3317 chap 1 lect 0810 student
TRANSCRIPT
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Chapter 1Chapter 1
Complex VectorsComplex VectorsECE 3317
Dr. Stuart Long
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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
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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?
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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
~
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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
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1-8Complex Numbers
real imaginary magnitude phase
1
c |c|j
a j b e
2h |h|j
f j g e
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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
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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
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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
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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.
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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]
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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
Time Harmonic Quantities
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( ) ( ) +
(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
Time Harmonic Quantities
B
A
V(t)
t
c cA
R e V
Im V
B
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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
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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.
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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
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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
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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