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6.013/ESD.013J Electromagnetics and Applications, Fall 2005
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Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.013 Electromagnetics and Applications
Problem Set #11
Fall Term 2005
Issued: 11/29/2005
Due: 12/9/05
Suggested Reading Assignment: Staelin, Sections 6.1-6.4, 10.1, 10.2, 10.4
Final Exam: Wednesday, Dec. 21, 2005, 1:30-4:30pm.
Problem 11.1
A popular 1-MHz AM radio station in the middle of Kansas has a single transmitting antenna on a flat
prairie that radiates 100kW isotropically (equally in all directions) over the upper 2 steradians (i.e., this
station has no underground audience.) The matched input impedance (the radiation resistance Rr ) of this
antenna is ~70 ohms, and it is driven by V0 sin t volts at maximum power.
a) What is V0 [Volts] ?
b) What is the radiated intensity I [W / m 2 ] 50 kilometers from this antenna?
c) What is the maximum power Pr that can be received from this station by an antenna 50 km away
with an effective area A = 10 m2 ?
Problem 11.2
A short dipole antenna, 10 cm in length and aligned along the zˆaxis, is driven uniformly along its length
with a sinusoidal current of peak value 1 amp.
a) What is the electric field E ( r , , t ) in the far field?
b) At what frequency would this antenna radiate 1 watt of power?
c) If a receiver with effective area A = 0.1 m2 needed 10-20 watts for successful reception, how far
away could it be and still receive signals from the 1 watt dipole? In what direction?
Problem 11.3
An antenna consists of two short dipoles, oriented along the z-axis and separated along the y-axis by a
distance a. They are driven in phase, each with a current I 0 and an effective length deff , ( deff  ), at
an angular frequency of . (Assume that each antenna radiates as it would in the absence of the other.)
a
a) What is the intensity of the radiation in the far field as a function of angle in the x-y plane?
b) For a 2, at what angles max and min is the intensity a relative maximum or zero?
Page 1 of 7
Problem 11.4
A "turnstile" antenna consists of two short Hertzian dipoles driven at an angular frequency 
and oriented
at right angles to each other as shown in the figure below. One dipole, oriented along the x-axis is driven
ˆ
ˆ
ˆˆ
ˆ
with a current I1 Iˆ
0 x and the other, oriented along the y-axis is driven with I2 jI0 y . Both have the
same effective length d eff .
a) Find the complex amplitude of the total electric field on the +z axis in the far field. (Express your
answer in Cartesian coordinates with unit vectors xˆˆ
, y, and zˆ.)
b) Why is the result of part (a) called left-handed circular polarization (LHCP) for +z directed waves
along the +z axis?
c) What is the complex amplitude of the magnetic field on the +z axis in the far field?
d) What is the intensity of the radiation on the z axis in the far field?
1
ˆ ˆ
Hint: S  Re 
E H 




2
Problem 11.5
Sketch the far field radiation patterns in the x-y plane for each of the following short dipole antenna
arrays. The identical dipoles are directed in either the +z or -z directions, as indicated, and the

so that its complex
2
amplitude is j. In each case find the angles corresponding to nulls (
n ) and peaks ( 
p ). If the peaks
currents have equal amplitudes of 
1 . In part (b) one current has a phase of
are unequal, also evaluate their relative values.
Page 2 of 7
Problem 11.6
Using the format of Problem 11.5 design two-dipole arrays that could produce the far field antenna gain
patterns illustrated below. The two dipoles have the same current amplitude but may differ in phase.
Find the spacing a between the two dipoles and their relative phase that results in the radiation patterns
shown in parts (a) - (c).
Page 3 of 7
6.013 Final Exam Formula Sheet
December 21, 2005
Cartesian Coordinates (x,y,z):





xˆ yˆ zˆ 

x

y

z
A y Az
A

A x 

x
y
z
A  

Ay 
Az 
Ax 
A
A 
A xˆ
 y
yˆ
 z
zˆ
 x





z  z
x  
x
y 
 y
2
2
2
2  

x2 y2 z2
Cylindrical coordinates (r,,z):
 
 z 
ˆ1 
rˆ
ˆ
r
r 


z
rA r  1 A Az
1 

A


r 
r
r  z
ˆ zˆ
rˆ r


rA

1 Az A 
A

A



A


1
1
ˆ r  z zˆ 
A rˆ


 r  det 
r  
z
r 
z 
z

r  r 
r
 r
  
 
A r rA  A z
 
2
2
1  
1 

2
r


2
2
2
r r r
r  
z
Spherical coordinates (r,,):
ˆ1  
ˆ 1 
rˆ 
r
r  r sin 

A  12
r
 
 r 2A r

sin A  1 A
 1

r
r sin 

r sin  


sin A  
 1 
rA  1 
A  ˆ
A r 1 
Ar 
ˆ  rA 
A rˆ 1 
 




r sin  


  r sin 
 r 
r  r 
r


ˆ r sin 
ˆ
rˆ r
1
2
det 
r  
r sin 
A
rA
r sin 
A
r
 




2
2
1  2 
1


1

 
r

sin 

2 
2
2
2

r
 r sin 2
r r
r sin 
Gauss’ Divergence Theorem:
G dv 
Gnˆda
V
A
Stokes’ Theorem:
G

nˆda 
G
d


A
C
Vector Algebra:
xˆ
x yˆ
y zˆ
z
A B Ax Bx Ay By Az Bz
 A 0

 A
 A 2 A

Page 4 of 7
Basic Equations for Electromagnetics and Applications
Fundamentals

f q 
E v o H 
N (Force on point charge)
E1//  E 2 // 0
E 
B
t
H1//  H 2 // Js nˆ
d
E ds  B da


c
dt A
H J D 
t
B1  B2  0
nˆ
D1 D2 s
d
Electromagnetic Quasistatics
E 

r , 
r 
r / 4
r
r 

dv 
V

f
2


C = Q/V = A/d [F]
L = /I
i(t) = C dv(t)/dt
v(t) = L di(t)/dt = d/dt
2
2
we = Cv (t)/2; wm = Li (t)/2
2
Lsolenoid = N A/W
= RC, = L/R
B da (per turn)
A
J t
-1
E = electric field (Vm )
-1
H = magnetic field (Am )
-2
D = electric displacement (Cm )
B = magnetic flux density (T)
-2
Tesla (T) = Weber m = 10,000 gauss
= charge density (Cm -3)
J = current density (Am -2)
-1
= conductivity (Siemens m )
KCL : i Ii (t) 0 at node
-1
J s = surface current density (Am )
KVL : i Vi (t) 0 around loop
-2
s = surface charge density (Cm )

o = 8.85 10
-7
Q 0 wT / Pdiss 0 / 
-1
Fm
0.5
0 LC 
-1
o = 410 Hm
c = (
o
o)
-0.5
8
3 10 ms
V 2 
t / R kT
-1
e = -1.60 10-19 C
o 377 ohms = ( o/o)0.5
Electromagnetic Waves
  t E 0 [Wave Eqn.]
2
2
2 2 t 2 E 0 [Wave Eqn.]
2
Ey(z,t) = E+ (z-ct) + E-(z+ct) = Re{Ey (z)ejt}
Hx(z,t) = o- 1[E+ (z-ct)-E-(z+ct)] [or(t-kz) or (t-z/c)]

E 2 H 2dv
E H da 
d dt 

A
V

E J dv (Poynting Theorem)
V
2
2

2 k 2 
Eˆ0, EˆEˆo e jk r
k = (
)0.5 = /c = 2/
2
2
2
2
2
kx + ky + kz = ko = 
vp = /k, vg = (k/)-1
r i
sint / sini ki / kt ni / nt
Media and Boundaries
D 
E P
o
c sin1 
nt / ni 
D f , 
B tan 1 
t / i  for TM
ˆ ˆ xjkz z
  E E Te
0.5

f p
oE 
c
P 
p , J E
t
i
k k 
jk 
T 1
B H o 
H M 



1 p  , p Ne m (plasma)
TTE 2 / 
1 
i cost / t cosi 

eff 
1j/ 
TTM 2 / 
1 
t cost / i cosi 

2
2
2
1
2
0 = if = 
H ds 
J da  D da


c
A
dt A
D  
D da 
dv
A
V
B 0  B da 0
A
-12
nˆ
0.5
Skin depth = 
2 /  
m
0.5
Page 5 of 7
Radiating Waves
Wireless Communications and Radar
1 A
2 A  2 2 J f
c 
t
2

1 
2

 f
2
2
c 
t

2
G(,) = P r/(P R/4r2 )
PR 
P

, , r
r2 sin d
d
4 r
J f 
t rQP / c 
dV 
A 
V
4rQP
Prec = Pr (,)Ae(,)
f 
t rQP / c 
dV 

V
4
rQP
Ae (
, ) G(
, )2 4

A
E 
 , B A

t
r
ˆ(r ) 


ˆ(r )e jkr 
/ 4
r 'r dV 
V


ˆ  
A(r)
r e jk r ' r 4r 'r dV '
Jˆ
V'
ˆ  H
ˆ 
ˆ 4r 
E
jkId
ejkr sin 
ff
 ff
2 ˆ
2
ˆ x, y, z 
ˆ 




ˆ, x, y, z, t Re 

e jt 


ˆ
2 ˆ
2
j t 
ˆ
ˆ

 A 
A J , A 
x, y, z, t Re A 
x, y, z e




G (
, ) 1.5sin2  (Hertzian Dipole)
i2 (t)
R r PR
jkx x jky y
jkr
Eff 
0 je r Et (x, y)e
dxdy
A
ˆ  a Eejkri (element factor)(array f)
E
z
i i
Ebit ~4 10
-20
[J]
Z12 Z21 if reciprocity
At o , we  wm
Forces, Motors, and Generators
J 
E v B
 2 
2
ˆ 4

H
dv
V
w e  Eˆ 4 dv
V
wm
F I B 
Nm-1 (force per unit length)
Q n n w Tn Pn n 2 n
E v B inside perfectly conducting wire 
 
f mnp 
c 2 
m a 
n b 
p d
2
2
-2
Max f/A = B /2, D /2[Nm ]
dw
vi  T f dz
dt
dt
f = ma = d(mv)/dt

2
sn = j n - n
Acoustics
P = fv = T(Watts)
P Po p, U Uo u
T = I d/dt
p 
u t
o
I i mi ri
u 
1
Po 

p 
t
2
FE E 
Nm 1 

Force per unit length on line charge 
2 k 22 t 2 p 0
1 2
1 q2
WM 
, x 
; WE 
q, x 
2 L 
x
2 C 
x
2
k 2 2 cs 
Po
o 
W
fM 
, x  M
x
W
fE ( q, x)  E
x

2 0.5
2
q
x
1
d
1 dL 
 2
x  I 2
1/ L 
2
dx
2
dx

x
1
d
1 dC 
 q2
1/ C 
x  v2

2 dx
2
dx
2
cs vp vg 

Po o  or 
K o 
0.5
0.5
s = p/u = oc s = (o
P o)
s = (oK)
Optical Communications
0.5
gases
solids, liquids
p, u continuous at boundaries
E = hf, photons or phonons
p = p+e
hf/c = momentum [kg ms -1]
uz = s (p+e
dn 2 dt 
An 2 B 
n 2 n 1 



0.5
-jkz
-1
+jkz
+ p- e
-jkz
– p-e
+jkz
)
up da 
d dt 
o u


A
V
2

2 p 2 2 
Po dV
Page 6 of 7
Transmission Lines
Time Domain
v(z,t)/z = -Li(z,t)/t
i(z,t)/z = -Cv(z,t)/t
2
2
2
2
v/z = LC v/t
v(z,t) = V+ (t – z/c) + V-(t + z/c)
i(z,t) = Yo [V+ (t – z/c) – V-(t + z/c)]
c = (LC)
-0.5
-0.5
= (
)
Zo = Y o-1 = (L/C)0.5
L = V-/V+ = (R L – Zo)/(RL + Zo)
Frequency Domain
ˆ =0
(d 2 /dz 2 +2 LC)V(z)
ˆ = Vˆe -jkz + V
ˆe +jkz , v( z, t ) Re 
V(z)
Vˆ
( z )e j t 
+


ˆ
ˆe-jkz - Vˆe +jkz ], i( z, t) Re 
ˆz) e j t 
I(z) = Y0[V
I(
+


k = 2/= /c = ()0.5
ˆ ˆ
Z(z) V(z)
I(z) Zo Zn (z)
Zn (z) 
1(z)
1 (z)R n jX n
(z) 
V V
e
2 jkz

Z n (z) 1
Z n (z) 1
Z(z) Zo ZL jZo tan kz 
Zo jZL tan kz 
VSWR Vmax Vmin
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