Vector Formulas = ( ) = a b c b c a c a b = ( ) a b c a c b a b c = ( ) a b c d a c b d a d =0 ( )=0 a )= ( ( a b c
) a
2 a
( ) = + a a a ) = + ( a a a + = ( ) + a a b a b a b b = b ( a b = a a Cartesian Coordinates +y +z x y z Ax Ay Az A = + + x y z A
ECE 222b
Applied Electromagnetics
Notes Set 3a
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego
1
Uniform Plane Waves (1)
Consider Maxwells equations:
In a lossless medium, and are rea
ECE 222b
Applied Electromagnetics
Notes Set 4a
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego
1
Dielectric Slab Waveguide (1)
y
h
x
Assumptions: 1. No variation along x
2. Wave propa
Problem 1
Problem 2
Problem 3
Problem 4:
a) The propagation constant in TL2 is 2 2 f 4 c , where
of light in TL2. The impedance seen at AA is:
Z Z 2 tan 2l2
Z AA Z 2 L
0.405 j 56.133
Z 2 Z L tan 2l2
The reflection coefficient at AA is
AA
c2 c
4 is
th
ECE 222b
Applied Electromagnetics
Notes Set 1
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego
1
Maxwells Equations (1)
Ampere (1826)
Faraday (1831)
C B dl 0 S J ds
d
c E dl
ECE 222b
Applied Electromagnetics
SET 2
Instructor: Dr. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego, IL 61801-2991
1
Review of Transmission Line (1)
Consider a segment of a transmission line:
Kirch
ECE 222b: Homework #4
Problem 1
An incident plane wave with the electric field E = z e jkx excites an impedance cylinder of
radius a and surface impedance Zs . Derive Mie series expressions for the scattered
electric and magnetic fields.
Problem 2
Give Mi
ECE 222b, Homework 2
Problem 1
A parallel polarized uniform plane wave is incident from the free space (from the right)
obliquely on a lossless dielectric slab with and . The slab is bounded (from the left)
by the PEC (perfect electric conductor) material
ECE 222b
Applied Electromagnetics
Notes Set 5
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego
1
Auxiliary Potential Functions (1)
2
Auxiliary Potential Functions (2)
3
Auxiliary Poten
ECE 222b
Applied Electromagnetics
Notes Set 3b
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego
1
Reflection and Transmission (1)
1. Normal incidence
i
E = xE0 e j1z
r
E = xRE0 e j1z
t
Problem 1:
The incident electric field is:
n=+
Einc = z e jkx = z j n J n (k )e jn .
n=
The incident magnetic field is:
H inc = y
1 jkx
1 n=+ n
e = y
j J n (k )e jn .
0
0 n=
Since the impedance boundary condition is related to the tangential component, th
ECE 222b
Applied Electromagnetics
Notes Set 6
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego
1
Uniqueness Theorem (1)
A solution is said to be unique when it is the only one
possible
Excitation and Orthogonality of Modes (based on section 7.3, Electromagnetic Waves by Staelin et
al., Prentice Hall, 1994)
The excitation of electromagnetic waves within waveguides is achieved by the judicious placement of one
or more current probes. In t
Vector Formulas = ( ) = a b c b c a c a b = ( ) a b c a c b a b c = ( ) a b c d a c b d a d =0 ( )=0 a )= ( ( a b c
) a
2 a
( ) = + a a a ) = + ( a a a + = ( ) + a a b a b a b b = b ( a b = a a Cartesian Coordinates +y +z x y z Ax Ay Az A = + + x y z A
ECE 222B Reection and refraction at a planar discontinuity, nite conducting limit We wish to analyze the standard reection-refraction problem under the following assumptions: 1. The incident region (z < 0) is ideal (no dissipation), with real permittivity
Some Bessel Function Identities and Integrals In the following, n denotes an integer, while is an unrestricted number. The identities are written for J, but apply equally to Y . Jn (x) = (1)n Jn (x) d J0 (x) = J1 (x) dx
d d (x J (x) = x J1 (x) x J (x) = x
Vector Formulas - - = - (- - ) = - - - - a b c b c a c a b - - - = ( - ) - - - - - - a b c a c b a b c - - - = ( - ) - - - - - - - a b c d a c b d a d - - b c
=0 ( -)=0 a - ) = ( - ) - 2- a ( a a -)=- + - ( a a a -)= - + - ( a a a - +- - - = ( ) - + - -
1
Anisotropic Propagation in electromagnetic Media
Five quantities that characterize propagation of electromagnetic waves in dielectric and/or magnetic media are: Homogeneous or inhomogeneous, i.e., do the properties of the medium depend on position? isot
1
1.1
electrostatic field from charge distributions
Aside: Dirac Delta Function (Impulse Function)
The Dirac Delta Function (- ), or impulse function is a means representing a r unit impulse response. In one dimension it can be considered to be an infini
ECE 222b
Applied Electromagnetics
Notes Set 4c
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego
1
Cylindrical Wave Functions (1)
0
Helmoholtz equation: 2 + k 2 =
2
2
+
1
+
1 2
2
ECE 222b: Homework #3
Problem 1
Consider a current sheet with current distribution given by J s = yJ 0 e jhx .
a) Find the corresponding electric field for z > 0 and < 0
b) What is the polarization of the field ( TM z or TE z ) ?
c) What condition on h wo
Solution:
(1) Since the electric current flows in the y-direction, the magnetic field can have only x- and
z- components, which in turn indicates that the electric field has only y-component. The
source has an e jhx dependence, then all field quantities s
ECE 222b
Applied Electromagnetics
Notes Set 4b
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego
1
Uniform Waveguide (1)
Wave propagation in the +z direction:
z =[et ( x, y ) + ze
z (
ECE 222b, Homework 1
Problem 1
Starting from the Faraday and Amper-Maxwell laws
B
E=
M
t
D
H=
+J
t
and from the electric and magnetic continuity equations
J =
t
M = m
t
derive the electric and magnetic Gauss laws
D =
B = m
Problem 2
Consider two conce