ECE Department, TUP Manila
Transmission Media and Antenna Systems
Guided Problem Solving No. 1
1. Calculate the minimum length of a pair of conductors to be considered as a transmission line at a
frequency of 80 MHz.
2. A parallel-wire transmission line is constructed of #6 AWG copper wire (d = 0.162 in) with a 12-inch
separation in air. Determine the per-meter values of L, C and R assuming an operating frequency of 1
MHz.
3. The specifications for rigid Teflon-dielectric coaxial line used in a radar set operating at 3 GHz are:
copper material, stub-supported at intervals to maintain the Teflon dielectric; outside diameter, 7/8
inch; wall thickness, 0.032 inch; inner-conductor diameter, 0.375 inch. Determine the per-meter values
of L, C and R.
4. Given the following primary line constants:
R = 0.1 ohm/ft
G = 0.05 S/ft
L = 76.8 nH/ft
C = 31.4 pF/ft
Determine the characteristic impedance of the line at (a) 60 Hz; (b) 1 MHz; (c) 2.45 GHz
5. A lossless transmission line has L = 32.5 nH/m and C = 12.2 pF/m. What value of load resistance
should this line be terminated to ensure maximum power transfer?
6. Derive the equation used for the characteristic impedance of coaxial and parallel-wire line
transmission line
7. Calculate the Zo for an air dielectric two – air parallel transmission line. The ratio of the distance
between the centers of the two conductors and the radius of the conductors is 12.3.
8. From the propagation constant, determine the value of the attenuation and phase shift coefficient.
The distributed parameters are: R = 0.1 ohm/m, G = 0 S/m, L = 76.8 nH/m and C = 22.2 pF/m. Assume
an operating frequency of 750 MHz.
9. Attenuation is one of the major concerns in the application of transmission lines in communications
engineering. How much attenuation, in dB, does a 100 m of air-dielectric parallel – wire transmission
line introduce? Assume that R = 0.1 ohm/m and Z o = 300 ohms.
10. A 2-wire transmission line consists of No. 12 wire AWG (81 mils). The distance between the wire
centers is 10 inches. What is the attenuation in dB per 100 ft of the line at 400 MHz?
ECE Department, TUP Manila
Transmission Media and Antenna Systems
Guided Problem Solving No. 2
1. A system with operating frequency of 750 MHz uses a lossless transmission line with C = 40 pF/m
and L = 50 μH/m, determine: a) the propagation velocity, b) the velocity factor, c) the phase shift
coefficient and d) the phase shift for an 8-m length of this line.
2. Determine the capacitance per meter of a transmission line having a characteristic impedance of 75
ohms and propagation velocity of 0.66c.
3. A communication system uses an RG-58U transmission line terminated at a load impedance of 50
– j25 ohms. Determine: a) the reflection coefficient, b) the SWR, c) the voltage minima if voltage
maxima is 15 V and d) the value of resistive load that will give the same form of standing waves.
4. Find the impedance of a λ/8 lossless transmission line terminated at a load of 400 ohms. Assume a
characteristic impedance of 50 ohms.
5. The impedance at point AA’ in the transmission line shown in the figure is Z = 20 – j80 ohms,
determine the load impedance, input impedance and the reflection coefficient at the load if Z o = 300
ohms. Assume lossless condition.
6. A 50-ohm short-circuited line operates at 500 MHz. Determine the equivalent reactance and
component of this line if it is: a) 0.1 λ and b) 0.375 λ long.
7. Suppose it is required to match a 73 ohm antenna to a 600 ohm feeder line at a frequency of 150
MHz. Determine the characteristic impedance of the quarter wave transformer that should be used.
8. A lossless transmission line with ZO = 50 ohms is terminated to a load impedance of 20 – j80 ohms.
What is the VSWR and the distance of the first voltage minima from the load?
9. A lossless transmission line having a characteristic impedance of 73 ohms (a typical characteristic
impedance to match the feed line impedance of half – wave dipole), the VSWR is 3.50. The distance
between successive minima of the pattern is 45.5 cm and the first voltage minimum is 36.8 cm from
the load terminals. Determine the value of the load impedance terminating the line.
10. When a 75-ohm lossless transmission line with VSWR of 2.8 was shorted, the voltage minima
moves 30° towards the generator. Determine the load impedance terminating this transmission line.
11. When a 180 o, 75-ohm lossless transmission line with VSWR of 2.8 was opened, the voltage maxima
moves 45° towards the load. Determine the sending impedance of this transmission line.
12. A transmitter delivers 50 W into a 600 ohm lossless line that is terminated with an antenna that
has an impedance of 275 ohms, resistive. How much power actually reaches the antenna?
13. A TDR display shows a discontinuity 1.4 us from the start. If the line has a velocity factor of 0.8,
how far is the fault from the reflectometer?
14. Two adjacent minima on a slotted line are 23 cm apart. Find the wavelength and frequency of the
TEM wave on the transmission line if the line has a typical velocity factor of a coaxial cable.
PROBLEMS
12-1. Determine the wavelengths for electromagnetic waves in free space with the following fre­
quencies: 1 kHz, 100 kHz, l MHz, and 1 GHz.
12-2. Determine the frequencies for electromagnetic waves in free space with the following wave­
lengths: 1 cm, I m, 10 m, 100 m, and 1000 m.
12-3. Determine the characteristic impedance for an air-dielectric transmission line with Dir ratio
of 8.8.
12-4. Determine the characteristic impedance for an air-filled concentric transmission line with Did
ratio of 4.
12-5. Determine the characteristic impedance for a coaxial cable with inductance L = 0.2 µH/ft and
capacitance C = 16 pF/ft.
12-6. For a given length of coaxial cable with distributed capacitance C = 48.3 pF/m and distrib­
uted inductance L = 241.56 nH/m, determine the velocity factor and velocity of propagation.
12-7. Determine the reflection coefficient for a transmission line with incident voltage E; = 0.2 V
and reflected voltage E,. = 0.0 I V.
12-8. Determine the standing-wave ratio for the transmission line described in problem 12-7.
12-9. Determine the SWR for a transmission line with maximum voltage standing-wave amplitude
Vmax = 6 V and minimum voltage standing-wave amplitude Vmin = 0.5.
12-10. Determine the SWR for a 50-Q transmission line that is terminated in a load resistance
ZL = 75 Q,
12-11. Determine the SWR for a 75-Q transmission line that is terminated in a load resistance
ZL = 50 Q,
12-12. Determine the characteristic impedance for a quarter-wavelength transformer that is used to
match a section of 75-Q transmission line to a 100-Q resistive load.
12-13. Using TDR, a pulse is transmitted down a cable with a velocity of propagation of 0.7c. The
reflected signal is received 1.2 µs later. How far down the cable is the impairment?
12-14. Using TDR, a transmission-line impairment is located 2500 m from the source. For a veloc­
ity of propagation of 0.95c, determine the elapsed time from the beginning of the pulse to the
reception of the echo.
12-15. Using TDR, a transmission-line impairment is located 100 m from the source. If the elapsed
time from the beginning of the pulse to the reception of the echo is 833 ns, determine the ve­
locity factor.
12-16. Determine the wavelengths for electromagnetic waves with the following frequencies: 5 kHz,
50 kHz, 500 kHz, and 5 MHz.
12-17. Determine the frequencies for electromagnetic waves with the following wavelengths: 5 cm,
50 cm, 5 m, and 50 m.
12-18. Determine the characteristic impedance for an air-dielectric transmission line with a Dir ratio
of 6.8.
12-19. Determine the characteristic impedance for an air-filled concentric transmission line with a
Did ratio of 6.
12-20. Determine the characteristic impedance for a coaxial cable with inductance L = 0.15 µH/ft
and capacitance C = 20 pF/ft.
12-21. For a given length of coaxial cable with distributed capacitance C = 24.15 pF/m and distrib­
uted inductance L = 483.12 nH/m, determine the velocity factor and velocity of propagation.
12-22. Determine the reflection coefficient for a transmission line with incident voltage E; = 0.4 V
and reflected voltage E,. = 0.002 V.
12-23. Determine the standing-wave ratio for the transmission line described in problem 12-22.
12-24. Determine the SWR for a transmission line with a maximum voltage standing-wave ampli­
tude Vmax = 8 V and a minimum voltage standing-wave amplitude Vmin = 0.8 V.
12-25. Determine the SWR for a 50-Q transmission line that is terminated in a load resistance
ZL = 60 Q.
12-26. Determine the SWR for a 60-Q transmission line that is terminated in a load resistance
ZL = 50 Q.
12-27. Determine the characteristic impedance for a quarter-wave transformer that is used to match
a section of 50-Q transmission line to a 60-Q resistive load.
Problems
1. Compute the wavelength in meters at a frequency of
350 MHz. ◆
2. At what frequency does a line 3.5 in long represent
one-half wavelength?
3. A coaxial cable has a capacitance of 30 pF/ft and an
inductance of 78 nH/ft. What is the characteristic impedance? ◆
4. A coaxial cable has a capacitance of 17 pF/ft. The characteristic impedance is 75 V. What is the equivalent
inductance per foot?
5. What is the velocity factor of a coaxial cable with a
dielectric constant of 2.5? ◆
6. A manufacturer’s data sheet states that the velocity
factor for a speciic coaxial cable is 0.7. What is the
dielectric constant?
7. Determine the length in feet of a one-quarter wavelength coaxial cable with a velocity factor of 0.8 at a
frequency of 49 MHz. ◆
8. What is the time delay introduced by a coaxial cable with
the speciications given in Prob. 7 if the length is 65 ft?
9. What is the time delay of 120 ft of a coaxial cable with
a dielectric constant of 0.7? ◆
10. How much phase shift is introduced by a coaxial cable
15 ft long to a 10-MHz sine wave signal? The dielectric
constant is 2.9.
11. What is the attenuation of 350 ft of RG-11 U coaxial
cable at 100 MHz?
12. What is the approximate attenuation of 270 ft of type
9913 coaxial cable at 928 MHz?
13. A transmitter has an output power of 3 W that is fed to
an antenna through an RG-58A/U cable 20 ft long.
How much power reaches the antenna?
14. What should be the transmission line impedance for optimum transfer of power from a generator to a 52-V load?
15. A 52-V coaxial cable has a 36-V antenna load. What is
the SWR?
16. If the load and line impedances of the cable in Prob. 15
are matched, what is the SWR?
17. The maximum voltage along a transmission line is
170 V, and the minimum voltage is 80 V. Calculate the
SWR and the relection coeficient.
18. The relection coeficient of a transmission line is 0.75.
What is the SWR?
19. What are the relection coeficient and SWR of an open
or shorted transmission line?
20. A transmission line has an SWR of 1.65. The power
applied to the line is 50 W. What is the amount of relected power?
21. A power meter inserted into a transmission line reads
1150 W of forward power and 50 W of relected power.
What is the SWR?
22. What must be the value of the load impedance on a
transmission line if the voltage variations along the line
go to zero at regular intervals?
23. At what frequency does an open-wire transmission line
6 in long act as a parallel resonant circuit? (The
dielectric is air.)
24. A coaxial cable has a velocity factor of 0.68. How
long is one-half wavelength of this cable at
133 MHz?
25. Calculate the impedance of a microstrip line whose
dimensions are h 5 0.05 in, w 5 0.125 in, and t 5
0.002 in. The dielectric constant is 4.5.
26. What is the length of one-quarter wavelength of the
microstrip in Prob. 25 at a frequency of 915 MHz?
27. Plot, in normalized form, the following impedances on
a Smith chart: Z1 5 80 1 j 25 and Z2 5 35 2 j 98, assuming a prime center value of 52 V.
28. Plot the load impedance 104 2 j58 on a coaxial line on
a Smith chart. Assume a characteristic impedance value
of 75 V. Then ind the SWR.
29. If the operating frequency is 230 MHz and the cable
length is 30 ft for Prob. 28, what is the impedance at the
generator if the velocity factor is 0.66? Use the Smith
chart and show all work.
30. What does the SWR circle on a Smith chart look like if
a 52-V load is connected to a 52-V coaxial cable?
◆ Answers
to Selected Problems follow Chap. 22.
Critical Thinking
1. Describe the operation of a transmission line, one-quarter
or one-half wavelength at the operating frequency, if the
line is shorted or open. How can such a line be used?
2. At what frequencies do transmission lines implemented
on a printed-circuit board become practical?
3. Compare the characteristics of microstrip and stripline.
Which is preferred and why?
4. A 22-ft coaxial cable with an impedance of 50 V has a
velocity factor of 0.78. The operating frequency is
400 MHz. The impedance of the antenna load combined with the cable impedance as measured at the
generator end of the cable is 74 1 j 66. What is the
antenna impedance as determined on the Smith chart?
What is the SWR?
5. Assume that a square wave of 10 MHz is applied to a
200-ft-long RG-58A/U coaxial cable. It is properly terminated. Describe the signal at the load, and explain
why the signal appears as it does. (See Fig. 13-13.)
6. Explain how a coaxial transmission line (RG-59/U)
could be used as a ilter to eliminate interference at the
input to a receiver. Assume a frequency of 102.3 MHz.
Design the ilter and calculate size.
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PROBLEMS
13.1. Explain what is meant by the characteristic impedance of a transmission line. Explain why you
would expect an infinitely long uniform line to have an input impedance equal to the characteristic
impedance.
13.2. The primary constants (which may be assumed independent of frequency) for a transmission line are
R 0.5 /m, L 250 nH/m, C 100 pF/m, and G 10 6S/m. Plot the variation of Z0 against frequency for this line.
13.3. For the transmission line in Problem 13.2, calculate the attenuation coefficient and the phase-shift
coefficient at a frequency of 20 MHz.
13.4. Explain what is meant by the propagation coefficient of a transmission line. The propagation coeffi1
cient for a transmission line is given as (0.0005 + j/10)m . Determine, for a matched-line length
of 50 m, (a) the total attenuation in nepers, (b) the total attenuation in decibels, and (c) the total phase
shift.
13.5. The propagation coefficient for a transmission line is given by 0.01 + j(/10). The line is 32 m
long and is correctly terminated to avoid reflections. Plot the phasor diagram showing the variation of
current over the length of line.
13.6. For the transmission line in Problem 13.4, determine the attenuation, in nepers, and in decibels, of
15 km of line.
13.7. Explain what is meant by the group delay timeand why this should be constant for a transmission
line. Given that LC, L 1H/m, and C 11.11 pF/m, calculate (a) the group delay time
and (b) the group velocity.
13.8. Explain what is meant by the voltage reflection coefficient for a transmission line. The voltage reflection coefficient on a 50- line is 0.730 . Determine the load impedance terminating the line.
13.9. A 300- line is terminated in a load impedance of 100 + j200 . Determine the voltage reflection
coefficient.
13.10. Explain how standing waves may be set up on a transmission line, and state the relationship between
voltage and current standing waves for a line whose characteristic impedance is purely resistive.
13.11. Calculate the voltage standing-wave ratio for a lossless line of characteristic impedance 50 when
it is terminated in (a) 100 resistive, and (b) (30 j50) .
13.12. Sketch the variation of input impedance with length for a lossless transmission line terminated in a
short circuit. A section of 50- line, 0.15 in length, is terminated in a short circuit. Determine the
effecttive input reactance of the section, showing clearly whether it is inductive or capacitive.
13.13. Calculate the shortest shorted stub length required to produce a reactance equal to (a) Z0 and (b) Z0.
Express the length as a fraction of a wavelength.
13.14. A 600- lossless line is terminated in a 5-pF capacitor. The line is operated at a frequency that makes
it –14 long. Determine the nature of the input impedance of the line, giving the equivalent inductance
or capacitance value.
13.15. Determine, using a Smith chart, the admittance value corresponding to an impedance of 150 + j80 .
Give the admittance value in Siemens, and also give the equivalent parallel component values in
ohms.
13.16. A load impedance of 80
j70 is connected to a 50- transmission line. Find (a) the VSWR,
(b) the distance, in terms of wavelength, to the first voltage minimum, and (c) the impedance at this
minimum.
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13.17. Measurements on a 50- slotted line gave the following results: VSWR 3.0; distance from load to
first voltage minimum 0.4 . Determine, using a Smith chart, the value of the load impedance.
13.18. An impedance 100 - j50 is used to terminate a 50- lossless line. The line is 0.6 long. Determine,
using a Smith chart, (a) the VSWR, (b) the input impedance, and (c) the impedance at a voltage
maximum.
13.19. A load admittance of 4 j10 mS is to be matched to a 50- line system using single stub matching.
Determine the position for the stub to be connected in parallel with the main line, the length of the
stub, and whether the stub is to be open or short circuited.
13.20. Repeat Problem 13.19 for a series-connected stub.
13.21. A double stub matching section is arranged as shown in Fig. 13.12.6(a) with the distance 0.1 .
Determine which of the following normalized admittances can be matched: (a) 0.6 j1.2, (b) 0.6 j1.2,
(c) 1.1 j1.3, (d) 3 j2
13.22. A load impedance of 75 j80 lis to be matched to a 50- line using the double stub section of
Problem 13.21. Determine the required stub lengths in wavelengths.
13.23. In a time-domain reflectometer measurement on a terminated transmission line, the reflected voltage
was found to be a vertical downward step, of magnitude one-half the input step. If the line can
be assumed lossless and has a characteristic impedance of 100 , determine the value of load
impedance.
13.24. Sketch the form of trace expected on a TDR measuring set when applied to a line terminated in a
series RL circuit. Give equations for the asymptotic values on the trace.
13.25. A transmission line is made of two parallel copper No. 18 AWG wires (1.024 mm in diameter) spaced
10 mm apart and embedded in a plastic spacer made of polyvinyl chloride (PVC). The wire has a
resistance of 1.984 /km, and the insulation has a leakage conductivity of 2.5 10 10 S/km. Find
the primary line constants. For PVC, εr 3.3. Find the characteristic impedance for the line.
13.26. Repeat Problem 13.25 for a coaxial line made of the same wire and insulating material, with a copper sheath of inside diameter of 2 cm. Assume the leakage to have the same value.
13.27. A microwave amplifier requires to be fed from a source impedance of 12 + j8 . The actual source
impedance is 50 . Determine the impedance values for a suitable stripline matching section consisting of a /4 transformer section and a parallel-connected 3 /8 stub, and state whether the stub
should be open or short circuited.
( Problems
1. A 75-Ω transmission line is terminated with a load having an impedance of 45 − j30 Ω. Find:
(a) the distance (in wavelengths) from the load to the closest place at
which a quarter-wave transformer could be used to match the line
(b) the characteristic impedance that should be used for the quarterwave transformer
2. A 50-Ω coaxial transmission line with a velocity factor of 0.8 is connected to a load of 89 − j47 Ω. The system operates at 300 MHz. Find the
proper position and component value to match this line using:
(a) a series capacitor
(b) a series inductor
(c) a transformer
3. Find the SWR for each setup of problem 2, at a frequency of 350 MHz.
4. A transmitter supplies 100 W to a 50-Ω lossless line that is 5.65 wavelengths long. The other end of the line is connected to an antenna with
a characteristic impedance of 150 + j25 Ω.
(a) Find the SWR and the magnitude of the reflection coefficient.
(b) How much of the transmitter power reaches the antenna?
(c) Find the best place at which to insert a shorted matching stub on
the line. Give the answer in wavelengths from the load.
(d) Find the proper length for the stub (in wavelengths).