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Fundamentals of Applied Electromagnetics 6eby
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli
Tables
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapters
Chapter 1 Introduction: Waves and Phasors
Chapter 2 Transmission Lines
Chapter 3 Vector Analysis
Chapter 4 Electrostatics
Chapter 5 Magnetostatics
Chapter 6 Maxwell’s Equations for Time-Varying Fields
Chapter 7 Plane-Wave Propagation
Chapter 8 Wave Reflection and Transmission
Chapter 9 Radiation and Antennas
Chapter 10 Satellite Communication Systems and Radar Sensors
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 1 Tables
Table 1-1 Fundamental SI units.
Table 1-2 Multiple and submultiple prefixes.
Table 1-3 The three branches of electromagnetics.
Table 1-4 Constitutive parameters of materials.
Table 1-5 Time-domain sinusoidal functions z(t) and their cosine-reference phasor-domaincounterparts Z, where z(t) = Re[Ze jωt ].
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 1-1: Fundamental SI units.
Dimension Unit SymbolLength meter mMass kilogram kgTime second sElectric Current ampere ATemperature kelvin KAmount of substance mole mol
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 1-2: Multiple and submultiple prefixes.
Prefix Symbol Magnitudeexa E 1018
peta P 1015
tera T 1012
giga G 109
mega M 106
kilo k 103
milli m 10−3
micro µ 10−6
nano n 10−9
pico p 10−12
femto f 10−15
atto a 10−18
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 1-3: The three branches of electromagnetics.
Branch Condition Field Quantities (Units)
Electrostatics Stationary charges Electric field intensity E (V/m)(∂q/∂ t = 0) Electric flux density D (C/m2)
D = εE
Magnetostatics Steady currents Magnetic flux density B (T)(∂ I/∂ t = 0) Magnetic field intensity H (A/m)
B = µH
Dynamics Time-varying currents E, D, B, and H(Time-varying fields) (∂ I/∂ t 6= 0) (E,D) coupled to (B,H)
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 1-4: Constitutive parameters of materials.
Parameter Units Free-space Value
Electrical permittivity ε F/m ε0 = 8.854×10−12 (F/m)
' 136π×10−9 (F/m)
Magnetic permeability µ H/m µ0 = 4π×10−7 (H/m)
Conductivity σ S/m 0
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 1-5: Time-domain sinusoidal functions z(t) and their cosine-reference phasor-domain counterparts Z, where z(t) = Re[Ze jωt ].
z(t) Z
Acosωt AAcos(ωt +φ0) Ae jφ0
Acos(ωt +βx+φ0) Ae j(βx+φ0)
Ae−αx cos(ωt +βx+φ0) Ae−αxe j(βx+φ0)
Asinωt Ae− jπ/2
Asin(ωt +φ0) Ae j(φ0−π/2)
ddt
(z(t)) jωZ
ddt
[Acos(ωt +φ0)] jωAe jφ0
∫z(t)dt
1jω
Z
∫Asin(ωt +φ0)dt
1jω
Ae j(φ0−π/2)
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 2 Tables
Table 2-1 Transmission-line parameters R′, L′, G′, and C ′ for three types of lines.
Table 2-2 Characteristic parameters of transmission lines.
Table 2-3 Magnitude and phase of the reflection coefficient for various types of loads.
Table 2-4 Properties of standing waves on a lossless transmission line.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 2-1: Transmission-line parameters R′, L′, G′, and C ′ for three types of lines.
Parameter Coaxial Two-Wire Parallel-Plate Unit
R′Rs
2π
(1a
+1b
)2Rs
πd2Rs
wΩ/m
L′µ
2πln(b/a)
µ
πln[(D/d)+
√(D/d)2−1
]µhw
H/m
G′2πσ
ln(b/a)πσ
ln[(D/d)+
√(D/d)2−1
] σwh
S/m
C ′2πε
ln(b/a)πε
ln[(D/d)+
√(D/d)2−1
] εwh
F/m
Notes: (1) Refer to Fig. ?? for definitions of dimensions. (2) µ,ε , and σ pertain to theinsulating material between the conductors. (3) Rs =
√π f µc/σc. (4) µc and σc pertain
to the conductors. (5) If (D/d)2 1, then ln[(D/d)+
√(D/d)2−1
]' ln(2D/d).
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 2-2: Characteristic parameters of transmission lines.
Propagation Phase CharacteristicConstant Velocity Impedance
γ = α + jβ up Z0
General case γ =√
(R′+ jωL′)(G′+ jωC ′) up = ω/β Z0 =
√(R′+ jωL′)(G′+ jωC ′)
Lossless α = 0, β = ω√
εr/c up = c/√
εr Z0 =√
L′/C ′(R′ = G′ = 0)
Lossless coaxial α = 0, β = ω√
εr/c up = c/√
εr Z0 = (60/√
εr) ln(b/a)
Lossless α = 0, β = ω√
εr/c up = c/√
εr Z0 = (120/√
εr)two-wire · ln[(D/d)+
√(D/d)2−1]
Z0 ' (120/√
εr) ln(2D/d),if D d
Lossless α = 0, β = ω√
εr/c up = c/√
εr Z0 = (120π/√
εr)(h/w)parallel-plateNotes: (1) µ = µ0, ε = εrε0, c = 1/
√µ0ε0, and
√µ0/ε0' (120π) Ω, where εr is the relative permittivity
of insulating material. (2) For coaxial line, a and b are radii of inner and outer conductors. (3) Fortwo-wire line, d = wire diameter and D = separation between wire centers. (4) For parallel-plate line,w = width of plate and h = separation between the plates.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 2-3: Magnitude and phase of the reflection coefficient for various types of loads. In general, zL = ZL/Z0 = (R+ jX)/Z0 = r + jx,where r = R/Z0 and x = X/Z0 are the real and imaginary parts of the normalized load impedance zL, respectively.
Reflection Coefficient Γ = |Γ|e jθr
Load |Γ| θr
ZL = (r + jx)Z0
[(r−1)2 + x2
(r +1)2 + x2
]1/2
tan−1(
xr−1
)− tan−1
(x
r +1
)0 (no reflection) irrelevant
(short) 1 ±180 (phase opposition)
(open) 1 0 (in-phase)
jX = jωL 1 ±180−2tan−1 x
jX =− jωC
1 ±180+2tan−1 x
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 2-4: Properties of standing waves on a lossless transmission line.
Voltage maximum |V |max = |V +0 |[1+ |Γ|]
Voltage minimum |V |min = |V +0 |[1−|Γ|]
Positions of voltage maxima (alsopositions of current minima)
dmax =θrλ
4π+
nλ
2, n = 0,1,2, . . .
Position of first maximum (alsoposition of first current minimum)
dmax =
θrλ
4π, if 0≤ θr ≤ π
θrλ
4π+
λ
2, if −π ≤ θr ≤ 0
Positions of voltage minima (alsopositions of current maxima)
dmin =θrλ
4π+
(2n+1)λ4
, n = 0,1,2, . . .
Position of first minimum (alsoposition of first current maximum)
dmin =λ
4
(1+
θr
π
)
Input impedance Zin = Z0
(zL + j tanβ l
1+ jzL tanβ l
)= Z0
(1+Γl
1−Γl
)Positions at which Zin is real at voltage maxima and minima
Zin at voltage maxima Zin = Z0
(1+ |Γ|1−|Γ|
)Zin at voltage minima Zin = Z0
(1−|Γ|1+ |Γ|
)Zin of short-circuited line Zsc
in = jZ0 tanβ l
Zin of open-circuited line Zocin =− jZ0 cotβ l
Zin of line of length l = nλ/2 Zin = ZL, n = 0,1,2, . . .
Zin of line of length l = λ/4+nλ/2 Zin = Z20/ZL, n = 0,1,2, . . .
Zin of matched line Zin = Z0
|V +0 |= amplitude of incident wave; Γ = |Γ|e jθr with −π < θr < π; θr in radians; Γl = Γe− j2β l .
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 3 Tables
Table 3-1 Summary of vector relations.
Table 3-2 Coordinate transformation relations.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 3-1: Summary of vector relations.
Cartesian Cylindrical SphericalCoordinates Coordinates Coordinates
Coordinate variables x,y,z r,φ ,z R,θ ,φ
Vector representation A = xAx + yAy + zAz rAr + φφφAφ + zAz RAR + θθθAθ + φφφAφ
Magnitude of A |A|= +√
A2x +A2
y +A2z
+√
A2r +A2
φ+A2
z+√
A2R +A2
θ+A2
φ
Position vector −→OP1 = xx1 + yy1 + zz1, rr1 + zz1, RR1,
for P(x1,y1,z1) for P(r1,φ1,z1) for P(R1,θ1,φ1)
Base vectors properties x · x = y · y = z · z = 1 r · r = φφφ ·φφφ = z · z = 1 R · R = θθθ ·θθθ = φφφ ·φφφ = 1x · y = y · z = z · x = 0 r ·φφφ = φφφ · z = z · r = 0 R ·θθθ = θθθ ·φφφ = φφφ · R = 0
x××× y = z r××× φφφ = z R××× θθθ = φφφ
y××× z = x φφφ××× z = r θθθ××× φφφ = Rz××× x = y z××× r = φφφ φφφ××× R = θθθ
Dot product A ·B = AxBx +AyBy +AzBz ArBr +Aφ Bφ +AzBz ARBR +Aθ Bθ +Aφ Bφ
Cross product A×××B =
∣∣∣∣∣∣x y zAx Ay AzBx By Bz
∣∣∣∣∣∣∣∣∣∣∣∣
r φφφ zAr Aφ AzBr Bφ Bz
∣∣∣∣∣∣∣∣∣∣∣∣
R θθθ φφφ
AR Aθ Aφ
BR Bθ Bφ
∣∣∣∣∣∣Differential length dl = x dx+ y dy+ z dz r dr + φφφr dφ + z dz R dR+ θθθR dθ + φφφRsinθ dφ
Differential surface areas dsx = x dy dzdsy = y dx dzdsz = z dx dy
dsr = rr dφ dzdsφ = φφφ dr dzdsz = zr dr dφ
dsR = RR2 sinθ dθ dφ
dsθ = θθθRsinθ dR dφ
dsφ = φφφR dR dθ
Differential volume dv = dx dy dz r dr dφ dz R2 sinθ dR dθ dφ
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 3-2: Coordinate transformation relations.
Transformation Coordinate Variables Unit Vectors Vector Components
Cartesian to r = +√
x2 + y2 r = xcosφ + ysinφ Ar = Ax cosφ +Ay sinφ
cylindrical φ = tan−1(y/x) φφφ =−xsinφ + ycosφ Aφ =−Ax sinφ +Ay cosφ
z = z z = z Az = Az
Cylindrical to x = r cosφ x = rcosφ − φφφsinφ Ax = Ar cosφ −Aφ sinφ
Cartesian y = r sinφ y = rsinφ + φφφcosφ Ay = Ar sinφ +Aφ cosφ
z = z z = z Az = Az
Cartesian to R = +√
x2 + y2 + z2 R = xsinθ cosφ AR = Ax sinθ cosφ
spherical + ysinθ sinφ + zcosθ +Ay sinθ sinφ +Az cosθ
θ = tan−1[ +√
x2 + y2/z] θθθ = xcosθ cosφ Aθ = Ax cosθ cosφ
+ ycosθ sinφ − zsinθ +Ay cosθ sinφ −Az sinθ
φ = tan−1(y/x) φφφ =−xsinφ + ycosφ Aφ =−Ax sinφ +Ay cosφ
Spherical to x = Rsinθ cosφ x = Rsinθ cosφ Ax = AR sinθ cosφ
Cartesian + θθθcosθ cosφ − φφφsinφ +Aθ cosθ cosφ −Aφ sinφ
y = Rsinθ sinφ y = Rsinθ sinφ Ay = AR sinθ sinφ
+ θθθcosθ sinφ + φφφcosφ +Aθ cosθ sinφ +Aφ cosφ
z = Rcosθ z = Rcosθ − θθθsinθ Az = AR cosθ −Aθ sinθ
Cylindrical to R = +√r2 + z2 R = rsinθ + zcosθ AR = Ar sinθ +Az cosθ
spherical θ = tan−1(r/z) θθθ = rcosθ − zsinθ Aθ = Ar cosθ −Az sinθ
φ = φ φφφ = φφφ Aφ = Aφ
Spherical to r = Rsinθ r = Rsinθ + θθθcosθ Ar = AR sinθ +Aθ cosθ
cylindrical φ = φ φφφ = φφφ Aφ = Aφ
z = Rcosθ z = Rcosθ − θθθsinθ Az = AR cosθ −Aθ sinθ
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 4 Tables
Table 4-1 Conductivity of some common materials at 20C.
Table 4-2 Relative permittivity (dielectric constant) and dielectric strength of commonmaterials.
Table 4-3 Boundary conditions for the electric fields.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 4-1: Conductivity of some common materials at 20C.
Material Conductivity, σ (S/m)
ConductorsSilver 6.2×107
Copper 5.8×107
Gold 4.1×107
Aluminum 3.5×107
Iron 107
Mercury 106
Carbon 3×104
SemiconductorsPure germanium 2.2Pure silicon 4.4×10−4
InsulatorsGlass 10−12
Paraffin 10−15
Mica 10−15
Fused quartz 10−17
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 4-2: Relative permittivity (dielectric constant) and dielectric strength of common materials.
Material Relative Permittivity, εr Dielectric Strength, Eds (MV/m)
Air (at sea level) 1.0006 3Petroleum oil 2.1 12Polystyrene 2.6 20Glass 4.5–10 25–40Quartz 3.8–5 30Bakelite 5 20Mica 5.4–6 200
ε = εrε0 and ε0 = 8.854×10−12 F/m.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 4-3: Boundary conditions for the electric fields.
Field Component Any Two Media Medium 1Dielectric ε1
Medium 2Conductor
Tangential E E1t = E2t E1t = E2t = 0
Tangential D D1t/ε1 = D2t/ε2 D1t = D2t = 0
Normal E ε1E1n− ε2E2n = ρs E1n = ρs/ε1 E2n = 0
Normal D D1n−D2n = ρs D1n = ρs D2n = 0
Notes: (1) ρs is the surface charge density at the boundary; (2) normalcomponents of E1, D1, E2, and D2 are along n2, the outward normal unit vectorof medium 2.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 5 Tables
Table 5-1 Attributes of electrostatics and magnetostatics.
Table 5-2 Properties of magnetic materials.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 5-1: Attributes of electrostatics and magnetostatics.
Attribute Electrostatics Magnetostatics
Sources Stationary charges ρv Steady currents J
Fields and Fluxes E and D H and B
Constitutive parameter(s) ε and σ µ
Governing equations• Differential form
• Integral form
∇ ·D = ρv∇×××E = 0
n∫
SD ·ds = Q
n∫
CE ·dl = 0
∇ ·B = 0∇×××H = J
n∫
SB ·ds = 0
n∫
CH ·dl = I
Potential Scalar V , with Vector A, withE =−∇V B = ∇×××A
Energy density we = 12 εE2 wm = 1
2 µH2
Force on charge q Fe = qE Fm = qu×××B
Circuit element(s) C and R L
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 5-2: Properties of magnetic materials.
Diamagnetism Paramagnetism FerromagnetismPermanent magnetic No Yes, but weak Yes, and strongdipole momentPrimary magnetization Electron orbital Electron spin Magnetizedmechanism magnetic moment magnetic moment domains
Direction of induced Opposite Same Hysteresismagnetic field [see Fig. ??](relative to external field)Common substances Bismuth, copper, diamond, Aluminum, calcium, Iron,
gold, lead, mercury, silver, chromium, magnesium, nickel,silicon niobium, platinum, cobalt
tungsten
Typical value of χm ≈−10−5 ≈ 10−5 |χm| 1 and hystereticTypical value of µr ≈ 1 ≈ 1 |µr| 1 and hysteretic
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 6 Tables
Table 6-1 Maxwell’s equations.
Table 6-2 Boundary conditions for the electric and magnetic fields.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 6-1: Maxwell’s equations.
Reference Differential Form Integral Form
Gauss’s law ∇ ·D = ρv n∫
SD ·ds = Q (6.1)
Faraday’s law ∇×××E =−∂B∂ t
n∫
CE ·dl =−
∫S
∂B∂ t·ds (6.2)∗
No magnetic charges ∇ ·B = 0 n∫
SB ·ds = 0 (6.3)
(Gauss’s law for magnetism)
Ampere’s law ∇×××H = J+∂D∂ t
n∫
CH ·dl =
∫S
(J+
∂D∂ t
)·ds (6.4)
∗For a stationary surface S.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 6-2: Boundary conditions for the electric and magnetic fields.
Field Components General Form Medium 1Dielectric
Medium 2Dielectric
Medium 1Dielectric
Medium 2Conductor
Tangential E n2××× (E1−E2) = 0 E1t = E2t E1t = E2t = 0Normal D n2 ·(D1−D2) = ρs D1n−D2n = ρs D1n = ρs D2n = 0Tangential H n2××× (H1−H2) = Js H1t = H2t H1t = Js H2t = 0Normal B n2 · (B1−B2) = 0 B1n = B2n B1n = B2n = 0Notes: (1) ρs is the surface charge density at the boundary; (2) Js is the surface current density at the boundary;(3) normal components of all fields are along n2, the outward unit vector of medium 2; (4) E1t = E2t implies thatthe tangential components are equal in magnitude and parallel in direction; (5) direction of Js is orthogonal to(H1−H2).
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 7 Tables
Table 7-1 Expressions for α , β , ηc, up, and λ for various types of media.
Table 7-2 Power ratios in natural numbers and in decibels.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 7-1: Expressions for α , β , ηc, up, and λ for various types of media.
Lossless Low-loss GoodAny Medium Medium Medium Conductor Units
(σ = 0) (ε ′′/ε ′ 1) (ε ′′/ε ′ 1)
α = ω
µε ′
2
√1+(
ε ′′
ε ′
)2
−1
1/2
0σ
2
õ
ε
√π f µσ (Np/m)
β = ω
µε ′
2
√1+(
ε ′′
ε ′
)2
+1
1/2
ω√
µε ω√
µε√
π f µσ (rad/m)
ηc =√
µ
ε ′
(1− j
ε ′′
ε ′
)−1/2 √µ
ε
õ
ε(1+ j)
α
σ(Ω)
up = ω/β 1/√
µε 1/√
µε√
4π f /µσ (m/s)λ = 2π/β = up/ f up/ f up/ f up/ f (m)
Notes: ε ′ = ε; ε ′′ = σ/ω; in free space, ε = ε0, µ = µ0; in practice, a material is considered alow-loss medium if ε ′′/ε ′ = σ/ωε < 0.01 and a good conducting medium if ε ′′/ε ′ > 100.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 7-2: Power ratios in natural numbers and in decibels.
G G [dB]10x 10x dB
4 6 dB2 3 dB1 0 dB0.5 −3 dB0.25 −6 dB0.1 −10 dB
10−3 −30 dB
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 8 Tables
Table 8-1 Analogy between plane-wave equations for normal incidence and transmission-lineequations, both under lossless conditions.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 8-1: Analogy between plane-wave equations for normal incidence and transmission-line equations, both under lossless conditions.
Plane Wave [Fig. ??(a)] Transmission Line [Fig. ??(b)]
E1(z) = xE i0(e− jk1z +Γe jk1z) (8.5a) V1(z) = V +
0 (e− jβ1z +Γe jβ1z) (8.5b)
H1(z) = yE i
0η1
(e− jk1z−Γe jk1z) (8.6a) I1(z) =V +
0Z01
(e− jβ1z−Γe jβ1z) (8.6b)
E2(z) = xτE i0e− jk2z (8.7a) V2(z) = τV +
0 e− jβ2z (8.7b)
H2(z) = yτE i
0η2
e− jk2z (8.8a) I2(z) = τV +
0Z02
e− jβ2z (8.8b)
Γ = (η2−η1)/(η2 +η1) Γ = (Z02−Z01)/(Z02 +Z01)
τ = 1+Γ τ = 1+Γ
k1 = ω√
µ1ε1 , k2 = ω√
µ2ε2 β1 = ω√
µ1ε1 , β2 = ω√
µ2ε2
η1 =√
µ1/ε1 , η2 =√
µ2/ε2 Z01 and Z02 depend ontransmission-line parameters
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 9 Tables
There are no Tables in Chapter 9.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Chapter 10 Tables
Table 10-1 Communications satellite frequency allocations.
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall
Table 10-1: Communications satellite frequency allocations.
Downlink Frequency Uplink FrequencyUse (MHz) (MHz)
Fixed ServiceCommercial (C-band) 3,700–4,200 5,925–6,425Military (X-band) 7,250–7,750 7,900–8,400Commercial (K-band)
Domestic (USA) 11,700–12,200 14,000–14,500International 10,950–11,200 27,500–31,000
Mobile ServiceMaritime 1,535–1,542.5 1,635–1,644Aeronautical 1,543.5–1,558.8 1,645–1,660
Broadcast Service2,500–2,535 2,655–2,690
11,700–12,750
Telemetry, Tracking, and Command137–138, 401–402, 1,525–1,540
Fawwaz T. Ulaby, Eric Michielssen, and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c©2010 Prentice Hall