14.4: Basic Equations for Electromagnetics and Applications
( \newcommand{\kernel}{\mathrm{null}\,}\)
Fundamentals
→f=q(→E+→v×μo→H)[N]
∇×→E=−∂→B/∂t
∮c→E∙d→s=−ddt∫A→B∙d→a
∇×→H=→J+∂→D/∂t
∮c→H∙d→s=∫A→J∙d→a+ddt∫A→D∙d→a
∇∙→D=ρ→∮A→D∙d→a=∫Vρdv
∇∙→B=0→∮A→B∙d→a=0
∇∙→J=−∂ρ/∂t
→E= electric field (Vm−1)
→H= magnetic field (Am−1)
→D= electric displacement (Cm−2)
→B= magnetic flux density (T) Tesla (T)= Weber m−2=10,000 gauss
ρ= charge density (Cm−3)
→J= current density (Am−2)
σ= conductivity (Siemens m−1)
→Js= surface current density (Am−1)
ρs= surface charge density (Cm−2)ε0=8.85×10−12 Fm−1
μo=4π×10−7 Hm−1
c=(εoμo)−0.5≅3×108 ms−1
e=−1.60×10−19 C
Ey(z,t)=E+(z−ct)+E−(z+ct)=Re{E_y(z)ejωt}
Hx(z,t)=η−1o[E+(z−ct)−E−(z+ct)] [ or (ωt−kz) or (t−z/c)]
∫A(→E×→H)∙d→a+(d/dt)∫V(ε|→E|2/2+μ|→H|2/2)dv=−∫V→E∙→J dv (Poynting Theorem)
Media and Boundaries
→D=εo→E+→P
∇∙→D=ρf, τ=ε/σ
∇∙εo→E=ρf+ρp
∇∙→P=−ρp, →J=σ→E
→B=μ→H=μo(→H+→M)
ε=εo(1−ω2p/ω2)
ωp=(Ne2/mεo)0.5
εeff=ε(1−jσ/ωε)
Electromagnetic Quasistatics
∇2Φ=0
KCL:∑iIi(t)=0 at node
KVL:∑iVi(t)=0 around loop
C=Q/V=Aε/d[F]
L=Λ/I
i(t)=Cdv(t)/dt
v(t)=Ldi(t)/dt=dΛ/dt
Cparallel =C1+C2
Cseries =(C−11+C−12)−1
we=Cv2(t)/2;wm=Li2(t)/2
Lsolenoid =N2μA/W
τ=RC,τ=L/R
Λ=∫A→B∙dˉa (per turn)
→f=q(→E+→v×μo→H)[N]
fz=−dwT/dz
→F=→I×μo→H[Nm−1]
→Ee=−→v×μo→H inside wire
P=ωT=WTdVolume /dt[W]
Max f/A=B2/2μo,D2/2εo[Nm−2]
vi =dwTdt+fdzdt
Electromagnetic Waves
(∇2−με∂2/∂t2)→E=0 [ Wave Eqn.]
(∇2+k2)→E_=0,→E_=→E_0e−j→k∙r
k=ω(με)0.5=ω/c=2π/λ
k2x+k2y+k2z=k2o=ω2με
vp=ω/k, vg=(∂k/∂ω)−1
θr=θi
sinθt/sinθi=ki/kt=ni/nt
θc=sin−1(nt/ni)
θ>θc⇒→E_t=→E_iT_e+αx−jkzz
→k_=→k′−j→k′′
Γ_=T_−1
T_TE=2/(1+[ηocosθt/ηtcosθi])
T_TM=2/(1+[ηtcosθt/ηicosθi])
θB=tan−1(εt/εi)0.5 for TM
Pd≅|→J_S|2/2σδ [Wm−2]
→E=−∇ϕ−∂→A/∂t,→B=∇×→A
Φ_(r)=∫V′(ρ_(→r)e−jk|→r′−→r|/4πεo|→r′−→r|)dv′
→A_(r)=∫V′μo(→J_(→r)e−jk|→r′−→r|/4π|→r′−→r|)dv′
→E_ff=ˆϑ(jηokI_d/4πr)e−jkrsinθ
∇2Φ_+ω2μ0ε0Φ_=−ρ/ε0
∇2→A_+ω2μoεo→A_=−μo→J_
Forces, Motors, and Generators
→f=q(→E+→v×μo→H)[N]
fz=−dwT/dz
→F=→I×μo→H[Nm−1]
→Ee=−→v×μo→H inside wire
P=ωT=WTdVolume /dt[W]
Max f/A=B2/2μo, D2/2εo[Nm−2]
vi=dwTdt+fdzdt
f=ma=d(mv)/dt
x=xo+vot+at2/2
P=fv [W]=Tω
wk=mv2/2
T=I dω/dt
I=∑imir2i
Circuits
KCL:∑iIi(t)=0 at node
KVL:∑iVi(t)=0 around loop
C=Q/V=Aε/d[F]
L=Λ/I
i(t)=C dv(t)/dt
v(t)=L di(t)/dt=dΛ/dt
Cparallel =C1+C2
Cseries =(C−11+C−12)−1
we=Cv2(t)/2; wm=Li2(t)/2
Lsolenoid =N2μA/W
τ=RC,τ=L/R
Λ=∫A→B∙d→a (per turn)
Z_ series=R+jωL+1/jωC
Y_par =G+jωC+1/jωL
Q=ωowT/Pdiss=ωo/Δω
ωo=(LC)−0.5
⟨v2(t)⟩/R=kT
Limits to Computation Speed
dv(z)/dz=−Ldi(z)/dt
di(z)/dz=−Cdv(z)/dt
d2v/dz2=LC d2v/dt2
v(z,t)=f+(t−z/c)+f−(t+z/c)=g+(z−ct)+g−(z−ct)
i(t,z)=Yo[f+(t−z/c)−f−(t+z/c)]
c=(LC)−0.5=1/√με
Zo=Y−1o=(L/C)0.5
ΓL=f/f+=(RL−Zo)/(RL+Zo)
v(z,t)=g+(z−ct)+g−(z+ct)
VTh=2f+(t), RTh=Zo
Power Transmission
(d2/dz2+ω2LC)V_(z)=0
V_(z)=V_+e−jkz+V_−e+jkz
I_(z)=Yo[V_+e−jkz−V_−e+jkz]
k=2π/λ=ω/c=ω(με)0.5
Z_(z)=V_(z)/I_(z)=ZoZ_n(z)
Z_n(z)=[1+Γ_(z)]/[1−Γ_(z)]=Rn+jXn
Γ_(z)=(V_−V_+)e2 jkz=[Z_n(z)−1]/[Z_n(z)+1]
Z_(z)=Z0(Z_L−jZ0tankz)/(Z_0−jZLtankz)
VSWR=|V_max|/|V_min|=Rmax
Wireless Communications and Radar
G(θ,ϕ)=Pr/(PR/4πr2)
PR=∫4πPr(θ,ϕ,r)r2sinθ dθdϕ
Prec=Pr(θ,ϕ)Ae(θ,ϕ)
Ae(θ,ϕ)=G(θ,ϕ)λ2/4π
Rr=PR/⟨i2(t)⟩
Eff(θ≅0)=(jejkr/λr)∫AEt(x,y)ejkxx+jkyydxdy
Prec=PR(Gλ/4πr2)2σs/4π
→E_=∑iai→E_ie−jkr1=( element factor )( array f)
Ebit ≥∼4×10−20[J]
Z_12=Z_21 if reciprocity
(d2/dz2+ω2LC)V_(z)=0
V_(z)=V_+e−jkz+V_−e+jkz
I_(z)=Yo[V_+e−jkz−V_−e+jkz]
k=2π/λ=ω/c=ω(με)0.5
Z_(z)=V_(z)/I_(z)=ZoZ_n(z)
Z_n(z)=[1+Γ_(z)]/[1−Γ_(z)]=Rn+jXn
Γ_(z)=(V_−/V_+)e2jkz=[Z_n(z)−1]/[Z_n(z)+1]
Z_(z)=Z0(Z_L−jZ0tankz)/(Z_0−jZLtankz)
VSWR=|V_max|/|V_min|=Rmax
θr=θi
sinθt/sinθi=ki/kt=ni/nt
θc=sin−1(nt/ni)
θ>θc⇒→E_t=→E_iT_e+αx−jkzz
→k_=→k′−j→k′′
Γ_=T_−1
Atω0, ⟨we⟩=⟨wm⟩
⟨we⟩=∫V(ε|→E_|2/4)dv
⟨wm⟩=∫V(μ|→H_|2/4)dv
Qn=ωnWTn/Pn=ωn/2αn
fmnp=(c/2)([m/a]2+[n/b]2+[p/d]2)0.5
Sn=jωn−αn
Optical Communications
E=hf, photons or phonons
hf/c= momentum [kg ms−1]
dn2/dt=−[An2+B(n2−n1)]
Acoustics
P=Po+p, →U=→Uo+u (→Uo=0 here )
∇p=−ρo∂→u/∂t
∇∙→u=−(1/γPo)∂p/∂t
(∇2−k2∂2/∂t2)p=0
k2=ω2/c2s=ω2ρo/γPo
cs=vp=vg=(γPo/ρo)0.5 or (K/ρo)0.5
ηs=p/u=ρocs=(ρoγPo)0.5 gases
ηs=(ρoK)0.5 solids, liquids
p,→u⊥ continuous at boundaries
p_=p_+e−jkz+p_−e+jkz
u_z=η−1s(p_+e−jkz−p_−e+jkz)
∫A→up∙d→a+(d/dt)∫V(ρo|→u|2/2+p2/2γPo)dV
Mathematical Identities
sin2θ+cos2θ=1
cosα+cosβ=2cos[(α+β)/2]cos[(α−β)/2]
H_(f)=∫+∞−∞h(t)e−jωtdt
ex=1+x+x2/2!+x3/3!+…
sinα=(ejα−e−jα)/2j
cosα=(ejα+e−jα)/2
Vector Algebra
∇=ˆx∂/∂x+ˆy∂/∂y+ˆz∂/∂z
ˉA∙ˉB=AxBx+AyBy+AzBz
∇2ϕ=(∂2/∂x2+∂2/∂y2+∂2/∂z2)ϕ
∇∙(∇×ˉA)=0
∇×(∇×ˉA)=∇(∇∙ˉA)−∇2ˉA
Gauss and Stokes’ Theorems
\oiint_{\mathrm{V}}(\nabla \bullet \overrightarrow{\mathrm{G}}) \mathrm{d} \mathrm{v}=\oiint_{\mathrm{A}} \overrightarrow{\mathrm{G}} \bullet \mathrm{d} \overrightarrow{\mathrm{a}} \nonumber
\oiint_{\mathrm{A}}(\nabla \times \overrightarrow{\mathrm{G}}) \bullet \mathrm{d} \overrightarrow{\mathrm{a}}=\oint_{\mathrm{c}} \overrightarrow{\mathrm{G}} \bullet \mathrm{d} \overrightarrow{\mathrm{s}} \nonumber
Complex Numbers and Phasors
\mathrm{v}(\mathrm{t})=\mathrm{R}_{\mathrm{e}}\left\{\mathrm{\underline{V}e}^{\mathrm{j} \omega \mathrm{t}}\right\} \text { where } \underline{\mathrm{V}}=|\mathrm{V}| \mathrm{e}^{\mathrm{j} \phi} \nonumber
\mathrm{e}^{\mathrm{j} \omega \mathrm{t}}=\cos \omega \mathrm{t}+\mathrm{j} \sin \omega \mathrm{t} \nonumber
Spherical Trigonometry
\int_{4 \pi} \mathrm{r}^{2} \sin \theta \ \mathrm{d} \theta \mathrm{d} \phi=4 \pi \nonumber