732 COMMUNICATION SYSTEMSThe simple concept of relative phase relationships is exploited by engineers and scientists in
several systems that are beneficial to the general public. For example, remote measurements of
the height of Greenland’s ice cap (by using high-quality GPS receivers) are being made to assess
the possible effects of global warming.Problems
15.1.1A coaxial cable with polyethylene dielectric (εr=
2 .26) connects an antenna to a receiver 30 m away.
Determine the velocity of wave propagation in
the cable and the delay of the cable. (Hint: See
Example 15.1.1.)
15.1.2Ifbandaare the radii of the outer and inner
conductors, respectively, of a coaxial cable using
a polyethylene dielectric (εr= 2 .26), what ratio
b/ais needed for the cable to have a characteristic
impedanceZ ̄ 0 =R 0 =( 60 /
√
εr)ln(b/a)of 50
?
*15.1.3A rigid 50-coaxial transmission line has air as
dielectric. If the radius of the outer conductor is 1
cm, find the cutoff frequency
fc=
c
π
√
εr(a+b)
Note thatZ ̄ 0 =R 0 =( 60 /
√
εr)ln(b/a).
15.1.4If the line of Problem 15.1.3 is made of copper
whose resistivityρ= 1. 72 × 10 −^8 ·m, determine
the maximum length that can be used if losses are
not to exceed 3 dB whenf =3 GHz. For the
expression of attenuation, see Example 15.1.1.
15.1.5For an RG-290/U aluminum rectangular wave-
guide,a= 58 .42 cm andb= 29 .21 cm. Compute
the theoretical and practical frequency ranges of
operation for the guide. (See Example 15.1.2.)
15.1.6A rectangular air-filled RG-52/U is made of brass
(ρ = 3. 9 × 10 −^8 ·m) and has dimensions
a= 22 .86 mm andb= 10 .16 mm.
(a) DetermineZ ̄ 0 (=R 0 )at the limits of the prac-
tical operating frequency range of the guide.
(See Example 15.1.2.)
(b) Compute the attenuation given by
( 0. 458 × 10 −^4 )
√
fρ
[
1 +( 2 b/a)(fc/f )^2]b√
1 −(fc/f )^2dBper unit length corresponding to those limiting
frequencies.
15.1.7By using the expression for attenuation given in
Problem 15.1.6(b), find the attenuation of the air-filled waveguide of Problem 15.1.5 ifρaluminum=
2. 83 × 10 −^8 ·m, at frequencies of 1.25fcand
1.9fc.
*15.1.8An RG-139/U rectangular waveguide is given to
have dimensionsa=0.8636 mm andb=0.4318
mm. Compute the theoretical and practical fre-
quency ranges of operation for the guide. (See
Example 15.1.2.)
15.1.9Find the diameter of a circular waveguide that will
have a lower cutoff frequency of 10 GHz and also
specify its largest usable frequency. (See Example
15.1.2.)
15.1.10It is desired to cut aλ/4 length of RG58A/U cable
(εr= 2 .3) at 150 MHz. What is the physical length
of this cable?
15.1.11A transmission line with air dielectric is 25 m
long. Find, in wavelengths, how long the line is
at frequencies of 1 kHz, 10 MHz, and 100 MHz.
15.1.12A transmission line with a dielectric (εr= 3 .5) is
100 m long. At a frequency of 10 GHz, how many
wavelengths long is the line?
*15.1.13Comment briefly on the following:
(a) Why are waveguides not used at low frequen-
cies?
(b) Why are open-wire lines not generally used as
guiding structures at very high frequencies?
(c) What is the velocity of wave propagation in a
Teflon (εr= 2 .1) coaxial transmission line?
15.1.14(a) What is the difference between a TEM mode
and a TE mode?
(b) Explain the terms “cutoff wavelength” and
“dominant mode” as applied to waveguides.
Find the cutoff wavelength for an air-filled
rectangular waveguide for the propagation by
the dominant mode.
15.1.15The cutoff frequency of a dominant mode in an
air-filled rectangular waveguide is 3 GHz. What
would the cutoff frequency be if the same wave-
guide were filled with a lossless dielectric having
anεr= 3 .24?