Notes 2 Complex Vectors

ECE 3317Prof. Ji Chen

Adapted from notes by Prof. Stuart A. Long

Spring 2014

1

• V(t) is a time-varying function.

• V is a phasor (complex number).

• A bar underneath indicates a vector: V(t), V.

Notation

Appendices A, B, C, and D in the Shen & Kong text book list frequently used symbols and their units.

2

Complex Numbers

Real part

| | jc a j b c e

MagnitudeImaginary part

Phase (always in radians)

1j

cos sinje j Euler's identity:

cos

sin

a c

b c

Im

c a jb

sinc

cosc

c

b

Rea

2 2 c a b

3

Complex Numbers

** - - | | jc a jb a jb c e

Im

* c a jb

Re

a

b

Im

c a jb

Re

a

b

Complex conjugate

4

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

5

1

1 2 1 2

( )11

2 2

j

j

c c c c e

cce

c c

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

6

1/2

/2

j

j

c c e

c e

Square Root

where n is an integer

The complex square root will have two possible values.

(The principal branch is unique.)

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)

Principle square root

7

0( ) = cos( )V t V t

0V jV e Define the phasor :

Time-Harmonic Quantities

2

f

Amplitude Angular Phase

Frequency

We then have

( ) = Re V j tV t e

0( ) = Re j j tV t V e e

From Euler’s identity: 1 T

f

8

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

9

B

B

A

V(t)

t

C CA

Re

Im

Graphical Illustration

0( ) = cos( )V t V t

The complex number V

0V

0( ) = Re

Re V

j t

j t

V t V e

e

0V = jV e

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( ) ( ) U+V

( ) V

( ) ( ) UV

U t V t

V t jt

U t V t

Note :

However,

Time-Harmonic Quantities (cont.)

( ) VV t

There are no time derivatives in the phasor domain!

All phasors are complex numbers, but not all complex numbers are phasors!

This assumes that the two sinusoidal signals are at the same frequency.

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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 a complex vector.

Complex Vectors

ˆˆV ˆ zx yy zx

j jj

V Ve V ee

yx z

ˆ ˆ ˆ( ) cos( ) cos( ) cos( )

ˆ ˆ ˆRe

Re V

yx z

x x y y z z

jj j j tx y z

j t

V t V t V t zV t

V e V e V e e

e

x y

x y z

12

To see this:

( ) VV t Hence

Example 1.15 (Shen & Kong)

( ) 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

xωt = 0

The vector rotates with time!

ˆ ˆA jx + yAssume

Find the corresponding time-domain vector

ˆ ˆ( ) cos sinA t x t y t

2 2 2 2( ) cos sin 1x yA t A t tA

13

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).

For a fixed value of z, the electric field vector rotates with time.

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E (z,t)

z

Example 1.16 (Shen & Kong)

ˆ ˆ ˆ ˆA ( ) cos sin

ˆ ˆ ˆ ˆ ˆ ˆ B ( ) ( ) sin cos

ˆA B (A B A B )

ˆ (1)( 1) ( )( )

ˆ ( 1) ( 1)

0

A B

and

x y y x

j A t t t

j j j B t t t

z

z j j

z

Recall from example 1.15

Let

x y x y

x y x y x y

2 2

0 ( ) ( ) 0

ˆ ˆ ( ) ( ) cos s

in

A t B t

A t B t z t t z

However,

We have to be careful about drawing conclusions from cross and dot products in the phasor domain!

15

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

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

16

Hence

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

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:

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Sinusoidal (time ave = 0)

**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

Next, consider

Hence,

Recall that

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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

Hence *1Re D E

2D t E t

, , , ,Re D j tx y z x y zD t e etc.

Consider:

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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.

, , , Re D , , j tD x y z t x y z e etc.

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Time Average of Time-Harmonic Quantities (cont.)

To illustrate, consider the time-average stored electric energy for a sinusoidal field.

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

21

(from ECE 2317)

Time Average of Time-Harmonic Quantities (cont.)

Similarly,

*1Re B H

4HU t

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