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Notes 5 Notes 5 Complex VectorsComplex Vectors
ECE 3317
Prof. D. R. Wilton
Adapted from notes by Prof. Stuart A. Long
• 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.
• V(t) is a time-varying function.
• V is a phasor (complex number).
• A bar underneath indicates a vector: V(t), V.
Notation
Complex Numbers
Real part
| | jc a j b c e
MagnitudeImaginary part
Phase
Im
c a jb
sinc
cosc
c
b
Rea
1j
cos sinje j Euler's identity:
2 2
1
cos
sin
tan
a c
b c
c a b
b a
Hence
Complex Numbers
1- * - | | jc a jb c e
Im
* c a jb
Re
a
b
Im
c a jb
Re
a
b
complex conjugate
1 2 1 2 1 2
1 2 1 2 1 2
( ) ( )
( ( )
)
c c a a j b b
c c a a j b b
addition
subtraction
Complex Algebra
1
2
1 1 1 1
2 2 2 2
| |
| |
j
j
c a jb c e
c a jb c e
Im
1 1 1 c a jb
Re1a
1b
2b2a
2 2 2 c a jb
1 2c c
Im
1c
Re2c
1 2c c
1
1 2 1 2
1 2 1 2 1 2 2 1
( )11
2 2
*1 2 1 2 2 1 1 21 2
2 2 22 22
j
j
c c c c ea a b b j a b a b
cce
c c
a a b b j a b a bc c
a bc
multiplication
division
Complex Algebra (cont.)
1
2
1 1 1 1
2 2 2 2
| |
| |
j
j
c a jb c e
c a jb c e
1/2
/2
j
j
c c e
c e
Square Root
where n is an integerNote: the complex square root will have two possible values.
1/21/2
1/22
/2
/2
j
j n
j n
j jn
c c e
c e
c e
c e e
c
jc c e
(principal branch)
Principal square root
(principal root)
0
0 0
0
( ) = cos( )
= Re{ cos( ) sin( ) }
( ) = Re
( ) = Re V
j j t
j t
V t V t
V t jV t
V t V e e
V t e
0 0 0V cos sinjV e V jV Define the :complex phasor
Time-Harmonic Quantities
2
f
Amplitude Angular Phase
Frequency
[1.4]
[1.5]
Charles Steinmetz
0( ) = cos( )V t V t
Time-Harmonic Quantities (cont.)
( ) VV t
( ) = Re V j tV t e
0V = jV e
time-domain phasor domain
going from time-domain to phasor domain
going from phasor domain to time domain
B
B
A
V(t)
t
CΦ
CA
Re V
Im V
Graphical Illustration
0( ) = cos( )V t V t
The complex number V
0V
0( ) = Re j tV t V e
Note: ( ) ( ) U V
( ) U
( ) V
However,
( ) ( )
U t V t
AU t A
V t jt
U t V t
UV
Time-Harmonic Quantities
( ) VV t
ˆ cos( ) ( ˆ cosˆ c ))os( ) (y zy zx xVV t t V tVt z yx
Transform each component of a time-harmonic vector function into complex vector.
Complex Vectors
ˆˆV ˆ zx yy zx
j jj
V Ve V ee
yx z
( ) Re V
ˆ ˆ ˆRe V V V
ˆ ˆ ˆRe
ˆ ˆ ˆcos( ) cos( ) cos( )
yx z
j t
j tx y z
jj j j tx y z
x x y y z z
V t e
e
V e V e V e e
V t V t zV t
= x y z
x y z
= x y
Example 1.15
( ) Re A
ˆ ˆ( ) Re ( )
ˆ ˆ( ) Re ( )(cos sin )
ˆ ˆ( ) cos sin
j t
j t
A t e
A t x jy e
A t x jy t j t
A t x t y t
ωt = 3π/2
ωt = π
ωt = π/2
y
x
ωt = 0, 2π
[Fig. 1.8]
[p.17]
The vector rotates with time!
ˆ ˆA j x yAssume
Find the corresponding time-domain vector
ˆ ˆ( ) cos sinA t x t y t
Example 1.15 (cont.)
ˆ ˆ( , ) cos sinE t z x t kz y t kz
Practical application:A circular-polarized plane wave (discussed later).
z
For a fixed value of z, the electric field vector rotates with time.
E (z,t)
For a fixed value of t, the electric field vector appears to spiral in space as shown at left.
Example 1.16[p.17]
ˆ ˆ ˆ ˆA ( ) cos sin
ˆ ˆ ˆ ˆ ˆ ˆ B ( )
Re
( ) sin cos
ˆA B (A B A B )
(1)( 1) ( )( )
( 1) ( 1)
call from example 1.15
and
Let an
0
A B 0
d
x y y x
j A t t t
j j j B t t t
z
j j
x y x y
x y x y x y
2 2
( ) ( ) 0
ˆ ˆ ( ) ( ) cos si
Howeve
n
r, A t B t
A t B t z t t z
We have to be careful about drawing conclusions from cross and dot products in the phasor domain!
2 2 200
22 0
0
2 22 0 0
1( ) cos ( )
1 cos[2( )]( )
2
( )2 2
T
T
V t V t dtT
V tV t dt
T
V VTV t
T
[p.17]
Time Average of Time-Harmonic Quantities
00
1( ) cos( ) 0
TV t V t dt
T
21T =
f
22 0( )
2
VV t
0( ) cos( )V t V t
0 0 0
0 00
0 0
0 0
1( ) ( ) cos( ) cos( )
cos cos 2
2
cos
2
cos
2
T
T
V t I t V I t t dtT
tV Idt
T
V IT
T
V I
[p.17]
Time Average of Time-Harmonic Quantities (cont.)
0 00
1( ) cos( ) cos(
TV t I t V t I t dt
T
Next, consider the time average of a product of sinusoids:
**0 0
0 0
0 0
VI j j
j j
j
V e I e
V e I e
V I e
Time Average of Time-Harmonic Quantities (cont.)
*0 0Re VI cosV I
0 0
cos( ) ( )
2V t I t V I
1( ) ( ) Re VI
2*V t I t
(from previous slide)
Hence
Now consider
Time Average of Time-Harmonic Quantities (cont.)
The results directly extend to vectors that vary sinusoidally in time.
* * *
* * *
1 1 1Re D E Re D E Re D E
2 2 21
Re D E + D E + D E2
x x y y z z
x x y y z z
x x y y z z
D E = D E + D E + D E
x x y y z zD t E t D E + D E + D E
or *1Re D E
2D t E t
, , , ,Re D j tx y z x y zD t e etc.
Consider:
Time Average of Time-Harmonic Quantities (cont.)
*1Re D E
2D t E t
*1Re E H
2E t H t
The result holds for both dot product and cross products.In summary,
1( ) ( ) Re VI
2*V t I t
Time Average of Time-Harmonic Quantities (cont.)
To illustrate, consider the time-average stored electric energy.
1
2EU t D t E t
*1Re D E
4EU t
*
1
21 1
Re D E2 2
EU t D t E t