14.4: Basic Equations for Electromagnetics and Applications

Fundamentals

\[\overline{\mathrm{f}}=\mathrm{q}\left(\overline{\mathrm{E}}+\overline{\mathrm{v}} \times \mu_{\mathrm{o}} \overline{\mathrm{H}}\right)[\mathrm{N}] \nonumber\]

\[\nabla \times \overline{\mathrm{E}}=-\partial \overline{\mathrm{B}} / \partial \mathrm{t} \nonumber\]

\[\oint_{\mathrm{c}} \overline{\mathrm{E}} \bullet \mathrm{d} \overline{\mathrm{s}}=-\frac{\mathrm{d}}{\mathrm{dt}} \int_{\mathrm{A}} \overline{\mathrm{B}} \bullet \mathrm{d} \overline{\mathrm{a}} \nonumber\]

\[\nabla \times \overline{\mathrm{H}}=\overline{\mathrm{J}}+\partial \overline{\mathrm{D}} / \partial \mathrm{t} \nonumber\]

\[\oint_{\mathrm{c}} \overline{\mathrm{H}} \bullet \mathrm{d} \overline{\mathrm{s}}=\int_{\mathrm{A}} \overline{\mathrm{J}} \bullet \mathrm{d} \overline{\mathrm{a}}+\frac{\mathrm{d}}{\mathrm{dt}} \int_{\mathrm{A}} \overline{\mathrm{D}} \bullet \mathrm{d} \overline{\mathrm{a}} \nonumber\]

\[\nabla \bullet \overline{\mathrm{D}}=\rho \rightarrow \oint_{\mathrm{A}} \overline{\mathrm{D}} \bullet \mathrm{d} \overline{\mathrm{a}}=\int_{\mathrm{V}} \rho \mathrm{d} \mathrm{v} \nonumber\]

\[\nabla \bullet \overline{\mathrm{B}}=0 \rightarrow \oint_{\mathrm{A}} \overline{\mathrm{B}} \bullet \mathrm{d} \overline{\mathrm{a}}=0 \nonumber\]

\[\nabla \bullet \overline{\mathrm{J}}=-\partial \rho / \partial \mathrm{t} \nonumber\]

\[ \overline{\mathrm{E}}=\text { electric field }\left(\mathrm{Vm}^{-1}\right) \nonumber \]

\[ \overline{\mathrm{H}}=\text { magnetic field }\left(\mathrm{Am}^{-1}\right) \nonumber \]

\[ \overline{\mathrm{D}}=\text { electric displacement }\left(\mathrm{Cm}^{-2}\right) \nonumber \]

\[\overline{\mathrm{B}}=\text { magnetic flux density (T) } \text {Tesla }(\mathrm{T})=\text { Weber } \mathrm{m}^{-2}=10,000 \text { gauss } \nonumber \]

\[ \rho=\text { charge density }\left(\mathrm{Cm}^{-3}\right) \nonumber \]

\[\overline{\mathrm{J}}=\text { current density }\left(\mathrm{Am}^{-2}\right) \nonumber \]

\[ \sigma=\text { conductivity (Siemens } \left.\mathrm{m}^{-1}\right) \nonumber \]

\[ \overline{\mathrm{J}}_{\mathrm{s}}=\text { surface current density }\left(\mathrm{Am}^{-1}\right) \nonumber \]

\[\rho_{\mathrm{s}}=\text { surface charge density }\left(\mathrm{Cm}^{-2}\right) \varepsilon_{0}=8.85 \times 10^{-12} \ \mathrm{Fm}^{-1} \nonumber \]

\[\mu_{\mathrm{o}}=4 \pi \times 10^{-7} \ \mathrm{Hm}^{-1} \nonumber \]

\[ \mathrm{c}=\left(\varepsilon_{\mathrm{o}} \mu_{\mathrm{o}}\right)^{-0.5} \cong 3 \times 10^{8} \ \mathrm{ms}^{-1} \nonumber \]

\[ \mathrm{e}=-1.60 \times 10^{-19} \ \mathrm{C} \nonumber \]

\[ \mathrm{E}_{\mathrm{y}}(\mathrm{z}, \mathrm{t})=\mathrm{E}_{+}(\mathrm{z}-\mathrm{ct})+\mathrm{E}_{-}(\mathrm{z}+\mathrm{ct})=\mathrm{R}_{\mathrm{e}}\left\{\mathrm{\underline E}_{\mathrm{y}}(\mathrm{z}) \mathrm{e}^{\mathrm{j} \omega \mathrm{t}}\right\} \nonumber \]

\[\mathrm{H}_{\mathrm{x}}(\mathrm{z}, \mathrm{t})=\eta_{\mathrm{o}}^{-1}\left[\mathrm{E}_{+}(\mathrm{z}-\mathrm{ct})-\mathrm{E}_{-}(\mathrm{z}+\mathrm{ct})\right] \ [\text { or }(\omega \mathrm{t}-\mathrm{kz}) \text { or }(\mathrm{t}-\mathrm{z} / \mathrm{c})] \nonumber \]

\[ \int_{\mathrm{A}}(\overline{\mathrm{E}} \times \overline{\mathrm{H}}) \bullet \mathrm{d} \overline{\mathrm{a}}+(\mathrm{d} / \mathrm{d} \mathrm{t}) \int_{\mathrm{V}}\left(\varepsilon|\overline{\mathrm{E}}|^{2} / 2+\mu|\overline{\mathrm{H}}|^{2} / 2\right) \mathrm{d} \mathrm{v} =-\int_{\mathrm{V}} \overline{\mathrm{E}} \bullet \overline{\mathrm{J}} \ \mathrm{d} \mathrm{v} \text { (Poynting Theorem) } \nonumber \]

Media and Boundaries

\[\overline{\mathrm{D}}=\varepsilon_{\mathrm{o}} \overline{\mathrm{E}}+\overline{\mathrm{P}} \nonumber \]

\[ \nabla \bullet \overline{\mathrm{D}}=\rho_{\mathrm{f}}, \ \tau=\varepsilon / \sigma \nonumber \]

\[ \nabla \bullet \varepsilon_{\mathrm{o}} \overline{\mathrm{E}}=\rho_{\mathrm{f}}+\rho_{\mathrm{p}} \nonumber \]

\[ \nabla \bullet \overline{\mathrm{P}}=-\rho_{\mathrm{p}}, \ \overline{\mathrm{J}}=\sigma \overline{\mathrm{E}} \nonumber \]

\[ \overline{\mathrm{B}}=\mu \overline{\mathrm{H}}=\mu_{\mathrm{o}}(\overline{\mathrm{H}}+\overline{\mathrm{M}}) \nonumber \]

\[ \varepsilon=\varepsilon_{\mathrm{o}}\left(1-\omega_{\mathrm{p}}^{2} / \omega^{2}\right) \nonumber \]

\[\omega_{\mathrm{p}}=\left(\mathrm{Ne}^{2} / \mathrm{m} \varepsilon_{\mathrm{o}}\right)^{0.5} \nonumber \]

\[ \varepsilon_{\mathrm{eff}}=\varepsilon(1-\mathrm{j} \sigma / \omega \varepsilon) \nonumber \]

Electromagnetic Quasistatics

\[\nabla^{2} \Phi=0 \nonumber\]

\[\mathrm{KCL}: \sum_{\mathrm{i}} \mathrm{I}_{\mathrm{i}}(\mathrm{t})=0 \text { at node } \nonumber\]

\[\mathrm{KVL}: \sum_{\mathrm{i}} \mathrm{V}_{\mathrm{i}}(\mathrm{t})=0 \text { around loop } \nonumber\]

\[\mathrm{C}=\mathrm{Q} / \mathrm{V}=\mathrm{A} \varepsilon / \mathrm{d}[\mathrm{F}] \nonumber\]

\[\mathrm{L}=\Lambda / \mathrm{I} \nonumber\]

\[\mathrm{i}(\mathrm{t})=\mathrm{C} \mathrm{dv}(\mathrm{t}) / \mathrm{dt} \nonumber\]

\[\mathrm{v}(\mathrm{t})=\mathrm{L} \operatorname{di}(\mathrm{t}) / \mathrm{dt}=\mathrm{d} \Lambda / \mathrm{dt} \nonumber\]

\[ \mathrm{C}_{\text {parallel }}=\mathrm{C}_{1}+\mathrm{C}_{2} \nonumber\]

\[\mathrm{C}_{\text {series }}=\left(\mathrm{C}_{1}^{-1}+\mathrm{C}_{2}^{-1}\right)^{-1} \nonumber\]

\[ \mathrm{w}_{\mathrm{e}}=\mathrm{Cv}^{2}(\mathrm{t}) / 2 ; \mathrm{w}_{\mathrm{m}}=\mathrm{Li}^{2}(\mathrm{t}) / 2 \nonumber\]

\[ \mathrm{L}_{\text {solenoid }}=\mathrm{N}^{2} \mu \mathrm{A} / \mathrm{W} \nonumber\]

\[\tau=\mathrm{RC}, \tau=\mathrm{L} / \mathrm{R} \nonumber\]

\[ \Lambda=\int_{\mathrm{A}} \overline{\mathrm{B}} \bullet \mathrm{d \bar{a}}\text { (per turn) } \nonumber\]

\[\overline{\mathrm{f}}=\mathrm{q}\left(\overline{\mathrm{E}}+\overline{\mathrm{v}} \times \mu_{\mathrm{o}} \overline{\mathrm{H}}\right)[\mathrm{N}] \nonumber\]

\[ \mathrm{f}_{\mathrm{z}}=-\mathrm{d} \mathrm{w}_{\mathrm{T}} / \mathrm{dz} \nonumber\]

\[ \overline{\mathrm{F}}=\overline{\mathrm{I}} \times \mu_{\mathrm{o}} \overline{\mathrm{H}}\left[\mathrm{Nm}^{-1}\right] \nonumber\]

\[\overline{\mathrm{E}}_{\mathrm{e}}=-\overline{\mathrm{v}} \times \mu_{\mathrm{o}} \overline{\mathrm{H}} \ \text { inside wire } \nonumber\]

\[ \mathrm{P}=\omega \mathrm{T}=\mathrm{W}_{\mathrm{T}} \mathrm{d} \mathrm{V}_{\text {olume }} / \mathrm{dt}[\mathrm{W}] \nonumber\]

\[ \text {Max } \mathrm{f} / \mathrm{A}=\mathrm{B}^{2} / 2 \mu_{\mathrm{o}}, \mathrm{D}^{2} / 2 \varepsilon_{\mathrm{o}}\left[\mathrm{Nm}^{-2}\right] \nonumber\]

\[\text {vi }=\frac{\mathrm{d} \mathrm{w}_{\mathrm{T}}}{\mathrm{dt}}+\mathrm{f} \frac{\mathrm{d} \mathrm{z}}{\mathrm{dt}} \nonumber\]

Electromagnetic Waves

\[\left(\nabla^{2}-\mu \varepsilon \partial^{2} / \partial \mathrm{t}^{2}\right) \overline{\mathrm{E}}=0 \ [\text { Wave } \mathrm{Eqn} .] \nonumber \]

\[\left(\nabla^{2}+\mathrm{k}^{2}\right) \overline{\mathrm{\underline E}}=0, \overline{\mathrm{\underline E}}=\overline{\mathrm{\underline E}}_{0} \mathrm{e}^{-\mathrm{j} \overline{\mathrm{k}} \bullet \mathrm{r}} \nonumber \]

\[\mathrm{k}=\omega(\mu \varepsilon)^{0.5}=\omega / \mathrm{c}=2 \pi / \lambda \nonumber \]

\[ \mathrm{k}_{\mathrm{x}}^{2}+\mathrm{k}_{\mathrm{y}}^{2}+\mathrm{k}_{\mathrm{z}}^{2}=\mathrm{k}_{\mathrm{o}}^{2}=\omega^{2} \mu \varepsilon \nonumber \]

\[ \mathrm{v}_{\mathrm{p}}=\omega / \mathrm{k}, \ \mathrm{v}_{\mathrm{g}}=(\partial \mathrm{k} / \partial \omega)^{-1} \nonumber \]

\[ \theta_{\mathrm{r}}=\theta_{\mathrm{i}} \nonumber \]

\[\sin \theta_{\mathrm{t}} / \sin \theta_{\mathrm{i}}=\mathrm{k}_{\mathrm{i}} / \mathrm{k}_{\mathrm{t}}=\mathrm{n}_{\mathrm{i}} / \mathrm{n}_{\mathrm{t}} \nonumber \]

\[\theta_{\mathrm{c}}=\sin ^{-1}\left(\mathrm{n}_{\mathrm{t}} / \mathrm{n}_{\mathrm{i}}\right) \nonumber \]

\[ \theta>\theta_{\mathrm{c}} \Rightarrow \overline{\mathrm{\underline E}}_{\mathrm{t}}=\overline{\mathrm{\underline E}}_{\mathrm{i}} \mathrm{\underline{T}e}^{+\alpha x-\mathrm{j} \mathrm{k}_{\mathrm{z}} \mathrm{z}} \nonumber \]

\[\overline{\mathrm{\underline k}}=\overline{\mathrm{k}}^{\prime}-\mathrm{j} \overline{\mathrm{k}}^{\prime \prime} \nonumber \]

\[ \underline{\Gamma}=\underline{\mathrm{T}}-1 \nonumber \]

\[ \underline{\mathrm{T}}_{\mathrm{TE}}=2 /\left(1+\left[\eta_{\mathrm{o}} \cos \theta_{\mathrm{t}} / \eta_{\mathrm{t}} \cos \theta_{\mathrm{i}}\right]\right) \nonumber \]

\[\underline{\mathrm{T}}_{\mathrm{TM}}=2 /\left(1+\left[\eta_{\mathrm{t}} \cos \theta_{\mathrm{t}} / \eta_{\mathrm{i}} \cos \theta_{\mathrm{i}}\right]\right) \nonumber \]

\[\theta_{\mathrm{B}}=\tan ^{-1}\left(\varepsilon_{\mathrm{t}} / \varepsilon_{\mathrm{i}}\right)^{0.5} \text { for } \mathrm{TM} \nonumber \]

\[ \mathrm{P}_{\mathrm{d}} \cong\left|\overline{\mathrm{\underline J}}_{\mathrm{S}}\right|^{2} / 2 \sigma \delta \ \left[\mathrm{Wm}^{-2}\right] \nonumber \]

\[ \overline{\mathrm{E}}=-\nabla \phi-\partial \overline{\mathrm{A}} / \partial \mathrm{t}, \quad \overline{\mathrm{B}}=\nabla \times \overline{\mathrm{A}} \nonumber \]

\[\underline{\Phi}(\mathrm{r})=\int_{\mathrm{V'}}\left(\underline{\rho}(\overline{\mathrm{r}}) \mathrm{e}^{-\mathrm{j} \mathrm{k}\left|\overline{\mathrm{r}}^{\prime}-\overline{\mathrm{r}}\right|} \Big/ 4 \pi \varepsilon_{\mathrm{o}}\left|\overline{\mathrm{r}}^{\prime}-\overline{\mathrm{r}}\right|\right) \mathrm{d} \mathrm{v}^{\prime} \nonumber \]

\[ \overline{\mathrm{\underline{ A}}}(\mathrm{r})=\int_{\mathrm{V'}} \mu_{\mathrm o}\left(\underline{\mathrm {\overline {J}}}(\overline{\mathrm{r}}) \mathrm{e}^{-\mathrm{j} \mathrm{k}\left|\overline{\mathrm{r}}^{\prime}-\overline{\mathrm{r}}\right|} \Big/ 4 \pi \left|\overline{\mathrm{r}}^{\prime}-\overline{\mathrm{r}}\right|\right) \mathrm{d} \mathrm{v}^{\prime} \nonumber \]

\[ \overline{\mathrm{\underline E}}_{\mathrm{ff}}=\hat{\vartheta}\left(\mathrm{j} \eta_{\mathrm{o}} \mathrm{k} \underline{\mathrm{I}} \mathrm{d} / 4 \pi \mathrm{r}\right) \mathrm{e}^{-\mathrm{j} \mathrm{kr}} \sin \theta \nonumber \]

\[ \nabla^{2} \underline{\Phi} +\omega^{2} \mu_{0} \varepsilon_{0} \underline{\Phi}=-\rho / \varepsilon_{0} \nonumber \]

\[ \nabla^{2} \overline{\mathrm{\underline A}}+\omega^{2} \mu_{\mathrm{o}} \varepsilon_{\mathrm{o}} \overline{\mathrm{\underline A}}=-\mu_{\mathrm{o}} \overline{\mathrm{\underline J}} \nonumber \]

Forces, Motors, and Generators

\[ \overline{\mathrm{f}}=\mathrm{q}\left(\overline{\mathrm{E}}+\overline{\mathrm{v}} \times \mu_{\mathrm{o}} \overline{\mathrm{H}}\right)[\mathrm{N}] \nonumber\]

\[\mathrm{f}_{\mathrm{z}}=-\mathrm{d} \mathrm{w}_{\mathrm{T}} / \mathrm{d} \mathrm{z} \nonumber\]

\[ \overline{\mathrm{F}}=\overline{\mathrm{I}} \times \mu_{\mathrm{o}} \overline{\mathrm{H}}\left[\mathrm{Nm}^{-1}\right] \nonumber\]

\[ \overline{\mathrm{E}}_{\mathrm{e}}=-\overline{\mathrm{v}} \times \mu_{\mathrm{o}} \overline{\mathrm{H}} \ \text { inside wire } \nonumber\]

\[ \mathrm{P}=\omega \mathrm{T}=\mathrm{W}_{\mathrm{T}} \mathrm{d} \mathrm{V}_{\text {olume }} / \mathrm{dt}[\mathrm{W}] \nonumber\]

\[ \text {Max } \mathrm{f} / \mathrm{A}=\mathrm{B}^{2} / 2 \mu_{\mathrm{o}}, \ \mathrm{D}^{2} / 2 \varepsilon_{\mathrm{o}}\left[\mathrm{Nm}^{-2}\right] \nonumber\]

\[ \mathrm{vi}=\frac{\mathrm{d} \mathrm{w}_{\mathrm{T}}}{\mathrm{dt}}+\mathrm{f} \frac{\mathrm{d} \mathrm{z}}{\mathrm{dt}} \nonumber\]

\[ \mathrm{f}=\mathrm{ma}=\mathrm{d}(\mathrm{mv}) / \mathrm{dt} \nonumber\]

\[ \mathrm{x}=\mathrm{x}_{\mathrm{o}}+\mathrm{v}_{\mathrm{o}} \mathrm{t}+\mathrm{at}^{2} / 2 \nonumber\]

\[ \mathrm{P}=\mathrm{fv} \ [\mathrm{W}]=\mathrm{T} \omega \nonumber\]

\[ \mathrm{w}_{\mathrm{k}}=\mathrm{mv}^{2} / 2 \nonumber\]

\[ \mathrm{T}=\mathrm{I} \ \mathrm{d} \omega / \mathrm{dt} \nonumber\]

\[ \mathrm{I}=\sum_{\mathrm{i}} \mathrm{m}_{\mathrm{i}} \mathrm{r}_{\mathrm{i}}^{2} \nonumber\]

Circuits

\[ \mathrm{KCL}: \sum_{\mathrm{i}} \mathrm{I}_{\mathrm{i}}(\mathrm{t})=0 \text { at node } \nonumber \]

\[ \mathrm{KVL}: \sum_{\mathrm{i}} \mathrm{V}_{\mathrm{i}}(\mathrm{t})=0 \text { around loop } \nonumber \]

\[ \mathrm{C}=\mathrm{Q} / \mathrm{V}=\mathrm{A} \varepsilon / \mathrm{d}[\mathrm{F}] \nonumber \]

\[ \mathrm{L}=\Lambda / \mathrm{I} \nonumber \]

\[ \mathrm{i}(\mathrm{t})=\mathrm{C} \ \mathrm{dv}(\mathrm{t}) / \mathrm{d} \mathrm{t} \nonumber \]

\[ \mathrm{v}(\mathrm{t})=\mathrm{L} \ \mathrm{di}(\mathrm{t}) / \mathrm{dt}=\mathrm{d} \Lambda / \mathrm{dt} \nonumber \]

\[ \mathrm{C}_{\text {parallel }}=\mathrm{C}_{1}+\mathrm{C}_{2} \nonumber \]

\[\mathrm{C}_{\text {series }}=\left(\mathrm{C}_{1}^{-1}+\mathrm{C}_{2}^{-1}\right)^{-1} \nonumber \]

\[\mathrm{w}_{\mathrm{e}}=\mathrm{Cv}^{2}(\mathrm{t}) / 2 ; \ \mathrm{w}_{\mathrm{m}}=\mathrm{Li}^{2}(\mathrm{t}) / 2 \nonumber \]

\[ \mathrm{L}_{\text {solenoid }}=\mathrm{N}^{2} \mu \mathrm{A} / \mathrm{W} \nonumber \]

\[ \tau=\mathrm{RC}, \tau=\mathrm{L} / \mathrm{R} \nonumber \]

\[ \Lambda=\int_{\mathrm{A}} \overline{\mathrm{B}} \bullet \mathrm{d \overline{a}} \text { (per turn) } \nonumber \]

\[ \underline{\mathrm{Z}}_\text { series}=\mathrm{R}+\mathrm{j} \omega \mathrm{L}+1 / \mathrm{j} \omega \mathrm{C} \nonumber \]

\[ \mathrm{\underline{Y}_{\text {par }}=G+j \omega C+1 / j \omega L} \nonumber \]

\[ \mathrm{Q=\omega_{o} w_{T} / P_{d i s s}=\omega_{o} / \Delta \omega} \nonumber \]

\[ \mathrm{\omega_{o}=(L C)^{-0.5} } \nonumber \]

\[ \mathrm{\left\langle v^{2}(t)\right\rangle / R=k T} \nonumber \]

Limits to Computation Speed

\[\operatorname{dv}(\mathrm{z}) / \mathrm{dz}=-\operatorname{Ldi}(\mathrm{z}) / \mathrm{dt} \nonumber \]

\[\operatorname{di}(\mathrm{z}) / \mathrm{dz}=-\operatorname{Cdv}(\mathrm{z}) / \mathrm{dt} \nonumber \]

\[ \mathrm{d}^{2} \mathrm{v} / \mathrm{dz}^{2}=\mathrm{LC} \ \mathrm{d}^{2} \mathrm{v} / \mathrm{dt}^{2} \nonumber \]

\[\begin{aligned} \mathrm{v(z, t)} &=\mathrm{f_{+}(t-z / c)+f_{-}(t+z / c)} \\ &=\mathrm{g_{+}(z-c t)+g_{-}(z-c t)} \end{aligned}\]

\[\mathrm{i}(\mathrm{t}, \mathrm{z})=\mathrm{Y}_{\mathrm{o}}\left[\mathrm{f}_{+}(\mathrm{t}-\mathrm{z} / \mathrm{c})-\mathrm{f}_{-}(\mathrm{t}+\mathrm{z} / \mathrm{c})\right] \nonumber \]

\[ \mathrm{c=(L C)^{-0.5}=1 / \sqrt{\mu \varepsilon}} \nonumber \]

\[\mathrm Z_{\mathrm{o}}=\mathrm{Y}_{\mathrm{o}}^{-1}=(\mathrm{L} / \mathrm{C})^{0.5} \nonumber \]

\[\Gamma_{\mathrm{L}}=\mathrm{f} / \mathrm{f}_{+}=\left(\mathrm{R}_{\mathrm{L}}-\mathrm{Z}_{\mathrm{o}}\right) /\left(\mathrm{R}_{\mathrm{L}}+\mathrm{Z}_{\mathrm{o}}\right) \nonumber \]

\[\mathrm{v(z, t)=g_{+}(z-c t)+g_{-}(z+c t)} \nonumber \]

\[\mathrm{V}_{\mathrm{Th}}=2 \mathrm{f}_{+}(\mathrm{t}), \ \mathrm{R}_{\mathrm{Th}}=\mathrm{Z}_{\mathrm{o}} \nonumber \]

Power Transmission

\[\left(\mathrm{d}^{2} / \mathrm{dz}^{2}+\omega^{2} \mathrm{LC}\right) \mathrm{\underline V}(\mathrm{z})=0 \nonumber \]

\[\underline{\mathrm{V}}(\mathrm z)=\underline{\mathrm V}_{+} \mathrm{e}^{-\mathrm{jkz}}+\underline{\mathrm{V}}_{-} \mathrm{e}^{+\mathrm{j} \mathrm{k} \mathrm{z}} \nonumber \]

\[\mathrm{\underline{I}(z)=Y_{o}\left[\underline{V}_{+}e^{-j k z}-\underline{V}_{-} e^{+j k z}\right]} \nonumber \]

\[\mathrm{k}=2 \pi / \lambda=\omega / \mathrm{c}=\omega(\mu \varepsilon)^{0.5} \nonumber \]

\[\mathrm{\underline{Z}(z)=\underline{V}(z) / \underline{I}(z)=Z_{o} \underline{Z}_{n}(z)} \nonumber \]

\[\underline{\mathrm Z}_{\mathrm{n}}(\mathrm{z})=[1+\underline{\Gamma}(\mathrm{z})] /[1-\underline{\Gamma}(\mathrm{z})]=\mathrm{R}_{\mathrm{n}}+\mathrm{j} \mathrm{X}_{\mathrm{n}} \nonumber \]

\[\underline{\Gamma}(\mathrm{z})=\left(\underline{\mathrm{V}}-\underline{\mathrm{V}}_{+}\right) \mathrm{e}^{2 \ \mathrm{jkz}}=\left[\underline{\mathrm{Z}}_{\mathrm{n}}(\mathrm{z})-1\right] /\left[\underline{\mathrm{Z}}_{\mathrm{n}}(\mathrm{z})+1\right] \nonumber \]

\[\mathrm{\underline{Z}(z)=Z_{0}\left(\underline{Z}_{L}-j Z_{0} \tan k z\right) /\left(\underline{Z}_{0}-j Z_{L} \tan k z\right)} \nonumber \]

\[\mathrm{VSWR}=\left|\underline{\mathrm{V}}_{\max }\right| \big/\left|\underline{\mathrm{V}}_{\min }\right|=\mathrm{R}_{\max } \nonumber \]

Wireless Communications and Radar

\[\mathrm{G}(\theta, \phi)=\mathrm{P}_{\mathrm{r}} /\left(\mathrm{P}_{\mathrm{R}} / 4 \pi \mathrm{r}^{2}\right) \nonumber \]

\[\mathrm{P_{R}=\int_{4 \pi} P_{r}(\theta, \phi, r) r^{2} \sin \theta \ d \theta d \phi} \nonumber \]

\[\mathrm P_{\mathrm{rec}}=\mathrm{P}_{\mathrm{r}}(\theta, \phi) \mathrm{A}_{\mathrm{e}}(\theta, \phi) \nonumber \]

\[\mathrm{A}_{\mathrm{e}}(\theta, \phi)=\mathrm{G}(\theta, \phi) \lambda^{2} / 4 \pi \nonumber \]

\[\mathrm{R}_{\mathrm{r}}=\mathrm{P}_{\mathrm{R}} \big/\left\langle\mathrm{i}^{2}(\mathrm{t})\right\rangle \nonumber \]

\[\mathrm{E}_{\mathrm{ff}}(\theta \cong 0)=\left(\mathrm{je}^{\mathrm{jkr}} / \lambda \mathrm{r}\right) \int_{\mathrm{A}} \mathrm{E}_{\mathrm{t}}(\mathrm{x}, \mathrm{y}) \mathrm{e}^{\mathrm{jk}_{\mathrm{x}}{\mathrm{x}+\mathrm{jk}}_{ \mathrm{y}}{\mathrm{y}}} \mathrm{dxdy} \nonumber \]

\[\mathrm P_{\mathrm{rec}}=\mathrm{P}_{\mathrm{R}}\left(\mathrm{G} \lambda / 4 \pi \mathrm{r}^{2}\right)^{2} \sigma_{\mathrm{s}} / 4 \pi \nonumber \]

\[\underline{\mathrm{\overline E}}=\sum_{\mathrm{i}} \mathrm{a}_{\mathrm{i}} \overline{\mathrm{\underline E}}_{\mathrm{i}} \mathrm{e}^{-\mathrm{jk} \mathrm{r}_{1}}=(\text { element factor })(\text { array } \mathrm{f}) \nonumber \]

\[\mathrm{E}_{\text {bit }} \geq \sim 4 \times 10^{-20}[\mathrm{J}] \nonumber \]

\[\underline{\mathrm Z}_{12}=\underline{\mathrm Z}_{21} \text { if reciprocity } \nonumber \]

\[\left(\mathrm{d}^{2} / \mathrm{d} \mathrm z^{2}+\omega^{2} \mathrm{LC}\right) \underline{\mathrm{V}}(\mathrm{z})=0 \nonumber \]

\[\mathrm{\underline V}(\mathrm{z})=\underline{\mathrm{V}}_{+}\mathrm{e}^{-\mathrm{jk} \mathrm{z}}+\underline{\mathrm{V}}_{-} \mathrm{e}^{+\mathrm{jk} \mathrm{z}} \nonumber \]

\[\mathrm{\underline{I}(z)=Y_{o}\left[\underline{V}_{+}e^{-j k z}-\underline{V}_{-} e^{+j k z}\right]} \nonumber \]

\[\mathrm{k}=2 \pi / \lambda=\omega / \mathrm{c}=\omega(\mu \varepsilon)^{0.5} \nonumber \]

\[\mathrm{\underline{Z}(z)=\underline{V}(z) / \underline{I}(z)=Z_{o} \underline{Z}_{n}(z)} \nonumber \]

\[\mathrm{\underline{Z}_{n}(z)=[1+\underline{\Gamma}(z)]/[1-\underline{\Gamma}(z)]=R_{n}+j X_{n}} \nonumber \]

\[\underline{\Gamma}(\mathrm{z})=\left(\underline{\mathrm{V}}_{-}/\underline{\mathrm{V}}_{+}\right) \mathrm{e}^{2 \mathrm{jkz}}=\left[\underline{\mathrm{Z}}_{\mathrm{n}}(\mathrm{z})-1\right]/\left[\mathrm{\underline Z}_{\mathrm{n}}(\mathrm{z})+1\right] \nonumber \]

\[\mathrm{\underline{Z}(z)=Z_{0}\left(\underline{Z}_{L}-j Z_{0} \tan k z\right) /\left(\underline{Z}_{0}-j Z_{L} \tan k z\right)} \nonumber \]

\[\mathrm{VSWR}=\left|\underline{\mathrm{V}}_{\max }\right| /\left|\underline{\mathrm{V}}_{\min }\right|=\mathrm{R}_{\max } \nonumber \]

\[\theta_{\mathrm{r}}=\theta_{\mathrm{i}} \nonumber \]

\[\sin \theta_{\mathrm{t}} / \sin \theta_{\mathrm{i}}=\mathrm{k}_{\mathrm{i}} / \mathrm{k}_{\mathrm{t}}=\mathrm{n}_{\mathrm{i}} / \mathrm{n}_{\mathrm{t}} \nonumber \]

\[\theta_{\mathrm{c}}=\sin ^{-1}\left(\mathrm{n}_{\mathrm{t}} / \mathrm{n}_{\mathrm{i}}\right) \nonumber \]

\[\theta>\theta_{\mathrm{c}} \Rightarrow \overline{\mathrm{\underline E}}_{\mathrm{t}}=\overline{\mathrm{\underline E}}_{\mathrm{i}} \mathrm{\underline{T}e}^{+\alpha x-\mathrm{j} \mathrm{k}_{\mathrm{z}^{\mathrm{z}}}} \nonumber \]

\[\underline{\mathrm{\overline k}}=\overline{\mathrm{k}}^{\prime}-\mathrm{j} \overline{\mathrm{k}}^{\prime \prime} \nonumber \]

\[\underline{\Gamma}=\underline{\mathrm{T}}-1 \nonumber \]

\[\operatorname{At} \omega_{0}, \ \left\langle\mathrm{w}_{\mathrm{e}}\right\rangle=\left\langle\mathrm{w}_{\mathrm{m}}\right\rangle \nonumber \]

\[\left\langle\mathrm{w}_{\mathrm{e}}\right\rangle=\int_{\mathrm{V}}\left(\varepsilon|\overline{\mathrm{\underline E}}|^{2} \big/ 4\right) \mathrm{d} \mathrm{v} \nonumber \]

\[\left\langle\mathrm{w}_{\mathrm{m}}\right\rangle=\int_{\mathrm{V}}\left(\mu|\overline{\mathrm{\underline H}}|^{2} \big/ 4\right) \mathrm{d} \mathrm{v} \nonumber \]

\[\mathrm{Q}_{\mathrm{n}}=\omega_{\mathrm{n}} \mathrm{W}_{\mathrm{Tn}} / \mathrm{P}_{\mathrm{n}}=\omega_{\mathrm{n}} / 2 \alpha_{\mathrm{n}} \nonumber \]

\[\mathrm{f}_{\mathrm{mnp}}=(\mathrm{c} / 2)\left([\mathrm{m} / \mathrm{a}]^{2}+[\mathrm{n} / \mathrm{b}]^{2}+[\mathrm{p} / \mathrm{d}]^{2}\right)^{0.5} \nonumber \]

\[\mathrm{S}_{\mathrm{n}}=\mathrm{j} \omega_{\mathrm{n}}-\alpha_{\mathrm{n}} \nonumber \]

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

\[\mathrm{P}=\mathrm{P}_{\mathrm{o}}+\mathrm{p}, \ \overline{\mathrm{U}}=\overline{\mathrm{U}}_{\mathrm{o}}+\mathrm{u} \ \ \left(\overline{\mathrm{U}}_{\mathrm{o}}=0 \text { here }\right) \nonumber \]

\[\nabla \mathrm{p}=-\rho_{\mathrm{o}} \partial \overline{\mathrm{u}} / \partial \mathrm{t} \nonumber \]

\[\nabla \bullet \overline{\mathrm{u}}=-\left(1 / \gamma \mathrm{P}_{\mathrm{o}}\right) \partial \mathrm{p} / \partial \mathrm{t} \nonumber \]

\[\left(\nabla^{2}-\mathrm{k}^{2} \partial^{2} / \partial \mathrm{t}^{2}\right) \mathrm{p}=0 \nonumber \]

\[\mathrm{k}^{2}=\omega^{2} / \mathrm{c}_{\mathrm{s}}^{2}=\omega^{2} \rho_{\mathrm{o}} / \gamma \mathrm{P}_{\mathrm{o}} \nonumber \]

\[\mathrm{c}_{\mathrm{s}}=\mathrm{v}_{\mathrm{p}}=\mathrm{v}_{\mathrm{g}}=\left(\gamma \mathrm{P}_{\mathrm{o}} / \rho_{\mathrm{o}}\right)^{0.5} \text { or }\left(\mathrm{K} / \rho_{\mathrm{o}}\right)^{0.5} \nonumber\]

\[\eta_{\mathrm{s}}=\mathrm{p} / \mathrm{u}=\rho_{\mathrm{o}} \mathrm{c}_{\mathrm{s}}=\left(\rho_{\mathrm{o}} \gamma \mathrm{P}_{\mathrm{o}}\right)^{0.5} \text { gases } \nonumber \]

\[\eta_{\mathrm{s}}=\left(\rho_{\mathrm{o}} \mathrm{K}\right)^{0.5} \text { solids, liquids } \nonumber \]

\[\mathrm{p}, \overline{\mathrm{u}}_{\perp} \text { continuous at boundaries } \nonumber \]

\[\mathrm{\underline p}=\mathrm{\underline p}_{+} \mathrm{e}^{-\mathrm{jkz}}+\mathrm{\underline p}_{-} \mathrm{e}^{+\mathrm{jkz}} \nonumber \]

\[\mathrm{\underline{ u}_{z}=\eta_{s}^{-1}\left(\underline{p}_{+} e^{-j k z}-\underline{p}_{-}e^{+j k z}\right)} \nonumber \]

\[\mathrm{\int_{A} \overline{u} p \bullet d \overline{a}+(d / d t) \int_{V}\left(\rho_{o}|\overline{u}|^{2} / 2+p^{2} / 2 \gamma P_{o}\right) d V} \nonumber \]

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

\[\oiint_{\mathrm{V}}(\nabla \bullet \overline{\mathrm{G}}) \mathrm{d} \mathrm{v}=\oiint_{\mathrm{A}} \overline{\mathrm{G}} \bullet \mathrm{d} \overline{\mathrm{a}} \nonumber \]

\[\oiint_{\mathrm{A}}(\nabla \times \overline{\mathrm{G}}) \bullet \mathrm{d} \overline{\mathrm{a}}=\oint_{\mathrm{c}} \overline{\mathrm{G}} \bullet \mathrm{d} \overline{\mathrm{s}} \nonumber \]

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

\[\int_{4 \pi} \mathrm{r}^{2} \sin \theta \ \mathrm{d} \theta \mathrm{d} \phi=4 \pi \nonumber \]