# Problems

# Sections 2.1 to 2.4: Transmission-Line Model

1. 2.1 A transmission line of length l connects a load to a sinusoidal voltage source with an oscillation frequency f. Assuming that the velocity of wave propagation on the line is c, for which of the following situations is it reasonable to ignore the presence of the transmission line in the solution of the circuit?

   1. *a.l=20 cm, f=20 kHz

   2. b.l=50 km , f=60 Hz

   3. *c.l=20 cm, f=600 MHz

   4. d.l=1 mm, f=100 GHz

2. 2.2 A two-wire copper transmission line is embedded in a dielectric material with ε r=2.6 and σ=2×10−6 S/m. Its wires are separated by 3 cm and their radii are 1 mm each.

   1. Calculate the line parameters R′, L′, G′, and C′ at 2 GHz.

   2. Compare your results with those based on Module 2.1. Include a printout of the screen display.

3. 2.3 Show that the transmission-line model shown in Fig. P2.3 yields the same telegrapher’s equations given by Eqs. (2.14) and (2.16).

   # Figure P2.3 Transmission-line model for Problem 2.3.

   Figure P2.3 Full Alternative Text

4. *2.4 A 1 GHz parallel-plate transmission line consists of 1.2-cm wide copper strips separated by a 0.15-cm thick layer of polystyrene. Appendix B gives μ c=μ 0=4 π×10−7 (H/m) and σ c=5.8×10 7 (S/m) for copper, and ε r=2.6 for polystyrene. Use Table 2-1 to determine the line parameters of the transmission line. Assume that μ=μ 0 and σ≈0 for polystyrene.

5. 2.5 For the parallel-plate transmission line of Problem 2.4, the line parameters are given by R′=1 Ω/m, L′=167 nH/m, G′=0, and C′=172 pF/m. Find α, β, u p, and Z 0 at 1 GHz.

6. 2.6 A coaxial line with inner and outer conductor diameters of 0.5 cm and 1 cm, respectively, is filled with an insulating material with ε r=4.5 and σ=10−3 S/m. The conductors are made of copper.

   1. Calculate the line parameters at 1 GHz.

   2. Compare your results with those based on Module 2.2. Include a printout of the screen display.

7. 2.7 Find α, β, u p, and Z 0 for the coaxial line of Problem 2.6. Verify your results by applying Module 2.2. Include a printout of the screen display.

8. *2.8 Find α, β, u p, and Z 0 for the two-wire line of Problem 2.2. Compare results with those based on Module 2.1. Include a printout of the screen display.

# Section 2.5: Microstrip

1. 2.9 A lossless microstrip line uses a 1-mm wide conducting strip over a 1-cm thick substrate with ε r=2.5. Determine the line parameters, ε eff, Z 0, and β at 10 GHz. Compare your results with those obtained by using Module 2.3. Include a printout of the screen display.

2. *2.10 Use Module 2.3 to design a 100 Ω microstrip transmission line. The substrate thickness is 1.8 mm and its ε r=2.3. Select the strip width w, and determine the guide wavelength λ at f=5 GHz. Include a printout of the screen display.

3. 2.11 A 50 Ω microstrip line uses a 0.6 mm alumina substrate with ε r=9. Use Module 2.3 to determine the required strip width w. Include a printout of the screen display.

4. 2.12 Generate a plot of Z 0 as a function of strip width w over the range from 0.05 mm to 5 mm for a microstrip line fabricated on a 0.7-mm thick substrate with ε r=9.8.

# Section 2.6: Lossless Line

1. 2.13 In addition to not dissipating power, a lossless line has two important features: (1) it is dispersionless (u p is independent of frequency), and (2) its characteristic impedance Z 0 is purely real. Sometimes, it is not possible to design a transmission line such that R′≪ω L′ and G′≪ω C′, but it is possible to choose the dimensions of the line and its material properties to satisfy the condition

   R′C′=L′G′(distortionless line)

   Such a line is called a distortionless line, because despite the fact that it is not lossless, it nonetheless possesses the previously mentioned features of the lossless line. Show that for a distortionless line,

   α=R′C′L′=R′G′,β=ω L′C′,Z 0=L′C′

2. *2.14 For a distortionless line (see Problem 2.13) with Z 0=50 Ω, α=20 (mNp/m), and u p=2.5×10 8 (m/s), find the line parameters and λ at 100 MHz.

3. 2.15 Find α and Z 0 of a distortionless line whose R′=2 Ω/m and G′=2×10−4 S/m.

4. *2.16 A transmission line operating at 125 MHz has Z 0=40 Ω, α=0.02 (Np/m), and β=0.75 rad/m. Find the line parameters R′, L′, G′, and C′.

5. 2.17 Using a slotted line, the voltage on a lossless transmission line was found to have a maximum magnitude of 1.5 V and a minimum magnitude of 0.6 V. Find the magnitude of the load’s reflection coefficient.

6. *2.18 Polyethylene with ε r=2.25 is used as the insulating material in a lossless coaxial line with a characteristic impedance of 50 Ω. The radius of the inner conductor is 1.2 mm.

   1. What is the radius of the outer conductor?

   2. What is the phase velocity of the line?

7. 2.19 A 50 Ω lossless transmission line is terminated in a load with impedance Z L=(30−j 50) Ω. The wavelength is 8 cm. Determine:

   1. The reflection coefficient at the load.

   2. The standing-wave ratio on the line.

   3. The position of the voltage maximum nearest the load.

   4. The position of the current maximum nearest the load.

   5. Verify quantities in parts (a)–(d) using Module 2.4. Include a printout of the screen display.

8. 2.20 A 300 Ω lossless air transmission line is connected to a complex load composed of a resistor in series with an inductor, as shown in Fig. P2.20. At 5 MHz, determine: (a) Γ, (b) S, (c) location of voltage maximum nearest to the load, and (d) location of current maximum nearest to the load.

   # Figure P2.20 Circuit for Problem 2.20.

9. *2.21 On a 150 Ω lossless transmission line, the following observations were noted: distance of first voltage minimum from the load =3 cm; distance of first voltage maximum from the load =9 cm; and S=3. Find Z L.

10. 2.22 Using a slotted line, the following results were obtained: distance of first minimum from the load =4 cm; distance of second minimum from the load =14 cm; and voltage standing-wave ratio =1.5. If the line is lossless and Z 0=50 Ω, find the load impedance.

11. *2.23 A load with impedance Z L=(25−j 50) Ω is to be connected to a lossless transmission line with the characteristic impedance Z 0. Z 0 is chosen such that the standing-wave ratio is the smallest possible. What should Z 0 be?

12. 2.24 A 50 Ω lossless line terminated in a purely resistive load has a voltage standing-wave ratio of 3. Find all possible values of Z L.

13. 2.25 Apply Module 2.4 to generate plots of the voltage standing-wave pattern for a 50 Ω line terminated in a load impedance Z L=(100−j 50) Ω. Set V g=1 V, Z g=50 Ω, ε r=2.25, l=40 cm, and f=1 GHz. Also determine S, d m a x, and d min.

14. 2.26 A 50 Ω lossless transmission line is connected to a load composed of a 75 Ω resistor in series with a capacitor of unknown capacitance (Fig. P2.26). If at 10 MHz the voltage standing-wave ratio on the line was measured to be 3, determine the capacitance C.

    # Figure P2.26 Circuit for Problem 2.26.

# Section 2.7: Wave and Input Impedance

1. *2.27 At an operating frequency of 300 MHz, a lossless 50 Ω air-spaced transmission line 2.5 m in length is terminated with an impedance Z L=(40+j 20) Ω. Find the input impedance.

2. 2.28 A lossless transmission line of electrical length l=0.35 λ is terminated in a load impedance as shown in Fig. P2.28. Find Γ, S, and Z in. Verify your results using Modules 2.4 or 2.5. Include a printout of the screen’s output display.

   # Figure P2.28 Circuit for Problem 2.28.

3. 2.29 Show that the input impedance of a quarter-wavelength lossless line terminated in a short circuit appears as an open circuit.

4. 2.30 Show that, at the position where the magnitude of the voltage on the line is a maximum, the input impedance is purely real.

5. 2.31 A voltage generator with

   υ g(t)=5 cos(2 π×10 9 t) V

   and internal impedance Z g=50 Ω is connected to a 50 Ω lossless air-spaced transmission line. The line length is 5 cm and the line is terminated in a load with impedance Z L=(100−j 100) Ω. Determine:

   1. *a.Γ at the load.

   2. b.Z in at the input to the transmission line.

   3. c.The input voltage V˜i and input current I˜i.

   4. d.The quantities in parts (a)–(c) using Modules 2.4 or 2.5.

6. 2.32 A 6 m section of 150 Ω lossless line is driven by a source with

   v g(t)=5 cos(8 π×10 7 t−30°)(V)

   and Z g=150 Ω. If the line, which has a relative permittivity ε r=2.25, is terminated in a load Z L=(150−j 50) Ω, determine:

   1. a.λ on the line,

   2. *b.the reflection coefficient at the load,

   3. c.the input impedance,

   4. d.the input voltage V˜i,

   5. e.the time-domain input voltage v i(t), and

   6. f.quantities in parts (a)–(d) using Modules 2.4 or 2.5.

7. 2.33 Two half-wave dipole antennas, each with an impedance of 75 Ω, are connected in parallel through a pair of transmission lines, and the combination is connected to a feed transmission line, as shown in Fig. P2.33.

   # Figure P2.33 Circuit for Problem 2.33.

   Figure P2.33 Full Alternative Text

   All lines are 50 Ω and lossless.

   1. *a.Calculate Z in 1, the input impedance of the antenna-terminated line, at the parallel juncture.

   2. b.Combine Z in 1 and Z in 2 in parallel to obtain Z L′, the effective load impedance of the feedline.

   3. c.Calculate Z in of the feedline.

8. 2.34 A 50 Ω lossless line is terminated in a load impedance Z L=(30−j 20) Ω.

   1. Calculate Γ and S.

   2. It has been proposed that by placing an appropriately selected resistor across the line at a distance d m a x from the load (as shown in Fig. P2.34(b)), where d m a x is the distance from the load of a voltage maximum, then it is possible to render Z i=Z 0, thereby eliminating reflection back to the end. Show that the proposed approach is valid and find the value of the shunt resistance.

   # Figure P2.34 Circuit for Problem 2.34.

   Figure P2.34 Full Alternative Text

9. *2.35 For the lossless transmission line circuit shown in Fig. P2.35, determine the equivalent series lumped-element circuit at 400 MHz at the input to the line. The line has a characteristic impedance of 50 Ω and the insulating layer has ε r=2.25.

   # Figure P2.35 Circuit for Problem 2.35.

# Section 2.8: Special Cases of the Lossless Line

1. 2.36 At an operating frequency of 300 MHz, it is desired to use a section of a lossless 50 Ω transmission line terminated in a short circuit to construct an equivalent load with reactance X=40 Ω. If the phase velocity of the line is 0.75 c, what is the shortest possible line length that would exhibit the desired reactance at its input? Verify your result using Module 2.5.

2. *2.37 A lossless transmission line is terminated in a short circuit. How long (in wavelengths) should the line be for it to appear as an open circuit at its input terminals?

3. 2.38 The input impedance of a 31-cm long lossless transmission line of unknown characteristic impedance was measured at 1 MHz. With the line terminated in a short circuit, the measurement yielded an input impedance equivalent to an inductor with inductance of 0.064 μ H, and when the line was open-circuited, the measurement yielded an input impedance equivalent to a capacitor with capacitance of 40 pF. Find Z 0 of the line, the phase velocity, and the relative permittivity of the insulating material.

4. *2.39 A 75 Ω resistive load is preceded by a λ/4 section of a 50 Ω lossless line, which itself is preceded by another λ/4 section of a 100 Ω line. What is the input impedance? Compare your result with that obtained through two successive applications of Module 2.5.

5. 2.40 A 100 MHz FM broadcast station uses a 300 Ω transmission line between the transmitter and a tower-mounted half-wave dipole antenna. The antenna impedance is 73 Ω. You are asked to design a quarter-wave transformer to match the antenna to the line.

   1. Determine the electrical length and characteristic impedance of the quarter-wave section.

   2. If the quarter-wave section is a two-wire line with D=2.5 cm and the wires are embedded in polystyrene with ε r=2.6, determine the physical length of the quarter-wave section and the radius of the two wire conductors.

6. 2.41 A 50 Ω lossless line of length l=0.375 λ connects a 300 MHz generator with V˜g=300 V and Z g=50 Ω to a load Z L. Determine the time-domain current through the load for:

   1. a.Z L=(50−j 50) Ω

   2. *b.Z L=50 Ω

   3. c.Z L=0 (short circuit)

   For part (a), verify your results by deducing the information you need from the output products generated by Module 2.4.

# Section 2.9: Power Flow on Lossless Line

1. 2.42 A generator with V˜g=300 V and Z g=50 Ω is connected to a load Z L=75 Ω through a 50 Ω lossless line of length l=0.15 λ.

   1. *a.Compute Z in, the input impedance of the line at the generator end.

   2. b.Compute I˜ i and V˜i.

   3. c.Compute the time-average power delivered to the line, P in=1 2 R e[V˜i I˜ i∗].

   4. d.Compute V˜L, I˜L, and the time-average power delivered to the load, P L=1 2 R e[V˜L I˜ L∗]. How does P in compare with P L? Explain.

   5. e.Compute the time-average power delivered by the generator, P g, and the time-average power dissipated in Z g. Is conservation of power satisfied?

2. 2.43 If the two-antenna configuration shown in Fig. P2.43 is connected to a generator with V˜g=250 V and Z g=50 Ω, how much average power is delivered to each antenna?

   # Figure P2.43 Antenna configuration for Problem 2.43.

   Figure P2.43 Full Alternative Text

3. *2.44 For the circuit shown in Fig. P2.44, calculate the average incident power, the average reflected power, and the average power transmitted into the infinite 100 Ω line. The λ/2 line is lossless and the infinitely long line is slightly lossy. (Hint: The input impedance of an infinitely long line is equal to its characteristic impedance so long as α≠0.)

   # Figure P2.44 Circuit for Problem 2.44.

   Figure P2.44 Full Alternative Text

4. 2.45 The circuit shown in Fig. P2.45 consists of a 100 Ω lossless transmission line terminated in a load with Z L=(50+j 100) Ω. If the peak value of the load voltage was measured to be |V˜L|=12 V, determine:

   1. *a.the time-average power dissipated in the load,

   2. b.the time-average power incident on the line, and

   3. c.the time-average power reflected by the load.

   # Figure P2.45 Circuit for Problem 2.45.

5. 2.46 An antenna with a load impedance

   Z L=(75+j 25) Ω

   is connected to a transmitter through a 50 Ω lossless transmission line. If under matched conditions (50 Ω load) the transmitter can deliver 20 W to the load, how much power can it deliver to the antenna? Assume that Z g=Z 0.

# Section 2.10: Smith Chart

1. 2.47 Use the Smith chart to find the reflection coefficient corresponding to a load impedance of

   1. a.Z L=3 Z 0

   2. *b.Z L=(2−j 2)Z 0

   3. c.Z L=−j 2 Z 0

   4. d.Z L=0 (short circuit)

2. 2.48 Repeat Problem 2.47 using Module 2.6.

3. 2.49 Use the Smith chart to find the normalized load impedance corresponding to a reflection coefficient of

   1. Γ=0.5

   2. Γ=0.5 60°

   3. Γ=−1

   4. Γ=0.3−30°

   5. Γ=0

   6. Γ=j

4. *2.50 Use the Smith chart to determine the input impedance Z in of the two-line configuration shown in Fig. P2.50.

   # Figure P2.50 Circuit for Problem 2.50.

   Figure P2.50 Full Alternative Text

5. 2.51 Repeat Problem 2.50 using Module 2.6.

6. *2.52 On a lossless transmission line terminated in a load Z L=100 Ω, the standing-wave ratio was measured to be 2.5. Use the Smith chart to find the two possible values of Z 0.

7. 2.53 A lossless 50 Ω transmission line is terminated in a load with Z L=(50+j 25) Ω. Use the Smith chart to find the following:

   1. a.The reflection coefficient Γ.

   2. *b.The standing-wave ratio.

   3. c.The input impedance at 0.35 λ from the load.

   4. d.The input admittance at 0.35 λ from the load.

   5. e.The shortest line length for which the input impedance is purely resistive.

   6. f.The position of the first voltage maximum from the load.

8. 2.54 Repeat Problem 2.53 using Module 2.6.

9. *2.55 A lossless 50 Ω transmission line is terminated in a short circuit. Use the Smith chart to determine:

   1. The input impedance at a distance 2.3 λ from the load.

   2. The distance from the load at which the input admittance is Y in=−j 0.04 S.

10. 2.56 Repeat Problem 2.55 using Module 2.6.

11. *2.57 Use the Smith chart to find y L if z L=1.5−j 0.7.

12. 2.58 A lossless 100 Ω transmission line 3 λ/8 in length is terminated in an unknown impedance. If the input impedance is Z in=−j 2.5 Ω,

    1. Use the Smith chart to find Z L.

    2. Verify your results using Module 2.6.

13. 2.59 A 75 Ω lossless line is 0.6 λ long. If S=1.8 and θ r=−60°, use the Smith chart to find |Γ|, Z L, and Z in.

14. 2.60 Repeat Problem 2.59 using Module 2.6.

15. *2.61 Using a slotted line on a 50 Ω air-spaced lossless line, the following measurements were obtained: S=1.6 and |V˜|m a x occurred only at 10 cm and 24 cm from the load. Use the Smith chart to find Z L.

16. 2.62 At an operating frequency of 5 GHz, a 50 Ω lossless coaxial line with insulating material having a relative permittivity ε r=2.25 is terminated in an antenna with an impedance Z L=150 Ω. Use the Smith chart to find Z in. The line length is 30 cm.

# Section 2.11: Impedance Matching

1. *2.63 A 50 Ω lossless line 0.6 λ long is terminated in a load with Z L=(50+j 25) Ω. At 0.3 λ from the load, a resistor with resistance R=30 Ω is connected as shown in Fig. P2.63. Use the Smith chart to find Z in.

   # Figure P2.63 Circuit for Problem 2.63.

   Figure P2.63 Full Alternative Text

2. 2.64 Use Module 2.7 to design a quarter-wavelength transformer to match a load with Z L=(100−j 200) Ω to a 50 Ω line.

3. 2.65 Use Module 2.7 to design a quarter-wavelength transformer to match a load with Z L=(50+j 10) Ω to a 100 Ω line.

4. 2.66 A 200 Ω transmission line is to be matched to a computer terminal with Z L=(50−j 25) Ω by inserting an appropriate reactance in parallel with the line. If f=800 MHz and ε r=4, determine the location nearest to the load at which inserting:

   1. A capacitor can achieve the required matching, and the value of the capacitor.

   2. An inductor can achieve the required matching, and the value of the inductor.

5. 2.67 Repeat Problem 2.66 using Module 2.8.

6. 2.68 A 50 Ω lossless line is to be matched to an antenna with Z L=(75−j 20) Ω using a shorted stub. Use the Smith chart to determine the stub length and distance between the antenna and stub.

7. *2.69 Repeat Problem 2.68 for a load with

   Z L=(100+j 50) Ω.

8. 2.70 Repeat Problem 2.68 using Module 2.9.

9. 2.71 Repeat Problem 2.69 using Module 2.9.

10. 2.72 Determine Z in of the feed line shown in Fig. P2.72. All lines are lossless with Z 0=50 Ω.

    # Figure P2.72 Network for Problem 2.72.

    Figure P2.72 Full Alternative Text

11. *2.73 Repeat Problem 2.72 for the case where all three transmission lines are λ/4 in length.

12. 2.74 A 25 Ω antenna is connected to a 75 Ω lossless transmission line. Reflections back toward the generator can be eliminated by placing a shunt impedance Z at a distance l from the load (Fig. P2.74). Determine the values of Z and l.

    # Figure P2.74 Circuit for Problem 2.74.

    Figure P2.74 Full Alternative Text

# Section 2.12: Transients on Transmission Lines

1. 2.75 Generate a bounce diagram for the voltage V(z,t) for a 1-m long lossless line characterized by Z 0=50 Ω and u p=2 c/3 (where c is the velocity of light) if the line is fed by a step voltage applied at t=0 by a generator circuit with V g=60 V and R g=100 Ω. The line is terminated in a load R L=25 Ω. Use the bounce diagram to plot V(t) at a point midway along the length of the line from t=0 to t=25 ns.

2. 2.76 Repeat Problem 2.75 for the current I(z,t) on the line.

3. 2.77 In response to a step voltage, the voltage waveform shown in Fig. P2.77 was observed at the sending end of a lossless transmission line with R g=50 Ω, Z 0=50 Ω, and ε r=2.25. Determine:

   1. the generator voltage,

   2. the length of the line, and

   3. the load impedance.

4. *2.78 In response to a step voltage, the voltage waveform shown in Fig. P2.78 was observed at the sending end of a shorted line with Z 0=50 Ω and ε r=4. Determine V g, R g, and the line length.

   # Figure P2.78 Voltage waveform of Problem 2.78.

   Figure P2.78 Full Alternative Text

5. 2.79 Suppose the voltage waveform shown in Fig. P2.77 was observed at the sending end of a 50 Ω transmission line in response to a step voltage introduced by a generator with V g=15 V and an unknown series resistance R g. The line is 1 km in length, its velocity of propagation is 1×10 8 m/s, and it is terminated in a load R L=100 Ω.

   # Figure P2.77 Voltage waveform for Problems 2.77 and 2.79.

   Figure P2.77 Full Alternative Text

   1. Determine R g.

   2. Explain why the drop in level of V(0,t) at t=6 μ s cannot be due to reflection from the load.

   3. Determine the shunt resistance R f and location of the fault responsible for the observed waveform.

6. 2.80 A generator circuit with V g=200 V and R g=25 Ω was used to excite a 75 Ω lossless line with a rectangular pulse of duration τ=0.4 μ s. The line is 200 m long, its u p=2×10 8 m/s, and it is terminated in a load R L=125 Ω.

   1. Synthesize the voltage pulse exciting the line as the sum of two step functions, V g 1(t) and V g 2(t).

   2. For each voltage step function, generate a bounce diagram for the voltage on the line.

   3. Use the bounce diagrams to plot the total voltage at the sending end of the line.

   4. Confirm the result of part (c) by applying Module 2.10.

7. 2.81 For the circuit of Problem 2.80, generate a bounce diagram for the current and plot its time history at the middle of the line.

8. *2.82 In response to a step voltage, the voltage waveform shown in Fig. P2.82 was observed at the midpoint of a lossless transmission line with Z 0=50 Ω and u p=2×10 8 m/s. Determine: (a) the length of the line, (b) Z L, (c) R g, and (d) V g.

   # Figure P2.82 Circuit for Problem 2.82.

   Figure P2.82 Full Alternative Text

9. 2.83 The transmission-line circuit of Fig. P2.83 is excited by a 1-ns long pulse that starts at t=0. The line is 1 m long and the phase velocity of the line is 2×10 8 m/s. Generate:

   1. a plot of the voltage across the line at t=6 ns,

   2. a plot of the voltage across the line at t=12 ns,

   3. a plot of the voltage as a function of time at the sending end of the line, and

   4. a plot of the voltage as a function of time at the midpoint of the line.

   # Figure P2.83 Circuit for Problem 2.83.

   Figure P2.83 Full Alternative Text

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