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Fundamentals of ElectromagneticsFundamentals of Electromagneticsfor Teaching and Learning:for Teaching and Learning:
A Two-Week Intensive Course for Faculty inA Two-Week Intensive Course for Faculty inElectrical-, Electronics-, Communication-, and Electrical-, Electronics-, Communication-, and
Computer- Related Engineering Departments in Computer- Related Engineering Departments in Engineering Colleges in IndiaEngineering Colleges in India
byby
Nannapaneni Narayana RaoNannapaneni Narayana RaoEdward C. Jordan Professor EmeritusEdward C. Jordan Professor Emeritus
of Electrical and Computer Engineeringof Electrical and Computer EngineeringUniversity of Illinois at Urbana-Champaign, USAUniversity of Illinois at Urbana-Champaign, USADistinguished Amrita Professor of EngineeringDistinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, IndiaAmrita Vishwa Vidyapeetham, India
Program for Hyderabad Area and Andhra Pradesh FacultySponsored by IEEE Hyderabad Section, IETE Hyderabad
Center, and Vasavi College of EngineeringIETE Conference Hall, Osmania University Campus
Hyderabad, Andhra PradeshJune 3 – June 11, 2009
Workshop for Master Trainer Faculty Sponsored byIUCEE (Indo-US Coalition for Engineering Education)
Infosys Campus, Mysore, KarnatakaJune 22 – July 3, 2009
4-3
Module 4Wave Propagation
in Free Space
4.1 Uniform Plane Waves in Time Domain4.2 Sinusoidally Time-Varying Uniform Plane Waves4.3 Polarization4.4 Poynting Vector and Energy Storage
4-4
Instructional Objectives
23. Write the expression for a traveling wave function for a set of specified characteristics of the wave24. Obtain the electric and magnetic fields due to an infinite plane current sheet of an arbitrarily time-varying uniform current density, at a location away from it as a function of time, and at an instant of time as a function of distance, in free space25. Find the parameters, frequency, wavelength, direction of propagation of the wave, and the associated magnetic (or electric) field, for a specified sinusoidal uniform plane wave electric (or magnetic) field in free space26. Write expressions for the electric and magnetic fields of a uniform plane wave propagating away from an infinite plane sheet of a specified sinusoidal current density, in free space
4-5
Instructional Objectives (Continued)
27. Obtain the expressions for the fields due to an array of infinite plane sheets of specified spacings and sinusoidal current densities, in free space28. Write the expressions for the fields of a uniform plane wave in free space, having a specified set of characteristics, including polarization 29. Express linear polarization and circular polarization as superpositions of clockwise and counterclockwise circular polarizations 30. Find the power flow and the electric and magnetic stored energies associated with electric and magnetic fields
4.1 Uniform Plane Wavesin Time Domain
(EEE, Sec. 3.4; FEME, Secs. 4.1, 4.2, 4.4, 4.5)
4-7
Infinite Plane Current Sheet Source
for 0z
Example:
S S xJ tJ a
0 cos t S S xt J J a
4-8
For a current distribution having only an x-component of current density that varies only with z,
t
B
× Et
D
× H J +
0 0 0 0
x y z x y z
x y z x y z
z t z tE E E H H H
a a a a a a
B DJ +
4-9
The only relevant equations are:
Thus, ,x xE z tE a ,y yH z tH a
y xE B
z t
yxBE
z t
0 zB
t
yxBE
z t
y xx
H DJ
z t
yxDH
z t
0 zD
t
Dy xx
HJ
z t
4-10
In the free space on either side of the sheet, Jx = 0
Combining, we get
Wave Equation
0y yx
B HE
z t t
0y x x
H D E
z t t
2
02
0
0 0
2 2
0 02 2
yx
y
x
x x
HE
z z t
H
t z
E
t t
E E
z t
4-11
Solution to the Wave Equation
4-12
4-13
0 0 0 0
0 0 0 0
0 0 0 0
2
0 0 0 0 0 02
2
0 0 2
,
x
x
x
x
E z t Af t z Bg t z
EA f t z
z
B g t z
EAf t z g t z
z
E
t
4-14
represents a traveling wave propagating in the +z-direction.
represents a traveling wave propagating in the
–z-direction.
8
0 0
13 10 m/s = c,pv
Where velocity of light
pg t z v
pf t z v
,xp p
z zE z t Af t Bg t
v v
4-15
E4.1: Examples of Traveling Waves
25pf t z v t z
15 m/s
1 5pv
f
1
25
1
5t 0t
1z
1 20
4-16
2 2 2t z t zptg z v e e
12 m/s
1 2pv
1z
1 20 2 3
g
0t
1
2t
1
4-17
0
0
0
0 0 0
From ,0
1
1
1,
where Intrinsic impedance
120 377
yx
y x
p p p
yp p
HE
z tH E
t z
z zAf t Bg t
v v v
z zH z t Af t Bg t
v v
4-18
Thus, the general solution is
For the particular case of the infinite plane current sheet in thez = 0 plane, there can only be a () wave for z > 0 and a ()wave for z < 0. Therefore,
,xp p
z zE z t Af t Bg t
v v
0
1,y
p p
z zH z t Af t Bg t
v v
for 0,
for 0
p x
p x
Af t z v zz t
Bg t z v z
aE
a
0
0
for 0
, for 0
p y
p y
Af t z v z
z tB
g t z v z
a
Ha
4-19
Applying Faraday’s law in integral formto the rectangular closed path abcda in the limit that the sides bc and da0,
0 00x xz z
ab E dc E
say, Af t Bg t F t
Lim Lim0 00 0
E l E l B Sb d
bc bca c abcdada da
dd d d
dt
4-20
Therefore,
Now, applying Ampere’s circuital law in integral form to therectangular closed path efgha in the limit that the sides fg and he0,
, for 0xp
zz t F t z
v
E a
0
1, for 0y
p
zz t F t z
v
H a
Lim00
H l H lf h
fge ghe
d d
Lim00
J S D Sfghe efghe efghe
dd d
dt
4-21
Uniform plane waves propagating away from the sheet toeither side with velocity vp = c.
Thus, the solution is
0 0
1 1SF t F t J t
0 0y y Sz z
ef H hg H ef J t
0
2 SF t J t
0, for 02 S x
p
zz t J t z
v
E a
1, for 0
2 S yp
zz t J t z
v
H a
4-22
In practice, there are no uniform plane waves. However,many practical situations can be studied based on uniformplane waves. For example, at large distances from physicalantennas and ground, the waves can be approximated asuniform plane waves.
4-23
x
y z
z = 0
S tJ
0,2 S x
p
zz t J t
v
E a
1,
2 S yp
zz t J t
v
H a
0,2 S x
p
zz t J t
v
E a
1,
2H aS y
p
zz t J t
v
4-24
x
y z
z = 0
S tJ
z > 0z < 0
z
E4.2
4-25
a for 300 mxE t z
b for 450 myH t z
4-26
c for 1 sxE z t
d for 2.5 syH z t
4-27
Review Questions
4.1. Outline the procedure for obtaining from the two Maxwell’s equations the particular differential equations for the special case of J = Jx(z, t)ax.4.2. State the wave equation for the case of E = Ex(z, t)ax. Describe the procedure for its solution. 4.3. What is a uniform plane wave? Why is the study of uniform plane waves important?4.4. Discuss by means of an example how a function f(t – z/vp) represents a traveling wave propagating in the positive z-direction with velocity vp.4.5. Discuss by means of an example how a function g(t + z/vp) represents a traveling wave propagating in the negative z-direction with velocity vp.
4-28
Review Questions (Continued)
4.6. What is the significance of the intrinsic impedance of free space? What is its value?4.7. Summarize the procedure for obtaining the solution for the electromagnetic field due to the infinite plane sheet of uniform time-varying current density. 4.8. State and discuss the solution for the electromagnetic field due to the infinite plane sheet of current density Js(t) = – Js(t)ax for z = 0.
4-29
Problem S4.1. Writing expressions for traveling wave functions for specified time and distance variations
4-30
Problem S4.2. Plotting field variations for a specified infinite plane-sheet current source
4-31
Problem S4.3. Source and more field variations from a given field variation of a uniform plane wave
4.2 Sinusoidally Time-Varying Uniform Plane Waves
(EEE, Sec. 3.5; FEME, Secs. 4.1, 4.2, 4.4, 4.5)
4-33
Sinusoidal function of time
4-34
Sinusoidal Traveling Waves
cos
cos
p
p
f t z v
t z v
t z
cos
cos
p
p
g t z v
t z v
t z
0 0where pv
4-35
, cosf z t t z
1
0 z
1
f2
t
4
t
0t
2
4-36
, cosg z t t z
2t
4
t
0t
2
g
1
0
1
z
4-37
The solution for the electromagnetic field is
0For cos for 0,S S xt J t z J a
><
><
><
0 0
0 0
0
0
cos for 02
= cos for 02
cos for 02
= cos for 02
E a
a
H a
a
Sp x
Sx
Sp y
Sy
Jt z v z
Jt z z
Jt z v z
Jt z z
><
0 0where pv
4-38
Three-dimensional depiction of wave propagation
4-39
Parameters and Properties
1. Phase, t z
frequency2
= number of 2 radians of phase change
per second
f
2. radian frequency =
rate of change of phase with time
for a fixed value of (movie)
t
z
4-40
3. phase constant =
= magnitude of rate of change of phase with
distance for a fixed value of (still photograph)
z
z t
4. phase velocity =
velocity with which a constant phase progresses
along the direction of propagation
follows from 0
pv
d t z
4-41
25. = wavelength =
distance in which the phase changes by 2
for a fixed t
6. Note that
2
2
in m in MHz = 300
p
fv f
f
07.
= Ratio of the amplitude of to the amplitude
of for either wave
E
H
x x
y y
E E
H H
4-42
8. (Poynting Vector, )
for (+) wave
for ( ) wave
is in the direction of propagation.
x y z
x y z
E × H P
a × a a
a × a a
x
H
P
E
z
y
x
E
H P
y
z
4-43
E4.3
Direction of propagation is –z.
Then8 8
88
6 10 , 3 102
22 , 1 m
6 103 10 m s
2p
f Hz
v
8Consider 37.7 cos 6 10 2 V m.ayE t z
80.1 cos 6 10 2 A mH axt z
4-44
E4.4 Array of Two Infinite Plane Current Sheets
1SJ 2SJ
0z 4z
1For ,SJ
1 0
2 0
cos for 0
sin for 4S S x
S S x
J t z
J t z
J a
J a
0 0
10 0
cos for 02
cos for 02
aE
a
Sx
Sx
Jt z z
Jt z z
4-45
2For ,SJ
0 0
2
0 0
0 0
0 0
0 0
sin for 2 4 4
sin for 2 4 4
sin for 2 2 4
sin for 2 2 4
cos for 2
a
E
a
a
a
a
Sx
Sx
Sx
Sx
Sx
Jt z z
Jt z z
Jt z z
Jt z z
Jt z z
0 0
4
cos for 2 4
aSx
Jt z z
4-46
For both sheets,
No radiation to one side of the array. “Endfire” radiation pattern.
1 2
1 20 4
1 20 4
1 20 4
0 0
0 0
for 4
for 0 4
for 4
cos for 4
sin sin for 0 4
0 for 0
E = E E
E E
E E
E E
a
a
z z
z z
z z
S x
S x
z
z
z
J t z z
J t z z
z
4-47
Depiction of superposition of the two waves
4-48
Review Questions
4.9. Why is it important to give special consideration for sinusoidal functions of time and hence sinusoidal waves?4.10. Discuss the quantities ω, β, and vp associated with sinusoidally time-varying uniform plane waves. 4.11. Define wavelength. What is the relationship among wavelength, frequency, and phase velocity? What is the wavelength in free space for a frequency of 15 MHz?4.12. How is the direction of propagation of a uniform plane wave related to the directions of its fields?4.13. What is the direction of the magnetic field of a uniform plane wave having its electric field in the positive z- direction and propagating in the positive x-direction?
4-49
Review Questions (Continued)
4.14. Discuss the principle of antenna array, with the aid of an example.4.15. What should be the spacing and the relative phase angle of the current densities for an array of two infinite, plane, parallel current sheets of uniform densities, equal in amplitude, to confine the radiation to the region between the two sheets?
4-50
Problem S4.4. Finding parameters and the electric field for a specified sinusoidal uniform plane wave magnetic field
4-51
Problem S4.5. Apparent wavelengths of a uniform plane wave propagating in an arbitrary direction
4-52
Problem S4.5. Apparent wavelengths of a uniform plane wave propagating in an arbitrary direction (Continued)
4-53
Problem S4.5. Apparent wavelengths of a uniform plane wave propagating in an arbitrary direction (Continued)
4-54
Problem S4.6. Ratio of amplitudes of the electric field on either side of an array of two infinite plane current sheets
4-55
4.3 Polarization(EEE, Sec. 3.6; FEME, Sec. 1.4, 4.5)
4-56
Sinusoidal function of time
4-57
Polarization is the characteristic which describes how the position of the tip of the vector varies with time.
Linear Polarization:Tip of the vectordescribes a line.
Circular Polarization:Tip of the vector describes a circle.
4-58
Elliptical Polarization:Tip of the vectordescribes an ellipse.
(i) Linear Polarization
Linearly polarized in the x direction.
F1 F1 cos (t ) ax
Direction remainsalong the x axis
Magnitude variessinusoidally with time
4-59
Linear polarization
4-60
F2 F2 cos (t ) ay Direction remainsalong the y axisMagnitude varies
sinusoidally with time
Linearly polarized in the y direction.
If two (or more) component linearly polarized vectors are in phase, (or in phase opposition), then their sumvector is also linearly polarized.
Ex: 1 2cos cosx y ( ) ( )F t F tF a a
4-61
Sum of two linearly polarized vectors in phase is a linearly polarized vector
4-62
(ii) Circular PolarizationIf two component linearly polarized vectors are(a) equal to amplitude(b) differ in direction by 90˚(c) differ in phase by 90˚,then their sum vector is circularly polarized.
tan–1 F2 cos (t )F1 cos (t )
tan–1 F2F1
constant
y
x
F1
F2 F
4-63
Circular Polarization
4-64
Example:
1 1
2 2
1 1
1
1 1
1
1
cos sin
cos sin
, constant
sin tan
cos
tan tan
x yF t F t
F t F t
F
F t
F t
t t
F a a
F
1F
2FF
x
y
4-65
(iii) Elliptical PolarizationIn the general case in which either of (i) or (ii) is not satisfied, then the sum of the two component linearly polarized vectors is an elliptically polarized vector.
Example: F F1 cos t ax F2 sin t ay
1F
2FF
x
y
4-66
x–F0
–F0
F0
F0F1
F2 F
/4
y
Example: 0 0cos cos 4F a ax yF t F t
4-67
D3.17
F1 and F2 are equal in amplitude (= F0) and differ in direction by 90˚. The phase difference (say ) depends on z in the manner –2z – (–3z) = z.
(a) At (3, 4, 0), = (0) = 0.
(b) At (3, –2, 0.5), = (0.5) = 0.5 .
81 0
82 0
cos 2 10 2
cos 2 10 3
x
y
F t z
F t z
F a
F a
1 2 is linearly polarized.F F
1 2 is circularly polarized. F F
4-68
(c) At (–2, 1, 1), = (1) = .
(d) At (–1, –3, 0.2) = = (0.2) = 0.2.
1 2 is linearly polarized.F F
1 2 is elliptically polarized. F F
4-69
Clockwise and Counterclockwise Polarizations
In the case of circular and elliptical polarizations for the field of a propagating wave, one can distinguish between clockwise (cw) and counterclockwise (ccw) polarizations. If the field vector in a constant phase plane rotates with time in the cw sense, as viewed along the direction of propagation of the wave, it is said to be cw- or right-circularly (or elliptically) polarized. If it rotates in the ccw sense, it is said to be ccw- or left- circularly (or elliptically) polarized.
4-70
For example, consider the circularly polarized electric field of a wave propagating in the +z-direction, given by
Then, considering the time variation of the field vector in the z = 0 plane, we note that for and for
0 0cos sinE a E ax yE t z t z
00, E a ,xt E 02 ., E ayt E
0 0If cos sin , then the
polarization is ccw- or left-circular.
E a E ax yE t z t z
Since the polarization is cw- or right-circular. a × a a ,x y z
4-71
Review Questions
4.16. A sinusoidally time-varying vector is expressed in terms of its components along the x-, y-, and z- axes. What is the polarization of each of the components?4.17. What are the conditions for the sum of two linearly polarized sinusoidally time-varying vectors to be circularly polarized?4.18. What is the polarization for the general case of the sum of two sinusoidally time-varying linearly polarized vectors having arbitrary amplitudes, phase angles, and directions?4.19. Discuss clockwise and counterclockwise circular and elliptical polarizations associated with sinusoidally time-varying uniform plane waves.
4-72
Problem S4.7. Expressing uniform plane wave field in terms of right- and left- circularly polarized components
4-73
Problem S4.8. Finding the polarization parameters for an elliptically polarized uniform plane wave field
4-74
4.4 Power Flowand Energy Storage(EEE, Sec. 3.7; FEME, Sec. 4.6)
4-75
Consider the quantity . Then, from a vector identity,E× H
E× H H × E E × H
Substituting
0
B× E
D D× H J J
t
t t
0where represents source current density, we haveJ
0
2 20 0 0
1 12 2
t t
E Ht t
D BE× H E J E H
E J E× H
4-76
Performing volume integration on both sides, and using thedivergence theorem for the last term on the right side, we get
where we have defined , known as the Poynting vector. The equation is known as the Poynting’s Theorem.
P E × H
20 0
20
12
1 2
E J
P S
V V
V S
dv E dvt
H dv dt
4-77
Poynting’s Theorem
2 20 0 0
1 12 2
E J P SV V V S
dv E dv H dv dt t
Source power density,
(power per unit volume),
W/m3
Electric stored energy density,
J/m3
Magnetic storedenergy density,
J/m3 Power flowout of S
4-78
Interpretation of Poynting’s Theorem
Poynting’s Theorem says that the power delivered to the volume V by the current source J0 is accounted for by the sum of the time rates of increase of the energies stored in the electric and magnetic fields in the volume, plus another term, which we must interpret as the power carried by the electromagnetic field out of the volume V, for conservation of energy to be satisfied. It then follows that the Poynting vector P has the meaning of power flow density vector associated with the electromagnetic field. We note that the units of E x H are volts per meter times amperes per meter, or watts per square meter (W/m2) and do indeed represent power flow density.
4-79
In the case of the infinite plane sheet of current, note that theelectric field adjacent to and on either side of it is directed opposite to the current density. Hence, some work has to bedone by an external agent (source) for the current to flow, and represents the power density (per unit volume) associated with this work.
2 30
1 Energy density J m stored2
in the electric field
ew E
2 30
1 Energy density J m stored2
in the magnetic field
mw H
2 Power flow density W m associated
with the electromagnetic field
P E × H
0E J
4-80
Review Questions
4.20. What is the Poynting vector? What is the physical interpretation of the Poynting vector over a closed surface? 4.21. State Poynting’s theorem. How is it derived from Maxwell’s curl equations?4.22. Discuss the interpretation of Poynting’s theorem.4.23. What are the energy densities associated with electric
and magnetic fields?4.24. Discuss how fields far from a physical antenna vary inversely with distance from the antenna.
4-81
Problem S4.9. Finding the Poynting vector and power radiated for specified radiation fields of an antenna
4-82
Problem S4.10. Finding the electric field and magnetic field energies stored in a parallel-plate resonator
4-83
Problem S4.10. (Continued)
4-84
Problem S4.11. Finding the work associated with rearranging a charge distribution
The End