BiotSavart Law
Consider two current loops, C1 and C2, with currents I1 and I2. Each loop
can be subdivided into current elements, characterized by the product of the
current and the vector differential length, such as I1d 1 and I2d 2.
It has been experim
NAME:
QULZ
".A^"Aq\,1.
ECEN 322
I
Solve the differential equation
5 by using the phasor technique.
x
1o6
fi* t i.: usin(106t)
osw't'e
W*r*
Ras k
p
(c
4
T
I
.a1r'
: 43 ?4, (/ o6* y^) =R*l 3;i.Y' '* ta'4] Cr\) 4oG *y, \
"* /d* +)acl
dL
^
l
(
a
6r t;6
ECEN 322 Homework 1
1. Given two vectors
A =x 3 y + 2z
B=
3 x + 4 y z
(a) Calculate the smaller angle between the two vectors.
(b) Calculate the projection of A (scalar component of A) along B.
(c) Find a unit vector perpendicular to the plane containing
ECEN 322
Homework #9
1. A transmission line has the following per unit length parameters: L = 0.2 H/m, C = 300
pF/m, R = 5 /m, and G = 0.01 S/m.
(a) Calculate the propagation constant and characteristic impedance of the line at 500
MHz.
(b) Recalculate th
ECEN 322
Homework #3
Continue reading the sections of the book given in the syllabus up through Section 7.3
1. Starting with Maxwells equations in the frequency domain derive the vector Helmholtz equation
for the magnetic field H. At each step where you u
ECEN 322
Homework #5
1. A plane wave propagating in the +xdirection and polarized in the ydirection, with an
amplitude of 10 V/m in a nonmagnetic ( r = 1) region with = 1.0 S/m, r = 12 .
(a) Calculate the exact and approximate attenuation ( ) and propa
ECEN 322
Homework #7
1. A measurement shows that a certain material with a loss tangent greater than 100 has a
permeability = o , an intrinsic impedance of 28.1 e
(a) The conductivity of the material
(b) The wavelength in the material
(c) The phase veloci
ECEN 322
Homework 6
(1) Suppose a plane wave from region 1 is normally incident, i = 0 , on a planar boundary as
shown in the figure below:
(a) If both regions contain the same materials, ie 1 = 2 , 1 = 2 , =
1 =
2 0 , starting
with the formulas of normal
ECEN 322
Homework #2
1. Convert the following time domain expressions for the electric and magnetic field into frequency
domain expressions:
(a)=
E (t ) 5cos(t 30 z ) x V/m
(b) E (t=
) 3sin(t + 4 z )x + 2 cos(t + 4 z )y V/m
(c) E (t ) 12e 200 z cos(t 400
ECEN 322
Homework #4
1. Given a 3 GHz wave traveling in free space at angles = 30o , = 60o , = 90o with respect to the x,
y, and z axes. Find an expression for (a) the wave vector k (b) the wave number k
2. Assume that E (2x 4 y 4z )e j (2 x 3 y 4 z ) V/m
NAME:
QUIZ
1
ECEN 322
Find the component of the vector drawn from (5,0,3) to (3,3,2) along the direction of the vector
drawn frorn (6, 2,4) to (3, 3,6). Show your work.
*,P.,*Ar7z ( 7,v,2 )
frT,
(1,),6>
l
cfw_*
qnt5,o, \>
F)^L
t
e
ttnTz
'9
r, c6 r?, *)
.
NAME: SOLUTION
QUIZ 4
ECEN 322
Given
a vector field E = x
y + y x, evaluate the line integral
R
E d` from P1 (0, 0, 1) to P2 (1, 1, 1)
(a) along the straight line joining the two points, and
(b) along the parabola y = x2
Is this E a conservative field? E
NAME:
QU1Z
Given the
ECEN 322
fields
Er
find .E
: Et *
:
0.b sin(o,t)a"
, Ez: 0.b cos(wtrf\)a,
Ezby using the phasor technique. Show your work.
9".e,'ut[*r
n =
0,6 (Ft cut,.Yt 9x  g, = 0.6 ;%*n
x
'rcfw_E
Er= o'q 2 *x
E =E^u Er= 0,6 GiT/\ "a"A> t*

.
So P^rAqv
QULZ s
ECEN 322
An infinite current sheet Js : ar5 coinciding with the ryplane separates air (region l, z > 0)
from a medlmyith ltr, : 2 (region 2, z < 0). Given that II1 : ar30 + a, 40 + a"20
[A/m], fincfw_,'lf2, /where IL and.El2 denote the m
S,8'^ !+a'7w
NAME:
ECEN 322
QULZ 5
For the vector field ,4 : ar(z
*
L)
(o) Evaluate $, A. ds, where S is a closed surface of a hemispherical dome of radius
R:5, centered at the origin, with its flat base in the ryplane.,S is oriented out of
the volume
I
ECEN 322
Homework 10
1. Given an air filled transmission line with a characteristic impedance is Z=
50 , a
o
frequency of 3 GHz, a physical length of 1 cm, and a load impedance Z L =22 + j 34 ,what
is the (a) the input impedance Zin (b) input voltage (c)
ECEN 322
Homework #8
1. For the case of oblique incidence of a uniform plane wave with parallel polarization incident on a
plane boundary (see Fig. 1) with=
2.25 o and =
1 =
=
o , assume H oi = 0.053 A/m,
o , 2
1
2
f = 100 MHz, and i = 30o .
a. Find the r
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', cfw_.a',*r
'. M"  .r\rwry^ryF 1
d
q f'dl 
n
. V
lq.p8s
dre
+
,
,
I
J
g,^
\
ffi
e16.\
@t
^t^(
4j'^r"
&d^,
re
""'+
S'cfw_aFG"R
^legOla!
e^ttl<*a
lrg P
+*\ ^
lvofl=W
l,6zA* lSA"ez /@)
g
fl
Yxg'
f
S
S a^f"^":
De
K
DB
C
r_.dg
@sWvuar
'
Input Impedance of a Transmission Line
We define the TL impedance at z (looking towards the load) as
V (z)
Z(z) =
[]
I(z)
Recall that for a lossless line
V (z)
= V0+
j z
e
+e
j z
= V0+e j z
1+e
j2 z
V+
+
V
j
z
j
z
j
z
j2
z
0
0
=
e
e
=
e
1e
I(z)
Z0
Z0
w
Review of Complex Numbers
A complex number z is an ordered pair
(x, y) of two real numbers, x and y, where
x = Re(z) (real part of z) and y = Im(z)
(imaginary part of z). Can be represented
as a vector in the complex plane. The
Cartesian form is
z = x + j
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8
ECEN 322
The magnetic field of a plane wave propagating in a lossless dielectric
S.!t^A.,n
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QUIZ4
4
vn,
le'AS
ECEN 322
Consider jr.lossless,50ohm transmission line, using apo
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QUIZ 3 ECEN 322
Measurements on a 50ohm slotted line indicate that the VSWR S = 2.5, the d
SLlArAvl'
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solemnly swear that
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quiz.
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ECEN 322
QUIZ s
Compute the gradient of the scalar field
V
:2r2 + Ba + at/Z
at the
ge
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is expressly prohibited.
On my honor,
I
solemnly swear that
I
have neither received nor given unauthorized aid on this
exam.
Signatu
WW
Rules: You may refer to your own halfpage formula sheet, but nothing else. In particular, the use
of any old quiz/homework/exam solutions is expressly prohibited.
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quiz.
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QUIZ 2 ECEN 322
A 22.51 cm long, lossless, 50ohm transmission line is terminated in a load impedance of 50+j25
ohms. The signal
NAME: S9 W M
EXAM 2 ECEN 322
Problem 1
Given a vector eld A = a? my 3) 2:3, verify the Stokes theorem over a quartercircular disk with
a radius a = 3 in the rst quadrant of the Cartesian coordinate system.
Problem 2
A direct current I ows in free space i
NAME:
EXAM 2
Problem 1
For the vector eld E 2 cfw_Easy 32(332 + 23/2)
conrm the Stokes theorem on triangular
surface and contour shown in Fig. 1.
Problem 2
Two halfspace regions are separated by the
surface S , which is the z = 0 plane (i.e., the
st: y pl
NAME:
Exam 2 ECEN 322
Problem 1
Two lossless homogeneous media with 6n = 12, m1 = 2 and 67.2 = 9, #72 = 1 are in contact at the
a: = 0 plane. The electric and magnetic elds in medium 1 just above the boundary are measured
as E1 = 30.35 + 20.1, 60.2 and H1
NAME:
Exam 2 ECEN 322
Problem 1
Two lossless homogeneous media with 6n = 12, m1 = 2 and 67.2 = 9, #72 = 1 are in contact at the
a: = 0 plane. The electric and magnetic elds in medium 1 just above the boundary are measured
as E1 = 30.35 + 20.1, 60.2 and H1
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Rules: Closed books, closed notes. However, you may use a onepage formula sheet prepared by
yourself. This formula sheet must not comprise any problem solutions or solution procedures.
On my honor, I solemnly swear that I have