90
Chapter 2: Transmission Line Theory
REFERENCES
S. Ramo, J. R. Winnery, and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd
edition, John Wiley & Sons, New York, 1994.
[2] J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941.
[3] H. A. Wheeler, “Reflection Charts Relating to Impedance Matching,” IEEE Transactions on Microwave Theory and Techniques, vol. MTT-32, pp. 1008—1021, September 1984.
[4] P. H. Smith, “Transmission Line Calculator,” Electronics, vol. 12, No. 1, pp. 29-31, January 1939.
[5] P. J. Nahin, Oliver Heaviside. Sage in Solitude, IEEE Press, New York, 1988.
[6] H. A. Wheeler, “Formulas for the Skin Effect,” Proceedings of the IRE, vol. 30, pp. 412W24,
September 1942.
[7] T. C. Edwards, Foundations for Microstrip Circuit Design, John Wiley & Sons, New York, 1987.
PROBLEMS
2.1 A 75 f2 coaxial line has a current i (t, z) 1.8 cos(3.77 x 109 i — 18. l3z) mA. Determine (a) the
frequency, (b) the phase velocity, (c) the wavelength, (d) the relative permittivity of the line, (e) the
phasor form of the current, and (f) the time domain voltage on the line.
2.2 A transmission line has the following per-unit-length parameters: L 0.5 pH/m, C = 200 pF/m,
fi = 4.0 D/m, and G = 0.02 S/m. Calculate the propagation constant and characteristic impedance
of this line at 800 MHz. If the line is 30 cm long, what is the attenuation in dB? Recalculate these
quantities in the absence of loss (fi = G 0).
2.3 RG-402U semirigid coaxial cable has an inner conductor diameter of 0.91 mm and a dielectric diameter (equal to the inner diameter of the outer conductor) of 3.02 mm. Both conductors are copper, and
the dielectric material is Teflon. Compute the fi, L, G, and C parameters of this line at 1 GHz, and
use these results to find the characteristic impedance and attenuation of the line at 1 GHz. Compare
your results to the manufacturer's specifications of 50 f2 and 0.43 dB/m, and discuss reasons for the
difference.
2.4 Compute and plot the attenuation of the coaxial line of Problem 2.3, in dB/m, over a frequency range
of 1 MHz to 100 GHz. Use log-log graph paper.
2.5 For the parallel plate line shown in the accompanying figure, derive the fi, L, G, and C parameters.
Assume YI d.
d
2.6 For the parallel plate line of Problem 2.5, derive the telegrapher equations using the field theory
approach.
2.7 Show that the Z'-model of a transmission line shown in the accompanying figure also yields the
telegrapher equations derived in Section 2.1.
Problems
2
2
91
2
2
2.8 A lossless transmission line of electrical length é = 0.3X is terminated with a complex load impedance
as shown in the accompanying figure. Find the reflection coe&cient at the load, the SWR on the line,
the reflection coe&cient at the input of the line, and the input impedance to the line.
»
/ = 0.33
Zp
Z — 30 — y20 f2
2.9 A 75 f2 coaxial transmission line has a length of 2.0 cm and is terminated with a load impedance
of 37.5 + y75 f2. If the relative permittivity of the line is 2.56 and the frequency is 3.0 GHz, find
the input impedance to the line, the reflection coe&cient at the load, the reflection coe&cient at the
input, and the SWR on the line.
2.10 A terminated transmission line with Z0 = 60 f2 has a reflection coefficient at the load of r
0.4 60°.
(a) What is the load impedance? (b) What is the reflection coe&cient 0.3X away from the load? (c)
What is the input impedance at this point?
2.11 A 100 f2 transmission line has an effective dielectric constant of 1.65. Find the shortest open-circuited
length of this line that appears at its input as a capacitor of 5 pF at 2.5 GHz. Repeat for an inductance
of 5 nH.
2.12 A lossless transmission line is terminated with a 100 f2 load. If the SWR on the line is 1.5, find the
two possible values for the characteristic impedance of the line.
2.13 Let Zsc be the input impedance of a length of coaxial line when one end is short-circuited, and let
Z be the input impedance of the line when one end is open-circuited. Derive an expression for the
characteristic impedance of the cable in terms of Zsc and Z z.
2.14 A radio transmitter is connected to an antenna having an impedance 80 + y40 f2 with a 50 f2 coaxial
cable. If the 50 f2 transmitter can deliver 30 W when connected to a 50 f2 load, how much power is
delivered to the antenna?
2.15 Calculate standing wave ratio, reflection coe&cient magnitude, and return loss values to complete
the entries in the following table:
sws
r/
1.00
1.01
—
1.05
0.00
—
0.01
—
RL (dB)
30.0
1.10
1.20
0.10
1.50
10.0
2.00
2.50
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Chapter 2: Transmission Line Theory
2.16 The transmission line circuit in the accompanying figure has Vg 15 V rms, Zg 75 f2, Z0 = 75 f2,
Z p 60 — y40 f2, and I = 0.7X. Compute the power delivered to the load using three different
techniques:
(a) Find r and compute
1
2
Z0
— r)2).
(b) find Zinand compute
P¿ ——
Re (Zin) '
(c) find V¿ and compute
Re (Z¿ } .
Discuss the rationale for each of these methods. Which of these methods can be used if the line is
not lossless?
2.17 For a purely reactive load impedance of the form Z¿ = JJ, show that the reflection coefficient
magnitude |F | is always unity. Assume that the characteristic impedance Zo is real.
2.18 Consider the transmission line circuit shown in the accompanying figure. Compute the incident
power, the reflected power, and the power transmitted into the infinite 75 f2 line. Show that power
conservation is satisfied.
50 f2
10 V
inc
ref “
trans
2.19 A generator is connected to a transmission line as shown in the accompanying figure. Find the voltage
as a function of z along the transmission line. Plot the magnitude of this voltage for —é < z < 0.
100 f2
/ = 1.51
^
Z¿ — 80 —y40 f2
Z0 = 100 Al
10 V
—1
0
2.20 Use the Smith chart to find the following quantities for the transmission line circuit shown in the
accompanying figure:
(a) The SWR on the line.
(b) The reflection coefficient at the load.
(c) The load admittance.
(d) The input impedance of the line.
(e) The distance from the load to the first voltage minimum.
Problems
93
(f) The distance from the load to the first voltage maximum.
f = 0.4â
»
Zp — 60 +J50 f2
Zin
2.21 Use the Smith chart to find the shortest lengths of a short-circuited 75 f2 line to give the following
input impedance:
(a)
Zin
0•
(b)
Zin'
-
(Q) Zin '
(d) Zin'
(e) Zin'
75 f2.
50 f2 •
10 f2.
2.22 Repeat Problem 2.21 for an open-circuited length of 75 f2 line.
2.23 A slotted-line experiment is performed with the following results: distance between successive minima = 2.1 cm; distance of first voltage minimum from load = 0.9 cm; SWR of load = 2.5. If
Z0 = 50 f2, find the load impedance.
2.24 Design a quarter-wave matching transformer to match a 40 D load to a 75 f2 line. Plot the SWR for
0.5 < /// < 2.0, where / is the frequency at which the line is X/4 long.
2.25 Consider the quarter-wave matching transformer circuit shown in the accompanying figure. Derive
expressions for K+ and K , the respective amplitudes of the forward and reverse traveling waves on
the quarter-wave line section, in terms of Ki, the incident voltage amplitude.
0
2.26 Derive equation (2.71) from (2.70).
2.27 In Example 2.7, the attenuation of a coaxial line due to finite conductivity is
s
2p ln b/a
1
a+ b
Show that ap is minimized for conductor radii such that x lnx = 1 + x, where x = b/a. Solve this
equation for x, and show that the corresponding characteristic impedance for *r 1 is 77 f2.
2.28 Compute and plot the factor by which attenuation is increased due to surface roughness, for rms
roughness ranging from 0 to 0.01 mm. Assume copper conductors at 10 GHz.
2.29 A 50 D transmission line is matched to a 10 V source and feeds a load Z¿
100 f2. If the line is
2.3X long and has an attenuation constant n = 0.5 dB/Z, find the powers that are delivered by the
source, lost in the line, and delivered to the load.
2.30 Consider a nonreciprocal transmission line having different propagation constants, Q+ and Q*, for
propagation in the forward and reverse directions, with corresponding characteristic impedances Z0
and Z0. (An example of such a line could be a microstrip transmission line on a magnetized ferrite
94
Chapter 2: Transmission Line Theory
substrate.) If the line is terminated as shown in the accompanying figure, derive expressions for the
reflection coefficient and impedance seen at the input of the line.
2.31 Plot the bounce diagram for the transient circuit shown in the accompanying figure. Include at least
three reflections. What is the total voltage at the midpoint of the line (z = //2), at time = 3 /Up*
10 V
Z0 = 50 f2
100 f2