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