ECE440 Problem Set #2 EM Plane Wave Propagation

Misr International University Faculty of Engineering Department of Electronics and Communication Course: ECE440 Microwave Engineering Instructors: Prof. Fawzy Ibrahim and Eng. Lamaa Kashan

Problem Set #2 Electromagnetic Plane Wave Propagation Question #2.1

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The magnetic field, H in free space is given by:

 E  0.1 cos(2 x108 t  k o x )aˆ y A / m

a) Find the direction of wave propagation. b) Calculate wave number or propagation constant, ko, the wavelength, λo and the period, T. c) Calculate the time, t1 it takes to travel a distance of λ /8. d) Sketch the wave at time, t1.

Question #2.2 [HW] A uniform plane wave in free space its electric field intensity is given by  E s  200e j 30 e  j 250 z aˆ x V/m. Find: a) The propagation constant, ko. b) The radian frequency, ω. c) The wavelength, λ. d) The intrinsic impedance η.   e) The magnetic field intensity, H s f) E at z = 8 mm, t = 6 ps. o

Question #2.3 a) Starting from Maxwell’s equations do the following: i) Derive the wave equations or Helmholtz equations. ii) Write the solution of theses equations in free space. iii) Derive basic plane wave parameters: Phase velocity, vp, The Wavelength, λ and the Wave or intrinsic impedance, η. b) The electric field of 30MHz plane wave traveling along +Z direction in air and directed along X-direction. If the peak value of Ex is 10 [mV/m] and Ex is maximum at t=0 and Z = 1.5 m. Obtain the expressions for the instantaneous and phasor values   of electric magnetic field intensities E and H .

Question #2.4 A uniform plane wave is propagating in free space along the +ve direction, do the following:   a) If the electric field intensity is given by E  aˆ x E cos( t  k o z ) Determine: (i) The magnetic field intensity. (ii) The time average Poynting vector. b) If the magnetic field intensity is given by Find and calculate: (i) The electric field intensity. MIU

 H  aˆ x10 cos( 2000t  ko z )

(ii) The wavelength, o.

ECE440 Problem Set #2

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(iii) The propagation constant, ko.

(iv) The average power density.

Question #2.5 [HW] The electric field intensity of a uniform plane wave in air has amplitude of 800 V/m and is in the x-direction. If the wave is propagating in the z-direction and has a wavelength of 60 cm, find: a) the frequency, b) the value of ko if the field is expressed in the form A cos(t  k o z ) ,  c) the amplitude of H Question #2.6 [Hayt 12.3]  8 If the magnetic field intensity in free space is given by H ( x, t )  aˆ y 10 cos(10 t  k o x) A / m Find and calculate: a) The propagation constant, ko. b) The wavelength, o. c) The electric field intensity E s ( x, t ) at P(0.1, 0.2, 0.3) at t = 1 ns. Question #2.7 [Hayt 12.4] [HW] In phasor form, the electric field intensity of a uniform plane wave in free space is given  by E ( z )  aˆ (40  j 30 )e  j 20 z V / m Find and calculate the following: s x a) The propagation constant, ko. c) The wave frequency, f. e) The magnetic field intensity in phasor form

b) The radian frequency, . d) The wavelength, o. H s ( z , t ) at P(6, -1, 0.07) and t = 71 ps.

Question #2.8 [Hayt 12.5] A150 MHz uniform plane wave in free space is described by

 H s ( z )  ( 4  j10)( 2aˆ x  jaˆ y )e  jk o z

Find and calculate: a) The numerical values for ,  and ko. b) H(z, t) at t = 1.5 ns and z = 20 cm. c) | E | max . Question #2.9 [HW]



o

j 20 If H  [ (5e ) aˆ x  3  j aˆ y ] e  instantaneous amplitude of H at: a) (0,0,0) at t=0, b) (2,5,8) at t=0,

 jk o z

A/m in free space, and f = 6 MHz. Find the c) (0,0,0) at t=0.1 s, d) (2,5,8) at t=0.1 s.

Question #2.10 [Pozar 1.4] A plane wave traveling along the z-axis in a dielectric medium with r = 2.55 has an electric field intensity in free space is given by

 E ( z , t )  aˆ y 30 cos( t  kz ) V / m

If the wave frequency is 2.4 GHz Find and calculate: a) The amplitude and direction of the magnetic field. MIU

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b) The velocity vp and wavelength, . c) The phase shift between the positions z1 = 0.5 m and z2 = 1.7 m. Question #2.11  A wave propagating in a lossless dielectric has the components, E  5 cos(10 7 t  z )aˆ x V/m  and H  1.1cos(107 t   z )aˆ y mA/m. If the wave is traveling at phase velocity vp = 0.5c, find: a)  r b)  r c) β; d) λ: e) η; Question #2.12 [HW] A 9.4 GHz uniform plane wave is propagating in polyethylene (r=2.26, r=1). If the  amplitude of the magnetic field intensity, H is 7 mA/m and the material is assumed to be lossless, find: a) the velocity of propagation, vp. b) the wavelength in polyethylene, . c) phase constant, . d) theintrinsic impedance, . e) the amplitude of the electric field intensity, H .

Question #2.13 a) To study the wave propagation in a conducting medium of conductivity σ, permittivity ε, permeability µ and charge free ( = 0). Derive the expression of: i) The propagation constant. ii) The intrinsic impedance.

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4 z

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cos(10 t  4 z ) aˆ x V/m. Determine the b) A plane wave is given by: E ( z , t )  0.5 e following: i) The propagation constant and the wave parameters (Vp, ,  and s)  ii) The magnetic field, H associated with the wave in both phasor and time domain representations. Question #2.14 [Hayt 12.7] In phasor form, the magnetic field intensity for a 400 MHz uniform plane wave in a  certain lossless material is given by H ( z )  ( 2 aˆ  j 5aˆ )e  j 25 x A / m s

y

z

Knowing that the maximum amplitude of E is 1500 V/m, Find and calculate:  a) , , , r and r , b) H ( x, y, z, t ) Question #2.15 [HW]  A plane wave is given by: E ( x, y, z, t )  3e  z sin(109 t  9 z ) aˆ x  4e  z cos(109 t  9 z )aˆ y V/m. Find the following: a) the velocity of the wave, vp and the direction of propagation, b) the dielectric constant, r. c) the conductivity of the medium, . d) the phasor representing of the wave,  e) the magnetic field associated with the wave, H . Question #2.16[Hayt 12.24] Most microwave ovens operate at 2.45 GHz, assume the  = 1.2x106 S/m and r=500 for stainless steel interior. If Es = 500o at the surface, find: MIU

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a) The depth of penetration or skin depth s. b) The amplitude of electric field intensity Es as a function of the angle and plot this curve as the field propagates in the stainless steel. Question #2.17 [HW] A plane wave of amplitude 1mV/m and frequency 30 MHz is normally incident from air onto a medium with r = 81, r =1, and  = 0.02 S/m. Calculate a) The reflection coefficient and transmission coefficient T. b) The skin depth s. Question #18 [Hayt 13.1] A uniform plane wave in air, Ex (z, t) = E+ cos(1010t - z) V/m, is normally incident on a copper surface at z = 0. Calculate the: a) Intrinsic impedance  of the copper. b) Reflection coefficient and transmission coefficient T. c) Percentage of the incident power density transmitted into the copper. Question #2.19 [HW] For a uniform plane at normal incidence on a surface between two lossless dielectric media determine: a) The condition under which the magnitude of the reflection coefficient equals to that of transmission coefficient. b) The ratio of the transmitted power to the incident power. Question #2.20 In a nonmagnetic medium (o and ) if the electric field intensity of a plane wave is  given by: 9

E ( x, t )  5 sin( 2x10 t  0.75 x) aˆ z V / m

Determine and calculate the following: a) r and .  b) The time-average power density, Pdav of the wave .  c) The total time-average power, Ptav crossing an area, A = 100 cm2 of plane x +2 y = 4. Question #2.21 Consider a plane wave normally incident from free space on a half-space of a medium 2 with parameters (µr = 2 and εr =5) as shown in Fig. 20. If the wave frequency, f = 2 GHz and the incident electric field in the phasor form, for z < 0 is:

 Eis ( z )  20e  jko z aˆ x mV / m

Find the instantaneous and phase values and the average power density of the following:    a) The incident wave [ Ei , H i and S i (z) ].    b) The reflected wave [ E r , H r and S r (z) ].    c) The transmitted wave [ Et , H t and S t (z ) ].

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Fig. 2.21 Plane wave reflection.

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