Problems and solution for chapter 1,2 PDF

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Problems 45REFERENCES[1] T. S. Sarkar, R. J. Mailloux, A. A. Oliner, M. Salazar-Palma, and D. Sengupta,History of Wireless, John Wiley & Sons, Hoboken, N., 2006. [2] A. A. Oliner, “Historical Perspectives on Microwave Field Theory,”IEEE Transactions on Mi- crowave Theory and Techniques, vol....

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Problems

45

REFERENCES [1] T. S. Sarkar, R. J. Mailloux, A. A. Oliner, M. Salazar-Palma, and D. Sengupta, History of Wireless, John Wiley & Sons, Hoboken, N.J., 2006. [2] A. A. Oliner, “Historical Perspectives on Microwave Field Theory,” IEEE Transactions on Microwave Theory and Techniques, vol. MTT-32, pp. 1022–1045, September 1984 [this special issue contains other articles on the history of microwave engineering]. [3] F. Ulaby, Fundamentals of Applied Electromagnetics, 6th edition, Prentice-Hall, Upper Saddle River, N.J., 2010. [4] J. D. Kraus and D. A. Fleisch, Electromagnetics, 5th edition, McGraw-Hill, New York, 1999. [5] S. Ramo, T. R. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics, 3rd edition, John Wiley & Sons, New York, 1994. [6] R. E. Collin, Foundations for Microwave Engineering, 2nd edition, Wiley-IEEE Press, Hoboken, N.J., 2001. [7] C. A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, New York, 1989. [8] D. M. Pozar, Microwave and RF Design of Wireless Systems, John Wiley & Sons, Hoboken N.J., 2001.

PROBLEMS 1.1 Who invented radio? Guglielmo Marconi often receives credit for the invention of modern radio, but there were several important developments by other workers before Marconi. Write a brief summary of the early work in wireless during the period of 1865–1900, particularly the work by Mahlon Loomis, Oliver Lodge, Nikola Tesla, and Marconi. Explain the difference between inductive communication schemes and wireless methods that involve wave propagation. Can the development of radio be attributed to a single individual? Reference [1] may be a good starting point. 1.2 A plane wave traveling along the x-axis in a polystyrene-filled region with ǫr = 2.54 has an electric field given by E y = E 0 cos(ωt − kx). The frequency is 2.4 GHz, and E 0 = 5.0 V/m. Find the following: (a) the amplitude and direction of the magnetic field, (b) the phase velocity, (c) the wavelength, and (d) the phase shift between the positions x 1 = 0.1 m and x 2 = 0.15 m. 1.3 Show that a linearly polarized plane wave of the formE ¯= E 0 (a xˆ + b yˆ )e− j k0 z , where a and b are

real numbers, can be represented as the sum of an RHCP and an LHCP wave. 1.4 Compute the Poynting vector for the general plane wave field of (1.76).

1.5 A plane √ wave is normally incident on a dielectric slab of permittivity ǫr and thickness d, where d = λ0 /(4 ǫr ) and λ0 is the free-space wavelength of the incident wave, as shown in the accompanying figure. If free-space exists on both sides of the slab, find the reflection coefficient of the wave reflected from the front of the slab.

1

T

Γ 0

r 0

0

d

0

d

z

1.6 Consider an RHCP plane wave normally incident from free-space (z < 0) onto a half-space (z > 0) consisting of a good conductor. Let the incident electric field be of the form E¯ i = E 0 (xˆ − j yˆ )e− j k0 z ,

46

Chapter 1: Electromagnetic Theory and find the electric and magnetic fields in the region z > 0. Compute the Poynting vectors for z < 0 and z > 0 and show that complex power is conserved. What is the polarization of the reflected wave? 1.7 Consider a plane wave propagating in a lossy dielectric medium for z < 0, with a perfectly conducting plate at z = 0. Assume that the lossy medium is characterized by ǫ = (5 − j 2)ǫ0 , µ = µ0 , and that the frequency of the plane wave is 1.0 GHz, and let the amplitude of the incident electric field be 4 V/m at z = 0. Find the reflected electric field for z < 0 and plot the magnitude of the total electric field for −0.5 ≤ z ≤ 0.

1.8 A plane wave at 1 GHz is normally incident on a thin copper sheet of thickness t. (a) Compute the transmission losses, in dB, of the wave at the air–copper and the copper–air interfaces. (b) If the sheet is to be used as a shield to reduce the level of the transmitted wave by 150 dB, what is the minimum sheet thickness? 1.9 A uniform lossy medium with ǫr = 3.0, tan δ = 0.1, and µ = µ0 fills the region between z = 0 and z = 20 cm, with a ground plane at z = 20 cm, as shown in the accompanying figure. An incident plane wave with an electric field −γ z V/m E¯ i = x100e ˆ

is present at z = 0 and propagates in the +z direction. The frequency is 3.0 GHz.

(a) Compute Si , the power density of the incident wave, and Sr , the power density of the reflected wave, at z = 0. (b) Compute the input power density, Sin , at z = 0 from the total fields at z = 0. Does Sin = Si − Sr ?

r = 3.0 tan  = 0.1

Ei

Er

l = 20 cm z

0

1.10 Assume that an infinite sheet of electric surface current densityJs ¯= J0 xˆ A/m is placed on the z = 0 plane between free-space for z < 0 and a dielectric with ǫ = ǫr ǫ0 for z > 0, as in the accompanying figure. Find the resultingE ¯and H¯ fields in the two regions. HINT: Assume plane wave solutions propagating away from the current sheet, and match boundary conditions to find the amplitudes, as in Example 1.3. x 0

r 0

ˆ 0 A/m Js = xJ

0

z

ˆ − jβ x A/m, where 1.11 Redo Problem 1.10, but with an electric surface current density ofJs¯ = J0 xe β < k0 .

Problems

47

1.12 A parallel polarized plane wave is obliquely incident from free-space onto a magnetic material with permittivity ǫ0 and permeability µ0 µr . Find the reflection and transmission coefficients. Does a Brewster angle exist for this case where the reflection coefficient vanishes for a particular angle of incidence? 1.13 Repeat Problem 1.12 for the perpendicularly polarized case. 1.14 An artificial anisotropic dielectric material has the tensor permittivity [ǫ] given as follows:  1 [ǫ] = ǫ0 −3 j 0

3j 2 0

0 0 4

¯ 3xˆ − 2 yˆ + 5ˆz . What is D¯ at At a certain point in the material the electric field is known to beE = this point? 1.15 The permittivity tensor for a gyrotropic dielectric material is   ǫ jκ 0 r

[ǫ] = ǫ0

−jκ 0

ǫr 0

0 . 1

Show that the transformations E+ = E x − j E y , D+ = Dx − j Dy , E− = E x + j E y , D− = Dx + j Dy , allow the relation between E¯ and D¯ to be written as   E  D+ + D− = [ǫ ′ ] E − , Ez Dz where [ǫ ′ ] is now a diagonal matrix. What are the elements of [ǫ ′ ]? Using this result, derive wave equations for E + and E − and find the resulting propagation constants. 1.16 Show that the reciprocity theorem expressed in (1.157) also applies to a region enclosed by a closed surface S, where a surface impedance boundary condition applies. 1.17 Consider an electric surface current density ofJs¯ = yˆ J0 e−β x A/m located on the z = d plane. If a perfectly conducting ground plane is located at z = 0, use image theory to find the total fields for z > 0. 1.18 Let E¯ = E ρ ρˆ + E φ φˆ + E z zˆ be an electric field vector in cylindrical coordinates. Demonstrate that 2 ¯in cylindrical coordinates as ρ∇ ˆ 2 Eφ + ˆ 2 E ρ +φ∇ it is incorrect to interpret the expression ∇E ¯ − ∇ 2 E¯ for the given zˆ ∇ 2 E z by evaluating both sides of the vector identity ∇ × ∇ ×E¯ = ∇(∇ · E) electric field.

Chapter 1 This is an open-ended question where the focus of the answer may be largely chosen by the student or the instructor. Some of the relevant historical developments related to the early days of radio are listed here (as cited from T. S. Sarkar, R. J. Mailloux, A. A. Oliner, M. Salazar-Palma, and D. Sengupta, History of Wireless, Wiley, N.J., 2006): 1.1

1865: James Clerk Maxwell published his work on the unification of electric and magnetic phenomenon, including the introduction of the displacement current and the theoretical prediction of EM wave propagation. 1872: Mahlon Loomis, a dentist, was issued US Patent 129,971 for “aerial telegraphy by employing an ‘aerial’ used to radiate or receive pulsations caused by producing a disturbance in the electrical equilibrium of the atmosphere”. This sounds a lot like radio, but in fact Loomis was not using an RF source, instead relying on static electricity in the atmosphere. Strictly speaking this method does not involve a propagating EM wave. It was not a practical system. 1887-1888: Heinrich Hertz studied Maxwell’s equations and experimentally verified EM wave propagation using spark gap sources with dipole and loop antennas. 1893: Nikola Tesla demonstrated a wireless system with tuned circuits in the transmitter and receiver, with a spark gap source. 1895: Marconi transmitted and received a coded message over a distance of 1.75 miles in Italy. 1894: Oliver Lodge demonstrated wireless transmission of Morse code over a distance of 60 m, using coupled induction coils. This method relied on the inductive coupling between the two coils, and did not involve a propagating EM wave. 1897: Marconi was issued a British Patent 12,039 for wireless telegraphy. 1901: Marconi achieved the first trans-Atlantic wireless transmission. 1943: The US Supreme Court invalidated Marconi’s 1904 US patent on tuning using resonant circuits as being superseded by prior art of Tesla, Lodge, and Braun. So it is clear that many workers contributed to the development of wireless technology during this time period, and that Marconi was not the first to develop a wireless system that relied on the propagation of electromagnetic waves. On the other hand, Marconi was very successful at making radio practical and commercially viable, for both shipping and land-based services.

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Chapter 2: Transmission Line Theory

REFERENCES [1] 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. 412–424, September 1942. [7] T. C. Edwards, Foundations for Microstrip Circuit Design, John Wiley & Sons, New York, 1987.

PROBLEMS 2.1 A 75  coaxial line has a current i (t, z) = 1.8 cos(3.77 × 109 t − 18.13z) 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 µH/m, C = 200 pF/m, R = 4.0 /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 (R = 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 R, 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  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 R, L, G, and C parameters. Assume W ≫ d.

y  W ฀r

d

x

z

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 T -model of a transmission line shown in the accompanying figure also yields the telegrapher equations derived in Section 2.1.

Problems i(z, t)

R∆ z 2

L∆z 2

R∆ z 2

L∆z 2

+

91

i(z + ∆z, t) +

G∆z

v(z, t)

C∆ z

v(z + ∆z, t)

–

– ∆z

2.8 A lossless transmission line of electrical length ℓ = 0.3λ is terminated with a complex load impedance as shown in the accompanying figure. Find the reflection coefficient at the load, the SWR on the line, the reflection coefficient at the input of the line, and the input impedance to the line. l = 0.3 Z in

ZL ZL = 30 ⫺ j 20 Ω

Z 0 = 75 Ω

2.9 A 75  coaxial transmission line has a length of 2.0 cm and is terminated with a load impedance of 37.5 + j 75 . 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 coefficient at the load, the reflection coefficient at the input, and the SWR on the line. 2.10 A terminated transmission line with Z 0 = 60  has a reflection coefficient at the load of Ŵ = 0.4 60◦ . (a) What is the load impedance? (b) What is the reflection coefficient 0.3λ away from the load? (c) What is the input impedance at this point? 2.11 A 100  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  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 Z sc be the input impedance of a length of coaxial line when one end is short-circuited, and let Z oc 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 Z sc and Z oc . 2.14 A radio transmitter is connected to an antenna having an impedance 80 + j 40  with a 50  coaxial cable. If the 50  transmitter can deliver 30 W when connected to a 50  load, how much power is delivered to the antenna? 2.15 Calculate standing wave ratio, reflection coefficient magnitude, and return loss values to complete the entries in the following table: SWR

|Ŵ|

RL (dB)

1.00 1.01 — 1.05 — 1.10 1.20 — 1.50 — 2.00 2.50

0.00 — 0.01 — — — — 0.10 — — — —

∞ — — — 30.0 — — — — 10.0 — —

92

Chapter 2: Transmission Line Theory 2.16 The transmission line circuit in the accompanying figure has Vg = 15 V rms, Z g = 75 , Z 0 = 75 , Z L = 60 − j 40 , and ℓ = 0.7λ. Compute the power delivered to the load using three different techniques: (a) Find Ŵ and compute PL =



 Vg 2 1 (1 − |Ŵ|2 ); Z0 2

(b) find Z in and compute   2  Vg  Re {Z in } ; P L =  Z g + Z in 

(c) find VL and compute

   VL 2  Re {Z L } . P L =  ZL 

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 L = j X, show that the reflection coefficient magnitude |Ŵ| is always unity. Assume that the characteristic impedance Z 0 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  line. Show that power conservation is satisfied. 50 Ω

/2 Z 0 = 50 Ω

10 V

Pinc Pref

Z 1 = 75 Ω Ptrans

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 −ℓ ≤ z ≤ 0. 100 Ω

l = 1.5λ

Z L = 80 – j40 Ω

Z 0 = 100 Ω

10 V

–l

0

z

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. l = 0.4

Zin

Z 0 = 50 Ω

Z L = 60 + j 50 Ω

2.21 Use the Smith chart to find the shortest lengths of a short-circuited 75  line to give the following input impedance: (a) Z in = 0. (b) Z in = ∞. (c) Z in = j 75 . (d) Z in = − j 50 . (e) Z in = j 10 . 2.22 Repeat Problem 2.21 for an open-circuited length of 75  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 Z 0 = 50 , find the load impedance. 2.24 Design a quarter-wave matching transformer to match a 40  load to a 75  line. Plot the SWR for 0.5 ≤ f / f o ≤ 2.0, where f o is the frequency at which the line is λ/4 long. 2.25 Consider the quarter-wave matching transformer circuit shown in the accompanying figure. Derive expressions for V + and V − , the respective amplitudes of the forward and reverse traveling waves on the quarter-wave line section, in terms of V i , the incident voltage amplitude. V+ V–

Vi

Z0

/4

Z 0 RL

RL

–l

0

z

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 αc =

Rs 2η ln b/a



 1 1 + . a b

Show that αc is minimized for conductor radii such that x ln x = 1 + x, where x = b/a. Solve this equation for x, and show that the corresponding characteristic impedance for ǫr = 1 is 77 . 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  transmission line is matched to a 10 V source and feeds a load Z L = 100 . If the line is 2.3λ long and has an attenuation constant α = 0.5 dB/λ, 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, β + and β − , for propagation in the forward and reverse directions, with corresponding characteristic impedances0 Z + and Z − . (An example of such a line could be a microstrip transmission line on a magnetized ferrite 0

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 = l/2), at time t = 3ℓ/v p ? 25 Ω + 10 V

t=0

100 Ω

Z0 = 50 Ω

– 0

l

z

Chapter 2 2.1
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