Chapter 8
Electromagnetic
Wave
Prof. Seong-Ook Park
1
8. Plane Plane Electrom Electromagn agnetic etic Waves Waves (co (conte ntents nts)) 8-1 Introduction uniform plane wave, wavefront
8-2 Plane Wave in Lossless Media free-space wavenumber, phase velocity, intrinsic impedance, Doppler effect, red shift, TEM wave, wave, wavenumber vector, polarization polarization (linear, elliptically, circularly, right-hand right-hand & left-hand)
8-3 Plane Waves in Lossy Media attenuation constant, phase constant, skin depth, (plasma frequency, plasma oscillation, cutoff frequency) frequency)
8-4 Group Velocity dispersion, group velocity
8-5 Flow of Electromagnetic Power and the Poynting Vector Poynting vector, instantaneous & average power densities 8-6 Normal Incidence at a Plane Conducting Boundary 8-7 Oblique Incidence at a Plane Conducting Boundary 8-8 Normal Incidence at a Plane Dielectric Boundary 8-9 Normal Incidence at Multiple Dielectric Interfaces 8-10 Oblique Incidence at a Plane Dielectric Boundary
2
Plane Plan e Wave in Lossy Lossy Media – Com Complex plex Propaga Propagation tion Constan Constantt In a source-free lossy medium the homogeneous vector Helmholt’s equation to be solved is ∇ 2 E + k c2 E = 0, (8 − 42 ) where the wavenumber k c = ω µε c is a complex number, as given in Eq. (7-114). The derivations and discussions pertaining to plane waves in a lossless medium in Section 8-2 can be modified to apply to wave wave propagation in a lossy medium medium by simply replacing k with kc. However, in an effort to conform with the conventional notation used in transmission-line theory, it is customary to define a propagation constant, γ , such that γ = jck = ωj µ ε c (8 − 43 ) ( m -1 ) Since γ is complex, we write, with the help of Eq. (7-110), γ
= α + j β =
⎞ ⎟ jω ε ⎠
⎛ µ ε ' ⎜ 1 − ⎝
ε '' ⎞ j ⎟ ε ' ⎠
jω µ ε
or, from Eq.(7-114), γ
=α +
βj
=
ωj
1/ 2
⎛ ⎜ 1 + ⎝
σ
,
(8 − 44 )
,
(8 − 45 )
1/ 2
where α and β are the real and imaginary imaginary parts of γ , respectively. respectively. Their physical significance will be explained presently. For a lossless medium
3
In general case :
Since α and β are both positi positive, ve, we we finally finally get get :
γ = α + j β
= (α + j β )2 = jωµ (σ + jωε ) (1) α 2 − β 2 = −ω 2 µε 2αβ = ωµσ (2) γ 2
The negative α corresponding to an exponentially increasing function. This is physically impossible and so is discarded.
Sub (1) into (2), we have 2 σ ⎞ ⎤ 1⎡ ⎛ 2 2 2 α = ⎢− ω µε + ω µε 1 + ⎜ ⎟ ⎥ 2⎢ ⎝ ωε ⎠ ⎥⎦ ⎣ 2 ⎤ ⎡ σ 1 ⎛ ⎞ β 2 = ⎢ω 2 µε + ω 2 µε 1 + ⎜ ⎟ ⎥ 2⎢ ⎝ ωε ⎠ ⎥⎦ ⎣
α =
β =
4
1
ω µε ⎡ 2
2 ⎤2 σ ⎞ ⎛ ⎢ 1 + ⎜ ⎟ − 1⎥ ⎢⎣ ⎥⎦ ⎝ ωε ⎠
ω µε ⎡ 2
1
⎤2 σ ⎞ ⎛ ⎢ 1 + ⎜ ⎟ + 1⎥ ⎢⎣ ⎥⎦ ⎝ ωε ⎠ 2
Revision on Taylor’s series expansion :
(1 + z )n = 1+ n C 1 z + n C 2 z 2 + + n C n z n )(n − 2 ) (n − r − 1) n(n − 1)( where n C n
=
r !
Special cases : 1
3 2 z + z + 2 8 1 ≈ 1 + z 2 1 1 = 1 − z − z 2 + 2 8 1 1 ≈ 1 − z − z 2 2 8
(1 − z )− 2 = 1 +
1
(1 − z )2
1
(1 − z )−1 = 1 + z − z 2 + = 1 + z
( z << 1) ( z << 1) ( z << 1)
5
Plane Wave in Lossy Media – E – E of Uniform Plane WaveThe Helmholtz's equation, Eq. (8-42), becomes
(8 − 46 )
∇ 2 E − γ 2 E = 0
The solution of Eq. (8-46), representing a uniform plane wave propagating in the +zdirection, is E = a x E x = a x E0 e− γ z , (8 − 47 ) where we have have assumed that that the wave is linearly polarized in the x-direction. The − γ z propagation factor e can be written as a product of two factors: E x
=
−α
E0 e
z
− jβ z
e
.
As we shall see, both α and β are positive quantities. The first factor,e − α z, decreases as z increases and thus is an attenuation factor, and α is called an attenuation constant. The Sl unit of the attenuation constant is neper per meter (Np/m). The − j β z second factor,e is a phase factor β is called a phase constant and is expressed in radians per meter (rad/m). The phase constant expresses the amount of phase shift that occurs as the wave travels one meter. General expressions of α and β in terms of ω and the constitutive parameters-ε,μ, and σ -of the medium are rather involved (see Problem p.8-9). In the following paragraphs we examine the approximate expressions for low-loss dielectrics, good conductors. And ionized gases. 6
Low-Los Low -Loss s Dielectric Dielectricss- Pro Propaga pagation tion Consta Constant nt A low-loss dielectric is a good but imperfect insulator with a nonzero nonzero equivalent . 1Under conductivity, such that ε " ε ' o r σ / ω ε this condition,γ in Eq. (8-45) con be approximated by using the binomial expansion: γ
=α +
βj ωj
⎡ µε ' ⎢1 − ⎢⎣
ε " 1 ⎛ ε " ⎞ + ⎜ ⎟ j 2ε ' 8 ⎝ ε ' ⎠
2
⎤ ⎥, ⎥⎦
p ro p a g atio n co n s tan t
from which we obtain the attenuation constant α ≅
ω ε ''
µ
2
ε '
( Np/m )
(8 − 48 )
and the phase constant β≅ ω
⎡ 1 ⎛ ε " ⎞ 2 ⎤ µε ' ⎢1 + ⎜ ⎟ ⎥ 8 ' ε ⎝ ⎠ ⎥⎦ ⎢⎣
( rad /m ) .
(8 − 49 )
It is seen from Eq. (8-48) that the attenuation constant of a low-loss dielectric is a positive quantity and is approximately directly proportional to the frequency. The phase constant in Eq. (8-49) deviates only very slightly from the value ω µε for a perfect (lossless) dielectric.
7
Low-Lo Low -Loss ss Dielec Dielectri trics cs - Int Intrin rinsic sic Impe Impedan dance ce and Phas Phase e Velocit Velocity y The intrinsic Impedance of a low-loss low-loss dielectric is complex quantity. η c −1
(1 − x ) 2 ≈ 1 +
⎛1 − = ⎜ ε'⎝ µ
x
−1 / 2
⎟
ε ' ⎠
⎛ 1 + j ε " ⎞ ⎜ ⎟ 2ε ' ⎠ ε'⎝ µ
≅
2
j
ε " ⎞
(8 − 50 ) (Ω )
Since the intrinsic impedance is the ratio of Ex and Hy for a uniform plane wave, the electric and magnetic field intensities in a lossy dielectric are thus not in time phase, as they are in a lossless medium. The pha phase se velo velocit city y up is obtained from the ratio ω / β in a manner similar to that in Eq. (8-9). Using Eq. (8-49), we have u p
=
ω β
⎡ 1 ⎛ ε " ⎞ 2 ⎤ ≅ ⎢1 − ⎜ ⎟ ⎥ 8 ⎝ ε ' ⎠ ⎦⎥ µε ' ⎣⎢ 1
8
( m /s ) .
( 8 − 51 )
Good Goo d Conduct Conductors ors-- Pro Propaga pagation tion Cons Constant tant ωε>>1. Under this condition it is A good conductor is a medium for which σ / ωε convenient to use Eq. (8-44) and neglect 1 in comparison with the term σ /jωε. We write 1 + j σ γ ≅ jω µε ω µ σ propagation constant = j ω µσ = jω ε 2 or ( 8 − 52 ) γ = α + βj ≅ (1 + )j π fµσ ,
where we have used the relations j=
(
jπ / 2
e
1/ 2
)
=
jπ / 4
e
= (1 + )j /
2
and ω=2πf. Equation (8-52) indicates that α and β for a good conductor are approximately equal and both increase as f and σ . For a good conductor, α
=β =
π f µ σ .
attenuation constant and phase constant
9
(8 − 53 )
Good Conduc Conductorstors- Intri Intrinsic nsic Impedan Impedance ce and Phase Phase velocityvelocityThe intrinsic intrinsic impedan impedance ce of a good conduct conductor or is η c
=
µ εc
≅
jω µ
σ
= (1 + j )
π f µ σ
= (1 + j )
α σ
(Ω )
(8 − 54 )
which has a phase angle of 45°. Hence the magnetic field intensity lags behind the electric field intensity by 45°. The phase phase velocity velocity in a good conduc conductor tor is u p
=
ω
≅
2ω
β
µσ
which is proportional to f and 1 / σ
10
( m /s ) ,
(8 − 55 )
Good Goo d Conduc Conductor tors s - Ex Examp ample le of Coppe Copperr Consider copper as an example: σ = 5 .8 0 × 1 0 7
( S /m ) , µ = 4π × 1 0 −7 ( H /m ) , u p = 7 2 0 ( m /s ) a t 3 ( M H z ) ,
which is about twice the velocity of sound in i n air and is many orders of magnitude, slower than the velocity of light in air. The wavelength of a plane wave in a good conductor is u p 2π π (8 − 56 ) λ = = = (m ). β f µσ f For copper at 3 (MHz), λ= 0.24 (mm). As a comparison, a 3 (MHz) electromagnetic wave in air has a wavelength of 100 (m). At very high frequencies the attenuation constant α for a good conductor, as given by Eq. (8-53), tends to be very large. For copper at 3 (MHz), α
=
(
π 3 × 10 6
) ( 4π × 10 − ) ( 5.80 × 10 ) = 2.62 × 10 7
7
4
N p /m ) . ( Np
− α z Since the attenuation factor is e , the amplitude of a wave will be attenuated by a −1 factor of e = 0.368 when it travels a distance δ = 1/ α. For copper at 3 (MHz) this distance is (1/2.62) х 10-4 (m), or 0.038 (mm). At 10 (GHz) it is only 0.66 ( μm)-a very small distance indeed. Thus a high-frequency electromagnetic wave is attenuate very rapidly as it propagates in a good conductor. 11
Good Goo d Cond Conduct uctors ors - Ski Skin n Depth Depth The distance δ through through which the the amplitude amplitude of a traveling traveling plane wave wave decreases decreases by a factor of e-1 or 0.3 0.368 68 is called called the skin skin dept depth h or the dep depth th of of penet penetrat ration ion of a conductor: 1 1 δ = = ( m ). (8 − 57 ) α π f µσ Since α=β for a good conductor, δ can also be written as 1 λ δ = = ( m ). 2π β
(8 − 58 )
At microwave frequencies the skin depth or depth of penetration of a good conductor is so small that fields and currents can be considered as, for all practical purposes, confined in a very thin layer (that is, in the skin) of the conductor surface •The skin depth depth of material is the depth to which which a uniform plane wave can penetrate before it is attenuated by a factor of 1/ e. 1 1 •We have e −αδ = 1 ⇒ δ = •For a good conductor, we have δ = ≈ α α
12
2 ωµσ
Lossy Media Γb =
i
Sav
Srav
Stav
E1
η 2 − η1 η 2 + η1
= aˆ z
E 0
b
ε2
e −α 1
z
2η 1
T
2η 2 η1 + η 2
2
= −aˆ z Γ b = aˆ z
=
, T b
2
2
x
E 0
E0
E x
2
e
2η 1
+2α 1 z
Incident
Γb
2
e
2η 2
−2α 2 z
= aˆ z
T
b
2
η 1 E 0
2
η 2 2η 1
e
Reflected
Jc
−2α 2 z
ˆ x E0 e −α z e − j β z ⎡⎣1 + Γ b e +2α z e + j 2 β z ⎤⎦ = E i + E r = a 1
1
traveling wave
H1
= H i + H r = aˆ y
= 9ε 0 , µ 2 = µ 0 σ 2 = 10 −1 S / m
E 0
η 1
e
1
1
e
z y
J
= J 0 e − z / δ
δ
−α1z − j β1 z
J 0H y
standing wave
⎡⎣1 − Γb e +2α z e + j 2 β z ⎤⎦ 1
1
standing wave
traveling wave
13
Electric and magnetic field intensities, and electric current density distributions in a lossy earth.
Lossy Media Jc
−α z − j β z = σ E = aˆ x Tb E0σ e e
Jc
= σ E = aˆ x J 0 e−α z e− jβ z
Jc
=
J s
= ∫0
2
2
−α z − aˆ x J 0 e 2 e ∞
J c d z
j2 z β
2
J 0
= J 0 e−α
2
z
−α z
A/m 2
∞
J0 e
=
− z / δ
J0 e
∞
= ∫0 J 0 e−α z dz = J 0 ∫ 0 e− z / δ dz 2
∞ − z / δ 2
= J 0 ( −δ 2 ) ∫ 0 e J s
2
2
⎛ dz ⎞ − z / δ ⎜ − δ ⎟ = −δ 2 J 0 e ⎝ 2 ⎠
δ
∞
2
= −δ 2 J 0 [0 − 1] = δ 2 J 0
A/m
0 + E x ( z , t = 0 )
e − α z
A Ae
−1
≈
0 . 368 A
z
− A
δ =
1 α
=
δ
1
β z
π f µσ
π
14
2 π
3π
4 π
Good Conduc Conductors tors - Exam Examples ples of Skin Skin Depth Depth Values Values Table 8-1 Skin Depths, δ in (mm), of Various Materials Material
σ ( S /m )
f=60(Hz)
1(MHz)
1(GHz)
Silver
6 .1 7 × 1 0 7
8.27(mm)
0.064(mm)
0.0020(mm)
Cooper
5 .8 0 × 1 0 7
8.53
0.066
0.0021
Gold
4 .1 0 × 1 0 7
10.14
0.079
0.0025
Aluminum
3 .5 4 × 1 0 7
10.92
0.084
0.0027
1 .0 0 × 1 0 7
0.65
0.005
0.00016
4
32(m)
0.25(m)
Iron ( µ r 10 Seawater
3
)
The ε of seawater is approximately 7 2ε 0 . At f = 1 ( G H z ) , σ / ω ε 1 ( no t 1). Under these conditions, seawater is not a good conductor, and Eq. (8-57) is no linger l inger applicable 15
EM Wave Propagation into Seawater (1) Example 8-4 The electric field intensity of a linearly polarized uniform plane wave propagating in the +z-direction in seawater is E = a x 1 00 cos (10 7 π t ) ( V /m ) at z = 0. The constitutive parameters of seawater are ε r = 7 2, µ r = 1, and σ = 4 ( S /m ) . (a) Determine the attenuation constant, phase constant, intrinsic impedance, phase velocity, wavelength, and skin depth. (b) Find the distance at which the amplitude of E is 1% of its value at z = 0. (c) Write the expressions for E(z,t) and H(z, t) at z = 0.8 (m) as functions of t. Solution ω
= 10 7 π
f
=
σ ωε
=
ω 2π
( rad /s ) ,
= 5 × 10 6
σ ω ε 0ε r
( Hz ) , 4
=
⎛ × 10 −9 ⎞⎟ 72 ⎝ 36π ⎠
1 0 7 π ⎜
1
Hence we can use the formulas for good conductors. 16
= 200 1.
EM Wave Propagation into Seawater (2) a) Attenuation constant: α
=
5π 1 0 6 4π 1 0 −7 4
(
=
π f µσ
)
= 8 .8 9 ( N p / m )
Phase constant: β=
π f µσ = 8.89
( rad/m )
Intrinsic impedance: π f µ η c = (1 + j ) σ
= (1 + j ) Phase velocity: u p
Wavelength:
=
π ( 5 × 10 10 6 ) ( 4π × 1 0 −7 ) 4 ω β
=
λ =
2π
δ =
1
Skin depth:
β
α
10 7 π 8.89
= =
= 3 .5 3 × 1 0 6
2π
8.89
( m /s )
= 0 .7 0 7
( m ).
= 0 .1 1 2
( m ).
8.89 1
= π e jπ / 4
17
(Ω )
EM Wave Propagation into Seawater (3) b) Distance z1 at which the amplitude of wave decreases to 1% of its value at z=0: 1 α z − α z = 100, e 1 = 0.01 or e 1 = 0.01 z1
=
1
α
ln 1 0 0 =
4 .6 0 5 8.89
= 0 .5 1 8
( m ).
c) In phasor notation, E ( z) = a x 1 0 0 e−α z e−
jβ z
.
The instantaneous expression for E is E ( z, t) = R e ⎡⎣ E ( z) ejω t ⎤⎦
= R e ⎡⎣ a x 100 e −α z e j(ω t− β z) ⎤⎦ = a x100 e −α z cos (ω t − β z ) . At z=0.8 (m) we have E ( 0 .8, t ) = a x 1 0 0 e −0.8α co s 1 0 7 π t − 0 .8 β
(
= a x 0.082 cos (10 7 ω t − 7.11)
18
) ( V /m ) .
Ionized Gases (1) In the earth's upper atmosphere, roughly from 50 to 500 (km) in altitude, there exist layers of ionized gases called the ionosphere.. The ionosphere consists of free electrons and positive ions that are produced when the ultraviolet radiation from the sun if absorbed by the atoms and molecules in the upper atmosphere. The charged particles tend to be trapped by the earth's magnetic field. The altitude and character of the ionized layers depend both on the nature of the solar radiation and on the composition of the atmosphere. They change with the sunspot cycle, the season, and the hour of the day in a very complicated way. The electron and ion densities in the individual ionized layers are essentially equal. Ionized gases with equal electron and ion densities are called plasma. The ionosphere plays an important role in the propagation of electromagnetic waves and affects telecommunication. Because the electrons are much lighter than the positive ions, they are accelerated more by the electric fields of electromagnetic waves passing through the ionosphere. In our analysis we shall ignore the motion of the ions and regard the ionosphere as a free electron gas. Furthermore, we shall neglect the collisions between the electrons and the gas atoms and molecules.
19
Ionized Gases (2) An electron of charge –e and mass m in a time-harmonic electric field E in the xdiretion at an angular frequency ω experiences a force –eE, which displaces it from a positive ion by a distance x such that
− eE = m
d 2 x 2
dt
= − mω 2 x
(8 − 59 )
or x
=
e mω 2
E,
(8 − 60 )
where E and x are phasors, Such a displacement gives rise to an electric dipole moment: p = − ex. (8 − 61 ) If there are N electrons per unit volume, we have a volume density of electric dipole moment or polarization vector 2 Ne N e p = N p = − E. ( 8 − 62 ) mω 2
20
Ionized Gases (3) In writing Eq. (8-62) we have implicitly neglected the mutual effect of the induced dipole moments of the electrons on one another, From Eqs. (3-97) and (8-62) we obtain 2 ⎛ Ne N e ⎞ D = ε 0 E + P = ε 0 ⎜ 1 − ⎟E 2 ω ε m 0 ⎠ ⎝ (8 − 63 ) ⎛ ω p2 ⎞ = ε 0 ⎜⎜ 1 − 2 ⎟⎟ E, ⎝ ω ⎠ where ω p
Ne N e 2
=
mε 0
( rad/s )
(8 − 64 )
is called the plasma angular frequency, a characteristic of the ionized medium. The corresponding plasma frequency is f p
=
ω p 2π
=
1
Ne N e 2
2π
mε 0
21
( Hz ).
( 8 − 65 )
Ionized Gases (4) Thus the equivalent permittivity of the ionosphere or plasma pl asma is ε p
⎛ ω p2 ⎞ = ε 0 ⎜⎜ 1 − 2 ⎟⎟ ⎝ ω ⎠ ⎛ f p2 ⎞ = ε 0 ⎜⎜ 1 − 2 ⎟⎟ ⎝ f ⎠
(8 − 66 ) ( F /m ) .
On the basis of Eq. (8-66) we obtain the propagation constant as γ
= jω
⎛ f p ⎞ 1− ⎜ ⎟ ⎝ f ⎠
µ ε 0
2
,
( 8 − 67 )
and the intrinsic impedance as η p
η 0
=
⎛ f p ⎞ ⎟ ⎝ f ⎠
1− ⎜
where η 0
=
µ 0 ε 0 =/
π1 2 0 ( Ω )
. 22
2
,
( 8 − 68 )
Ionized Gases (5) From Eq. (8-66) we note the peculiar phenomenon of a vanishing ε as f approaches fp. When ε becomes zero, electric displacement D (which depends on free charges only) is zero even when electric field intensity E (which depends on both free and polarization charges) is not. In that case it would be possible for an oscillating E to exist in the plasma in the absence of free charges, leading to a socalled plasma oscillation. When f < f p , γ becomes purely real, indicating an attenuation without propagation; at the same time, η p becomes purely imaginary, indicating a reactive load with no transmission of power. Thus fp is also referred to as the cutoff frequency. We will discuss wave reflection and transmission under various conditions later in this chapter. When f > f p, γ is purely imaginary, imaginary , and electromagnetic waves propagate unattenuated in the plasma (assuming negligible collision losses).
23
Ionized Gases (6) If the value of e, m, and ε0 are substituted into Eq. (8-65), we find a very simple formula for the plasma (cutoff) frequency: f p
≅9
N
( Hz ).
(8 − 69 )
As we have mentioned before, N at a given altitude is not a constant; it varies with the time time of the day, day, the season season,, and other other factors. factors. The The electron electron density density of the ionosphere ranges from about 1010 /m3 in the lowest layer to 1012 /m3 in the highest layer. Using these values for N in Eq. (8-69), we find fp to vary from 0.9 to 9 (MHz). Hence, for communication with a satellite or a space station beyond the ionosphere we must use frequencies much higher than 9 (MHz) to ensure wave penetration through the layer with the largest N at any angle of incidence (see Problem p.8-14). Signals with frequencies lower than 0.9 (MHz) cannot penetrate into even the lowest layer of the ionosphere but may propagate very far around the earth by way of multiple reflections at the ionosphere's boundary and the earth's surface. Signals having frequencies between 0.9 and 9 (MHz) will penetrate partially into the lower ionospheric ionospheric layers layers but will will eventually eventually be turned turned back back where where N is large. large. We have given here only a very very simplified picture of wave propagation propagation in the ionosphere. The actual situation is complicated by the lack of distinct layers of constant electron densities and by the presence of the earth's magnetic field. Atomic reaction 24
Plasma created by Reentering Spacecraft Example 8-5 When a spacecraft reenters the earth's atmosphere, its speed and temperature ionize the surrounding atoms and molecules and create a plasma, It has been estimated that the electron density is in i n the neighborhood of 2× 108 per (cm3). Discuss the plasma's effect on frequency usage in radio communication between the spacecraft and the mission controllers on earth. Solution For
( ) p er ( m ) ,
N = 2 × 1 08 p er cm 3
= 2 × 1014
3
14 7 Eq. (8-69) gives f p = 9 × 2 × 1 0 = 1 2 .7 × 1 0 ( H z ) , o r 1 2 7 ( M H z ) . Thus, radio communication cannot be established for frequencies below 127 (MHz)
25
Dispersion In Section 8-2 we defined the phase velocity, up, of a single-frequency plane wave as the velocity of propagation of an equiphase wavefront. The relation between up and the phase constant, β, is u p
=
ω
( m/s ) .
β
(8 − 70)
µε is a linear function of As a For plane waves in a lossless medium, β = ω µε consequence, the phase velocity u p = 1 / ε is a constant that is independent of frequency. However, in some cases (such (such as wave propagation in a lossy dielectric, dielectric, as discussed previously, or along a transmission line, or in a waveguide to be discussed in later chapters) the phase constant is not a linear function of ω; waves of different different frequencies frequencies will propagat propagate e with different phase phase velocities. velocities. Inasmuch Inasmuch as all information-bearing signals consist of a band of frequencies, waves of the component frequencies travel with different phase velocities, causing a distortion in the signal wave shape. shape. The signal "disperses." "disperses." The phenomenon of signal distortion caused by a dependence of the phase velocity on frequency is called dispersion . In view of Eqs. (8-51) (8-51) and (7-115), we we conclude that a lossy lossy dielectric is obviously a dispersive medium . 26
Wave-P Wav e-Pack acket et - Su Sum m of Two Two Time Time-Ha -Harmo rmonic nic Trav Traveli eling ng Wave Waves sAn information-bearing signal normally has a small spread of frequencies (side-bands) around a high carrier frequency. Such a signal comprises a "group" of frequencies and forms a wave packet. A group velocity is the velocity velocity of propagation propagation of the wave-packet wave-packet envelope (of a group of frequencies). Wave-packet, energy, signal. Consider the simplest simplest case of a wave packet that consists of two traveling waves having having equal amplitude and slightly different angular frequencies ω0 + ∆ω and ω0 − ∆ω ∆ω ω 0 . The phase constants, being functions of frequency, will also be slightly different. Let the phase constants corresponding to the two frequencies be β0 + ∆β and β0 − ∆β We have E( z, t) = E0 cos ⎡⎣(ω0 + ∆ω ) t− ( β 0 + ∆β ) z⎤⎦
+
0
Ecos ⎡⎣(ω0 − ∆ω )
−t( β 0 − ∆β ) ⎤⎦ z
(8 − 71)
= 2 E0 cos ( t∆ω − z∆β ) cos (ω0 t− β0 z ) . Since ∆ω ω 0 , the expression in Eq. (8-71) represents a rapidly oscillating wave having an angular frequency ω0 and an amplitude that varies slowly with an angular frequency ∆ω. This is depicted in Fig. 8-6. envelope Fig 8-6 sum of two timeharmonic traveling waves of equal amplitude and slightly different frequencies at a given t . carrier 27
Group Velocity The wave inside the envelope propagates with a phase velocity found by setting ω0t − β 0 z = Constant: u p
=
dz dt
=
ω 0 β 0
.
The velocity of the envelope (the group velocity up ) can be determined by setting the argument of the first cosine factor in Eq. (8-71) equal to a constant: t ∆ω − z ∆β = Constant,
from which we obtain u p
=
dz dt
=
ω 0 β0
=
1
.
∆ β / ∆ ω
In the limit that ∆ω → 0, we have the formula formula for computing computing the group velocity velocity in a dispersive medium: u p
=
1 d β / d ω
( m/s ) .
(8 − 72 )
This is the velocity of a point on the envelope of the wave packet, as shown in Fig 86, and is identified as the velocity of the narrow-band signal. As we saw in Subsection 8-3.3, β is a function of ω. If ω is plotted versus β, an ω-β graph is obtained. The slope of the straight line drawn from the origin to a point on the graph gives the phase velocity, ω / β, and the local slope of the tangent to the graph at the point is the group velocity, dω / d β . 28
Group Delay The consumed time for the packet to propagate along a distance l is called group delay delay and is given given by the following following equatio equation: n:
τ =
l ug
=l
d β dω
=
d ( β l ) d ω
or τ
=
d φ d ω
with φ
= β l
(electrical length)
The last equation is an useful equation to calculate the group delay not only for propagation in space but for circuits.
29
ω-β Diagram for Dispersive Medium In fig. 8-7 an ω-β graph for wave propagation in an ionized medium is plotted, based of Eq. (8-67); 2
⎛ f p ⎞ ⎟ ⎝ f ⎠
β = ω µε µε 1 − ⎜
=
ω c
⎛ ω p ⎞ ⎟ ⎝ ω ⎠
1− ⎜
(8 − 73)
2
.
At ω = ω p (the cutoff angular frequency), β= 0. For ω> ω p , wave propagation is possible, and c ω u p
=
β
=
⎛ ω p ⎞ 1− ⎜ ⎟ ⎝ ω ⎠
2
.
(8 − 74 )
ω µε
Substituting Eq. (8-73) in Eq. (8-72), we have ug
=c
⎛ ω p ⎞ 1− ⎜ ⎟ ⎝ ω ⎠
⎛ f p ⎞ 1− ⎜ ⎟ ⎝ f ⎠
2
2
(8 − 75)
.
We note that u p ≥ c and u g ≤ c, and for wave propagation propagation in an ionized medium, u pu g = c2 . A similar situation exists in waveguides (Section 10-2). The ω-β diagram is used to express the characteristics of waveguides and periodic structures, too. 30
Fig 8-7 ω-β graph for ionized gas
General Relation between Group and Phase Velocitie A general relation between the group and phase velocities velocities may be obtained by combining combining Eqs. (8-70) sand (8-72). From Eq. (8-70) we have d β dω
=
⎛ ω⎞ 1 ωdu p ⎜⎜ ⎟⎟ = − 2 dω ⎝ u p⎠ u p u p d ω d
(u
p
is funcr uncrio ion n of ω )
Substitution of the above in Eq. (8-72) yield
ug
u p
= 1−
u p d ω
From Eq. (8-76) we see three thr ee possible cases: a) No dispersion:
du p d ω
=0
(u
p
du p d ω
<0
(u
p
c) anomalous dispersion:
du p d ω
>0
u p ug
p
is also inportant general relation.
= up .
The packet, energy, or signal propagates in the phase velocity in a non-dispersive medium.
decreasing decreasing with ω ) , ug
(u
= c2
independent of ω, β a linear function of ω ) , ug
b) Normal dispersion:
(8 − 76 )
ω du p
< up.
increasing with ω ) , ug
> up . 31
The group velocity is slower than the phase velocity. This is the usual case. The phase velocity is slower than the group velocity. This is the unusual case such as special case of periodic structure.
Narrow-Band Signal Propagates in Lossy Medium (1) Example 8-6 A narrow-band signal propagates in a lossy dielectric medium which has a loss tangent 0.2 at 550 (kHz), the carrier frequency of the signal. The dielectric constant of the medium is 2.5. (a) Determine α and β. (b) Determine up and ug. Is the medium dispersive? Solution 2 2 a) Since the loss tangent ε "/ ε ' = 0.2 and ε " / 8ε ' 1, Eqs. (8-48) and (8-49) can be used to determine α and β respectively. But first we find ε" from the loss tangent:
ε " = 0.2ε ' = 0.2 × 2.5ε 0
= 4.42 ×10−12 Thus,
α
=
ωε "
µ
2
ε '
β= ω
( F/m ) .
= π (550 ×103 ) × ( 4.42 ×10 −12 ) ×
⎡ 1 ⎛ ε " ⎞2 ⎤ µε µε ' ⎢1 + ⎜ ⎟ ⎥ ⎢⎣ 8 ⎝ ε ' ⎠ ⎥⎦
= 2π (550 ×103 )
⎡1 + 1 0.2 2 ⎤ ( )⎥ 3 ×108 ⎢⎣ 8 ⎦ 2.5
= 0.0182 ×1.005 = 0.0183
( rad/m ).
32
377 2.5
= 1.82 ×10 −3 ( Np/m ) ;
Narrow-Band Signal Propagates in Lossy Medium (2) b) Phase velocity (form Eq. 8-51):
u p
=
=
ω β
1
=
⎡ 1 ⎛ ε " ⎞2 ⎤ µε ' ⎢1 + ⎜ ⎟ ⎥ ⎣⎢ 8 ⎝ ε ' ⎠ ⎦⎥
3 ×108 2.5
⎡ 1 ⎛ ε " ⎞2 ⎤ ⎢1 − ⎜ ⎟ ⎥ µε ' ⎣⎢ 8 ⎝ ε ' ⎠ ⎦⎥ 1
⎡1 − 1 0.2 2 ⎤ = 1.888 ×108 ⎢⎣ 8 ( ) ⎥⎦
m/s ) . (m/
c) Group velocity (from Eq.8-49):
⎡ 1 ⎛ ε " ⎞2 ⎤ = µε ' ⎢1 + ⎜ ⎟ ⎥ . d ω ⎢⎣ 8 ⎝ ε ' ⎠ ⎥⎦
d β
ug
=
1
( d β / d ω )
≅
1
µε '
≅ up.
Thus a low-loss dielectric is nearly nondispersive. Here we have assumed ε" to be independent of frequency. For a high-loss dielectric, ε" will be a function of ω and may have a magnitude comparable to ε'. The approximation in Eq. (8-49) will no longer hold, and the medium will be dispersive.
33
Flow of of Electroma Electromagnetic gnetic Power and Poynting Poynting Vector Vector <8-5> Electromagnetic waves carry with them electromagnetic power. Energy is transported through space to distant receiving points by electromagnetic waves. We will now derive a relation between the rate of such energy transfer and the electric and magnetic field intensities associated with a traveling electromagnetic wave. We begin with the curl equations:
Faraday's law Ampere's cir ircciuta tall law
∂B ∂t ∂D ∇×H = J + ∂t
∇×E = −
( 7 − 53a ) (8 − 77 ) ( 7 − 53b ) ( 8 − 78)
The verification if the following identity of vector operations (see. Problem O. 233) is straightforward:
∇ ⋅ ( E × H ) = H ⋅ (∇ × E) − E ⋅ (∇ × H ) .
( 8 − 79)
Substitution of Eqs. (8-77) and (8-78) in Eq. (8-79) yields
∇ ⋅ ( E × H ) = −H ⋅
∂B ∂D − E ⋅ − E ⋅ J. ∂t ∂t
Physical meaning? 34
(8 − 80)
Flow of Electromagnetic Power and Poynting Vector - Flo Flow w of Electro Electromagn magneti etic c Power Power In a simple medium, whose constitutive parameters ε, μ, and, σ do not change with time, we have ∂ ( µ H ) 1 ∂ ( µ H ⋅ H ) ∂ ⎛ 1 ∂B Time differential of = H⋅ = = ⎜ µ H 2 ⎞⎟ , H⋅ Magnetic energy 2 ∂t ∂t ∂t ∂t ⎝ 2 ⎠
∂ ( ε E ) 1 ∂ ( ε E ⋅ E ) ∂ ⎛ 1 2 ⎞ ∂D = E⋅ = = ⎜ ε E ⎟ , E⋅ ∂t ∂t ∂t ∂t ⎝ 2 ⎠ 2 E ⋅ J = E ⋅ (σ E ) = σ E 2 . Equation (8-80) can then be written as
Divergence of EM power ∇ ⋅ ( E × H ) = −
∂ ⎛1 ε ∂t ⎜⎝ 2
Electric energy Dissipated power
E+ 2
1
⎞ ⎠
µ H2 ⎟ − σ E2 ,
(8 − 81)
2 which is a point-function relationship. An integral form of Eq. (8-81) is obtained by integrating both sides over the volume of concern: ∂ ⎛1 2 1 2⎞ 2 × ⋅ = − + − E H ε µ σ d s E H d v E dv, ( ) (8 − 82) ⎜ ⎟ S V ∂t V ⎝ 2 2 ⎠ where the divergence theorem has been applied to convert the volume integral of ∇ ⋅ ( E × H ) to the closed surface integral of ( E × H ) .
∫
∫
∫
35
Flow of Electromagnetic Power and Poynting Vector - Poy Poynti nting’ ng’s s Theore Theorem m (1) -
∂ ⎛1 E H ε × ⋅ = − d s ( ) ⎜ ∫ ∫ S V ∂t ⎝ 2
E+ 2
1 2
⎞ ⎠
µ H2 ⎟ dv−
∫ V
σ E2 dv,
(8 − 82)
We recognize that the first first and second second terms on on the right side of Eq. (8-82) represent represent the time-rate of change of the energy stored in the electric and magnetic fields fields,, respectively. [Compare with Eqs. (3-l76b) and (6-l72c).] The last term is the ohmic Power dissipated in the volume as a result of the flow of conduction current density σE in the presence of the electric field E. Hence we may interpret the right side of Eq. (8-82) as the rate of decrease of the electric and magnetic energies stored, subtracted by the ohmic power dissipated as heat in the volume V. V. To be consistent with the law of conservation of energy, this must equal the power (rate of energy) leaving the volume through its surface. Thus the quantity ( E × H ) is a vector representing the power flow per unit area. Define P
= (E × H)
( W/m ) . 2
( 8 − 83)
Quantity is known as the Poynting vector , which is a Power density vector associated with an electromagnetic field. The assertion that the the surface integral of over a closed surface, surface, as given by the left side of Eq. (8-82), (8-82), equals the power power leaving the enclosed enclosed volume is referred to as Poynting’s theorem . This assertion is not limited to plane waves.
36
Flow of Electromagnetic Power and Poynting Vector - Poy Poynt ntin ing’s g’s The Theore orem m (2) Equation (8-82) may be written in another form:
− ∫ S P ⋅ ds = Where
1
we
=
wm
=
pσ
=σ
2 1 2
ε E 2
=
µ H 2 E2
=
1
ε E ⋅ E*
2 1
=
∂ ( w + w ) dv + ∫ V pσ dv, ∂t ∫V e m
2
=
µ H ⋅ H*
J2/ σ
(8 − 84)
Electric energy density,
(8 − 85)
=
( 8 − 86)
Magnetic energy density,
= σE ⋅E * =J ⋅J
/ σ =Oh mic power density.
*
( 8 − 87 )
In words, Eq. (8-84) states that the total power power flowing info a closed surface at any instant equals the the sum of the rates of increase increase of the stored stored electric electric and and magnetic magnetic energi energies es and the ohmic ohmic power power dissipated within the enclosed volume. Two points concerning the Poynting vector are worthy of note. First, the power relations given in Eqs. (882) and (8-84) pertain to the total t otal power flow across a closed surface obtained by the surface integral of ( E × H ) . The definition of the Poynting vector in Eq. (8-83) as the power density vector at every point on the surface is an arvitrary, albeit useful, useful, concept. Second, the Poynting vector is in a direction normal to both E and H. If the region of concern is lossless ( σ = 0), then the last term in Eq. (8-84) vanishes, and the total power flowing into a closed surface is equal to the rate of increase of the stored electric and magnetic energies in the enclosed volume. In a static situation the first two terms on the right side of Eq. (8-84) vanish, and the total power flowing into a closed surface is equal to the ohmic power dissipated in the enclosed volume.
37
Review Questions (1) R.8-1 Define uniform plane wove. R.8-2 What is a wavefront? R.8-3 Write the homogeneous vector Hclmholtz's equation for E in free space. R.8-4 Define wavenumber. How is wavenumber related to wavelength? R.8-5 Define phase velocity. R.8-6 Define intrinsic impedance of a medium. What is the value of the intrinsic impedance of free space?
38
Review Questions (2) R.8-7 What is Doppler effect? R.8-8 What is a TEM wave? R.8-9 Write the phasor expressions for the electric and magnetic field intensity vectors of an x-polarized uniform plane wave propagating in the + z-direction. R.8-10 What is meant by the polarization of a wave? When is a wave linearly polarized? Circularly polarized? R.8-11 Two orthogonal linearly polarized waves are combined. State the conditions under which the resultant will be (a) another linearly polarized wave, (b) a circularly polarized wave, and (c) an elliptically polarized wave. R.8-12 How is the E-field from AM broadcast stations polarized? From television stations? From FM broadcast stations? R.8-13 Define (a) propagation constant, (b) attenuation constant, and (c) phase constant. R.8-14 What is meant by the skin depth of a conductor? How is it related to the attenuation constant? How does it depend on σ? On f? R.8-15 What is the constitution of the ionosphere? R.8-16 What is a plasma? R.8-17 What is the significance of plasma frequency? R.8-18 When does the equivalent permittivity of the ionosphere become negative? What is the significance of a negative permittivity in terms of wave propagation? 39
Proble lem ms (1) P.8-1 Obtain the wave equations governing the E and H fields in a source-free conducting medium with constitutive parameters ε, μ, and σ. P.8-2 Prove that the electric field intensity in Eq. (8-22) satisfies the homogeneous Helmholtz's equation provided that the condition in Eq. (8-23) (8- 23) is satisfied. P.8-3 A Doppler radar is used to determine the speed of a moving vehicle by measuring the frequency shift of the wave reflected from the vehicle. a) Assuming that the reflecting r eflecting surface of the vehicle can be represented by a perfectly conducting plane and that the transmitted signal is a time-harmonic uniform plane wave of a frequency f incident normally on the reflecting surface, find the relation between between the frequency shift ∆f and the speed u of the vehicle. b) Determine u both in (km/hr) and in (miles/hr) if ∆f = 2.33 (kHz) with f= 10.5 (GHz). P.8-4 For a harmonic uniform plane wave propagating in a simple medium, both E and H vary in accordance with the factor exp (-jk∙R) as indicated in Eq. (8-26). Show that the four Maxwell's equations for uniform plane wave in a source-free region reduce to the t he following:
k×E
= ω µ H . k × H = − ωε E , k ⋅ E = 0, k ⋅ H = 0. P.8-5 The instantaneous expression for the magnetic field intensity of a uniform plane wave propagating in the +y direction in air is given by
H
π = a z 4 × 10 −6 cos ⎛⎜ 10 7 π t − k 0 y + ⎞⎟ 4⎠ ⎝
a) Determine k0 and the location where Hz vanishes at t = 3 (ms). b) Write the instantaneous Expression for E. 40
( A/m ) .
Prob obllem ems s (2) P.8-6 the E-field of a uniform uniform plane wave propagating in a dielectric medium is given by
(
E( t, z) = a x 2 cos 1 0 8 t − z/ 3
)−a
y
(
si s i n 1 0 8 t − z/ 3
)
(V / m ) .
a) Determine the frequency and wavelength of the wave. b) What is the dielectric constant of the medium? c) Describe the polarization of the wave. d) Find the corresponding H-field. P.8-7 Show that a plane wave with an instantaneous expression for the electric field
E( z, t) = a x E10 sin (ω t− kz) − a r E20 sin (ω t− kz+ ψ ) is elliptically polarized. Find the polarization ellipse. P.8-8 Prove the following: a) An elliptically polarized plane wave can be resolved into right-hand and left-hand circularly polarized waves. b) A circularly polarized plane wave can be obtained obtained from a superposition of two oppositely oppositely directed elliptically polarized waves. P.8-9 Derive the following general expressions of the attenuation and phase constants for conducting media:
α
β
= ω
µε
= ω
µε
2
2
⎡ ⎢ ⎢⎣ ⎡ ⎢ ⎢⎣
⎛ ⎞ ⎟ ⎝ ω ε ⎠
2
⎛ σ ⎞ 1+ ⎜ ⎟ ⎝ ωε ⎠
2
1+ ⎜
σ
1/ 2
⎤ − 1⎥ ⎥⎦ 1/ 2 ⎤ + 1⎥ ⎥⎦
( N p/m ) . ( ra d / m ) .
P.8-10 Determine and compare the intrinsic impedance, attentuation constant (in both Np/m and dB/m), and skin depth of copper. ⎡σ cu = 5.80 × 10 7 ( S /m ) ⎤ , Silver ⎡σ ag = 6.15 × 10 7 ( S /m ) ⎤ , an d brass ⎣ ⎦ ⎣ ⎦ at the following frequencies: ⎡σ br = 1.59 × 10 7 ( S /m ) ⎤ ( a ) 6 0 ( H z ) , ( b ) 1 ( M H z ) , ( c ) 1 ( G H z ) .
⎣
⎦
41
Proble lem ms (3) P.8-11 A 3 (GHz), y-polarized uniform plane wave propagates in the +x-direction in a nonmagnetic medium having a dielectric constant 2.5 and a loss tangent 10-2. a) Determine the distance over which the amplitude of the propagating wave will be cut in half, b) Determine the intrinsic impedance, the wavelength, the phase velocity, and the group velocity of the wave in the medium. c) Assuming E = a y 5 0 sin 6π 109 t + π / 3 ( V /m ) at x = 0, write the instantaneous expression for H for all t and x. P.8-12 The magnetic field intensity of a linearly polarized uniform plane wave propagating in the +y-direction in seawater ⎡⎣ε r = 80, µ r = 1, σ = 4 ( S /m ) ⎤⎦ is
(
H
)
= a x 0.1 sin (10 1 010 π t − π / 3 )
( A/ m )
At f=0. a) Determine the attenuation constant, the phase constant, the intrinsic impedance, the phase velocity, the wavelength, and the skin depth. b) Find the location at which the amplitude am plitude of H is 0.01 (A/m). c) Write the expressions for E(y, t) and H(y, t) at y = 0.5 (m) as functions of t. P.8-13 Given that the skin depth for graphite at 100 (MHz) is 0.16 (mm) determine (a) the conductivity of graphite, and (b) the distance that a 1 (GHz) wave travels in graphite such that its field intensity is reduced by 30 (dB). P.8-14 Assume the ionosphere to be modeled by a plasma region with an electron density that increases with altitude from a low value at the lower boundary toward a value Nmax and decreases again as the altitude gets higher. A plane electromagnetic wave impinges on the lower boundary at an angle θt with the normal. Determine the highest frequency of the wave that will be turned back toward the earth. (Hint: Imagine the ionosphere to be stratified into layers of successively decreasing constant permittivities until the layer containing Nmax, The frequency to be determined corresponds to that for an emerging angle of π /2). 57
8. Plane Plane Electro Electromag magnet netic ic Waves Waves (Re (Revie view) w) 8-1 Introduction uniform plane wave, wavefront
8-2 Plane Wave in Lossless Media free-space wavenumber, phase velocity, intrinsic impedance, Doppler effect, red shift, TEM wave, wave, wavenumber vector, polarization polarization (linear, elliptically, circularly, right-hand right-hand & left-hand)
8-3 Plane Waves in Lossy Media attenuation constant, phase constant, skin depth, (plasma frequency, plasma oscillation, cutoff frequency)
8-4 Group Velocity dispersion, group velocity
8-5 Flow of Electromagnetic Power and the Poynting Vector Poynting vector, instantaneous & average power densities 8-6 Normal Incidence at a Plane Conducting Boundary 8-7 Oblique Incidence at a Plane Conducting Boundary 8-8 Normal Incidence at a Plane Dielectric Boundary 8-9 Normal Incidence at Multiple Dielectric Interfaces 8-10 Oblique Incidence at a Plane Dielectric Boundary 58
