fundamentals of applied electromagneticsem.eecs.umich.edu/pdf/ulaby_tables.pdf · ·...

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


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 ].


Table 1-1: Fundamental SI units.

Dimension Unit SymbolLength meter mMass kilogram kgTime second sElectric Current ampere ATemperature kelvin KAmount of substance mole mol


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


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)


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


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)


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.


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).


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)


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


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.


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


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 .


Chapter 3 Tables

Table 3-1 Summary of vector relations.

Table 3-2 Coordinate transformation relations.


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


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


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.


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


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.


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.


Chapter 5 Tables

Table 5-1 Attributes of electrostatics and magnetostatics.

Table 5-2 Properties of magnetic materials.


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


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


Chapter 6 Tables

Table 6-1 Maxwell’s equations.

Table 6-2 Boundary conditions for the electric and magnetic fields.


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.


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).


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.


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

√µ

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

√µ

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.


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


Chapter 8 Tables

Table 8-1 Analogy between plane-wave equations for normal incidence and transmission-lineequations, both under lossless conditions.


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


Chapter 9 Tables

There are no Tables in Chapter 9.


Chapter 10 Tables

Table 10-1 Communications satellite frequency allocations.


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


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