1 eee 498/598 overview of electrical engineering lecture 1: introduction to electrical engineering

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1 EEE 498/598 EEE 498/598 Overview of Electrical Overview of Electrical Engineering Engineering Lecture 1: Lecture 1: Introduction to Introduction to Electrical Engineering Electrical Engineering

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Page 1: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

1

EEE 498/598EEE 498/598Overview of Electrical Overview of Electrical

EngineeringEngineering

Lecture 1:Lecture 1:Introduction to Electrical Introduction to Electrical

EngineeringEngineering

Page 2: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

2Lecture 1

Lecture 1 ObjectivesLecture 1 Objectives

To provide an overview of packaging.To provide an overview of packaging. To review the electrical functions of a To review the electrical functions of a

package.package. To understand the foundations of To understand the foundations of

electrical engineering.electrical engineering. To become aware of the topics that will To become aware of the topics that will

be covered in this class.be covered in this class. To define the coordinate systems that To define the coordinate systems that

will be used in this class.will be used in this class.

Page 3: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

3Lecture 1

Overview of PackagingOverview of Packaging

Silicon Die Package Motherboard

~ .040”

~ .012“

Courtesy of Intel Corp.

Page 4: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

4Lecture 1

Overview of PackagingOverview of Packaging

Interposer

FCBGA

LSC

Interposer Caps

DSC

Die

Pins

IHS

C4 Balls

PTH Vias

BGA Balls

InterposerVias

FCBGA

Interposer

Pins

Die

Microvias

FCBGA

Interposer

Courtesy of Intel Corp.

Page 5: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

5Lecture 1

Overview of PackagingOverview of Packaging

““Packaging engineers today must solve Packaging engineers today must solve complex, coupled problems that require complex, coupled problems that require fundamental understanding of electrical, fundamental understanding of electrical, thermal, mechanical, material science, thermal, mechanical, material science, and manufacturing principles.”and manufacturing principles.”

Dr. Nasser Grayeli, Intel CorporationDr. Nasser Grayeli, Intel Corporation This course focuses on preparing students This course focuses on preparing students

to understand the electrical principles.to understand the electrical principles.

Page 6: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

6Lecture 1

Electrical Functions of the Electrical Functions of the PackagePackage

Power DeliveryPower Delivery Supply a clean power and reference voltage to Supply a clean power and reference voltage to

active devices on the die.active devices on the die.

Signal Input/OutputSignal Input/Output Transmit signals from the die to the Transmit signals from the die to the

motherboard faithfully and in minimum time.motherboard faithfully and in minimum time.

EMI/EMCEMI/EMC Minimize radiation of electromagnetic energy Minimize radiation of electromagnetic energy

into the environment, and the impact of ambient into the environment, and the impact of ambient electromagnetic energy on circuit performance.electromagnetic energy on circuit performance.

Page 7: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

7Lecture 1

Foundations of Electrical Foundations of Electrical EngineeringEngineering

Electrophysics.Electrophysics. Information Information

(Communications) (Communications) Theory.Theory.

Digital Logic.Digital Logic.

80 % of this course

20 % of this course

Not covered in detail in this class

Page 8: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

8Lecture 1

Foundations of Electrical Foundations of Electrical EngineeringEngineering

ElectrophysicsElectrophysics:: Fundamental theories of physics Fundamental theories of physics

and important special cases.and important special cases. Phenomenological/behavioral Phenomenological/behavioral

models for situations where the models for situations where the rigorous physical theories are too rigorous physical theories are too difficult to apply.difficult to apply.

Page 9: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

9Lecture 1

Hypothesis, Model, and Hypothesis, Model, and TheoryTheory

A A hypothesishypothesis is an idea or suggestion that has is an idea or suggestion that has been put forward to explain a set of been put forward to explain a set of observations. It may be expressed in terms of a observations. It may be expressed in terms of a mathematical mathematical modelmodel. The . The modelmodel makes a number makes a number of predictions that can be tested in of predictions that can be tested in experiments. After many tests have been experiments. After many tests have been made, if the made, if the modelmodel can be refined to correctly can be refined to correctly describe the outcome of all experiments, it describe the outcome of all experiments, it begins to have a greater status than a mere begins to have a greater status than a mere suggestion.suggestion.

A A theorytheory is a well-tested and well-established is a well-tested and well-established understanding of an underlying mechanism or understanding of an underlying mechanism or process.process.The material in this slide has been adapted from

material at http://www2.slac.stanford.edu/vvc/theory/modeltheory.html.

Page 10: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

10Lecture 1

Hypothesis, Model, and Hypothesis, Model, and TheoryTheory

Maxwell’s equations are ‘Maxwell’s equations are ‘just a theoryjust a theory’ and yet ’ and yet my cell phone works!my cell phone works!

At one time, a theory would have been At one time, a theory would have been referred to as a ‘law’.referred to as a ‘law’. Newton’s lawsNewton’s laws Boyle’s lawBoyle’s law

But remember no theory is a complete But remember no theory is a complete description of all reality; all theories are description of all reality; all theories are incomplete.incomplete.

Electrical engineers make use of a number of Electrical engineers make use of a number of theories – some of which are special cases of theories – some of which are special cases of others.others.

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11Lecture 1

Four Fundamental Forces Four Fundamental Forces of Physicsof Physics Gravitational ForceGravitational Force

Associated particle is graviton (hypothesized)Associated particle is graviton (hypothesized) Always attractive Always attractive Varies inversely as the square of the distanceVaries inversely as the square of the distance

Electromagnetic ForceElectromagnetic Force Associated particle is photonAssociated particle is photon 10104242 times stronger than gravity times stronger than gravity Force can be attractive or repulsiveForce can be attractive or repulsive Varies inversely as the square of the distanceVaries inversely as the square of the distance

Strong InteractionStrong Interaction Associated particle is gluonAssociated particle is gluon About 100X stronger than electromagnetic force but only About 100X stronger than electromagnetic force but only

acts over distances the size of an atomic nucleus acts over distances the size of an atomic nucleus Responsible for holding the protons and neutrons togetherResponsible for holding the protons and neutrons together

Weak InteractionWeak Interaction Associated particles are the weak gauge bosons (Z and W Associated particles are the weak gauge bosons (Z and W

particles)particles) Acts only over distances the size of an atomic nucleusActs only over distances the size of an atomic nucleus Responsible for certain types of radioactive decayResponsible for certain types of radioactive decay

Page 12: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

12Lecture 1

The Standard ModelThe Standard Model Physicists call the theoretical framework that Physicists call the theoretical framework that

describes the interactions between elementary describes the interactions between elementary building blocks (quarks and leptons) and the building blocks (quarks and leptons) and the force carriers (bosons) the force carriers (bosons) the Standard ModelStandard Model..

Most of the standard model is a Most of the standard model is a theorytheory; some of ; some of it is still it is still hypothesishypothesis..

Physicists use the Standard Model to explain Physicists use the Standard Model to explain and calculate a vast variety of particle and calculate a vast variety of particle interactions and quantum phenomena. High-interactions and quantum phenomena. High-precision experiments have repeatedly verified precision experiments have repeatedly verified subtle effects predicted by the Standard subtle effects predicted by the Standard Model. Model.

The material in this slide and in the following two slides has been adapted from material from www.fnal.gov (Fermilab).

Page 13: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

13Lecture 1

The Standard ModelThe Standard Model

The biggest success of the Standard The biggest success of the Standard Model is the unification of the Model is the unification of the electromagnetic and the weak forces into electromagnetic and the weak forces into the so-called the so-called electroweak forceelectroweak force..

Many physicists think it is possible to Many physicists think it is possible to eventually describe all forces with a eventually describe all forces with a Grand Unified Theory or a so-called Grand Unified Theory or a so-called Theory Theory of Everythingof Everything (ToE). (ToE). M-theory (a generalization of superstring M-theory (a generalization of superstring

theory) is the current embodiment of the ToE.theory) is the current embodiment of the ToE.

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14Lecture 1

Philosophical Implications Philosophical Implications of a ToEof a ToE

ReductionistReductionist point-of-view: everything from point-of-view: everything from the big bang to human emotions can be the big bang to human emotions can be obtained from the ToE given enough obtained from the ToE given enough computational power.computational power.

Another point of view: new types of Another point of view: new types of fundamental laws arise in complex systems fundamental laws arise in complex systems that cannot be derived from the ToE.that cannot be derived from the ToE.

Practical point of view: any ToE could never Practical point of view: any ToE could never be successfully applied to complex systems, be successfully applied to complex systems, and so it is irrelevant whether or not the laws and so it is irrelevant whether or not the laws of complex systems can be derived from the of complex systems can be derived from the ToE.ToE.

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15Lecture 1

Religion/Faith/Metaphysics Religion/Faith/Metaphysics v. Sciencev. Science

Science attempts to explain the Science attempts to explain the processes by which the universe processes by which the universe functions (i.e. the “how”).functions (i.e. the “how”).

Religion attempts to explain why Religion attempts to explain why the universe exists and to impart the universe exists and to impart meaning to its existence (i.e., meaning to its existence (i.e., the “why”).the “why”).

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16Lecture 1

Religion/Faith/Metaphysics Religion/Faith/Metaphysics v. Sciencev. Science

Science is absolutely neutral on the issue of Science is absolutely neutral on the issue of whether “God” exists or not.whether “God” exists or not. Consider the following: people of many different Consider the following: people of many different

faiths or no faith at all can work together to design faiths or no faith at all can work together to design complex systems such as a packaged integrated complex systems such as a packaged integrated circuit.circuit.

Millions of devoutly religious people accept Millions of devoutly religious people accept scientific theories as valid explanations for scientific theories as valid explanations for natural processes.natural processes.

Those who do not should imagine what life Those who do not should imagine what life was like for the average human before modern was like for the average human before modern scientific advances in medicine, engineering, scientific advances in medicine, engineering, etc.etc.

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17Lecture 1

EngineeringEngineering

With the exception of nuclear With the exception of nuclear engineering, the engineering engineering, the engineering disciplines (e.g., mechanical, disciplines (e.g., mechanical, aerospace, civil, etc.) deal with aerospace, civil, etc.) deal with phenomena that involve the forces of phenomena that involve the forces of gravity and electromagnetism.gravity and electromagnetism.

Much of electrical engineering Much of electrical engineering involves understanding phenomena involves understanding phenomena that result from the force of that result from the force of electromagnetism.electromagnetism.

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18Lecture 1

Hierarchy of Physics Theories Hierarchy of Physics Theories Involved in the Study of Involved in the Study of Electrical EngineeringElectrical Engineering

Quantum electrodynamicsQuantum electrodynamics Quantum mechanicsQuantum mechanics

Schrödinger equationSchrödinger equation Classical electromagneticsClassical electromagnetics

ElectrostaticsElectrostatics MagnetostaticsMagnetostatics Circuit theoryCircuit theory Geometric opticsGeometric optics

Page 19: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

19Lecture 1

Information TheoryInformation Theory Originally developed by Claude Originally developed by Claude

Shannon of Bell Labs in the 1940s.Shannon of Bell Labs in the 1940s. InformationInformation is defined as a symbol that is defined as a symbol that

is uncertain at the receiver.is uncertain at the receiver. The fundamental quantity in The fundamental quantity in

information theory is information theory is channel capacitychannel capacity – – the maximum rate that information the maximum rate that information can be exchanged between a can be exchanged between a transmitter and a receiver.transmitter and a receiver.

The material in this slide and the next has been adapted from material from www.lucent.com/minds/infotheory.

Page 20: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

20Lecture 1

Information TheoryInformation Theory

Defines relationships between Defines relationships between elements of a communications elements of a communications system. For example,system. For example, Power at the signal sourcePower at the signal source Bandwidth of the systemBandwidth of the system NoiseNoise InterferenceInterference

Mathematically describes the Mathematically describes the principals of data compression.principals of data compression.

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21Lecture 1

Exercise: What is Exercise: What is Information?Information?

Message with redunancy:Message with redunancy: ““Many students are likely to fail Many students are likely to fail

that exam.”that exam.” Message coded with less Message coded with less

redundancy:redundancy: ““Mny stdnts are lkly to fail tht Mny stdnts are lkly to fail tht

exm.”exm.”

Page 22: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

22Lecture 1

Digital LogicDigital Logic

Digital logic signals are really Digital logic signals are really analog signals, and digital analog signals, and digital circuits are ultimately designed circuits are ultimately designed using circuit theory.using circuit theory.

However, in many situations the However, in many situations the function of a digital circuit is function of a digital circuit is more easily synthesized using more easily synthesized using the principles of digital logic.the principles of digital logic.

Page 23: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

23Lecture 1

Digital LogicDigital Logic

Based on logic gates, truth Based on logic gates, truth tables, and combinational and tables, and combinational and sequential logic circuit design sequential logic circuit design

Uses Boolean algebra and Uses Boolean algebra and Karnaugh maps to develop Karnaugh maps to develop minimized logic circuits. minimized logic circuits.

Not explicitly addressed in this Not explicitly addressed in this class.class.

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24Lecture 1

EE SubdisciplinesEE Subdisciplines

Power SystemsPower Systems ElectromagneticsElectromagnetics Solid State Solid State Communication/Signal Communication/Signal

ProcessingProcessing ControlsControls Digital DesignDigital Design

Page 25: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

25Lecture 1

Power SystemsPower Systems

Generation of electrical energyGeneration of electrical energy Storage of electrical energyStorage of electrical energy Distribution of electrical energyDistribution of electrical energy Rotating machinery-generators, Rotating machinery-generators,

motorsmotors

Page 26: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

26Lecture 1

ElectromagneticsElectromagnetics

Propagation of electromagnetic Propagation of electromagnetic energyenergy

AntennasAntennas Very high frequency signalsVery high frequency signals Fiber opticsFiber optics

Page 27: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

27Lecture 1

Solid StateSolid State

DevicesDevices TransistorsTransistors Diodes (LED’s, Laser diodes)Diodes (LED’s, Laser diodes) PhotodetectorsPhotodetectors

Miniaturization of electrical Miniaturization of electrical devicesdevices

Integration of many devices on a Integration of many devices on a single chipsingle chip

Page 28: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

28Lecture 1

Communications/Signal Communications/Signal Proc.Proc.

Transmission of information Transmission of information electrically and opticallyelectrically and optically

Modification of signalsModification of signalsenhancementenhancementcompressioncompressionnoise reductionnoise reductionfilteringfiltering

Page 29: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

29Lecture 1

ControlsControls

Changing system inputs to Changing system inputs to obtain desired outputsobtain desired outputs

FeedbackFeedback StabilityStability

Page 30: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

30Lecture 1

Digital DesignDigital Design

Digital (ones and zeros) signals and Digital (ones and zeros) signals and hardwarehardware

Computer architecturesComputer architectures Embedded computer systemsEmbedded computer systems

MicroprocessorsMicroprocessors MicrocontrollersMicrocontrollers DSP chipsDSP chips Programmable logic devices (PLDs)Programmable logic devices (PLDs)

Page 31: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

31Lecture 1

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

ATCi Corporate Headquarters450 North McKemyChandler, AZ 85226 USA

SimulsatParabolic

Horn feed

Multiple horn feeds

Page 32: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

32Lecture 1

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

Incoming plane wave is focused by reflector at location of horn feed.

Geometric Optics

Page 33: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

33Lecture 1

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

Feed horn is designed so that it will illuminate the reflector in such a way as to maximize the aperture efficiency.

Maxwell’sequations

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34Lecture 1

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

Feed horn needs to be able to receive orthogonal linear polarizations (V-pol and H-pol) and maintain adequate isolation between the two channels.

V-pol

H-pol

Page 35: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

35Lecture 1

A planar orthomode transducer (OMT) is used to achieve good isolation between orthogonal linear polarizations.

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

Maxwell’s Equations (“Full-Wave Solution”)

Page 36: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

36Lecture 1

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

Horn

Feed waveguide (WR 229)

To LNB

Stripline circuit with OMT, ratrace and WR229 transitions

Maxwell’sequations

Page 37: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

37Lecture 1

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

Single-ended probe

Differential-pair probes

Ratrace hybrid

WR229Transitions

50 ohm transmission line

Layout of the stripline trace layer

Vias

Circuit Theory

Page 38: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

38Lecture 1

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

The two linear polarizations each are fed to a LNB (low noise block).

LNB

LNB

Page 39: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

39Lecture 1

Case Study: C/Ku Band Earthstation Case Study: C/Ku Band Earthstation Antennas Antennas

LNB:

LNA Mixer

IF Output:950-1750 MHz(To Receiver)

Local Oscillator

BPF

Circuit Theory,

Behavioral Models,

Information Theory

Page 40: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

40Lecture 1

Overall System Overall System PerformancePerformance

Carrier-to-noise ratio (CNR)Carrier-to-noise ratio (CNR) Bit error rate (BER)Bit error rate (BER)

Maxwell’s Equations, Circuit Theory, Behavioral Models,

Information Theory

Page 41: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

41Lecture 1

SI (International System SI (International System of) Unitsof) Units

QuantityQuantity UnitUnit AbbreviatiAbbreviationon

lengthlength metermeter mm

massmass kilogramkilogram kk

timetime secondsecond ss

currentcurrent ampereampere AA

temperatutemperaturere

kelvinkelvin KK

luminous luminous intensityintensity

candelacandela cdcd

Fundamental SI Units

Page 42: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

42Lecture 1

Why Do We Need Why Do We Need Coordinate Systems?Coordinate Systems?

The laws of electrophysics (like the laws of The laws of electrophysics (like the laws of physics in general) are independent of a physics in general) are independent of a particular coordinate system.particular coordinate system.

However, application of these laws to However, application of these laws to obtain the solution of a particular problem obtain the solution of a particular problem imposes the need to use a suitable imposes the need to use a suitable coordinate system.coordinate system.

It is the It is the shape of the boundaryshape of the boundary of the of the problem that determines the most suitable problem that determines the most suitable coordinate system to use in its solution.coordinate system to use in its solution.

Page 43: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

43Lecture 1

Orthogonal Right-Orthogonal Right-Handed Coordinate Handed Coordinate

SystemsSystems A A coordinate systemcoordinate system defines a set of three defines a set of three

reference directions at each and every reference directions at each and every point in space.point in space.

The The originorigin of the coordinate system is of the coordinate system is the reference point relative to which we the reference point relative to which we locate every other point in space.locate every other point in space.

A A position vectorposition vector defines the position of a defines the position of a point in space relative to the origin.point in space relative to the origin.

These three reference directions are These three reference directions are referred to as referred to as coordinate directionscoordinate directions, and , and are usually taken to be mutually are usually taken to be mutually perpendicular (perpendicular (orthogonalorthogonal).).

Page 44: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

44Lecture 1

Orthogonal Orthogonal Right-Right-HandedHanded Coordinate Coordinate

SystemsSystems Unit vectors along the coordinate Unit vectors along the coordinate

directions are referred to as directions are referred to as base vectorsbase vectors.. In any of the orthogonal coordinate In any of the orthogonal coordinate

systems, an arbitrary vector can be systems, an arbitrary vector can be expressed in terms of a superposition of expressed in terms of a superposition of the three base vectors. the three base vectors.

Consider base vectors such thatConsider base vectors such that

213

132

321

ˆˆˆ

ˆˆˆ

ˆˆˆ

aaa

aaa

aaa

1a

3a Such a coordinate system is called right-handed.

Page 45: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

45Lecture 1

Orthogonal Right-Orthogonal Right-Handed Coordinate Handed Coordinate

SystemsSystems Note that the base vectors can, Note that the base vectors can,

in general, point in different in general, point in different directions at different points in directions at different points in space.space.

Obviously, if they are to serve as Obviously, if they are to serve as references, then their directions references, then their directions must be known a priori for each must be known a priori for each and every point in space. and every point in space.

Page 46: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

46Lecture 1

Coordinate Systems Coordinate Systems Used in This ClassUsed in This Class

In this class, we shall solve In this class, we shall solve problems using three problems using three orthogonal right-handed orthogonal right-handed coordinate systems:coordinate systems: CartesianCartesian cylindricalcylindrical sphericalspherical

zyx ,,

z,,

,,r

Page 47: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

47Lecture 1

Cartesian CoordinatesCartesian Coordinates The point The point P(xP(x11,y,y11,z,z11)) is located as the is located as the

intersection of three mutually intersection of three mutually perpendicular perpendicular planesplanes: : x=xx=x11, , y=yy=y11, ,

z=zz=z11.. The base vectors areThe base vectors are The base vectors satisfy the The base vectors satisfy the

following relations:following relations:

zyx aaa ˆ,ˆ,ˆ

yxz

xzy

zyx

aaa

aaa

aaa

ˆˆˆ

ˆˆˆ

ˆˆˆ

za ya

xa

Page 48: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

48Lecture 1

Cylindrical CoordinatesCylindrical Coordinates The point The point PP((11,,11,,zz11)) is located as is located as

the intersection of three mutually the intersection of three mutually perpendicular surfaces: perpendicular surfaces: ==11 (a (a circular cylinder), circular cylinder), ==11 (a half- (a half-plane containing the z-axis), plane containing the z-axis), z=zz=z11 (a plane).(a plane).

The base vectors areThe base vectors are

zaaa ˆ,ˆ,ˆ

za

a

a

z increasing ofdirection in ther unit vecto a is ˆ

increasing ofdirection in ther unit vecto a is ˆ

increasing ofdirection in ther unit vecto a is ˆ

Page 49: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

49Lecture 1

Cylindrical Coordinates Cylindrical Coordinates (Cont’d)(Cont’d)

x

y

z

P(1,1,z1)z1

11

za

a

a

Page 50: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

50Lecture 1

Cylindrical Coordinates Cylindrical Coordinates (Cont’d)(Cont’d)

The base vectors satisfy the The base vectors satisfy the following relations:following relations:

aaa

aaa

aaa

z

z

z

ˆˆˆ

ˆˆˆ

ˆˆˆ

za a

a

zqpaa pqqp ,,,;ˆˆ

Page 51: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

51Lecture 1

Spherical CoordinatesSpherical Coordinates The point The point PP((rr11,,11,,11)) is located as is located as

the intersection of three mutually the intersection of three mutually perpendicular surfaces: perpendicular surfaces: r = rr = r11 (a (a sphere), sphere), 11 (a cone), and (a cone), and ==11 (a half-plane containing the z-(a half-plane containing the z-axis).axis).

The base vectors areThe base vectors are

aaar ˆ,ˆ,ˆ

increasing ofdirection in ther unit vecto a is ˆ

increasing ofdirection in ther unit vecto a is ˆ

increasing ofdirection in ther unit vecto a is ˆ

a

a

rar

Page 52: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

52Lecture 1

Spherical Coordinates Spherical Coordinates (Cont’d)(Cont’d)

x

y

z

P(r1,1,1)

1

r1 ara

a

1

Page 53: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

53Lecture 1

Spherical Coordinates Spherical Coordinates (Cont’d)(Cont’d)

The base vectors satisfy the The base vectors satisfy the following relations:following relations:

aaa

aaa

aaa

R

r

r

ˆˆˆ

ˆˆˆ

ˆˆˆ

a

ra

,,,;ˆˆ rqpaa pqqp

Page 54: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

54Lecture 1

Spherical Coordinates Spherical Coordinates (Cont’d)(Cont’d)

x

y

z

r

or20

0

0 r

Page 55: 1 EEE 498/598 Overview of Electrical Engineering Lecture 1: Introduction to Electrical Engineering

55Lecture 1

Position vector:Position vector:

Arbitrary function of position:Arbitrary function of position:

Position VectorPosition Vector

ra

zaa

zayaxar

r

z

zyx

ˆ

ˆˆ

ˆˆˆ

rf