c Appendix A: Numerical Constants A.1 Fundamental Constants velocity of light 2.998 × 108 m/s εo permittivity of free space 8.854 × 10-12 F/m μo permeability of free space 4π × 10-7 H/m ηo characteristic impedance of free space 376.7 Ω e charge of an electron, (-e.v./Joule) -1.6008 × 10-19 C m mass of an electron 9.1066 × 10-31 kg mp mass of a proton 1.6725 × 10-27 kg h Planck constant 6.624 × 10-34 J⋅s k Boltzmann constant 1.3805 × 10-23 J/K No Avogadro’s constant 6.022 × 1023 molec/mole R Universal gas constant 8. 31 J/mole⋅K A.2 Electrical Conductivity σ, S/m Silver 6.14 × 107 Monel 0.24 × 107 Copper 5.80 × 107 Mercury 0.1 × 107 Gold 4.10 × 107 Sea Water 3 – 5 Aluminum 3.54 × 107 Distilled Water 2 × 10-4 Tungsten 1.81 × 107 Bakelite 10-8 – 10-10 Brass 1.57 × 107 Glass 10-12 Nickel 1.28 × 107 Mica 10-11 – 10-15 Iron (pure) 1.0 × 107 Petroleum 10-14 Steel 0.5 – 1.0 × 107 Fused Quartz <2 × 10-17 Lead 0.48 × 107 - A421 -A.3 Relative Dielectric Constant ε/εo at 1 MHz Vacuum Styrofoam (25% filler) Firwood Paper Petroleum Paraffin Teflon Vaseline Rubber Polystyrene Sandy soil Plexiglas Fused quartz A.4 Relative Permeability μ/μo Vacuum Biological tissue Cold steel Iron (99.91%) Purified iron (99.95%) mu metal (FeNiCrCu) Supermalloy (FeNiMoMn) 1 1 2,000 5,000 180,000 100,000 800,000 1.00 Vycor glass 3.8 1.03 Low-loss glass 4.1 1.8 – 2.0 Ice 4.15 2.0 – 3.0 Pyrex glass 5.1 2.1 Muscovite (mica) 5.4 2.1 Mica 5.6 – 6.0 2.1 Magnesium silicate 5.7 – 6.4 2.16 Porcelain 5.7 2.3 – 4.0 Aluminum oxide 8.8 2.55 Diamond 16.5 2.6 Ethyl alcohol 24.5 2.6 – 3.5 Distilled water 81.1 3.78 Titanium dioxide 100 - A422 -Appendix B: Complex Numbers and Sinusoidal Representation Most linear systems that store energy exhibit frequency dependence and therefore are more easily characterized by their response to sinusoids rather than to arbitrary waveforms. The resulting system equations contain many instances of Acos(ωt + φ), where A, ω, and φ are the amplitude, frequency, and phase of the sinsusoid, respectively. Acos(ωt + φ) can be replaced by A using complex notation, indicated here by the underbar and reviewed below; it utilizes the arbitrary definition: j ≡−()0.51 (B.1) This arbitrary non-physical definition is exploited by De Moivre’s theorem (B.4), which utilizes a unique property of e = 2.71828: eφ=+1 φ + φ2 2! + φ3 3! +... (B.2) Therefore: ejφ=1+j φ− φ2 2! − jφ3 3!+ φ4 4! + jφ5 5! −... =−⎡1 2 ⎤⎡3 φ 2! + φ4 4! −... + jφ − jφ 3!+ jφ5 (B.3) 5!... ⎤⎣ ⎦⎣ ⎦ ejφ= cos φ + jsin φ (B.4) This is a special instance of a general complex number A: A = Ar + jA i (B.5) where the real part is Ar ≡ Re{A} and the imaginary part is Ai ≡ Im{A}. It is now easy to use (B.4) and (B.5) to show that76: Acos (ω+t φ)= Re{jt(ω+φ)Ae }= RAe j φωj t}=te{e Rjω e{A e }= Arcos ωt − Aisin ωt (B.6) where: A = Ae jφ= Acos φ+ jAsin φ = Ar + jA i (B.7) 76 The physics community differs and commonly defines Acos(ωt + φ) = R t + φ) e {Ae-j(ω} and Ai ≡ -Asinφ, where the rotational direction of φ is reversed in Figure B.1. Because phase is reversed in this alternative notation, the impedance of an inductor L becomes -jωL, and that of a capacitor becomes j/ωC. In this notation j is commonly replaced by -i. - B423 -When φ = 0 we have R {Aejωte} = Acosωt, and when φ = π/2 we have -Asinωt. Advances in time alter the phasor A in the same sense as advances in φ; the phasor rotates counterclockwise. The utility of this diagram is partly that the signal of interest, Re{Aejωt}, is simply the projection of the phasor Aejωt on the real axis. It also makes clear that: 0.5 A =(A2r + A2i)(B.9)φ=tan −1(Ai Ar ) (B.10)It is also easy to see, for example, that ejπ= -1, and that A = jA corresponds to -Asinωt. Examples of equivalent representations in the time and complex domains are: Acos ω↔t A −Asinωt ↔ jA Acos (ω+t φ)↔ Ae jφ (t ) jAe jφj(φ−π 2= Ae)Asin ω+φ ↔ − Complex numbers behave as vectors in some respects, where addition and multiplication are also illustrated in Figure B.1(b) and (c), respectively: Ar ≡ Acos φ, Ai ≡ Asin φ (B.8) The definition of A given in (B.8) has the useful geometric interpretation shown in Figure B.1(a), where the magnitude of the phasor A is simply the given amplitude A of the sinusoid, and the angle φ is its phase. (a) Im{A} (b) Im{A,B} (c) AB = AB ej(φAB+φ ) A B φφAC φB 0 Re{A,B} Im{A,B} A φ A = Ar + jAi = Aejφ A BA + B Asinφ j -1 0 AcosφRe{A} 01 Re{A,B} -j Figure B.1 Representation of phasors in the complex plane. - B424 -A +=B B + A = Ar + Br + j(Ai + Bi ) (B.11) ( )(j( )φAB = + A +φBA = A B − A B j A B + A B = AB e B )rr i i r i i r (B.12) A* = Ar − jA =−φj A i A e(B.13) We can easily solve for the real and imaginary parts of A: A r =(A + A*)2, A*i =(A − A )2 (B.14) Ratios of complex numbers can also be readily computed: AB =(AB ) e j ( φ−B* Aφ *B )= AB B = AB *B 2(B.15) Even an nth root of A = Aejφ can be simply found: A1n = A1nejφn (B.16) where n legitimate roots exist and are: A1n = A (1n)e(jφ n)e(j2 πm n ) (B.17) for m = 0, 1, …, n – 1. - B425 -- B426 -Appendix C: Mathematical Identities A = xˆAx+ yˆAy+ zˆAz AB i = Axx B + AyB y + AzB z = a ˆ× bˆA B cos θx ˆyˆ z ˆAB ×=de tAxA yA zB xB yB z= xˆ( AyzB − AB zy )+ yˆ(AB zx − AxB z )+ zˆ(AxB y − AyB x ) =×abˆ ˆA B sin θ ABi(×C ) = Bi( C × A )= Ci(A×B ) AB×( × C )=(Ai C) B −(Ai B) C (AB × )i( C× D )=(Ai C )( Bi D )−(Ai D )( Bi C ) ∇×∇Ψ = 0 ∇∇i(×A)= 0 ∇×( ∇×A )= ∇(∇iA) −∇2 A 1 −×A (∇× A)=(Ai∇)A − ∇(Ai A ) 2 ∇Ψ( Φ) =Ψ∇ Φ+Φ∇Ψ ∇Ψi( A )= Ai∇ Ψ +Ψ∇ iA ∇×( ΨA )= ∇Ψ× A +Ψ∇×A ∇Ψ2 =∇i∇ Ψ ∇(ABi )=(Ai∇) B +( B i∇) A+A ×∇( × B)+B ×∇(×A) ∇i(AB× )= Bi(∇× A)− Ai(∇× B) ∇×(A×B )= A(∇iB)− B(∇iA)+(Bi∇)A −( Ai∇)B - C427 -Cartesian Coordinates (x,y,z): ∂Ψ ∂Ψ ∂Ψ ∇Ψ = xˆ + yˆ + zˆ ∂x ∂y ∂z ∂A ∂A∇iA =x y ∂A + +z ∂x ∂y ∂z ⎛∂A∂Ay ⎞ ⎛∂A ∂A⎛∂A⎞∇×A =xˆ z − +yˆ x −z ⎞ y ∂Ax ⎜ ⎟ ⎜ ⎟+ zˆ ⎜−⎟ ∂y ∂z ∂x ∂x ∂y …