31zinc

Zn65.38

30copper

Cu63.546

29nickel

Ni58.693

28cobalt

Co58.933

27iron

Fe55.845

26manganese

Mn54.938

25chromium

Cr51.996

24vanadium

V50.942

23titanium

Ti47.867

22scandium

Sc44.956

21calcium

Ca40.078

20potassium

K39.098

19

magnesium

Mg24.305

12sodium

Na22.990

11

beryllium

Be9.0122

4lithium

Li6.941

3

hydrogen

H1.0079

1

xenon

Xe131.29

54iodine

I126.90

53tellurium

Te127.60

52antimony

Sb121.76

51tin

Sn118.71

50indium

In114.82

49cadmium

Cd112.41

48silver

Ag107.87

47palladium

Pd106.42

46rhodium

Rh102.91

45ruthenium

Ru101.07

44technetium

Tc[98]

43molybdenum

Mo95.96

42niobium

Nb92.906

41zirconium

Zr91.224

40yttrium

Y88.906

39strontium

Sr87.62

38rubidium

Rb85.468

37

radon

Rn[222]

86astatine

At[210]

85polonium

Po[209]

84bismuth

Bi208.98

83lead

Pb207.2

82

dysprosium

Dy162.50

66terbium

Tb158.93

65gadolinium

Gd157.25

64europium

Eu151.96

63samarium

Sm150.36

62promethium

Pm[145]

61neodymium

Nd144.24

60praseodymium

Pr140.91

59cerium

Ce140.12

58lanthanum

La138.91

57

barium

Ba137.33

56caesium

Cs132.91

55

roentgenium

Rg[272]

111darmstadtium

Ds[271]

110meitnerium

Mt[268]

109hassium

Hs[277]

108bohrium

Bh[264]

107seaborgium

Sg[266]

106dubnium

Db[262]

105rutherfordium

Rf[261]

104radium

Ra[226]

88francium

Fr[223]

87

lutetium

Lu174.97

71ytterbium

Yb173.05

70thulium

Tm168.93

69erbium

Er167.26

68holmium

Ho164.93

67

thallium

Tl204.38

81mercury

Hg200.59

80gold

Au196.97

79platinum

Pt195.08

78iridium

Ir192.22

77osmium

Os190.23

76rhenium

Re186.21

75tungsten

W183.84

74tantalum

Ta180.95

73hafnium

Hf178.49

72

berkelium

Bk[247]

97lawrencium

Lr[262]

103nobelium

No[259]

102mendelevium

Md[258]

101fermium

Fm[257]

100einsteinium

Es[252]

99californium

Cf[251]

98curium

Cm[247]

96americium

Am[243]

95plutonium

Pu[244]

94neptunium

Np[237]

93uranium

U238.03

92protactinium

Pa231.04

91thorium

Th232.04

90actinium

Ac[227]

89

Reason for Resistance

Li

9

ClCuSi

The Electron Bucket Brigade

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Example: Electron Drift Rate

11

Pause

Resistivity

12

Pause

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Finding Resistance

13

Used background to see how voltage and current relate moving through materials Introduced Ohms Law and its application Discussed the physical cause for resistance Described electron drift rate and calculated this

value in a case study Calculated resistance using the dimensions and

resistivity of a material

Summary

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Kirchhoffs Laws

Introduce Kirchhoffs Voltage Law (KVL) and apply to parallel circuitsIntroduce Kirchhoffs Current Law (KCL) and apply to series circuitsUse Kirchhoffs Laws to solve a simple circuit

Describe KVL and KCL Describe the voltage relationship of parallel

elements Describe the current relationship of series

elements Use Kirchhoffs Laws to find unknown values in a

simple circuit

Lesson Objectives

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Kirchhoffs Voltage Law (KVL)

6

KVL and Parallel Circuits

7

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Kirchhoffs Current Law (KCL)

8

What if?

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KCL and Series Circuits

10

Solving Values in Circuits

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Introduced KVL and KCL Applied KVL to parallel elements Applied KCL to series elements Gave a justification for KCL Solved a simple circuit using

Kirchhoffs Laws

Summary

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Resistors

Introduce resistors as a circuit elementConsider resistors in series and parallelCalculate equivalent resistance by combining parallel/series resistors

Apply Ohms Law and Kirchhoffs Laws to simple resistive circuits Calculate an equivalent resistance of resistors in

parallel/series Find equivalent resistance through successive

application of combining parallel and series resistors

Learning Objectives

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Resistors

8

Review

9

Ohms Law Kirchhoffs Voltage LawKirchhoffs Current Law

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Resistors in Series

10

Pause

Voltage Divider

11

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Resistors in Parallel

12

Pause

Current Divider

13

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Example

14

Introduced to resistors as a circuit element Combine series/parallel resistors Found an equivalent resistance using

successive application of series/parallel resistance

Summary

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Dr. Bonnie FerriProfessor and Associate ChairSchool of Electrical and Computer Engineering

Lab Demo: Introduction to ElectricalComponents

Demonstrate basic instruments and components.

Lab Demo: Introduction to Electrical Components

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Physical resistors Color codes Tolerances

Digital Multimeter (DMM) Measure voltage, current, resistance

Protoboard (breadboard) Ease of building circuits

Summary

5

Thanks to Marion Crowder (School of Electrical and Computer Engineering at Georgia Tech) for video-taping the experiment

DMM used in experiment is manufactured by Fluke Corporation

Credits

7

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Lab Demo: Resistors and Connections

Resistors in series and parallel, measuring voltage and current in circuits.

Demonstrate Series and parallel resistance Measure voltage and current

using the voltage divider law and Ohms Law

Lesson Objectives

4

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2

5

Review

Protoboard

Resistors in Series

R1

R2

R=R1+R2

R1

21

21

RR

RRR

+=

Resistors in Parallel

Lab Demo: Resistors and Connections

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Connect physical resistors in parallel and in series Measure voltages and currents in

a circuit, applying the voltage divider law and Ohms Law

Summary

7



Credits

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Linearity

Describe linearity, superposition, and homogeneity

Define linearity, superposition, and homogeneity Identify if a given function exhibits

superposition or homogeneity

Lesson Objectives

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Why is this course called linear circuits? What does the linear mean?

Linear Circuits

6

Linearity Defined

7

If both properties hold, the system is linear.

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Ohms Law: Linear

8

Examples and Counterexamples

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Introduced linear operators (superposition and homogeneity)

Identified if an operator is linear Used linear operators to generate new linear

operators

Summary

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Superposition

Use linearity (particularly superposition) to solve circuitsIdentify superposition as an important part of many analysis techniques

Given a complex system, generate a set of simple systems, each with a single independent source Using solution of simple systems, find the

complete behavior of the system

Lesson Objectives

5

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Isolating Independent Sources

6

Zero-out all independent sources Return sources one at a time and solve for

value of interest in simplified system Take the arithmetic sum of these values to

find the final quantity

Steps For Superposition

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Example 1

8

Example 1 (a)

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Example 1 (b)

10

v(a) = 1V

Example 1 (c)

11

v(a) = 1Vv(b) = 3V

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Dependent sources must be analyzed in each solution

Must be linear

Working with Dependent Sources

12

Example 2

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Example 2 (a)

14

Example 2 (b)

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Used superposition to solve circuits Independent sources only With dependent sources

Summary

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Systematic Solution Methods: Part 1

Introduce several ways of obtaining circuit equations.

Introduce Mesh analysis Node analysis

Thvenin equivalent circuit Norton equivalent circuit

Lesson Objective

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Ohms Law V = iR KVL sum of all voltages around any loop = 0

KCL sum of all currents out of any node = 0

Physical Behavior

6

Systematic Ways to Solve Circuit Problems

7

Method SummaryMesh Analysis Systematic KVL to obtain simultaneous

equations for currentsNode Analysis Systematic KCL to obtain simultaneous

equations for voltagesThvenin and Norton Equivalent Circuits

Reduce circuit to smaller equivalent Source transformations using graphical method

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Mesh Analysis

8

1. Define mesh currents, one for each non-inclusive loop

2. Do KVL around each loop

I1

I2

I3

is

-

+

-

+

v1vo

v2

R1

R2

R3

Ro

+

-

1. Select a ground node

2. Define node voltages for every node connected to 3 or more elements

3. Do KCL at every node

Node Analysis

9

is

-

+

-

+

v1vo

v2

R1

R2

R3

Ro

+

-

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Summary

10

Method Summary When to ApplyMesh Analysis Systematic KVL,

simultaneous equations for currents

Multiple currents are needed Current sources are present

Node Analysis Systematic KCL, simultaneous equations for voltages

Multiple voltages are needed Voltage sources are present

Thvenin and NortonEquivalent Circuits

Simple equivalent circuits, source transformations

Intermediate values notimportant; only output voltage or current

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Systematic Solution Methods: Part 2

Introduce several ways of obtaining circuit equations.

Demonstrate Thvenin equivalent and Norton equivalent circuits Source transformations

Lesson Objective

4

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2

Systematic solution Methods

5

Method Summary When to ApplyMesh Analysis KVL to obtain simultaneous

equations for currents Multiple currents are needed Current sources are present

Node Analysis KCL to obtain simultaneous equations for voltages

Multiple voltages are needed Voltage sources are present

Thvenin and NortonEquivalent Circuits

Simple equivalent circuits, source transformations

Intermediate values notimportant; only output voltage or current

Thvenin Equivalent

6

Replace circuit with equivalent resistance and voltage source

b

a

Circuit vTh -+vTh

RTh

aisc

b

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vTh : open circuit across a-b and find vab= vTh isc : short circuit across a-b and find isc

Thvenin Equivalent Circuit

7

scThTh iRv =

RTh : circuit resistance with voltage sources shorted and current sources open circuited (when no dependent sources are present)

b

a

Circuit vThb

a

Circuit

isc

Thvenin Equivalent Example

8

0.2A

-

+

-

+

1vvo

2v

2Ro

+

-4

10

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Norton Equivalent Circuit

9

-

+vThRTh

aisc

b

iscRTh

b

a

Source Transformation: these configurations are interchangeable in a circuit

Thvenin equivalent circuit Norton equivalent circuit

Source Transformation Example

10

0.2A

-

+

-

+

1vvo

2v

2Ro

+

-4

10

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Mesh and node analysis Systematic ways to find independent

simultaneous equations Thvenin and Norton methods Replace most of the circuit with a simple

equivalent circuit Source transformations Extra worked problems are given on

these methods

Summary

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Maximum Power Transfer

An introduction to linear electric circuit elements and a study of circuits containing such devices.

Find the load resistance that gives maximum power transfer to the load

Calculate this power consumed by the load resistor giving maximum power transfer

Lesson Objectives

5

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2

Two-Terminal Linear Circuits

6

Power Equations for Resistors

7

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3

Load Resistance

8

Maximum Power Transfer

9

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Specified power equations for resistors Matched load resistance to system resistance

for maximum power transfer Specified equation for maximum power

transfer

Summary

10

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Wye-Delta Transforms and the WheatstoneBridge

Transform resistors from a wye configuration to a delta configuration and vice-versaHow to use a wheatstone bridge to measure a resistance

Transform resistor circuits between wye and delta configurations Specify a test resistor which balances a

Wheatstone bridge Identify whether the resistor under test in a

Wheatstone bridge is below or above the target resistance

Learning Objectives

5

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2

Wye-Delta Transformation

6

Summary

7

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3

Example

8

Wheatstone Bridge

9

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4

Used Y- transform to simplify circuits Balanced a Wheatstone bridge Identified whether the resistor under test was

above or below balanced resistance based on current across the bridge

Summary

10

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Application: Resistors in Sensors

Show sensors that depend on variable resistance.

Sensor: device that converts a physical quantity to an electrical signal

Resistors in Sensors

4

Variable Resistors:R as pressure R as temperature R as strain gauge elongatesR varies with position

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Lab Demo: Variable Resistors in Sensors

5

Resistance often varies with physical properties

Sensors utilize this property to convert physical quantities to voltage

Summary

6

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3


Thanks for James Steinberg and Kevin Pham for technical assistance

Flexforce sensor manufactured by Tekscan

Credits

8

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Application: Wheatstone Bridge

An Wheatstone Bridge used in a sensor.

Wheatstone Bridge

4

-

+vs

R2R1

R3 Rx

ba

sx

xs vRR

RvRR

R+

=+ 231

3

Cancel the vs . Similarly

Divide both sides of these last equations to get

xRRR

RRR

+=

+ 2

2

31

1

Balance R2 and R3 so va =vb and apply the voltage divider law

21

3

RR

RR x=

Measure va - vb

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Lab Demo: Wheatstone Bridge

5

Wheatstone bridge is used to detect small changes in resistance

Four strain gauges in a Wheatstone configuration removes thermal effect

Summary

6

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3

Thanks to Sterling Skinner for building the flexible beam experimental platform and Dr. Aldo Ferri for expertise on that system (both of the George W. Woodruff School of Mechanical Engineering at Georgia Tech).



Credits

7

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Module 2Resistive Circuits Wrap Up

Summary of Resistive Circuits Module

Concept Map

3

Background

Reactive Circuits

Frequency Analysis

current, voltage, sources, resistance, circuits

Power

Resistive Circuits

Resistive Circuits Resistors Ohms Law Kirchoffs Laws Series and

parallel resistors

Superposition Circuit

equations Max Power Configurations Applications

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2

Be able to reduce resistive networks to a single equivalent resistance using parallel and series connections

Important Concepts and Skills

4

Understand Kirchoffs Voltage Law (KVL) and Kirchoffs Current Law (KCL)

Be able to apply KVL and KCL to circuits to obtain equations

Be able to compute voltages and currents from the voltage divider law and the current divider laws

Understand superposition and its application in circuits to find specific voltages and currents

Given a color chart, be able to identify physical resistor values and tolerances

Understand the purpose of a protoboard (breadboard) and its basic operation

Understand how current can be measured in a circuit using the voltage divider law


5

Given a circuit with multiple sources, be able to use the Superposition Principle to solve for circuit voltages and currents

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Be able to compute the load resistance that maximizes the power


6

Have a basic understanding of mesh analysis, node analysis, Thveninequivalent and Norton equivalent circuits and when to use one versus another

Be able to solve for specific voltages and currents in a given circuit

Know the transformation Understand that these configurations may be used

in different applications, such as 3 phase circuits


7

Know examples of resistors that vary with physical quantities

Understand how a potentiometer is used to measure position or angle

Know when a Wheatstone Bridge is used in a practical application

Be able to write equations for a Wheatstone Bridge

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4

Concept Map

8

Background Resistive Circuits

Reactive Circuits

Frequency Analysis

Power

1 2

3 4

5

Documents

Linear Circuits - ECE 2040: Circuit Analysisece2040.ece.gatech.edu/videos/Module2Handouts.pdf · 2 Why is this course called linear circuits? What does the linear mean? ... to solve