# Dragon Notes

UNDER CONSTRUCTION
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# EM: General Analysis Methods

• [Geom.] – identify the coordinate system that best (1) utilizes symmetry, and (2) facilitates integration & differentiation calculations
• [Geom.] – identify which quantities along integration path are (1) constant or (2) varying – including unit vectors
• [Geom.] – distinguish between (1) source, and (2) field quantities in used relations
• [Sym.] – identify any symmetries in (1) source or (2) field distributions in used coord. sys.
• [Calc.] – check whether used relations satisfy their underlying assumptions
• [Calc.] – identify real-time vs. complex functions – using phasor for latter:
•     >$$H_{y}(z,t)=E_{x}(z,t)/η_c$$ is a mistake, as $$η_c$$ is complex; convert to phasor first: $$H_{y}(z)=E_{x}(z)/η_{c}$$
• [BC] – identify all applicable boundary conditions across an interface between two media:

MEDIA

$$\boxed{\mathrm{lossless\ diel.}\mathrm{\to }{\rho }_s=0};\boxed{\mathrm{simple\ med.}\mathrm{\to }\mu ,\epsilon =\mathrm{space\ coord.\ indep.}}$$ $$\boxed{\mathrm{lossy\ med.}\mathrm{\to }\mathrm{\sigma }\mathrm{\neq }\mathrm{0}}\mathrm{;}\boxed{\sigma \neq \mathrm{const.}\mathrm{\to }\rho \neq 0};\boxed{\mathrm{source}\ \mathrm{free\ med.}\mathrm{\to }\rho =0,J_{\text{ext}}=0 }$$ $$\boxed{\mathrm{noncond./lossless\ med.\ }\mathrm{\to }\mathrm{\sigma }\mathrm{=0,\ }{\rho }_v=0,J_{int}=0,\alpha =0,\beta =k=\omega \sqrt{\mu \epsilon },{\epsilon }'=\epsilon ,{\epsilon }''=0}$$ $$\boxed{\mathrm{PEC}\mathrm{\to }\sigma =\infty ,\ \eta =0}$$

E/D-fields

$$\underline{{\rho }_s=\mathrm{any:}}\mathrm{\ }\boxed{E_{1t}=E_{2t}}\ \boxed{{\epsilon }_2D_{1t}={\epsilon }_1D_{2t}}\ \boxed{D_{1n}-D_{2n}={\rho }_s}$$ $$\underline{{\rho }_s=0:}\ \ \boxed{D_{1n}=D_{2n}}\ \boxed{E_{1n}=\left(\frac{{\epsilon }_2}{{\epsilon }_1}\right)E_{2n}}\ \boxed{\frac{{\mathrm{tan} \left({\phi }_2\right)\ }}{{\mathrm{tan} \left({\phi }_1\right)\ }}=\frac{{\epsilon }_2}{{\epsilon }_1}}\ \boxed{E_2=E_1{\left[{{\mathrm{sin}}^2 \left({\phi }_1\right)\ }+{\left(\frac{{\epsilon }_1}{{\epsilon }_2}{\mathrm{cos} \left({\phi }_1\right)\ }\right)}^2\right]}^{1/2}}$$ $$\underline{\left[\boldsymbol{\#}\boldsymbol{2}\right]\boldsymbol{=}\mathrm{cond.:}}\boldsymbol{\ }\boxed{E_{1t}=E_{2t}=0}\boldsymbol{\ }\boxed{D_{1n}={\rho }_s}\boldsymbol{\ \ }\boxed{D_{2n}=0}\boldsymbol{\ }\boxed{E_{1n}={\rho }_s/{\epsilon }_1\ }\boldsymbol{\ }$$ $$\underline{{\rho }_v=\mathrm{any}:}\ \ \boxed{{\boldsymbol{\mathrm{\nabla }}}^2V=-\rho /\epsilon };\ \ \underline{{\rho }_v=0:}\ \ \boxed{{\boldsymbol{\mathrm{\nabla }}}^2V=0}$$

J-field

$$\underline{{\rho }_s=\mathrm{any:}}\mathrm{\ }\boxed{J_{1n}=J_{2n}}\ \boxed{J_{1t}/J_{2t}={\sigma }_1/{\sigma }_2}\ \boxed{\frac{{\mathrm{tan} \left({\phi }_2\right)\ }}{{\mathrm{tan} \left({\phi }_1\right)\ }}=\frac{{\sigma }_2}{{\sigma }_1}}\ \boxed{J_2=J_1{\left[{\left(\frac{{\sigma }_2}{{\sigma }_1}{\mathrm{sin} \left({\phi }_1\right)\ }\right)}^2+{{\mathrm{cos}}^2 \left({\phi }_1\right)\ }\right]}^{1/2}}$$ $$\boxed{{\rho }_s=\left({\epsilon }_1\frac{{\sigma }_2}{{\sigma }_1}-{\epsilon }_2\right)E_{2n}=\left({\epsilon }_1-{\epsilon }_2\frac{{\sigma }_1}{{\sigma }_2}\right)E_{1n}},\ \widehat{\boldsymbol{\mathrm{n}}}=\mathrm{outward\ from\ }\left[\boldsymbol{\mathrm{\#}}\boldsymbol{\mathrm{2}}\right]\boldsymbol{;}\boldsymbol{\ }\underline{{\rho }_s=0\mathrm{:}}\ \boxed{{\sigma }_2/{\sigma }_1={\epsilon }_2/{\epsilon }_1}$$

B/H-fields

$$\underline{{\sigma }_1,{\sigma }_2\neq \infty \mathrm{:}}\mathrm{\ }\boxed{B_{1n}=B_{2n}}\ \boxed{H_{1n}=\frac{{\mu }_2}{{\mu }_1}H_{2n}}\ \boxed{H_{1t}=H_{2t}}\ \boxed{B_{1t}=\frac{{\mu }_1}{{\mu }_2}B_{2t}}\ \boxed{\frac{{\mathrm{tan} \left({\phi }_2\right)\ }}{{\mathrm{tan} \left({\phi }_1\right)\ }}=\frac{{\mu }_2}{{\mu }_1}}\ \boxed{J_{sn}=0}$$ $$\boxed{H_2=H_1{\left[{{\mathrm{sin}}^2 \left({\phi }_1\right)\ }+{\left(\frac{{\mu }_1}{{\mu }_2}{\mathrm{cos} \left({\phi }_1\right)\ }\right)}^2\right]}^{1/2}}$$ $$\underline{{\sigma }_2\to \infty :}\ \boxed{{\widehat{\boldsymbol{n}}}_2\times {\overrightarrow{\boldsymbol{H}}}_1={\overrightarrow{\boldsymbol{J}}}_s,\ H_{2t}=0}\boldsymbol{\ \ }\boxed{B_{1n}=0,\ B_{2n}=0}$$

• 1 - $$u_{1x}=\omega/\beta_{1x}=\omega/[\beta_1 \sin{\left(\theta_1 \right)}]$$; $$[λ_{1x}=2π/β_{1x}]$$.     2 - $${\overrightarrow{\boldsymbol{E}}}_1 = 0$$ for all $$x$$ when $$\sin{\left(\beta_1 z \cos{\left(\theta_i \right)}\right)} = 0$$

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