3317 Chap 1 Lect 0810 Student

8/3/2019 3317 Chap 1 Lect 0810 Student

1/19

1-1

Chapter 1Chapter 1

Complex VectorsComplex VectorsECE 3317

Dr. Stuart Long


2/19

1-2Why study electromagnetic waves?

Motivation:

Most basic of all ECE courses: Mathematically the most satisfying; allothers courses are just more specific cases (i.e. circuit theory

is the low

frequency special case)

Electromagnetics explains physical phenomena: Confirms

observations in areas of electricity and magnetism; closer to the

electrophysics

side of ECE, more like applied physics

Important to know about EM: Even if you are specializing in circuits,

computers, control, communications, biomed, or solid state devices


3/19

1-3

Applications:

Applied Electromagnetics: Radar; Antennas; Microwaves; Militaryuses; Electrical machinery; Aircraft

Wireless Communications: Cell phones; Bluetooth; Wireless routers;

Cordless handsets; EZ pass tags; RFIDs

Computer Applications: Electromagnetic Compatibility (EMC);

Electromagnetic Interference (EMI); Chip design beyond simple circuit

analysis (faster circuits mean smaller wavelengths).

Why study electromagnetic waves?


4/19

1-6Notation, Symbols, and Units

Real Scalars:

Indicated by italic type or Greek letters e.g. a or.

Complex Scalars:

Indicated by a ~

underneath letters e.g. c

Real Vectors:

Indicated by boldface italic type e.g.B.

Unit Vectors:

Indicated by the symbol ^

above a quantity and boldface italic

type or Greek letters e.g. x

~

Complex Vectors:

Indicated by bold type with a ~

underneath letters e.g. J

~


5/19

1-7

References to equations and pages in your book will be written in green.

Appendices A, B, C, and D

in the text book list frequently used symbols and

their units.

Notation, Symbols, and Units


6/19

1-8Complex Numbers

real imaginary magnitude phase

1

c |c|j

a j b e

2h |h|j

f j g e


7/19

1-9

1

1 1c | c | | c | cos | c | sinj

a j b e j

real imaginary magnitude phase

Rea

Im

c a jb

1c s i n

1c cos

c

1

b

Graphic Representation of

Complex Numbers

2 2

1

1

| c |

tan

a b

b a


8/19

1-10

1-*

c h ( ) ( )

c

-h ( - ) ( - )

c - | c |

j

a f j b g

a f j b g

a jb e

addition

subtraction

complex conju t ga e

Complex Algebra

1

2

c |c|

h |h|

j

j

a j b e

f j g e


9/19

1-11

1( )

c h | c | | h |

c |c|

h |h|

j

j

e

e

multiplication

division

1

2

c |c|

h |h|

j

j

a j b e

f j g e

Complex Algebra


10/19

1-12

1

2

c c

c c

j n

x j x n xx

e

e

square root

power

1 2

c |c|j n

e

Complex Algebra

Where n

is an integer

Note: square root will have

two possible values, one

for n=0 and one for n=1.


11/19

1-13

0

0

( ) = cos( )

( ) = Re

( ) = Re V

j j t

j t

V t V t

V t V e e

V t e

0V = jV e

Time Harmonic Quantities

2

f

Amplitude Angular Phase

Frequency

[1.4]

[1.5]


12/19

1-14

B

( , , , ) ( , , )x y z t x y z V

V

real, time

harmonic

quantity

complex

representation

B

A

V(t)

t

c cA

Re V

Im V



13/19

1-15

( ) ( ) +

(t)

( )

Note:

However,

)

(

t t

j

t

t t

V U

V

VU

V U

V

V U

( , , , ) ( , , )x y z t x y z V

V

real, time

harmonic

quantity

complex

representation


B

A

V(t)

t

c cA

R e V

Im V

B

1 16


14/19

1-16

cos

cos ( ( )( ) s )co ( )

x zy

x

x x

y

y zy

z

z

jj j

t

V eV e

V

V t

V

t

t

e

V

V x

xV

z

z

y

y

Transform each component of a time

harmonic vector function into complexform

( ) Re j tt e VV

Complex Vectors

1 17


15/19

1-17Example 1.15

Let

( ) Re

( ) Re ( )

( ) Re ( )(cos sin )

( ) cos sin

j t

j t

j

t e

t j e

t j t j t

t t t

A x y

A A

A x y

A x y

A x yt = 3/2

t =

t = /2

y

x

t = 0

[Fig. 1.8]

[p.16]

The direction

of the vector

varies with time.

1 18


16/19

1-18

1 19


17/19

1-19

Recall from example 1.15

and

Let and

( ) cos sin

( ) ( ) sin cos

(A B A B )

( )( ) ( )( )

( ) ( )

x y y x

j t t t

j j j t t t

j j

A

B

A B

x y A x y

x y x y B x y

z

1 1

1 1

0

(x-formation only for basic vector produc )N tsT O

( ) ( )

( ) (

Ho

) cos si

w

n

ever, t t

t t t t

A B

2 2

0 A B 0

A B z z 0

Example 1.16

[p.16]

1 20Ti A f H i Q titi


18/19

1-20

00

2 2 200

20

0

22 0

Not true for pro t 2duc s

1( ) cos( ) 0

1( ) cos ( )

1 cos[2( )]

2

( )2

T

T

T

T =

V t V t dt T

1f

V t V t dt T

V tdt

T

VV t

[p.17]

Time Average of Harmonic Quantities

[p.17]

[p.17]

1 21E l


19/19

1-21

time average po

2

wer

Usual 60 [Hz] power

A way to compute the time average of the cr

Quic oss products of

two time har

k

monic vectors is g

!!!

(t) 0 (t)but 0

V V

* time-average 1(t) (t) Re2

iven belo

rule

w

A B

A B [1.19]

Example

Documents

3317 Chap 1 Lect 0810 Student