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**Unformatted text preview: **5a I‘d-[on ECE604 Homework 12
Out: Tuesday, April 12, 2005
Due: Tuesday, April 19, 2005 Announcements: 0 Prof. Weiner will m); be available for office hours on Wednesday, April 13, 2005.
0 Final exam will be held Monday, May 2, 2005, 3:20—5:20pm, Room to be announced Problems given by number are from the text by Ramo, Whinnery and Van Duzer (3rd edition,
1994). ' 1. Consider the TEm,n and TMm,n modes of a metallic rectangular waveguide (m ¢ 0, 11 ¢ 0) . Show that for any specific values of m and n, the TEm,n and TMm,n
modes are orthogonal (use same m,n for TB and TM mode). More precisely show that 1 ~ ~ * ~ ” *
aw Mariam) ﬂaw?) ll=° Note: It is sufficient to consider only the z—component of (P) — i.e., the component of
power along the waveguide direction. 2. 10.3d _ Hint: Review section 6.4 and note equivalence between 6 and (1)8”
3. 10.4a
4. 10.40 5. In this problem we consider data transmission in two different metal waveguide
structuresi (I) the parallel-plate waveguide and (11) the square waveguide. Both
waveguides are taken to be air-filled and lossless. (I) 2:6;Qm I ‘ l
.. . I‘- ﬁ=30m—"
p arallel metal-palates- -'$5Q‘%1?ar¢'v mctﬁl ‘3”de
-(_‘separation.a=i6 cm) .(sliiiiie'aFSCIn) (a) We wish to transmit information using a baseband transmission scheme. Our data
rate is R (units of bits/second), and each bit has duration T = UK. For each data bit
equal to ‘1’, we simply transmit a constant DC voltage during time T. For each data
bit equal to ‘0’, the transmitted voltage is zero during the duration T of that bit. (See
sketch below, showing transmitted waveform for data ‘1011’). We desire to operate at a data rate R = 100 Mbit/sec (108 bits/sec). For very
fundamental reasons, one of the two waveguide structures is expected to give much better performance. Which one, and why? (A short answer is sufficient if properly
justified). (b) N ow we wish to transmit data Via binary amplitude modulation of a 6-GHz sinusoid.
For each data bit equal to ‘1’ , we transmit a 6—GHz sinusoidal burst of duration T.
. For each data bit equal to ‘0’, the transmitted voltage is zero during the duration T of
that bit. (See sketch below, showing transmitted waveform for data ‘1011’). Again we desire to operate at a data rate R = 100 Mbit/sec. Comment again on the
relative merits of the two waveguide structures for this data transmission scheme. Is
there a reason to prefer one waveguide structure over the other? . 6. We are interested in the use of thin wire current sources to excite the modes of a lossless,
air-filled, rectangular metal waveguide. The waveguide has dimensions a = 10 cm in the
x direction and b = 5 cm in the y direction, with propagation in the z direction. All the
current sources are speciﬁed to be at frequency f = 5 GHz. 0 ‘i' .
a#.'10-em ---I "f0 For your convenience, the modes of a_ rectangular waveguide are written out below: 1‘an TEmn
Ez = Asin kxx sin kyy Hz = BCOS'kxX COS kyy
jﬁk .. '00 k
Ex :_k2 x Acoskxxsmkyy Ex .—.J 2” y Bcoskxxsinkyy
,‘ c111,11 kcmm
. ' k ‘ jwuk .
Ey =— JE y ASinkxxcoskyy Ey =— k2 X BSlnkXXCOSkyy
kann cm,n
'coek jBk '
1-1;; = J 2 y Asinkxxcoskyy Hx =PiBsm kXXCOSkyy
kcm,n cm,n
' ~ 'mek . ' k
Hy =—J 2 X Acoskxxsmkyy H = “3 y Bcoskxxsink y
k y k2 3’
cm,n cm,n
2 _ 2 2 . _ m7: . _ 1m
kcm’n—kx+ky , kX—T ,ky—"6“ a) A single thin wire current source l0 is used to excite the waveguide at 5 GHz. The wire is located at y = b/2, z = 0, as shown. What waveguide mode or modes can be
driven by this current source? (You must give a brief justification for your answer.) a=100m I ,0 b) Nowapair ofthin wire current sources located at x=%,z=0,and x=§f,z=0 are used to excite the waveguide at SGHz. The wires carry equal current (equal current
phasors Io ) but in opposite directions, as shown. What waveguide mode or modes can be driven by this excitation scheme? (You must give a brief justiﬁcation for your
answer.) l E X l
3
(IOCm) 3/2 . . . . a
In this questlon a set of parallel, thin w1re current sources are located at x = — and z = 0, 2
d, 2d, 3d, 4d, etc. The wires all carry equal current (equal current phasors l0 ), as shown. We wish to drive the TE10 waveguide mode to as large an amplitude as possible. Give an expression for the wire spacing (d) that meets this objective for a drive frequency of
5 GHz. (You are not required to evaluate d numerically. However, your expression must
correctly indicate the dependence of d on frequency and waveguide dimensions). A simple air-filled transmission line resonantor may be constructed by shorting both ends
of a transmission line, as pictured below left, for ideal (lossless) resonator. transmission line:
char. impedance ZO,
velocity 0 Resonator with loss Ideal (lossless) resonator The ideal resonator supports resonant modes such that m half-wavelengths fit within the In)» resonator, i.e., 7 = L, where m is a positive integer. a) What is the frequency of the mth resonance? b) Now consider a resonator with loss, which we model by replacing one of the shorted
ends with a load RL, where RL << 20. Work out an approximate expression for the
energy storage time and the Q of the mth resonance. You may assume that the
resonant frequencies are approximately the same as for part (a). 0) Give numbers for the resonant frequencies and the Q’s for L = 3 meters, Zo = 509,
and RL = 19. 1. Consider the TB,“ and TMm modes of a metallic rectangular waveguide (m ¢ 0, n 7': 0).
Show that for any speciﬁc values of m and n, the TB,“ and TMm,n modes are orthogonal
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In this question a set of parallel, th1n Wire current sources are located at x = -2— and z = 0, d, 2 etc. The wires all carry equal current (equal current phasors l0 ), as shown.
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