1. Transmission line capacitor. (a) Design an open-circuited 50Ω air
transmission line with the shortest length that will provide the
impedance of a 4 nF capacitor at 10 GHz. (b) Redesign the same capacitor
using a short-circuited 50Ω air transmission line. (c) Which design
yields the shortest length and why? Get solution
2. Transmission line inductor. (a) Design an open-circuited 50Ω lossless microstrip transmission line with shortest electrical length that will provide the impedance of a 5 nH inductor at 5 GHz. (b) Repeat the same design using a short-circuited 50Ω microstrip line. (c) Which design resulted in a shortest length line and why? Get solution
3. Input impedance of a transmission line. Consider a short-circuited 50Ω lossless transmission line as shown in Figure. Find the shortest electrical length of the line such that (a) Zin = j 50Ω, (b) Zin = −j 150Ω, (c) Zin =∞, and (d) Zin = 0.Figure Short-circuited transmission line. Problem.... Get solution
4. Terminated transmission line. A 50Ω lossless transmission line is terminated with a capacitive load having a load impedance of ZL = 100 − j 100Ω as shown in Figure. (a) Find the load reflection coefficient, ΓL. (b) Find the standing wave ratio S on the line. (c) Find the input impedance of the line, Zin, for four different line lengths: l1 = 0.125λ, l2 = 0.25λ, l3 = 0.375λ, and l4 = 0.5λ, respectively.Figure Terminated transmission line. Problem.... Get solution
5. Input impedance. Find the input impedance Zin of the two cascaded lossless transmission line system shown in Figure for (a) l2 = 0.25λ, (b) l2 = 0.5λ, and (c) l2 = 0.125λ.Figure Two cascaded transmission lines. Problem.... Get solution
6. Source- and load-end voltages. For the lossless transmission line circuit shown in Figure, calculate the phasor-form source-end and load-end voltages Vs and VL.Figure Source- and load-end voltages. Problem.... Get solution
7. Capacitive termination An air transmission line with Z0 = 50Ω is terminated with a capacitive load impedance of 50 − j 90Ω. If the line is operated at 100 MHz, calculate (a) the load reflection coefficient ΓL, (b) the standing-wave ratio S on the line, (c) the distance from the load to the first voltage minimum, and (d) the distance from the load to the first voltage maximum. Get solution
8. A wireless communication antenna. The following table provides the approximate values at various frequencies of the feed-point impedance of a circularly polarized patch antenna used in the wireless industry for making cellular phone calls in difficult environments, such as sport arenas and office buildings:f(MHz)ZL(Ω)80021.5 − j 15.485038.5 + j 2.2490043.8 + j 9.7495055.2 − j 10.2100028.8 − j 7.40If this antenna is directly fed by a 50Ω transmission line, find and sketch the standing-wave ratioSas a function of frequency. Get solution
9. Resistive line impedance. A 100Ω transmission line is terminated with a load impedance of 30 + j 60Ω at z = 0. Find the minimum electrical length l/λ of the line at which the line impedance (i.e., Z (z =−l )) is purely resistive. What is the value of the resistive line impedance? Get solution
10. Real Zin position. Consider a 50Ω transmission line terminated with a load impedance of 50 + j 140Ω. (a) Find the load reflection coefficient ΓL and the standing wave ratio S on the line. (b) Find the electrical position on the line nearest to the load where the line impedance is purely real. (c) Find the value of the line impedance at the position found in part (b). Get solution
11. Cascaded transmission lines. Two lossless transmission lines are cascaded as shown in Figure. (a) Find the standing-wave ratio S on each line. (b) Find the time-average power delivered to the load.Figure Inductive termination. Problem.... Get solution
12. Inductive termination. A 75Ω lossless transmission line is terminated with an inductive load given by 45 + j 60Ω, as shown in Figure. Calculate (a) the load reflection coefficient ΓL, (b) the standing-wave ratio S on the line, (c) the percentage time-average incident power that is absorbed by the load, and (d) the Vmax and Vmin positions that are nearest to the load.Figure Inductive termination. Problem.... Get solution
13. Input impedance. The input impedance of an 80Ω transmission line terminated with an unknown load ZL and having an electrical length of 3/8 at frequency f1 is measured to be 64 − j 48. Find the new value of Zin at frequency f2=2f1. Assume ZL and Z0 values to stay the same with frequency. Get solution
14. Resistive load. A lossless line is terminated with a resistive load of 120Ω. If the line presents an impedance of 48 + j 36Ω at a position 3λ/8 away from the load, what is the characteristic impedance Z0 of the line? Get solution
15. Input impedance. For the lossless transmission-line system shown in Figure, find Zin for the following load impedances: (a) ZL=∞ (open circuit), (b) ZL = 0 (short circuit), and (c) ZL = Z0/2.Figure Input impedance. Problem.... Get solution
16. Input impedance. Find the input impedance Zin of the two cascaded lossless transmission lines as shown in Figure, where ZP = j 50Ω is a lumped impedance connected at the junction.Figure Cascaded transmission line. Problem.... Get solution
17. Input impedance. Repeat Problem for the circuit shown in Figure.Figure Input impedance. Problem....Input impedance. For the lossless transmission-line system shown in Figure, find Zin for the following load impedances: (a) ZL=∞ (open circuit), (b) ZL = 0 (short circuit), and (c) ZL = Z0/2.Figure Input impedance. Problem.... Get solution
18. Unknown load. Consider a 50Ω transmission line terminated with an unknown load. If the standing-wave ratio on the line is measured to be 4.2 and the nearest voltage minimum point on the line with respect to the load position is located at 0.21λ, find the following: (a) The load impedance ZL. (b) The nearest voltage maximum and the next voltage minimum positions with respect to the load. (c) The input impedance Zin at each position found in part (b). Get solution
19. Input impedance. For the lossless transmission-line system shown in Figure, what is the ratio Z01/Z02 if Zin = 225Ω?Figure Input impedance. Problem.... Get solution
20. Unknown termination. Consider a transmission line with Z0 = 50Ω terminated with an unknown load impedance ZL. (a) Show that...where lmin is the length from the load to the first voltage minimum and S is the standing wave ratio. (b) Measurements on a line with Z0 = 50Ω having an unknown termination ZL show that S =√3, lmin = 25 mm, and that the distance between successive minima is 10 cm. Find the load reflection coefficient ΓL and the unknown termination ZL. Get solution
21. Unknown load. A 50Ω air transmission line with a standing wave ratio of 3 has its first and second voltage maximums nearest to the load located at 0.1 m and 0.3 m respectively. Calculate (a) the operating frequency f and (b) the unknown load impedance ZL. Get solution
22. Unkown load. A 75Ω air transmission line with a standing wave ratio of 6 has a voltage maximum and an adjacent voltage minimum position located with respect to the load at 3 m and 4.5 m, respectively. Calculate (a) the operating frequency f and (b) the unknown load impedance ZL. Get solution
23. Distance to the first maximum. Derive a formula similar to that in Problem in terms of lmax, where lmax is the distance from the load to the first voltage maximum.Unknown termination. Consider a transmission line with Z0 = 50Ω terminated with an unknown load impedance ZL. (a) Show that...where lmin is the length from the load to the first voltage minimum and S is the standing wave ratio. (b) Measurements on a line with Z0 = 50Ω having an unknown termination ZL show that S =√3, lmin = 25 mm, and that the distance between successive minima is 10 cm. Find the load reflection coefficient ΓL and the unknown termination ZL. Get solution
24. Terminated transmission line. A 50Ω air transmission line is terminated with an inductive load impedance given by ZL = 100 + j 150 and excited by a sinusoidal voltage source as shown in Figure. Calculate: (a) the load reflection coefficient L, (b) the standing wave ratio S, (c) the time-average power delivered to the load, and (d) the first two Vmax and the first two Vmin positions on the transmission line nearest to the load.Figure Terminated transmission line. Problem.... Get solution
25. Terminated transmission line. A 50Ω, 10.5-m long air transmission line terminated with a load impedance of ZL = 70 + j 10Ω is excited by a sinusoidal voltage source, as shown in Figure. (a) Calculate the load reflection coefficient ΓL and the standing wave ratio S on the line. (b) Find all the Vmax and Vmin positions (in actual lengths) on the line. (c) Find all the Imax and Imin positions on the line. (d) Find the input impedance Zin seen at each Vmax and Vmin position. (e) Find the line impedance Z (z ) seen at the source end of the line and draw the equivalent lumped circuit with respect to the source end. (f) Find the phasor voltages Vs, V +, V–, and VL. (g) Find the Vmax and Vmin values. (h) Find the Imax and Imin values. (i) Find the time-average powers P+, P–, PRs , PL, and Psource. What percentage of the power carried by the incident wave reflects back towards the source? (j) Repeat parts (a) through (i) for a load impedance of ZL = 15 − j 35Ω.Figure Terminated transmission line. Problem.... Get solution
26. Power dissipation. For the lossless transmission line system shown in Figure, with Z0 = 100Ω, (a) calculate the time-average power dissipated in each load. (b) Switch the values of the load resistors (i.e., RL1 = 200Ω, RL2 = 50Ω), and repeat part (a).Figure Power dissipation. Problem.... Get solution
27. Power dissipation. Consider the transmission line system shown in Figure. (a) Find the time-average power dissipated in the load RL with the switch S open. (b) Repeat part (a) for the switch S closed. Assume steady state in each case.Figure Power dissipation. Problem.... Get solution
28. Power dissipation. Repeat Problem if the characteristic impedance of the transmission line on the source side is changed from 50Ω to 25 √2Ω.Power dissipation. Consider the transmission line system shown in Figure. (a) Find the time-average power dissipated in the load RL with the switch S open. (b) Repeat part (a) for the switch S closed. Assume steady state in each case.Figure Power dissipation. Problem.... Get solution
29. Cascaded transmission lines. Two cascaded lossless transmission lines are connected as shown in Figure. (a) Find the standing-wave ratio on each line. (b) Find the time-average powers delivered to impedances Z1 and Z2.Figure Cascaded transmission lines. Problem.... Get solution
30. Two antennas. Two antennas having feed-point impedances of ZL1 = 40 − j 30Ω and ZL2 = 100 + j 50Ω are fed with a transmission line system, as shown in Figure. (a) Find S on the main line. (b) Find the time-average power supplied by the sinusoidal source. (c) Find the time-average power delivered to each antenna. Assume lossless lines.Figure Two antennas. Problem.... Get solution
31. Power dissipation. For the transmission line network shown in Figure, calculate the time-average power dissipated in the load resistor RL.Figure Power dissipation. Problem.... Get solution
32. Three identical antennas. Three identical antennas A1, A2, and A3 are fed by a transmission line system, as shown in Figure. If the feed-point impedance of each antenna is ZL = 50 + j 50Ω, find the time-average power delivered to each antenna.Figure Three identical antennas. Problem.... Get solution
33. Power delivery. The transmission line system shown in Figure is to be used at a frequency f such that ω = 2πf = 0.5 × 109 rad-s−1. Determine the total time-average power supplied by the source and also the time-average power supplied to each of the three different 25Ω load resistances (connected to the ends of lines 2, 3, and 4).Figure Power delivery. Problem.... Get solution
34. Power delivery. Consider the transmission line system shown in Figure. (a) Calculate the time-average power dissipated in the load resistances RL1=50Ω and RL2=50Ω at f =f1. (b) Calculate the time-average power dissipated in the load resistances RL1=50Ω and RL2=50Ω at f =2f1. (c) Calculate the time-average power dissipated in the load resistances RL1=50Ω and RL2=50Ω at f =1.5f1Figure Power delivery. Problem.... Get solution
35. Matching with a single lumped element. The transmission line matching networks shown in Figure are designed to match a 100Ω load impedance to a 50Ω line. (a) For the network with a shunt element, find the minimum distance l from the load where the unknown shunt element is to be connected such that the input admittance seen at B–Bʹ has a conductance part equal to 0.02 S. (b) Determine the unknown shunt element and its element value such that the input impedance seen at A–Aʹis matched to the line (i.e., ZA–Aʹ = 50Ω) at 1 GHz. (c) For the matching network with a lumped series matching element, find the minimum distance l and the unknown element and its value such that a perfect match is achieved at 1 GHz. Assume vp = 30 cm-(ns)−1.Figure Matching with a single lumped element. Problem.... Get solution
36. Matching with series stub. A load impedance of 135 − j 90Ω is to be matched to a 75Ω lossless transmission line system, as shown in Figure If λ = 20 cm, what minimum length of transmission line l will yield a minimum length ls for the series stub?Figure Matching with a series shorted stub. Problem.... Get solution
37. Series stub matching. A series-shorted-stub matching network is designed to match a capacitive load of RL = 50Ω and CL = 10/(3π) pF to a 100Ω line at 3 GHz, as shown in Figure. (a) The stub is positioned at a distance of λ/4 away from the load. Verify the choice of this position and find the minimum electrical length of the stub to achieve a perfect match at the design frequency. (b) Calculate the standing-wave ratio S on the main line at 2 GHz. (c) Calculate S on the main line at 4 GHzFigure Series stub matching. Problem.... Get solution
38. Quarter-wave transformer. (a) Design a single-section quarter-wave matching transformer to match an RL = 20Ω load to a line with Z0 = 80Ω operating at 1.5 GHz. (b) Calculate the standing-wave ratio S of the designed circuit at 1.2 and 1.8 GHz. Get solution
39. Helical antenna. The feed-point impedance of an axial-mode helical antenna with a circumference C on the order of one wavelength is nearly purely resistive and is approximately given33 by RL ≃ 140(C/λ), with the restriction that 0.8λ ≤ C ≤ 1.2λ. Consider a helical antenna designed with a circumference of C = λ0 for operation at a frequency f0 and corresponding wavelength λ0. The antenna must be matched for use with a 50Ω transmission line at f0. (a) Design a single-stage quarter-wave transformer to realize the design objective. (b) Using the circuit designed in part (a), calculate the standing-wave ratio S on the 50Ω line at a frequency 15% above the design frequency. (c) Repeat part (b) at a frequency 15% below the design frequency. Get solution
40. Quarter-wave matching. Many microwave applications require very low values of S over a broad band of frequencies. The two circuits shown in Figure are designed to match a load of ZL = RL = 400Ω to a line with Z0 = 50Ω, at 900 MHz. The first circuit is an airfilled coaxial quarter-wave transformer, and the second circuit consists of two air-filled coaxial quarter-wave transformers cascaded together. (a) Design both circuits. Assume ZQ1ZQ2 = Z0ZL for the second circuit. (b) Compare the bandwidth of the two circuits designed by calculating S on each line at frequencies 15% above and below the design frequency.Figure Quarter-wave matching. Problem.... Get solution
41. Quarter-wave matching. A 75Ω coaxial line is connected directly to an antenna with a feed-point impedance of ZL = 156Ω. (a) Find the load-reflection coefficient and the standing- wave ratio. (b) An engineer is assigned the task of designing a matching network to match the feed-point impedance of the antenna (156Ω) to the 75Ω coaxial line. However, all he has available to use for this design is another coaxial line of characteristic impedance 52Ω. The engineer uses the quarter-wave matching technique to achieve the match. How? Get solution
42. L-section matching networks. A simple and practical matching technique is to use the lossless L-section matching network that consists of two reactive elements. (a) Two L-section matching networks marked A1 and A2, each consisting of a lumped inductor and a capacitor, as shown in Figure, are used to match a load impedance of ZL = 60 − j 80Ω to a 100Ω line. Determine the L section(s) that make(s) it possible to achieve the design goal, and calculate the appropriate values of the reactive elements at 800 MHz. (b) Repeat part (a) for the two L-section networks marked B1 and B2, consisting of two inductances and two capacitors, respectively.Figure L-section matching networks. Problem.... Get solution
43. Variable capacitor. A shunt stub filter consisting of an air-filled coaxial line terminated in a variable capacitor is designed to eliminate the FM radio frequencies (i.e., 88–108 MHz) on a transmission line with Z0 = 100Ω, as shown in Figure. If the stub length is chosen to be 25 cm, find the range of the variable capacitor needed to eliminate any frequency in the FM band. Assume the characteristic impedance of the stub to be also equal to 100Ω.Figure Variable capacitor. Problem.... Get solution
44. Impedance matching network design. Consider the transmission-line circuit as shown in Figure. As a design engineer, your task is to determine the electrical lengths of the two short-circuited stubs (each 50Ω) connected at the load position to match the load impedance ZL=25 − j 75Ω to the line characteristic impedance Z0=50Ω such that l1 + l2 is minimum.Figure Impedance matching network design. Problem.... Get solution
45. Matching with lumped reactive elements. Two variable reactive elements are positioned on a transmission line to match an antenna having a feed-point impedance of 100 + j 100Ω to a Z0=100Ω air-filled line at 5 GHz, as shown in Figure. (a) Determine the values of the two reactive elements to achieve matching. (b) If the reactive elements are to be replaced by shorted 50Ω air-filled stubs, determine the corresponding stub lengths.Figure Matching with lumped reactive elements. Problem.... Get solution
46. Standing-wave ratio. For the transmission line shown in Figure, calculate S on the main line at (a) 800 MHz, (b) 880 MHz, and (c) 960 MHz.Figure Standing-wave ratio. Problem.... Get solution
47. Quarter-wave transformer design. Consider the transmission-line circuit, as shown in Figure. Design the quarter-wave matching network shown (find its electrical position l/λ from the load and its characteristic impedance ZQ) to match the load impedance ZL to Z0 under the condition ZQ > Z0.Figure Quarter-wave transformer design. Problem.... Get solution
48. mpedance matching. A load of ZL=36 + j 40Ω is to be matched to a main line of characteristic impedance Z0=100 using a circuit as shown in Figure. (a) The open-circuited stub is positioned at a distance of l1=λ/4 away from the load. Determine the characteristic impedance ZQ of this quarter-wave segment and the stub length ls which would result in a match. (b) For the values found in part (a), calculate the standing wave ratio S on the quarter-wave line segment with the characteristic impedance ZQ. Also find the magnitude and the phase of the V1 at the open-circuited end of the stub.Figure Impedance matching. Problem.... Get solution
49. Unknown feed-point impedance. A 50Ω transmission line is terminated with an antenna that has an unknown feed-point impedance. An engineer runs tests on the line and measures the standing-wave ratio, wavelength, and a voltage minimum location away from the antenna feed point to be at 3.2 cm, 20 cm, and 74 cm, respectively. Use the Smith chart to find the feed-point impedance of the antenna. Get solution
50. A lossy high-speed interconnect. The per-unit line parameters of an IC interconnect at 5 GHz are extracted using a high-frequency measurement technique resulting in R = 143.5Ω-(cm)−1, L = 10.1 nH-(cm)−1, C = 1.1 pF-(cm)−1, and G = 0.014 S-(cm)−1, respectively. 34Find the propagation constant γ and the characteristic impedance Z0 of the interconnect at 5 GHz. Get solution
51. Characterization of a high-speed GaAs interconnect. The propagation constant γ and the characteristic impedance Z0 at 5 GHz of the GaAs coplanar strip interconnects considered in Example 3.27 are determined from the measurements to be γ ≃ 1.1 np-(cm)−1 + j 3 rad-(cm)−1 and Z0 ≃ 110 − j 40Ω, respectively. Using these values, calculate the per-unit length parameters (R, L, G, and C) of the coplanar strip line at 5 GHz. Get solution
52. A lossy high-speed interconnect. Consider a high-speed microstrip transmission line of length 10 cm used to connect a 1-V amplitude, 2-GHz, 50Ω sinusoidal voltage source to an integrated circuit chip having an input impedance of 50Ω. The per-unit parameters of the microstrip line at 2 GHz are measured to be approximately given by R = 7.5Ω-(cm) −1, L = 4.6 nH-(cm) −1, C = 0.84 pF-(cm) −1, and G =0, respectively. (a) Find the propagation constant γ and the characteristic impedance Z0 of the line. (b) Find the voltages at the source and the load ends of the line. (c) Find the time-average power delivered to the line by the source and the time-average power delivered to the load. What is the power dissipated along the line? Get solution
2. Transmission line inductor. (a) Design an open-circuited 50Ω lossless microstrip transmission line with shortest electrical length that will provide the impedance of a 5 nH inductor at 5 GHz. (b) Repeat the same design using a short-circuited 50Ω microstrip line. (c) Which design resulted in a shortest length line and why? Get solution
3. Input impedance of a transmission line. Consider a short-circuited 50Ω lossless transmission line as shown in Figure. Find the shortest electrical length of the line such that (a) Zin = j 50Ω, (b) Zin = −j 150Ω, (c) Zin =∞, and (d) Zin = 0.Figure Short-circuited transmission line. Problem.... Get solution
4. Terminated transmission line. A 50Ω lossless transmission line is terminated with a capacitive load having a load impedance of ZL = 100 − j 100Ω as shown in Figure. (a) Find the load reflection coefficient, ΓL. (b) Find the standing wave ratio S on the line. (c) Find the input impedance of the line, Zin, for four different line lengths: l1 = 0.125λ, l2 = 0.25λ, l3 = 0.375λ, and l4 = 0.5λ, respectively.Figure Terminated transmission line. Problem.... Get solution
5. Input impedance. Find the input impedance Zin of the two cascaded lossless transmission line system shown in Figure for (a) l2 = 0.25λ, (b) l2 = 0.5λ, and (c) l2 = 0.125λ.Figure Two cascaded transmission lines. Problem.... Get solution
6. Source- and load-end voltages. For the lossless transmission line circuit shown in Figure, calculate the phasor-form source-end and load-end voltages Vs and VL.Figure Source- and load-end voltages. Problem.... Get solution
7. Capacitive termination An air transmission line with Z0 = 50Ω is terminated with a capacitive load impedance of 50 − j 90Ω. If the line is operated at 100 MHz, calculate (a) the load reflection coefficient ΓL, (b) the standing-wave ratio S on the line, (c) the distance from the load to the first voltage minimum, and (d) the distance from the load to the first voltage maximum. Get solution
8. A wireless communication antenna. The following table provides the approximate values at various frequencies of the feed-point impedance of a circularly polarized patch antenna used in the wireless industry for making cellular phone calls in difficult environments, such as sport arenas and office buildings:f(MHz)ZL(Ω)80021.5 − j 15.485038.5 + j 2.2490043.8 + j 9.7495055.2 − j 10.2100028.8 − j 7.40If this antenna is directly fed by a 50Ω transmission line, find and sketch the standing-wave ratioSas a function of frequency. Get solution
9. Resistive line impedance. A 100Ω transmission line is terminated with a load impedance of 30 + j 60Ω at z = 0. Find the minimum electrical length l/λ of the line at which the line impedance (i.e., Z (z =−l )) is purely resistive. What is the value of the resistive line impedance? Get solution
10. Real Zin position. Consider a 50Ω transmission line terminated with a load impedance of 50 + j 140Ω. (a) Find the load reflection coefficient ΓL and the standing wave ratio S on the line. (b) Find the electrical position on the line nearest to the load where the line impedance is purely real. (c) Find the value of the line impedance at the position found in part (b). Get solution
11. Cascaded transmission lines. Two lossless transmission lines are cascaded as shown in Figure. (a) Find the standing-wave ratio S on each line. (b) Find the time-average power delivered to the load.Figure Inductive termination. Problem.... Get solution
12. Inductive termination. A 75Ω lossless transmission line is terminated with an inductive load given by 45 + j 60Ω, as shown in Figure. Calculate (a) the load reflection coefficient ΓL, (b) the standing-wave ratio S on the line, (c) the percentage time-average incident power that is absorbed by the load, and (d) the Vmax and Vmin positions that are nearest to the load.Figure Inductive termination. Problem.... Get solution
13. Input impedance. The input impedance of an 80Ω transmission line terminated with an unknown load ZL and having an electrical length of 3/8 at frequency f1 is measured to be 64 − j 48. Find the new value of Zin at frequency f2=2f1. Assume ZL and Z0 values to stay the same with frequency. Get solution
14. Resistive load. A lossless line is terminated with a resistive load of 120Ω. If the line presents an impedance of 48 + j 36Ω at a position 3λ/8 away from the load, what is the characteristic impedance Z0 of the line? Get solution
15. Input impedance. For the lossless transmission-line system shown in Figure, find Zin for the following load impedances: (a) ZL=∞ (open circuit), (b) ZL = 0 (short circuit), and (c) ZL = Z0/2.Figure Input impedance. Problem.... Get solution
16. Input impedance. Find the input impedance Zin of the two cascaded lossless transmission lines as shown in Figure, where ZP = j 50Ω is a lumped impedance connected at the junction.Figure Cascaded transmission line. Problem.... Get solution
17. Input impedance. Repeat Problem for the circuit shown in Figure.Figure Input impedance. Problem....Input impedance. For the lossless transmission-line system shown in Figure, find Zin for the following load impedances: (a) ZL=∞ (open circuit), (b) ZL = 0 (short circuit), and (c) ZL = Z0/2.Figure Input impedance. Problem.... Get solution
18. Unknown load. Consider a 50Ω transmission line terminated with an unknown load. If the standing-wave ratio on the line is measured to be 4.2 and the nearest voltage minimum point on the line with respect to the load position is located at 0.21λ, find the following: (a) The load impedance ZL. (b) The nearest voltage maximum and the next voltage minimum positions with respect to the load. (c) The input impedance Zin at each position found in part (b). Get solution
19. Input impedance. For the lossless transmission-line system shown in Figure, what is the ratio Z01/Z02 if Zin = 225Ω?Figure Input impedance. Problem.... Get solution
20. Unknown termination. Consider a transmission line with Z0 = 50Ω terminated with an unknown load impedance ZL. (a) Show that...where lmin is the length from the load to the first voltage minimum and S is the standing wave ratio. (b) Measurements on a line with Z0 = 50Ω having an unknown termination ZL show that S =√3, lmin = 25 mm, and that the distance between successive minima is 10 cm. Find the load reflection coefficient ΓL and the unknown termination ZL. Get solution
21. Unknown load. A 50Ω air transmission line with a standing wave ratio of 3 has its first and second voltage maximums nearest to the load located at 0.1 m and 0.3 m respectively. Calculate (a) the operating frequency f and (b) the unknown load impedance ZL. Get solution
22. Unkown load. A 75Ω air transmission line with a standing wave ratio of 6 has a voltage maximum and an adjacent voltage minimum position located with respect to the load at 3 m and 4.5 m, respectively. Calculate (a) the operating frequency f and (b) the unknown load impedance ZL. Get solution
23. Distance to the first maximum. Derive a formula similar to that in Problem in terms of lmax, where lmax is the distance from the load to the first voltage maximum.Unknown termination. Consider a transmission line with Z0 = 50Ω terminated with an unknown load impedance ZL. (a) Show that...where lmin is the length from the load to the first voltage minimum and S is the standing wave ratio. (b) Measurements on a line with Z0 = 50Ω having an unknown termination ZL show that S =√3, lmin = 25 mm, and that the distance between successive minima is 10 cm. Find the load reflection coefficient ΓL and the unknown termination ZL. Get solution
24. Terminated transmission line. A 50Ω air transmission line is terminated with an inductive load impedance given by ZL = 100 + j 150 and excited by a sinusoidal voltage source as shown in Figure. Calculate: (a) the load reflection coefficient L, (b) the standing wave ratio S, (c) the time-average power delivered to the load, and (d) the first two Vmax and the first two Vmin positions on the transmission line nearest to the load.Figure Terminated transmission line. Problem.... Get solution
25. Terminated transmission line. A 50Ω, 10.5-m long air transmission line terminated with a load impedance of ZL = 70 + j 10Ω is excited by a sinusoidal voltage source, as shown in Figure. (a) Calculate the load reflection coefficient ΓL and the standing wave ratio S on the line. (b) Find all the Vmax and Vmin positions (in actual lengths) on the line. (c) Find all the Imax and Imin positions on the line. (d) Find the input impedance Zin seen at each Vmax and Vmin position. (e) Find the line impedance Z (z ) seen at the source end of the line and draw the equivalent lumped circuit with respect to the source end. (f) Find the phasor voltages Vs, V +, V–, and VL. (g) Find the Vmax and Vmin values. (h) Find the Imax and Imin values. (i) Find the time-average powers P+, P–, PRs , PL, and Psource. What percentage of the power carried by the incident wave reflects back towards the source? (j) Repeat parts (a) through (i) for a load impedance of ZL = 15 − j 35Ω.Figure Terminated transmission line. Problem.... Get solution
26. Power dissipation. For the lossless transmission line system shown in Figure, with Z0 = 100Ω, (a) calculate the time-average power dissipated in each load. (b) Switch the values of the load resistors (i.e., RL1 = 200Ω, RL2 = 50Ω), and repeat part (a).Figure Power dissipation. Problem.... Get solution
27. Power dissipation. Consider the transmission line system shown in Figure. (a) Find the time-average power dissipated in the load RL with the switch S open. (b) Repeat part (a) for the switch S closed. Assume steady state in each case.Figure Power dissipation. Problem.... Get solution
28. Power dissipation. Repeat Problem if the characteristic impedance of the transmission line on the source side is changed from 50Ω to 25 √2Ω.Power dissipation. Consider the transmission line system shown in Figure. (a) Find the time-average power dissipated in the load RL with the switch S open. (b) Repeat part (a) for the switch S closed. Assume steady state in each case.Figure Power dissipation. Problem.... Get solution
29. Cascaded transmission lines. Two cascaded lossless transmission lines are connected as shown in Figure. (a) Find the standing-wave ratio on each line. (b) Find the time-average powers delivered to impedances Z1 and Z2.Figure Cascaded transmission lines. Problem.... Get solution
30. Two antennas. Two antennas having feed-point impedances of ZL1 = 40 − j 30Ω and ZL2 = 100 + j 50Ω are fed with a transmission line system, as shown in Figure. (a) Find S on the main line. (b) Find the time-average power supplied by the sinusoidal source. (c) Find the time-average power delivered to each antenna. Assume lossless lines.Figure Two antennas. Problem.... Get solution
31. Power dissipation. For the transmission line network shown in Figure, calculate the time-average power dissipated in the load resistor RL.Figure Power dissipation. Problem.... Get solution
32. Three identical antennas. Three identical antennas A1, A2, and A3 are fed by a transmission line system, as shown in Figure. If the feed-point impedance of each antenna is ZL = 50 + j 50Ω, find the time-average power delivered to each antenna.Figure Three identical antennas. Problem.... Get solution
33. Power delivery. The transmission line system shown in Figure is to be used at a frequency f such that ω = 2πf = 0.5 × 109 rad-s−1. Determine the total time-average power supplied by the source and also the time-average power supplied to each of the three different 25Ω load resistances (connected to the ends of lines 2, 3, and 4).Figure Power delivery. Problem.... Get solution
34. Power delivery. Consider the transmission line system shown in Figure. (a) Calculate the time-average power dissipated in the load resistances RL1=50Ω and RL2=50Ω at f =f1. (b) Calculate the time-average power dissipated in the load resistances RL1=50Ω and RL2=50Ω at f =2f1. (c) Calculate the time-average power dissipated in the load resistances RL1=50Ω and RL2=50Ω at f =1.5f1Figure Power delivery. Problem.... Get solution
35. Matching with a single lumped element. The transmission line matching networks shown in Figure are designed to match a 100Ω load impedance to a 50Ω line. (a) For the network with a shunt element, find the minimum distance l from the load where the unknown shunt element is to be connected such that the input admittance seen at B–Bʹ has a conductance part equal to 0.02 S. (b) Determine the unknown shunt element and its element value such that the input impedance seen at A–Aʹis matched to the line (i.e., ZA–Aʹ = 50Ω) at 1 GHz. (c) For the matching network with a lumped series matching element, find the minimum distance l and the unknown element and its value such that a perfect match is achieved at 1 GHz. Assume vp = 30 cm-(ns)−1.Figure Matching with a single lumped element. Problem.... Get solution
36. Matching with series stub. A load impedance of 135 − j 90Ω is to be matched to a 75Ω lossless transmission line system, as shown in Figure If λ = 20 cm, what minimum length of transmission line l will yield a minimum length ls for the series stub?Figure Matching with a series shorted stub. Problem.... Get solution
37. Series stub matching. A series-shorted-stub matching network is designed to match a capacitive load of RL = 50Ω and CL = 10/(3π) pF to a 100Ω line at 3 GHz, as shown in Figure. (a) The stub is positioned at a distance of λ/4 away from the load. Verify the choice of this position and find the minimum electrical length of the stub to achieve a perfect match at the design frequency. (b) Calculate the standing-wave ratio S on the main line at 2 GHz. (c) Calculate S on the main line at 4 GHzFigure Series stub matching. Problem.... Get solution
38. Quarter-wave transformer. (a) Design a single-section quarter-wave matching transformer to match an RL = 20Ω load to a line with Z0 = 80Ω operating at 1.5 GHz. (b) Calculate the standing-wave ratio S of the designed circuit at 1.2 and 1.8 GHz. Get solution
39. Helical antenna. The feed-point impedance of an axial-mode helical antenna with a circumference C on the order of one wavelength is nearly purely resistive and is approximately given33 by RL ≃ 140(C/λ), with the restriction that 0.8λ ≤ C ≤ 1.2λ. Consider a helical antenna designed with a circumference of C = λ0 for operation at a frequency f0 and corresponding wavelength λ0. The antenna must be matched for use with a 50Ω transmission line at f0. (a) Design a single-stage quarter-wave transformer to realize the design objective. (b) Using the circuit designed in part (a), calculate the standing-wave ratio S on the 50Ω line at a frequency 15% above the design frequency. (c) Repeat part (b) at a frequency 15% below the design frequency. Get solution
40. Quarter-wave matching. Many microwave applications require very low values of S over a broad band of frequencies. The two circuits shown in Figure are designed to match a load of ZL = RL = 400Ω to a line with Z0 = 50Ω, at 900 MHz. The first circuit is an airfilled coaxial quarter-wave transformer, and the second circuit consists of two air-filled coaxial quarter-wave transformers cascaded together. (a) Design both circuits. Assume ZQ1ZQ2 = Z0ZL for the second circuit. (b) Compare the bandwidth of the two circuits designed by calculating S on each line at frequencies 15% above and below the design frequency.Figure Quarter-wave matching. Problem.... Get solution
41. Quarter-wave matching. A 75Ω coaxial line is connected directly to an antenna with a feed-point impedance of ZL = 156Ω. (a) Find the load-reflection coefficient and the standing- wave ratio. (b) An engineer is assigned the task of designing a matching network to match the feed-point impedance of the antenna (156Ω) to the 75Ω coaxial line. However, all he has available to use for this design is another coaxial line of characteristic impedance 52Ω. The engineer uses the quarter-wave matching technique to achieve the match. How? Get solution
42. L-section matching networks. A simple and practical matching technique is to use the lossless L-section matching network that consists of two reactive elements. (a) Two L-section matching networks marked A1 and A2, each consisting of a lumped inductor and a capacitor, as shown in Figure, are used to match a load impedance of ZL = 60 − j 80Ω to a 100Ω line. Determine the L section(s) that make(s) it possible to achieve the design goal, and calculate the appropriate values of the reactive elements at 800 MHz. (b) Repeat part (a) for the two L-section networks marked B1 and B2, consisting of two inductances and two capacitors, respectively.Figure L-section matching networks. Problem.... Get solution
43. Variable capacitor. A shunt stub filter consisting of an air-filled coaxial line terminated in a variable capacitor is designed to eliminate the FM radio frequencies (i.e., 88–108 MHz) on a transmission line with Z0 = 100Ω, as shown in Figure. If the stub length is chosen to be 25 cm, find the range of the variable capacitor needed to eliminate any frequency in the FM band. Assume the characteristic impedance of the stub to be also equal to 100Ω.Figure Variable capacitor. Problem.... Get solution
44. Impedance matching network design. Consider the transmission-line circuit as shown in Figure. As a design engineer, your task is to determine the electrical lengths of the two short-circuited stubs (each 50Ω) connected at the load position to match the load impedance ZL=25 − j 75Ω to the line characteristic impedance Z0=50Ω such that l1 + l2 is minimum.Figure Impedance matching network design. Problem.... Get solution
45. Matching with lumped reactive elements. Two variable reactive elements are positioned on a transmission line to match an antenna having a feed-point impedance of 100 + j 100Ω to a Z0=100Ω air-filled line at 5 GHz, as shown in Figure. (a) Determine the values of the two reactive elements to achieve matching. (b) If the reactive elements are to be replaced by shorted 50Ω air-filled stubs, determine the corresponding stub lengths.Figure Matching with lumped reactive elements. Problem.... Get solution
46. Standing-wave ratio. For the transmission line shown in Figure, calculate S on the main line at (a) 800 MHz, (b) 880 MHz, and (c) 960 MHz.Figure Standing-wave ratio. Problem.... Get solution
47. Quarter-wave transformer design. Consider the transmission-line circuit, as shown in Figure. Design the quarter-wave matching network shown (find its electrical position l/λ from the load and its characteristic impedance ZQ) to match the load impedance ZL to Z0 under the condition ZQ > Z0.Figure Quarter-wave transformer design. Problem.... Get solution
48. mpedance matching. A load of ZL=36 + j 40Ω is to be matched to a main line of characteristic impedance Z0=100 using a circuit as shown in Figure. (a) The open-circuited stub is positioned at a distance of l1=λ/4 away from the load. Determine the characteristic impedance ZQ of this quarter-wave segment and the stub length ls which would result in a match. (b) For the values found in part (a), calculate the standing wave ratio S on the quarter-wave line segment with the characteristic impedance ZQ. Also find the magnitude and the phase of the V1 at the open-circuited end of the stub.Figure Impedance matching. Problem.... Get solution
49. Unknown feed-point impedance. A 50Ω transmission line is terminated with an antenna that has an unknown feed-point impedance. An engineer runs tests on the line and measures the standing-wave ratio, wavelength, and a voltage minimum location away from the antenna feed point to be at 3.2 cm, 20 cm, and 74 cm, respectively. Use the Smith chart to find the feed-point impedance of the antenna. Get solution
50. A lossy high-speed interconnect. The per-unit line parameters of an IC interconnect at 5 GHz are extracted using a high-frequency measurement technique resulting in R = 143.5Ω-(cm)−1, L = 10.1 nH-(cm)−1, C = 1.1 pF-(cm)−1, and G = 0.014 S-(cm)−1, respectively. 34Find the propagation constant γ and the characteristic impedance Z0 of the interconnect at 5 GHz. Get solution
51. Characterization of a high-speed GaAs interconnect. The propagation constant γ and the characteristic impedance Z0 at 5 GHz of the GaAs coplanar strip interconnects considered in Example 3.27 are determined from the measurements to be γ ≃ 1.1 np-(cm)−1 + j 3 rad-(cm)−1 and Z0 ≃ 110 − j 40Ω, respectively. Using these values, calculate the per-unit length parameters (R, L, G, and C) of the coplanar strip line at 5 GHz. Get solution
52. A lossy high-speed interconnect. Consider a high-speed microstrip transmission line of length 10 cm used to connect a 1-V amplitude, 2-GHz, 50Ω sinusoidal voltage source to an integrated circuit chip having an input impedance of 50Ω. The per-unit parameters of the microstrip line at 2 GHz are measured to be approximately given by R = 7.5Ω-(cm) −1, L = 4.6 nH-(cm) −1, C = 0.84 pF-(cm) −1, and G =0, respectively. (a) Find the propagation constant γ and the characteristic impedance Z0 of the line. (b) Find the voltages at the source and the load ends of the line. (c) Find the time-average power delivered to the line by the source and the time-average power delivered to the load. What is the power dissipated along the line? Get solution
