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Unformatted text preview: 564 PROBLEMS CHAPTER 1 1 TRANSMISSION LINES
EEBTIUN 1 1.2 TRANSMISSIDN LINE PARAMETERS 11.1 An airﬁlled planar line with w = 30 cm, d = 1.2 cm, t = 3 mm has conducting plates
with ac = 7 x 107 S/m. Calculate R, L, C, and G at 500 MHZ. 11.2 A coaxial cable has an inner conductor of radius a = 0.8 mm and an outer conductor of
radius b = 2.6 mm. The conductors have ac = 5.28 X 107 S/m, ,uc = Mo, and 86 = so; they
are separated by a dielectric material having 0' = 10'5 S/m, p. = ,uo, e = 3.5 30. At
80 MHZ, calculate the line parameters L, C, G, and R. 11.3 The copper leads of a diode are 16 mm in length and have a radius of 0.3 mm. They are
separated by a distance of 2 mm as shown in Figure 11.44. Find the capacitance between the leads and the ac resistance at 10 MHZ. EEETIDN I 1.13 TRANSMISSICIN LINE EQUATIONS *11.4 In Section 11.3, it was mentioned that the equivalent circuit of Figure 11.5 is not the only
possible one. Show that eqs. (11.4) and (11.6) would remain the same if the Htype and
Ttype equivalent circuits shown in Figure 11.45 were used. 16mm Figure 11.44 The diode of Problem 11.3. “2’ ’) RAZ 1(2 + A2, t)
+0 he j m :1 a . +
V Q l g (z, r) 132% gm 2 Az. . gm V(HAZ’ t)
0 Z {3 _
(a)
5. L112 5 L A
[(2, t) 2 AZ 2 AZ V. 1(2 + AZ, t)
V(z’ I) H CA2 V(z + Az, t)
O O (b) Figure 11.45 Equivalent circuits for Problem 11.4: (a) Htype, (b) Ttype. 11.5 (a) Show that at high frequencies (R << wL, G << wL), R C G L
2 — —+— — +'\/LC
7 (ﬁg 2 C) 1‘” (b) Obtain a similar formula for Z0.
11.6 For a lossless line, Z0 = 50 (2 and u = 2.8 X 108 m/s. Determine L and C for the line. 11.7 Measurements on a lossy transmission line at 800 MHZ indicate Z0 = 50 + jO S2,
or = 0.01 Np/m, and ,8 = 4 rad/m. Determine the line parameters R, L, C, and G. 11.8 A 78 (l lossless planar line was designed but did not meet a requirement. What fraction of the widths of the strip should be added or removed to get the characteristic impedance
of 75 9? 11.9 A telephone line has the following parameters:
R = 40 fl/m, G = 400 uS/m, L = 0.2 uH/m, C = 0.5 nF/m (a) If the line operates at 10 MHZ, calculate the characteristic impedance Z0 and velocity
u. (b) After how many meters will the voltage drop by 30 dB in the line? 11.10 A distortionless line operating at 120 MHZ has R = 20 Q/m, L = 0.3 uH/m, and
C = 63 pF/m. (a) Determine 'y, u, and Z0. (b) How far will a voltage wave travel before
it is reduced to 20% of its initial magnitude? (c) How far will it travel to suffer a 45°
phase shift? 11.11 A 0.64A length of 50 9 line is driven by a voltage source Vg = 12 A): and Zg = 50 Q. If the
line is terminated in ZL = 75 9., calculate the timeaverage power delivered to the line. 11.12 For a lossless two—wire transmission line, show that (a) The phase velocity u = c = l
VLC
. . . 120 “1 d
(b) The characterlstic impedance Z0 = —— cosh — V5; 2a Is part (a) true of other lossless lines? 11.13 A twisted line, which may be approximated by a two—wire line, is very useful in the tele—
phone industry. Consider a line comprising two copper wires of diameter 0.12 cm that
have a 0.32 cm centerto—center spacing. If the wires are separated by a dielectric mater— ial with e = 3.530, ﬁnd L, C, and Z0.
11.14 On a distortionless line, the voltage wave is given by
V(€’) = 1206000256, cos (108: + 26’) + 60e‘°~°°25‘” cos (108: ~ 26’) where {3’ is the distance from the load. If Z L = 300 9, ﬁnd (a) oz, [3, and u, (b) Z0 and 1(€ ’). 566 CHAPTER 1 1 TRANSMISSION LINES Figure 11.46 For Problem 11.21. 11.15 A distortionless line at 150 MHZ has ZO = 75 Q, a = 0.06 Np/m, and u = 2.8 X 108 m/s.
Calculate the line parameters R, G, C, and L. 11.16 A coaxial line 5.6 m long has distributed parameters R = 6.5 Q/m, L = 3.4 ,uH/m,
G = 8.4 mS/m, and C = 21.5 pF/m. If the line operates at 2 MHz, calculate the charac teristic impedance and the endtoend propagation time delay. SEBTIEJN 1 1.4 INPUT IMPEDANCE, STANDING WAVE RATIO, AND POWER 11.17 ' (a) Show that a transmission coefﬁcient may be deﬁned as n 22L = =1+r=
TL v: L ZL+zo (b) Find 7L when the line is terminated by (i) a load whose value is nZO, (ii) an open
circuit, (iii) a short circuit, (iv) ZL = Z0 (matched line). 11.18 A 50 Q coaxial cable feeds a 75 + j20 Q dipole antenna. Find F and 3. 11.19 A lossy transmission line has R = 3.5 Q/m, L = 2 MH/m, C = 120 pF/m, and G '~V 0. At
400 MHZ, determine a, ,8, Z0, and M. 11.20 Show that a lossy transmission line of length 6 has an input impedance ZSC = Z0 tanh 76
when shorted and Zoc = Z0 coth 7/5 when open. Conﬁrm eqs. (11.41) and (11.42). 11.21 Find the input impedance of a shortncircuited coaxial transmission line of Figure 11.46 if
Zo = 65 + j38 S2, 7 = 0.7 + j2.5/m, 6 = 0.8 m. 11.22 Refer to the lossless transmission line shown in Figure 11.47. (a) Find 1‘ and s. (b) Deter
mine Zin at the generator. Figure 11.47 For Problem 11.22. PROBLEMS 567 (a) (b) Figure 11.48 For Problem 11.26: (a) network, (b) lossy line. 11.23 A 50 (2 lossless line has VL = 1061250 V, ZL = 50ej3°°. Find the current at N8 from the
load. 11.24 A 60 (Z lossless line is connected to a source with Vg == lOﬂ Vrms and Zg = 50 —
1'40 (2 and terminated with a load of j40 Q. If the line is 100 m long and 6 = 0.25 rad/m,
calculate Zn and V at
(a) The sending end
(b) The receiving end
(c) 4 m from the load
(d) 3 m from the source 11.25 A lossless transmission line with a characteristic impedance of 75 S2 is terminated by a
load of 120 (2. The length of the line is 1.25)\. If the line is energized by a source of
100 V (rms) with an internal impedance of \50 9, determine (a) the input impedance and
(b) the magnitude of the load voltage. *11.26 Consider the two—port network shown in Figure ll.48(a). The relation between the input
and output variables can be written in matrix form as V1 2 A B V2
11 C D _12
For the lossy line in Figure ll.48(b), show that the ABCD matrix is cosh 76 Z0 sinh 76
l — sinh 78 cosh 76
20 BECTIDN 1 1.5 THE SMITH CHART 11.27 A quarter—wave lossless 100 (2 line is terminated by a load ZL = 210 Q. If the voltage at
the receiving end is 80 V, what is the voltage at the sending end? 11.28 A 50 (2 transmission line is terminated by a 100 + leO 9 load. How far from the load
will the line impedance be 50 + j 110 Q? 568 CHAPTER 1 1 TRANSMISSION LINES M2 . . . .. an .. .4 . NH...» .. _, M“... . .. Lanna“... . 3 ZL=60—j400 Figure 11.49 For Problem 11.29. 11.29 11.30 11.31 11.32 11.33 11.34 11.35 11.36 11.37 Three lossless lines are connected as shown in Figure 11.49. Determine the input
impedance. A 50 Q lossless line is 4.2 m long. At the operating frequency of 300 MHz, the input im
pedance at the middle of the line is 80 — j60 9. Find the input impedance at the genera
tor and the voltage reﬂection coefﬁcient at the load. Take it = 0.8c. A 60 9 air line operating at 20 MHZ is 10 m long. If the input impedance is 90 + j150 9, Calculate ZL, F, and s. Consider an 80 (2 lossless transmission line terminated in ZL = 100 — j 150 (2. Find .9 and
the shortest distance from the load at which the input impedance is purely resistive. Determine the normalized input impedance at N8 from the load if (a) its normalized im
pedance is 2 + j, (b) its normalized admittance is 0.2 — j0.5, (c) the reﬂection coefﬁ—
cient at the load is 0.3 + j0.4. A lossless 50 9 line is terminated by a load ZL = 75 + j60 (2. Using a Smith chart, deter
mine (a) the reﬂection coefﬁcient I‘, (b) the standing wave ratio 3, (c) the input imped—
ance at 0.2)» from the load, ((1) the location of the ﬁrst minimum voltage from the load,
(e) the shortest distance from the load at which the input impedance is purely resistive. A transmission line is terminated by a load with admittance Y L = (0.6 + j0.8)/Zo. Find
the normalized input impedance at N6 from the load. An 80 0 transmission line operating at 12 MHz is terminated by a load ZL. At 22 m from
the load, the input impedance is 100 — j120 Q. If u = 0.86, (a) Calculate I‘L, Zimmax, and Zimmin.
(b) Find ZL, s, and the input impedance at 28 m from the load. (0) How many Zimmax and Zimmin are there between the load and the 100 ~ j 120 0 input
impedance? ‘ An antenna, connected to a 150 (2 lossless line, produces a standing wave ratio of 2.6 If
measurements indicate that voltage maxima are 120 cm apart and that the last maximum _
is 40 cm from the antenna, calculate (a) The operating frequency (b) The antenna impedance (c) The reﬂection coefﬁcient (assume that u = c). P R D B LEM s 569 A Figure 11.50 Double section trans
former of Problem 11.41.
20 201 202 759.
; l
I A I
r———2—~—+——z———4 11.38 The observed standing wave ratio on a 100 Q lossless line is 8. If the ﬁrst maximum volt
age occurs at 0.3)\ from the load, calculate the load impedance and the voltage reﬂection
coefﬁcient at the load. 11.39 An 80 Q lossless line has ZL = j60 (2 and Zn = j40 (Z. (a) Determine the shortest length
of the line. (b) Calculate s and PL. 11.40 A 75 S2 lossless line is terminated by an unknown load impedance ZL. If at a distance 0.2%
from the load the voltage is V. = 2 + j V while the current is 10 mA, ﬁnd ZL and 5. 11.41 Two N4 transformers in tandem are to connect a 50 9 line to a 75 (2 load as in Figure 11.50. (a) Determine the characteristic impedance Zol if 202 = 30 Q and there is no reﬂected
wave to the left of A. (b) If the best results are obtained when [22]?“ = .221 z Zol ZoZ ZL
determine Z01 and Z02 for this case. 11.42 Two identical antennas, each with input impedance 74 (2, are fed with three identical
50 Q quarterwave lossless transmission lines as shown in Figure 11.51. Calculate the
input impedance at the source end. 11.43 If the lines in Figure 11.51 are connected to a voltage source of 120 V with an internal
impedance of 80 (2, calculate the average power delivered to either antenna. ‘ 74 0 Figure 11.51 For Problems 11.42 and 11.43.
M4 ,, 570 CHAPTER 1 1 TRANSMISSION LINES 200 9 Figure 11.52 For Problem 11.44. Short
circuit 11.44 Consider the three lossless lines in Figure 11.52. If Z0 = 50 (2, calculate: (a) Zn looking into line 1
(b) Zin looking into line 2
(0) Zn looking into line 3 11.45 A section of lossless transmission line is shunted across the main line as in Figure 11.53. If (51 = N4, £2 = M8, and 63 = 7N8, ﬁnd yin], yinz, and yin3 given that Z0 = 100 9,
Z L = 200 + leO (2. Repeat the calculations if the shorted section were open. 3EET‘IDN 1 1.6 SOME APPLIEATIDNS UF TRANSMISSION LINES 11.46 It is desired to match a 50 (2 line to a load impedance of 60 — j50 9. Design a 50 Q stub
that will achieve the match. Find the length of the line and how far it is from the load. 11.47 A stub of length 0.12% is used to match a 60 (Z lossless line to a load. If the stub is located
at 0.3)\ from the load, calculate
(a) The load impedance ZL
(b) The length of an alternative stub and its location with respect to the load
(0) The standing wave ratio between the stub and the load 11.48 A 50 Q lossless transmission line that is 20 m long is terminated into a 120 + j220 (2 load.
To perfectly match, what should be the length and location of a short—circuited stub line?
Assume an operating frequency of 10 MHz. Figure 11.53 For Problem 11.45. PRDBLEMS 571 Figure 11.54 For Problem 11.50. 11.49 On a lossless line, measurements indicate 3 = 4.2 with the ﬁrst maximum voltage at N4
from the load. Determine how far from the load a shortcircuited stub should be located
and calculate its length. 11.50 A 60 9 lossless line terminated by load ZL has a voltage wave as shown in Figure 11.54.
Find s, I‘, and ZL. 11.51 The following slottedline measurements were taken on a 50 (2 system. With load:
5 = 3.2, adjacent Vmin occurs at 12 cm and 32 cm (high numbers on the load side);
with short circuit: Vmin occurs at 21 cm. Find the operating frequency and the load
impedance. 11.52 A 50 9 air slotted line is applied in measuring a load impedance. Adjacent minima are
found at 14 cm and 22.5 cm from the load when the unknown load is connected, and
Vmax = 0.95 V and me = 0.45 V. When the load is replaced by a short circuit, the min— ima are 3.2 cm to the load. Determine s, f, F, and Z L. SECTION 1 1.7 TRANSIENTS DN TRANSMISSIDN LINES 11.53 A 50 (2 coaxial cable is connected to an 80 S2 resistive load and a dc source with zero inter
nal resistance. Calculate the voltage reﬂection coefﬁcients at the source and at the load. 11.54 The switch in Figure 11.55 is closed at t = 0. Sketch the voltage and current at the right
side of the switch for 0 < t < 66m. Take ZO = 50 S2 and {flu = 2 ,us. Assume a lossless
transmission line. H
O 220 t 27 V Z0, Y 0.5 Z0 Figure 11.55 For Problem 11.54. 572 CHAPTER 1 1 TRANSMISSION LINES 509 100V u=2><108mls 1509 200 m Figure 11.56 For Problem 11.55. 11.55 For the system shown in Figure 11.56, sketch V(€, t) and [(6, t) for 0 < t < 5 Ms. *11.56 Refer to Figure 11.57, where Zg = 25 Q, Zo = 50 Q, ZL = 150 (2, 6 = 150 m, u = 6.
Assume that at t = 0, the pulse shown in Figure 11.58 is incident on the line. (a) Draw the voltage and current bounce diagrams.
(b) Determine V(0, t), V(€, t), [(0, t), and I(€, t) for O < t < 8 as. 11.57 A 12 V battery with an internal resistance of 10 Q is connected to a 20 m length of 50 Q coaxial cable with phase velocity of 2 X 108 m/s. If the receiving end is shortcircuited,
sketch the sending voltage V(0, t) and the receiving—end voltage V(€, t). Figure 11.57 For Problem 11.56. Figure 11.58 Two rectangular pulses of Problem
11.56. t (#5) PROBLEMS 573 SEBTIDN 1 1.8 APPLIEATIEJN NDTES: MIBRDSTRIP AND
EHARABTERIZATHZ‘JN DF‘ DATA EABLES 11.58 Using eq. (11.70) and Matlab, plot the effective relative permittivity for the h/w ratio
varying from 0.1 to 10. Assume 8, = 13 (gallium arsenide). ~ 11.59 A microstrip line is 1 cm thick and 1.5 cm wide. The conducting strip is made of brass
(arc = 1.1 X 107 S/m), while the substrate is a dielectric material with a, = 2.2 and
tan 0 =0.02. If the line operates at 2.5 GHZ, ﬁnd (a) Z0 and Seff, (b) ac and ad, (0) the dis—
tance down the line before the wave drops by 20 dB. 11.60 A 50 (2 microstrip line has a phase shift of 45° at 8 GHZ. If the substrate thickness is
h = 8 mm with 3,. = 4.6, ﬁnd (a) the Width of the conducting strip, (b) the length of the
microstrip line. 11.61 An alumina substrate (8 = 9.680) of thickness 2 mm is used for the construction of a mi
crostrip circuit. If the circuit designer can choose a line width within 0.4 mm and 8.0 mm,
What is the range of characteristic impedance of the line? 11.62 Design a 75 Q microstrip line on a 1.2 mm thick duroid (8,. = 2.3) substrate. Find the
Width of the conducting strip and the phase velocity. 11.63 Find the return loss due to a 1.50 9 cable terminated by a 100 9 load. ...
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