ECE 450 HOMEWORK 12 - SOLUTION PROBLEM 1 Obtain the expression for the resultant pattern for the case of a linear array of five isotropic antennas carrying currents with amplitudes in the ratio 1:2:2:2:1, spaced /2 apart, and with progressive phase shift
ECE 450: Spring 2008
Homework Set 6
Due on Wednesday, March 5, 2008
1. The electric field of a time-harmonic electromagnetic wave of angular frequency in the air region
z ! 0 is given in phasor form by E = a y sin ! z exp (# j " x ) V/m, where , are real
1. a. The time-harmonic form of Maxwells equation in the air region is
E = j H
(1)
H = j E
(2)
From (1),
H =
(
1
1
E =
cos( z )e j x a x + j sin( z )d j x a z
j
j
Inserting this into (2), H =
)
(3)
1
( 2 + 2 ) sin( z )e j x a y , which must be equal
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ECE450 Spring 2008 HW#5
1. a. f =
4 108
=
= 2 108 Hz
2
2
b. z direction.
c. = 2
=
=
2
2
=
2 f
p
p = 2 108 m /s
= 1m
d. E ( x, y, z ) = a x e j 2 z ja y e j 2 z
H ( x, y , z ) =
1
( a ) E = a e
1
z
y
j 2 z
j
1
a x e j 2 z
2. a. E ( z ) = a x e j z + a y
1. (a) The normalized input impedance is zin = Z in / Z 0 = ( 25 j 75 ) / 50 = 0.5 j1.5 .
We locate zin in the Smith chart, and draw the constant circle. Then we move
0.3 from zin to the load (counter-clockwise) to find R = 0.6615 + j 0.3434
(b) z R = 0.1
ECE 450 Spring 2007
EXAM 2
March 29, 2007
This is a closed book/notes exam.
You are allowed to use only:
a) A single sheet of notes (written on both sides);
b) Scientific Calculator.
c) If you choose to use a Smith Chart to do any of the problems use only
ECE 450 Fall 2007
EXAM 2
October 30, 2007
This is a closed book/notes exam.
You are allowed to use only:
a) A single sheet of notes (written on both sides);
b) Scientific Calculator.
c) If you choose to use a Smith Chart to do any of the problems use only
ECE 450 Fall 2007
EXAM 1
September 25, 2007
This is a closed book/notes exam.
You are allowed to use only:
a) A single sheet of notes (written on both sides);
b) Scientific Calculator.
c) For work on Smith Chart problems use only the Smith Charts provided
ECE 450 Lecture Notes Fall 09, Erhan Kudeki
Maxwells equations :
1 Overview, Maxwells equations
ECE 329 introduced Maxwells equations and examined their circuit implications (inductance, capacitance) as well as TEM plane-wave
solutions in homogeneous med
ECE 450
Homework 14
Due: Tue, Dec 8, 2009, 5 PM
1. Even though dielectric slab waveguides are used for the transmission of optical frequencies, their application to the transmission of microwave and millimeter-wave frequencies has been exploited also. As
ECE 450
Homework 13
Due: Tue, Dec 1, 2009, 5 PM
1. Rectangular waveguides are being used to construct the feed network of antenna arrays. The picture on the left below depicts such a feed network for a Ka-band multi-beam antenna (courtesy Alenia Spazio).
ECE 450
Homework 12
Due: Tue, Nov 17, 2009, 5 PM
1. Rectangular waveguides are commonly used in X-band microwave circuitry. X-band is dened to be the 8.2 GHz f 12.4 GHz portion of the electromagnetic frequency spectrum. The standard X-band rectangular-wav
1.
5
E = ax exp j
( y z ) = ax exp( j r )
2 2
where r = ax x + a y y + az z , =
5
( a y az )
2 2
1
(a y az )
a. Propagation direction is = =
2
b. = =
5 2 f 2 f
=
=
2
c
c0
r
2
5c
Therefore, r = 0 = 1.5625
4f
= r 0 = 1.5625 8.854 1012 = 1.383 101
1. The cutoff frequency for TE10 mode is:
f c ,TE10
3 108
=
=
=
= 2.5 109 ( Hz )
2a 2a r r 2 0.02 9
1
c
And the operating frequency is:
f = 1.25 f c ,TE10 = 3.125 109 ( Hz )
The propagation constant is:
2
f
2 f
z = 1 c ,TE10 =
c
f
2
r r
f
1 c ,TE10 =
ECE 450: Spring 2008
Homework Set 7
Due on Wednesday, March 12, 2008
1. In class, the equations for the electric and magnetic fields for transverse magnetic (TE)
waves propagating in the z direction were derived. Using that derivation as a guide,
derive t
ECE 450: Spring 2008
Homework Set 5
Due on Wednesday, Feb 27, 2008
1. In a source-free region the electric field intensity of a time-harmonic plane wave is given by the
expression:
E( x, y, z , t ) = a x cos(4! "108 t + 2! z ) + a y sin(4! "108 t + 2! z )
ECE 450: Spring 2008
Homework Set 4
Due on Friday, Feb. 15, 2008
1. The input impedance of a terminated section of a loss-less transmission line is
measured to be Zin = 25 j75 Ohm. The line has characteristic impedance Z0 = 50
Ohm and the length of the se
ECE 450: Spring 2008
Homework Set 3
Due on Wednesday, Feb 6, 2008
1. Starting from the expression for the line impedance, Z (d ) , of a lossless transmission line given in
equation (7.32) in Raos 6th edition, and making use of the expressions for the gene
ECE 450 Spring 2008
EXAM 2
April 3, 2008
This is a closed book/notes exam.
You are allowed to use only the following:
1) Pen/pencil
2) A single 8.5x11 inch sheet of notes (written on both sides)
The test is scored out of a possible 100 points.
Show all of
ECE 450: Spring 2008
Homework Set 2
Due on Wednesday, Jan 30, 2008
1. A 100-Ohm loss-less transmission line is used to deliver power to a complex load impedance
of 100+jX Ohm.
(a) Calculate the percentage of the time-average incident power that gets refle
ECE 450: Spring 2008
Homework Set 1
Due on Wednesday, Jan 23, 2008
1. Consider the parallel connection of a resistor, R, inductor, L, and capacitor C, connected to a
time-harmonic current source.
a) Derive the time-dependent equation for the voltage acros
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(a) The wavenumber is o = 2 f /co = 24108 /(3108 ) = 8/3. Hence, Einc = 10x e j8z/3
and o Hinc = z Einc = 10y e j8z/3 .
inc
(b) Er = x Eo e j8z/3 where Eo = 10 V/m and = o = 1/41 = 3/5 since = o / r =
inc
+o
1/4+1
o /4. Thus,
Er = 6x e j8z/3 ,
and
o Hr
ECE 450 Spring 2007
EXAM 1
February 20, 2007
This is a closed book/notes exam.
You are allowed to use only:
a) A single sheet of notes (written on both sides);
b) Scientific Calculator.
c) For work on Smith Chart problems use only the Smith Charts provide
ECE450 Spring 2008 HW#10
1
1. For the upper and lower interfaces, denoted respectively by subscripts u and l
below, the critical angles are given by
2.5
= 0.456
sin c ,u =
12.0
4.0
= 0.577
12.0
Since max( c ,u , c ,d ) , the bouncing angle we choose must
ECE 450
Homework 11
Due: Tue, Nov 10, 2009, 5 PM
1. The electric eld phasor of TEm mode in air-lled parallel-plate waveguides is given as E = 2j y Eo ejkz z sin(kx x) for 0 < x < a subject to constraints 2 m 2 2 and kz = k2 kx = kx . a c2 The guide consis
ECE 450
Homework 10
Due: Tue, Nov 3, 2009, 5 PM
1. The propagation of ocean waves on the surface of deep water (depth much longer than a wavelength = 2k ) is governed by the dispersion relation = gk
where g represents gravitational acceleration and , k ar