Chapter 1
Review of Phasor Notation
ECE 130a
Introduction to Electromagnetics
e jq = cos q + j sin q
Complex Vectors and Time Harmonic Representation
(Chapter 1.5, 1.6 Text)
Imaginary
Complex Numbers
i) Rectangular/Cartesian Representation
c = a + jb
ii) Polar Representation
c = c e , where c =
jf
b
a +b
2
c
2
φ
a
Real
f = tan −1 b a
d i
c = a + j b = c e jf = c cos f + j c sin f
real part
imaginary part
magnitude phase
iii) Addition/Subtraction
Cartesian representation most convenient
c = a + jb
d = e + jh
Then: c + d = a + e + j b + h
a f a f
c − d = aa − ef + j ab − hf
iv) Multiplication/Division
Note that
a jf
2
= −1
cc * = a 2 + b 2 = c ,
2
where c * = a − jb , the complex conjugate.
v) Cartesian Representation
a
fa
f a
f a
f
cd = a + jb e + jh = ae − bh + j ah + be
ae + bh + j be − ah
c a + jb a + jb e − jh
=
=
=
d
e + jh
e + jh e − jh
e2 + h2
a
f a
f
vi) Polar Representation (more convenient for multiplication and division)
c= ce
jf 1
d= d e
jf 2
cd = c d e
j f 1 +f 2
d
i
c c j df 1−f 2 i
= e
d d
vii) Complex Representation of Time-harmonic Scalars
Consider:
af
b
v t = A cos w o t + f
amplitude
g
phase
angular/radian
frequency
magnitude
and ω = 2π f
radian
frequency
$ $*
a$ = aa
1
2
amplitude Re a$ 1
frequency
Sinusoidally Time Varying Fields (1.6 Text)
a
f
F
H
1) f = A cos w t + f = 110 cos 120 p t +
FNote:
H
p
radians = 45∞
4
p
4
I
K
I
K
is a scalar field that depends on time only, but not on position (location). This is not
a wave.
voltage
For example, this is the voltage in any electric household plug. Note that the
phase f simply alters the starting point.
F
H
t in units of
p
w
I
K
The circular frequency ω = 2π f , where f is the frequency in Hz. Here, f = 60
hertz (Hz) and ω = 120 π = 377 rad / sec
2) Waves -- travelling waves (main subject of this course)
F
H
I
K
f
z
v
= Vo cos w t - kz or Vo cos w t - b z
z
= Vo cos2p f t Repeats z → z + λ
l
w 2p f
2p
k or b ∫
=
=
propagation constant
v
v
l
v=λf
v = Vo cosw t -
a
F
H
a
f
I
K
a f
λ = wavelength
v = phase velocity
T≡
2π 1
≡ ≡ period
ω
f
2π
=λ
β
FG
H
z in units of
with t = 0
This is a scalar traveling wave.
p
b
IJ
K
3) Vector Travelling Waves -- electromagnetic waves
Examples:
r
y
E
= e$x Eo cos w t - k y
(i) = e$x Eo cosw t v
(x-oriented field, + y propagation)
F I
H K
IJ
K
FG
H
a
f
r
1
3
e$x +
e$y Eo cos w t - k z
(ii) E =
2
2
e$E
y
1
3
e$E = e$ x +
e$ y
3
2
2
1
2
1
2
a
f
x
r
(iii)E = e$ x E o cos w t - k z + 45 ∞
(x-oriented, + z propagation)
c
h
r
p
p
(iv) H = - e$x Ho sin
x cos w t - b z + e$z H 1 cos
x sin w t - b z
a
a
F I a
H K
f
F I a
H K
f
(waveguide example)
Example: (x-directed planewave, - y propagation)
r
E y , t = e$ x 100 cos w t + b y + 45 ∞
ω
a f
c
h
L y O
= e$ 100 cos Mw F t + I + 45 P
N H vK Q
∞
v
=β
x
This is a vector travelling wave field.
t=0
t=
p
4w
FG
H
y in units of
Propagates in the
direction
p
b
−y
IJ
K
c
At t = 0 : cos b y + 45 ∞
At t =
h
p
: cos b y + 90 ∞ = - sin b y
4w
c
h
Note that this vector field:
(i) always points in the e$x direction (reverses, of course).
(ii) propagates in the y direction.
(iii)spatially depends only on the y coordinate (which is the direction of propagation).
z
z
uniform
in x-y
plane
y
y
x
x
The magnitude E, that is, the density of these lines (or the length of the arrows), varies sinusoidally, both in time and in the variable y direction of propagation.
Ex
3π
π
100
3
ω
ω
At y = − λ
t
8
2π
3
At y = − λ :
8
At y = 0 :
ω
r
2p 3
p
E = e$x 100 cos w t l+
l 8
4
p
= e$x 100 cos w t = e$x 100 sin w t
2
F
H
F
H
r
p
E = e$x 100 cos w t +
4
F
H
100
I
K
I
K
I
K
3π
ω
π
ω
2π
ω
t
Method of Phasors (“Set your phasors on ‘Stun,’ Mr. Spock!”)
Background material needed:
Review of complex algebra
Eulers Law: e jx = cos x + j sin x
Re e jx = cos x , Im e jx = sin x
Suppose we have a field that varies sinusoidally in time and/or space. For a scalar
field, voltage, for example, we have:
v = Vo cos ω t
v = Vo cos w t - kz
a
(a sinusoidal voltage)
(a voltage wave on a transmission line)
f
These are real measurable quantities.
We represent them by writing as follows:
m
v = R e Vo e jw t
r
Now we do either of (a) or (b):
(a) Drop “ Re ”. The remaining v = Vo e jω t is called the complex voltage (including time dependence) or voltage phasor (including time dependence).
m
(b) v = R e Vo e jw t
r
Drop “ Re ” and “ e jω t ”
The resultant v = Vo is called the (complex) phasor. Note that, for this example,
the complex phasor happens to be real. (PHASORS are written in bold typeface.)
a
f
If v = Vo cos w t + f
= Re Vo e j a w t + f f = R e Vo e jw t e jf
= Re Vo e jf e jw t
Vo e jf is a complex quantity in polar representation.
e jf = cos f + j sin f
Im
Vo
e jf = cos 2 f + sin 2 f = 1
φ
Re
Graphical Picture of Complex Fields
Consider a vector field with harmonic time dependence:
LM
N
r
E = e$ x E o c o s w t + 3 0 ∞ ∫ R e e$ x E o e
c
h
jw t
e
j
p
6
OP
Q
e$ x E o – 30 ∞ e
This is another notation for
Complex form:
r
E = e$x Eo – 30 ∞ e jw t
j
p
6
.
phasor amplitude (also a vector)
If we picture the entire time dependent complex field in the complex plane, we see
that it rotates at angular velocity ω .
t=
t=
π
3ω
ω
Im
Eo
30 °
t=0
Re
π
ω
Note that the direction of the complex field on this complex plane has no relationship to its direction in real physical space, which here is constant e$x .
The entire complex field, including e jω t time dependence, is often called “Rotating
Phasor.” The real field is the projection on the Re axis.
Amplitude, Time Average, Mean Square and Root Mean Square
(a) Amplitude is the maximum value of the time oscillation (real number).
r
(i) E = e$x 100 cos w t - b z + f
real
r
E = e$x 100 e j f
phasor
a
f
Amplitude is 100
Amplitude ≡ magnitude of phasor
rr
≡ E = EE*
1
2
(ii) v = 100 e − jβ z − 100 e + jβ z
2 waves in opposite direction, phasor form
LMUse sin x = e
N
=-
jx
- e - jx
2j
OP
Q
200 j + jb z - jb z
e -e
= - 200 j sin b z ⇐ This is the phasor
2j
(simplified).
Amplitude of the total wave is
a- 200 j sin b z fa200 j sin b zf = 200 sin b z
v = - 200 j sin b z =
function of z!
a f
Now, real form v z, t = 200 sin b z sin w t because:
a f
v z , t = R e - 200 j sin b z e
jw t
= Re − 200 j sin β z cos ω t + 200 sin β z sin ω t
= 200sin β z sin ω t
arrows show
time oscillations
z
(b) Time average of a harmonic quantity is always zero.
af
a
v t = Vo cos w t + f
af
v t
=
1
T
z
T
0
f
v = Vo e jf ←
a
T=
f
Vo cos w t + f ∫ 0
phasor
1
f
(c) Mean square is not zero.
af
v2 t
z
z
1 T 2
Vo cos 2 w t + f dt
0
T
1 T 2 1 1
Vo
=
+ cos 2 w t + 2 f
T 0
2 2
Vo2
=
2
=
a
LM
N
f
a
fOPQ d t
Using phasors for v 2 cannot give correct answers since phasors are not valid for a
nonlinear situation. Try it and see:
v 2 = Vo2 cos 2 ω t , but v 2 = Vo2 e 2 jw t +2 jf
Does not give cos 2 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.)
v 2 (t ) = Re
1
1
V2
vv * = Re Vo e jf Vo e - jf = o
2
2
2
(This works for vector phasors, too!)
r
If: E = e$x E x + e$ y E y
r
2
1 r r
E t, z
= Re E E *
2
r
- jkz
E = Eo e
e$x
d◊ i
AA
a f
Vector “dot” product
phasor
The Mean Square is related to Average Power (averaged over whole cycles).
The same “trick” works for any other second order quantity.
afaf
v t i t
= Re
Note: The " Re" is necessary here,
since v and i may have a phase
difference.
1
vi*
2
(d) Root Mean Square (RMS)
af
v2 t
or
af
Et
2
For harmonic fields, RMS value =
1
amplitude.
2
Example:
r
If: E = e$r 100 cos w t - b z
cylindrical coordinates
r
H = e$f 0. 3 cos w t - b z - 60 ∞
r r
Calculate E ¥ H , using both real fields and phasors.
a
a
f
f
ar , f , z f
Solution:
(a) Using real fields
r r
E ¥ H = e$z 30 cos w t - b z cos w t - b z - 60 ∞
a
f a
f
1
cos A - B + cos A + B
Use product of cos A cos B =
2
r r
E ¥ H = e$z 15 cos 60 ∞ + cos 2w t - 2 b z - 60 ∞
r r
E ¥ H = e$z 7. 5 + 0
◊
m a
(b) Using phasors
r r
r r
1
E ¥ H = Re E ¥ H *
2
d
+ j
1
E ¥ H * = e$ z 15 e
2
◊
f
a
r
E = e$r 100
a
p
r
- j
$
H = ef 0 . 3 e 3
i
fr
f
Phasors corresponding
to
r
r
the real E t , z , H t, z
a f a f
p
3
1
Re E ¥ H * = e$z 15 cos 60 ∞ = e$z 7. 5
2
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
r
z
+ f = e$ x Eo cos w t - b z + f
Real field: E z, t = e$x Eo cos w t v
RS F
T H
a f
Propagation is in
I UV
K W
a
β=
+ z$ direction
ω
v
v=
f
ω
β
r
Complex field: E z, t = e$x Eo e j aw t - b z +f f
a f
j w t -b zf
= e$x Eo e jf e a
Phasor field: (two options possible)
(i) Drop entire e j aw t - b z f dependence
r
Phasor: E = e$x Eo e jf (algebraic) ∫ e$x Eo – f (schematic)
r
For example, E = e$ x 100 – 30 ∞ ∫ e$ x 100 e j 30 ∞
r
j dw t - b z + p 6 i
E t , z = R e e$ 100 e
a f
x
a
f
= e$x 100 cos w t - b z + 30∞
Another example:
a
f
j = cos f + j sin f , when f = 90 ∞
r
E = e$y j 50 = e$y 50 – 90 ∞
r
j w t -b z +p 2h
j w t -b zf
= e$ y R e 5 0 e c
E t , z = R e e$ y 5 0 j e a
a f
a
f
a
= - 50 e$y sin w t - b z = 50 e$y cos w t - b z + 90 ∞
r
c
f
h
Another example: E = e$ x - j e$ y 100
r
j w t -b zf
E t , z = Re 100 e$ x - j e$ y e a
a f
c
h
a
a
f
= 100 e$ x cos w t - b z + e$ y sin w t - b z
e
2
r r
= Re 2 E E * = 1 2 100
1
◊
f
a f ce$ - j e$ h◊ce$ + j e$ h
2
x
y
x
a f a1+1f = a100f
= 12 100
(ii) Drop only e jω t dependence
r
E z, t = e$x Eo cos w t - b z + f
r
E z , t = e$ x Re E o e j a w t - b z + f f
r
E = e$x Eo e - j b z + jf
a f
a
2
y
2
f
a f
We can see that − β z and f both are phases. The phase changes with propagation. The phase rotates in the complex plane.
For waves travelling in one direction only, we don’t usually use this. Instead,
we drop off of e j aw t - b z f -- usually (but not always).
Example:
(1b) Transmission line wave
Waves travelling in two opposite directions
a f
a
f
a
f
v t, z = Vo cos w t - b z + G Vo cos w t + b z + f
v (t, z) = Re Vo e - jb z e jw t + G Vo e jf e + jb z e jw t
We only have one option: Drop e jω t dependence, since β z dependence is not the
same on the two parts.
Phasor: v = Vo e - jb z + G e jf e + jb z
~
= Vo e − jβ z 1 + Γ e 2 jβ z
~
Γ = G e jf = complex reflection
coefficient
~
1 + Γ e 2 jb z
~
Γ = G e jf
Im part
1
Γ
φ
called “Crank
Diagram”
Re part
diagram of [ ] quantity in complex plane
(2) Space dependence may exist perpendicular to the direction of propagation;
for example, laser beams, guided electromagnetic waves, optical waves in
fibers. Not the subject of ECE 130A, but used in ECE 130B.
Example of (2) with (i) propagation in one direction: E.M. wave in rectangular
waveguide
a
F p xI e a
Ha K
f:
p
E = e$ E sin F xI
Ha K
f
E x, y, z, t = e$y Eo sin
j w t-b z
f
x< a
j w t-b z
Drop e a
y
o
x< a
Another example:
H = - e$x Ho sin
F p xI - j e$ H cos F p xI
Ha K
Ha K
z
1
x< a
Phasor:
Significance of j : x$ and z$ components are 90° out of phase
Real field is:
r
p
p
H = - e$x Ho sin
x cos w t - b z + e$z H1 cos
x sin w t - b z
a
a
F I a
H K
f
F I a
H K
f
Similar fields exist in any guided wave, such as waves in optical fibers!
(3) Modulated (time dependent amplitude) waves
All the examples brought in (1) and (2) were waves of a single sinusoidal frequency 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 modulation of the amplitude contains the information.
Example: Pulse code modulation
unmodulated carrier
r
z
E = e$x Eo cosw t v
F I
H K
t
z=0
a single pulse impressed onto a carrier
r
2
z
- t- z / 2 s 2
E = e$x Eo e b vg o cos wo t v
F
H
t
r
-t
At z = 0: E = e$x Eo e
2
/ 2 s o2
I
K
cos w o t
r
2
2
E = e$ x Eo e - t / 2 s o e jw o t
Phasor form:
carrier
amplitude envelope
This envelope function
is a Gaussian.
E = E⋅E * = E e
2
2
o
− t 2 / σ o2
t
The modulated wave is no longer a single frequency. By Fourier analysis, it is easy
to show that a band of frequencies about ω o are present.
Frequency Domain Representation
unmodulated wave
a single pulse modulated wave
E (w )
w
wo
a
f
E t, z = 0 =
1
2p
z
E (w )
+•
-•
a f
E w e jw t d w
w
wo
af
Ew =
z
+•
-•
af
E t e - jw t dt
a f
For the single pulse modulated field: E w =
af
Ew
2
2 p s o Eo e
-
= 2p s o2 Eo2 e
s o2
w -w o
2
b
g
2
b
g
2
- s o2 w - w o
Such a pulse is in fact a very good approximation to an actual one travelling in an
optical fiber part of an optical communication system.
Schematic of such a system:
Many km of optical fiber
Electronic
Modulator
Diode
Laser
Encoder
Information
square law detector
PhotoDetector
Amplifier
Processor
Responds to envelope of wave
only since light frequency is too fast
\ is demodulator