MIU EET416 Problem Set #2 Solution Page 1 of 8

Misr International University Faculty of Engineering Department of Electronics and Communication Course: EET 416 Microwave Engineering Instructors: Prof. Fawzy Ibrahim

Problem Set #2 Solution Electromagnetic Plane Wave Propagation

Question #2.1 The magnetic field, H

in free space is given by:

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 A uniform plane wave in free space its electric field intensity is given by

xzjj

s aeeEo

ˆ200 25030

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, sH

f) E

at z = 8 mm, t = 6 ps.

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

.

Solution a) As in Lecture notes.

b) i) The forward and backward propagating waves having the general form: For time harmonic case at frequency ω. In time domain, this result is written as: The forward propagating wave having the form of:

mAaxktxE yo /ˆ)102cos(1.0 8

])(Re[),( tjjxsx eezEtzE

)cos(),( 0zktEtzEx

zjkzjkxs eEeEzE 00)(

MIU EET416 Problem Set #2 Solution Page 2 of 8

)cos(ˆ zktEaE ox

)2000cos(10ˆ zktaH ox

At t = 0 and z =1.25 m, Ex = E+ or cos( - koz) = 1. Then = koz = 6.28 x 1.25 = 89.77o . The instantaneous value of the electric field intensity, E

is given by:

The phasor value of the electric field intensity, E

is given by:

Exs(z) = 10 e(- j 6.25 z + j 89.77o) [mV/m]

b) ii) The instantaneous value of the magnetic field intensity, H

is given by:

The phasor value of the magnetic field intensity, H

is given by: Hys(z) =26.5 e(- j 6.25 z + j 89.77o) [A/m]

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 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. (ii) The wavelength, o. (iii) The propagation constant, ko. (iv) The average power density.

Solution

mradxxxcwk oo /28.6)103/()10302(/ 860

]/[)28.677.89cos(24)cos(),( 0 mmVztzktEtzE ox

]/[)28.677.89cos(5.26

)28.677.89cos()120/10()cos()/(),( 00

mAzt

ztzktEtzH

o

oy

MIU EET416 Problem Set #2 Solution Page 3 of 8

mAxktatxH oy /)10cos(10ˆ),( 8

mVejazE zjxs /)3040(ˆ)( 20

zjkyxs

oeajajzH )ˆˆ2)(104()(

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

Question #2.6 [Hayt 12.3] If the magnetic field intensity in free space is given by Find and calculate: a) The propagation constant, ko. b) The wavelength, o. c) The electric field intensity ),( txEs at P(0.1, 0.2, 0.3) at t = 1 ns.

Solution

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 Find and calculate the following:

a) The propagation constant, ko. b) The radian frequency, . c) The wave frequency, f. d) The wavelength, o. e) The magnetic field intensity in phasor form ),( tzH s

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 Find and calculate: a) The numerical values for , and ko. b) H(z, t) at t = 1.5 ns and z = 20 cm. c) max|| E .

)cos( zktA o

MIU EET416 Problem Set #2 Solution Page 4 of 8

mVkztatzE y /)cos(30ˆ),(

Solution

Question #2.9 [HW]

If zjkyx

j oo

eajaeH ]ˆ3ˆ)5([ 20

A/m in free space, and f = 6 MHz. Find the

instantaneous amplitude of H

at: a) (0,0,0) at t=0, c) (0,0,0) at t=0.1 s, b) (2,5,8) at t=0, 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 If the wave frequency is 2.4 GHz Find and calculate:

a) The amplitude and direction of the magnetic field. b) The velocity vp and wavelength, . c) The phase shift between the positions z1 = 0.5 m and z2 = 1.7 m. Solution

MIU EET416 Problem Set #2 Solution Page 5 of 8

Question #2.11 A wave propagating in a lossless dielectric has the components, xaztE ˆ)10cos(5 7

V/m

and yaztH ˆ)10cos(1.1 7

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.

b) A plane wave is given by: xz aztetzE ˆ)410cos(5.0),( 94

V/m. Determine the 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. Solution a) As in Lecture notes.

b-i)

ii)

mradjThen /)44(4

222 )5050(

)44(

7104910

)44(

7104910 4

jej

j

xx

j

xxjj

ms 25.04

11

mm 57.124

22

sec/8105.2

4

910mxp

xazjezeEz

sE ˆ )( mV

xazjezez

sE / ˆ 445.0)(

yazjeze

Ez

sH ˆ )(

mA/m ˆ )

44(425.2ˆ

4222

5,0)(

ya

jzjeze

yazjeze

je

zs

H

my

aztzetzH /mA ˆ)4

4910cos(425.2),(

mVx

aztzetzE / ˆ)4910cos(45.0),(

MIU EET416 Problem Set #2 Solution Page 6 of 8

mAeajazH xjzys /)ˆ5ˆ2()( 25

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 Knowing that the maximum amplitude of E is 1500 V/m, Find and calculate: a) , , , r and r , b) ),,,( tzyxH

Solution

Question #2.15 [HW] A plane wave is given by: y

zx

z azteaztetzyxE ˆ)910cos(4ˆ)910sin(3),,,( 99

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 = 500o at the surface, find:

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.

MIU EET416 Problem Set #2 Solution Page 7 of 8

Solution

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 #2.18 [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.

Solution

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.

MIU EET416 Problem Set #2 Solution Page 8 of 8

Solution

Question #2.20 In a nonmagnetic medium (o and ) if the electric field intensity of a plane wave is given by: Determine and calculate the following: a) r and . b) The time-average power density, davP

of the wave .

c) The total time-average power, tavP

crossing an area, A = 100 cm2 of plane x +2 y = 4.

Solution Refer to Exemple 2.9 in the lecture notes. 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. 2.21. If the wave frequency, f = 2 GHz and the incident electric field in the phasor form, for z < 0 is: Find the instantaneous and phase values and the average power density of the following: a) The incident wave [ iE

, iH

and )(zSi

].

b) The reflected wave [ rE

, rH

and )(zSr

].

c) The transmitted wave [ tE

, tH

and )(zSt

].

Fig. 2.21 Plane wave reflection. Solution Refer to Exemples 2.11 and 2.12 in the lecture notes.

mmVaezE xzjk

iso /ˆ20)(

mVaxtxtxE z /ˆ)75.0102sin(5),( 9