Chapter 1

Review of Phasor Notation

ECE 130a

Introduction to Electromagnetics

e jjq q q= +cos sin

Complex Vectors and Time Harmonic Representation(Chapter 1.5, 1.6 Text)

Complex Numbers

i) Rectangular/Cartesian Representation

ii) Polar Representation

iii) Addition/SubtractionCartesian representation most convenient

iv) Multiplication/Division

v) Cartesian Representation

vi) Polar Representation (more convenient for multiplication and division)

c

Imaginary

Real

b

a

c a j b= +

c a j b c e c j cj= + = = +f f fcos sin

, where

real part imaginary part magnitude phase

c a b= +2 2c c e j= f

f = tan 1 bad i

Then:

c a jb= +d e jh= +

c d a e j b h+ = + + +a f a fc d a e j b h = + a f a f

Note that

where , the complex conjugate.,

ja f2 1= cc a b c* = + =2 2 2

c a jb* =

c

da jbe jh

a jbe jh

e jhe jh

ae bh j be ahe h

=

+

+=

+

+

=

+ +

+

a f a f2 2

cd a jb e jh ae bh j ah be= + + = + +a fa f a f a f

c c ej

=f 1 d d e j= f 2 c d c d e j= +f f1 2d i

c

dc

de

j=

f f1 2d i

vii) Complex Representation of Time-harmonic Scalars Consider:

Sinusoidally Time Varying Fields (1.6 Text)

1)

is a scalar field that depends on time only, but not on position (location). This is not a wave.

For example, this is the voltage in any electric household plug. Note that the phase simply alters the starting point.

The circular frequency where f is the frequency in Hz. Here, f = 60 hertz (Hz) and

pi= 2 f

v t A toa f b g= +cos w f

and

amplitude angular/radianfrequency

phase

radianfrequency frequency

$ $ $ *a aa=12

Re $a1

magnitudeamplitude

Note: radians p4

45= FHIK

f A t t= + = +FHIKcos cosw f p

pa f 110 1204

f

t in units of pw

FH

IK

vo

ltage

pi= 2 f , pi= =120 377 rad / sec

2) Waves -- travelling waves (main subject of this course)

This is a scalar traveling wave.

or

Repeats

propagation constant

wavelengthphase velocity period

v V t zo= -FH

IKcosw v

= -V t kzo cos wa f V t zo cos w b-a f= -FH

IKV f t

zo cos2p l z z +

k for b w p pl

a f = =v v

2 2

v = f

T f 2 1piv =

=

2pi =

z in units of pb

FHG

IKJ

t = 0with

3) Vector Travelling Waves -- electromagnetic wavesExamples:

Example: (x-directed planewave, - y propagation)

This is a vector travelling wave field.

rH e H

ax t z e H

ax t zx o z= -

FH

IK - +

FH

IK -$ sin cos $ cos sin

p w b p w ba f a f1

rE e E t k zx o= - +

$ cos w 45c h

rE e e E t k zx y o= +

FHG

IKJ -

12

32

$ $ cos wa f

rE e E t y e E t kyx o x o= -

FH

IK = -$ cos $ cosw wv a f

(ii)

(iii)

(iv)

(i)(x-oriented field, + y propagation)

(waveguide example)

(x-oriented, + z propagation)

$ $ $e e eE x y= +12

32

x

y

13212

$eE

v

=

rE y t e t yx, $ cosa f c h= + + 100 45w b

= +FHIK +

LNM

OQP

$ cose ty

x100 45wv

t = 0

Propagates in thedirection

y

y in units of pb

FHG

IKJ

t = pw4

At

At

Note that this vector field:(i) always points in the direction (reverses, of course).(ii) propagates in the y direction.(iii)spatially depends only on the y coordinate (which is the direction of propaga-

tion).

The magnitude E, that is, the density of these lines (or the length of the arrows), var-ies sinusoidally, both in time and in the variable y direction of propagation.

t y= + 0 45: cos bc ht y y= + = -p

wb b

490: cos sinc h

$ex

x

y

z

x

y

z

uniformin x-yplane

2pi

t

100 pi

3piAt y = 38

Ex

3pi

2pi

pi

At

100

t

At

:

:

y = 38

= -FHIK =$ cos $ sine t e tx x100 2 100w

p w

rE e tx= - +

FH

IK$ cos100

2 38 4

w pl

l p

y = 0rE e tx= +

FH

IK$ cos100 4w

p

Method of Phasors (Set your phasors on Stun, Mr. Spock!)Background material needed:

Review of complex algebraEulers Law:

Suppose we have a field that varies sinusoidally in time and/or space. For a scalar field, voltage, for example, we have:

These are real measurable quantities.

We represent them by writing as follows:

Now we do either of (a) or (b):(a) Drop . The remaining is called the complex voltage (includ-

ing time dependence) or voltage phasor (including time dependence).

(b) Drop and

The resultant is called the (complex) phasor. Note that, for this example, the complex phasor happens to be real. (PHASORS are written in bold type-face.)

If

is a complex quantity in polar representation.

e x j xjx = +cos sin Re Ime x e xjx jx= =cos , sin

(a sinusoidal voltage)(a voltage wave on a transmission line)

v V to= cosv V t kzo= -cos wa f

v V eoj t= Re wm r

Re v = V eo j t

v V eoj t= Re wm r Re e j t

v = Vo

v = +V to cos w fa f= =+Re ReV e V e eo

j to

j t jw f w fa f

= Re V e eo j j tf w

V eo jf

Vo

Re

Im e jjf f f= +cos sine jf f f= + =cos sin2 2 1

Graphical Picture of Complex Fields

Consider a vector field with harmonic time dependence:

If we picture the entire time dependent complex field in the complex plane, we see that it rotates at angular velocity

Note that the direction of the complex field on this complex plane has no relation-ship to its direction in real physical space, which here is constant

The entire complex field, including time dependence, is often called Rotating Phasor. The real field is the projection on the axis.Amplitude, Time Average, Mean Square and Root Mean Square(a) Amplitude is the maximum value of the time oscillation (real number).(i)

rE e E ex o

j t= $ 30 w$ .e E ex o

j 30 6

p

This is another notation for

Complex form:

phasor amplitude (also a vector)

rE e E t e E e ex o x o

j t j= + LNM

OQP

$ cos $w wp

30 6c h Re

.

Im

30

t = 0

t =pi

t =pi3

Re

Eo

$ .ex

e j t

Re

Amplitude magnitude of phasor

phasor

real

Amplitude is 100E =

r rE E *

12

rE e t zx= - +$ cos100 w b fa frE = $e ex

j100 f

(ii)

Amplitude of the total wave is

Now, real form because:

(b) Time average of a harmonic quantity is always zero.

(c) Mean square is not zero.

Using phasors for cannot give correct answers since phasors are not valid for a nonlinear situation. Try it and see:

2 waves in opposite direction, phasor form

This is the phasor

v = +100 100e ej z j z

Use sin x e ejjx jx

= -LNM

OQP

-

2

(simplified).= - - = -+ -2002 200

jj e e j z

j z j zb b bsin

function of z!

v = - = - =200 200 200 200j z j z j z zsin sin sin sinb b b ba fa f

v z t z t, sin sina f = 200 b w

arrows showtime oscillations

z

= 200sin sin z t= +Re 200 200j z t z tsin cos sin sin

v z t j z e j t, sina f = -Re 200 b w

phasor

T f=1

v t V toa f a f= +cos w f v = V eo jfv t

TV to

Ta f a f= + z1 00 cos w f

= Vo2

2

v tT

V t dtoT2 2

021a f a f= +z cos w f

= + +LNMOQPz1 12 12 2 220T V t dto

Tcos w fa f

v 2

butv V to2 2 2= cos , v 2 2 2 2= +V eo j t jw f

Does not give when real part is taken for the time average. But the following trick works:

For the time average: (Note: It gives only the time average.)

(This works for vector phasors, too!)

If:

The Mean Square is related to Average Power (averaged over whole cycles).The same trick works for any other second order quantity.

(d) Root Mean Square (RMS)

For harmonic fields, RMS value amplitude.

Example:

If:

Calculate , using both real fields and phasors.

Solution:

(a) Using real fields

cos 2

v t V e V e Vo j o j o221

212 2( ) = = =

-Re Revv * f f

r r rE t z, *a f d i2 12= Re E E Vector dot product

phasorAA

rE = +$ $e E e Ex x y y

rE E e eo

jkzx=

- $

Note: The is necessary here,since and may have a phasedifference.

v t i ta f a f = Re 12

v i *" "Re

v i

orv t2a f E ta f 2

=12

cylindrical coordinatesrE e t zr= -$ cos100 w ba f r z, ,fa frH e t z= - - $ . cosf w b0 3 60a f

r rE H

r rE H e t z t zz = - - - $ cos cos30 60w b w ba f a f

Use product of

(b) Using phasors

Phasor examples with waves

(1) Space dependence is due to propagation only(a) Uniform plane EM wave -- vector(b) Transmission line wave -- scalar voltage or current

(1a) Propagation is only in one direction

Real field:

Complex field:

Phasor field: (two options possible)(i) Drop entire dependence

cos cos cos cosA B A B A B = - + +12

a f a fm rr rE H e t zz = + - - $ cos cos15 60 2 2 60w ba f

r rE H ez = +$ .7 5 0

Phasors corresponding tothe real

r rE t z H t z, , ,a f a f

rE = $er100

r r r rE H = 1

2Re E H *d i

12

15 3E H = +* $e ez jp

12

15 60 7 5Re E H = =* $ cos $ .e ez z

rH =

-$ .e e

jf

p

0 3 3

directionPropagation is in + $z =v

v =

rE z t e E t z

ve E t zx o x o, $ cos $ cosa f a f= -FH

IK +

RSTUVW = - +w f w b f

rE z t e E ex o

j t z, $a f a f= - +w b f

= -$e E e ex oj j t zf w ba f

ej t zw b-a f

Phasor: (algebraic) (schematic)

= - + $ cose t zx 100 30w ba f

For example,rE t z e ex

j t z, $a f d i= - +Re 100 6w b p

rE = $ $e e ex x

j100 30 100 30

rE = $e E ex o

jf $e Ex o f

Another example:

Another example:

(ii) Drop only dependence

We can see that and both are phases. The phase changes with propaga-tion. The phase rotates in the complex plane.

For waves travelling in one direction only, we dont usually use this. Instead, we drop off of -- usually (but not always).Example:(1b) Transmission line waveWaves travelling in two opposite directions

We only have one option: Drop dependence, since dependence is not the same on the two parts.

Phasor:

= - - = - + 50 50 90$ sin $ cose t z e t zy yw b w ba f a frE t z e j e e ey j t z y j t z, $ $a f a f c h= =- - +Re Re5 0 5 0 2w b w b prE = = $ $e j ey y50 50 90

j j= + = cos sin ,f f fwhen 90a f

= - + -100 $ cos $ sine t z e t zx yw b w ba f a f

rE t z e j e ex y j t z, $ $a f c h a f= - -Re 100 w brE = -$ $e j ex yc h100

e2 12 12

2100= = - + Re r rE E e je e jex y x y* $ $ $ $a f c h c h= + =12

2 2100 1 1 100a f a f a fe j t

rE z t e E t zx o, $ cosa f a f= - +w b frE z t e E ex o

j t z, $a f a f= - +Re w b f

rE = - +$e E ex o

j z jb f

z f

ej t zw b-a f

v t z V t z V t zo o, cos cosa f a f a f= - + + +w b w b fGv t z V e e V e e eo j z j t o j j z j t,( ) = +- +Re b w f b wG

e j t z

= +V e eo j z j z 1 2~ ~ = complex reflection

coefficient

v = +- +V e e eo j z j j zb f bG= G e jf

(2) Space dependence may exist perpendicular to the direction of propagation; for example, laser beams, guided electromagnetic waves, optical waves in fibers. Not the subject of ECE 130A, but used in ECE 130B.Example of (2) with (i) propagation in one direction: E.M. wave in rectangular waveguide

Another example:

Phasor:Significance of j : and components are out of phase

Real field is:

Similar fields exist in any guided wave, such as waves in optical fibers!

(3) Modulated (time dependent amplitude) wavesAll the examples brought in (1) and (2) were waves of a single sinusoidal fre-

quency and amplitude constant in time.In communications, we impress information onto a single frequency wave by

modulating its amplitude in time. The original sinusoid is the carrier and the mod-ulation of the amplitude contains the information.

1

diagram of [ ] quantity in complex plane

Im part

Re part called Crank

Diagram

1 + ~~ =

e j z2 b

G e jf

:Drop

x a