14.4: Basic Equations for Electromagnetics and Applications
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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
Wireless Communications and Radar
\mathrm{G}(\theta, \phi)=\mathrm{P}_{\mathrm{r}} /\left(\mathrm{P}_{\mathrm{R}} / 4 \pi \mathrm{r}^{2}\right) \nonumber
\mathrm{P_{R}=\int_{4 \pi} P_{r}(\theta, \phi, r) r^{2} \sin \theta \ d \theta d \phi} \nonumber
\mathrm P_{\mathrm{rec}}=\mathrm{P}_{\mathrm{r}}(\theta, \phi) \mathrm{A}_{\mathrm{e}}(\theta, \phi) \nonumber
\mathrm{A}_{\mathrm{e}}(\theta, \phi)=\mathrm{G}(\theta, \phi) \lambda^{2} / 4 \pi \nonumber
\mathrm{R}_{\mathrm{r}}=\mathrm{P}_{\mathrm{R}} \big/\left\langle\mathrm{i}^{2}(\mathrm{t})\right\rangle \nonumber
\mathrm{E}_{\mathrm{ff}}(\theta \cong 0)=\left(\mathrm{je}^{\mathrm{jkr}} / \lambda \mathrm{r}\right) \int_{\mathrm{A}} \mathrm{E}_{\mathrm{t}}(\mathrm{x}, \mathrm{y}) \mathrm{e}^{\mathrm{jk}_{\mathrm{x}}{\mathrm{x}+\mathrm{jk}}_{ \mathrm{y}}{\mathrm{y}}} \mathrm{dxdy} \nonumber
\mathrm P_{\mathrm{rec}}=\mathrm{P}_{\mathrm{R}}\left(\mathrm{G} \lambda / 4 \pi \mathrm{r}^{2}\right)^{2} \sigma_{\mathrm{s}} / 4 \pi \nonumber
\underline{\mathrm{\vec E}}=\sum_{\mathrm{i}} \mathrm{a}_{\mathrm{i}} \overrightarrow{\mathrm{\underline E}}_{\mathrm{i}} \mathrm{e}^{-\mathrm{jk} \mathrm{r}_{1}}=(\text { element factor })(\text { array } \mathrm{f}) \nonumber
\mathrm{E}_{\text {bit }} \geq \sim 4 \times 10^{-20}[\mathrm{J}] \nonumber
\underline{\mathrm Z}_{12}=\underline{\mathrm Z}_{21} \text { if reciprocity } \nonumber
\left(\mathrm{d}^{2} / \mathrm{d} \mathrm z^{2}+\omega^{2} \mathrm{LC}\right) \underline{\mathrm{V}}(\mathrm{z})=0 \nonumber
\mathrm{\underline V}(\mathrm{z})=\underline{\mathrm{V}}_{+}\mathrm{e}^{-\mathrm{jk} \mathrm{z}}+\underline{\mathrm{V}}_{-} \mathrm{e}^{+\mathrm{jk} \mathrm{z}} \nonumber
\mathrm{\underline{I}(z)=Y_{o}\left[\underline{V}_{+}e^{-j k z}-\underline{V}_{-} e^{+j k z}\right]} \nonumber
\mathrm{k}=2 \pi / \lambda=\omega / \mathrm{c}=\omega(\mu \varepsilon)^{0.5} \nonumber
\mathrm{\underline{Z}(z)=\underline{V}(z) / \underline{I}(z)=Z_{o} \underline{Z}_{n}(z)} \nonumber
\mathrm{\underline{Z}_{n}(z)=[1+\underline{\Gamma}(z)]/[1-\underline{\Gamma}(z)]=R_{n}+j X_{n}} \nonumber
\underline{\Gamma}(\mathrm{z})=\left(\underline{\mathrm{V}}_{-}/\underline{\mathrm{V}}_{+}\right) \mathrm{e}^{2 \mathrm{jkz}}=\left[\underline{\mathrm{Z}}_{\mathrm{n}}(\mathrm{z})-1\right]/\left[\mathrm{\underline Z}_{\mathrm{n}}(\mathrm{z})+1\right] \nonumber
\mathrm{\underline{Z}(z)=Z_{0}\left(\underline{Z}_{L}-j Z_{0} \tan k z\right) /\left(\underline{Z}_{0}-j Z_{L} \tan k z\right)} \nonumber
\mathrm{VSWR}=\left|\underline{\mathrm{V}}_{\max }\right| /\left|\underline{\mathrm{V}}_{\min }\right|=\mathrm{R}_{\max } \nonumber
\theta_{\mathrm{r}}=\theta_{\mathrm{i}} \nonumber
\sin \theta_{\mathrm{t}} / \sin \theta_{\mathrm{i}}=\mathrm{k}_{\mathrm{i}} / \mathrm{k}_{\mathrm{t}}=\mathrm{n}_{\mathrm{i}} / \mathrm{n}_{\mathrm{t}} \nonumber
\theta_{\mathrm{c}}=\sin ^{-1}\left(\mathrm{n}_{\mathrm{t}} / \mathrm{n}_{\mathrm{i}}\right) \nonumber
\theta>\theta_{\mathrm{c}} \Rightarrow \overrightarrow{\mathrm{\underline E}}_{\mathrm{t}}=\overrightarrow{\mathrm{\underline E}}_{\mathrm{i}} \mathrm{\underline{T}e}^{+\alpha x-\mathrm{j} \mathrm{k}_{\mathrm{z}^{\mathrm{z}}}} \nonumber
\underline{\mathrm{\vec k}}=\overrightarrow{\mathrm{k}}^{\prime}-\mathrm{j} \overrightarrow{\mathrm{k}}^{\prime \prime} \nonumber
\underline{\Gamma}=\underline{\mathrm{T}}-1 \nonumber
\operatorname{At} \omega_{0}, \ \left\langle\mathrm{w}_{\mathrm{e}}\right\rangle=\left\langle\mathrm{w}_{\mathrm{m}}\right\rangle \nonumber
\left\langle\mathrm{w}_{\mathrm{e}}\right\rangle=\int_{\mathrm{V}}\left(\varepsilon|\overrightarrow{\mathrm{\underline E}}|^{2} \big/ 4\right) \mathrm{d} \mathrm{v} \nonumber
\left\langle\mathrm{w}_{\mathrm{m}}\right\rangle=\int_{\mathrm{V}}\left(\mu|\overrightarrow{\mathrm{\underline H}}|^{2} \big/ 4\right) \mathrm{d} \mathrm{v} \nonumber
\mathrm{Q}_{\mathrm{n}}=\omega_{\mathrm{n}} \mathrm{W}_{\mathrm{Tn}} / \mathrm{P}_{\mathrm{n}}=\omega_{\mathrm{n}} / 2 \alpha_{\mathrm{n}} \nonumber
\mathrm{f}_{\mathrm{mnp}}=(\mathrm{c} / 2)\left([\mathrm{m} / \mathrm{a}]^{2}+[\mathrm{n} / \mathrm{b}]^{2}+[\mathrm{p} / \mathrm{d}]^{2}\right)^{0.5} \nonumber
\mathrm{S}_{\mathrm{n}}=\mathrm{j} \omega_{\mathrm{n}}-\alpha_{\mathrm{n}} \nonumber
Optical Communications
\mathrm{E}=\mathrm{hf}, \text { photons or phonons } \nonumber
\mathrm{hf} / \mathrm{c}=\text { momentum }\left[\mathrm{kg} \ \mathrm{ms}^{-1}\right] \nonumber
\mathrm{dn}_{2} / \mathrm{dt}=-\left[\mathrm{An}_{2}+\mathrm{B}\left(\mathrm{n}_{2}-\mathrm{n}_{1}\right)\right] \nonumber
Acoustics
\mathrm{P}=\mathrm{P}_{\mathrm{o}}+\mathrm{p}, \ \overrightarrow{\mathrm{U}}=\overrightarrow{\mathrm{U}}_{\mathrm{o}}+\mathrm{u} \ \ \left(\overrightarrow{\mathrm{U}}_{\mathrm{o}}=0 \text { here }\right) \nonumber
\nabla \mathrm{p}=-\rho_{\mathrm{o}} \partial \overrightarrow{\mathrm{u}} / \partial \mathrm{t} \nonumber
\nabla \bullet \overrightarrow{\mathrm{u}}=-\left(1 / \gamma \mathrm{P}_{\mathrm{o}}\right) \partial \mathrm{p} / \partial \mathrm{t} \nonumber
\left(\nabla^{2}-\mathrm{k}^{2} \partial^{2} / \partial \mathrm{t}^{2}\right) \mathrm{p}=0 \nonumber
\mathrm{k}^{2}=\omega^{2} / \mathrm{c}_{\mathrm{s}}^{2}=\omega^{2} \rho_{\mathrm{o}} / \gamma \mathrm{P}_{\mathrm{o}} \nonumber
\mathrm{c}_{\mathrm{s}}=\mathrm{v}_{\mathrm{p}}=\mathrm{v}_{\mathrm{g}}=\left(\gamma \mathrm{P}_{\mathrm{o}} / \rho_{\mathrm{o}}\right)^{0.5} \text { or }\left(\mathrm{K} / \rho_{\mathrm{o}}\right)^{0.5} \nonumber
\eta_{\mathrm{s}}=\mathrm{p} / \mathrm{u}=\rho_{\mathrm{o}} \mathrm{c}_{\mathrm{s}}=\left(\rho_{\mathrm{o}} \gamma \mathrm{P}_{\mathrm{o}}\right)^{0.5} \text { gases } \nonumber
\eta_{\mathrm{s}}=\left(\rho_{\mathrm{o}} \mathrm{K}\right)^{0.5} \text { solids, liquids } \nonumber
\mathrm{p}, \overrightarrow{\mathrm{u}}_{\perp} \text { continuous at boundaries } \nonumber
\mathrm{\underline p}=\mathrm{\underline p}_{+} \mathrm{e}^{-\mathrm{jkz}}+\mathrm{\underline p}_{-} \mathrm{e}^{+\mathrm{jkz}} \nonumber
\mathrm{\underline{ u}_{z}=\eta_{s}^{-1}\left(\underline{p}_{+} e^{-j k z}-\underline{p}_{-}e^{+j k z}\right)} \nonumber
\mathrm{\int_{A} \overrightarrow{u} p \bullet d \overrightarrow{a}+(d / d t) \int_{V}\left(\rho_{o}|\overrightarrow{u}|^{2} / 2+p^{2} / 2 \gamma P_{o}\right) d V} \nonumber
Mathematical Identities
\sin ^{2} \theta+\cos ^{2} \theta=1 \nonumber
\cos \alpha+\cos \beta=2 \cos [(\alpha+\beta) / 2] \cos [(\alpha-\beta) / 2] \nonumber
\mathrm{\underline{H}(f)=\int_{-\infty}^{+\infty} h(t) e^{-j \omega t} d t} \nonumber
\mathrm{e^{x}=1+x+x^{2} / 2 !+x^{3} / 3 !+\ldots} \nonumber
\mathrm{\sin \alpha=\left(e^{j \alpha}-e^{-j \alpha}\right) / 2 j} \nonumber
\mathrm{\cos \alpha=\left(e^{j \alpha}+e^{-j \alpha}\right) / 2} \nonumber
Vector Algebra
\nabla=\mathrm{\hat{x} \partial / \partial x+\hat{y} \partial / \partial y+\hat{z} \partial / \partial z } \nonumber
\mathrm{\bar{A} \bullet \bar{B}=A_{x} B_{x}+A_{y} B_{y}+A_{z} B_{z}} \nonumber
\mathrm{\nabla^{2} \phi=\left(\partial^{2} / \partial x^{2}+\partial^{2} / \partial y^{2}+\partial^{2} / \partial z^{2}\right) \phi} \nonumber
\nabla \bullet (\nabla \times \bar{\mathrm A})=0 \nonumber
\nabla \times(\nabla \times \mathrm{ \bar{A})=\nabla(\nabla \bullet \bar{A})-\nabla^{2} \bar{A}} \nonumber
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