144 lines
4.6 KiB
TeX
144 lines
4.6 KiB
TeX
% DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
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% Version 2, December 2004
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%
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% Copyright (C) 2023 Aidan Sharpe <amsharpe102@pm.me>
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%
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% Everyone is permitted to copy and distribute verbatim or modified
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% copies of this license document, and changing it is allowed as long
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% as the name is changed.
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%
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% DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
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% TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
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%
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% 0. You just DO WHAT THE FUCK YOU WANT TO.
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\documentclass[journal]{IEEEtran}
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\usepackage{graphicx}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{esint}
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\usepackage{physics}
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\DeclareMathOperator\arctanh{arctanh}
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\setlength{\parindent}{0pt}
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\title{EEMAGS Equation Sheet}
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\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}}
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\newcommand{\veps}{\varepsilon}
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\newcommand{\vphi}{\varphi}
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\begin{document}
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\maketitle
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\section{Constants}
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$$\varepsilon_0 = 8.854 \times 10^{-12}$$
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$$\mu_0 = 1.257 \times 10^{-6}$$
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$$c = 3 \times 10^8$$
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$$\eta_0 = \sqrt{{\mu_0 \over \varepsilon_0}} = 377\Omega \approx 120\pi$$
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\section{Fifth's Bullet Points}
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Find $\veps_r$ given $\lambda$ and $f$:
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$$\veps_r = {c^2 \over \mu_r \lambda^2 f^2}$$
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Find $v_p$ given $\delta$, $\alpha$, and $\beta$ given a good conductor:
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$$v_p = {\omega \over \beta}$$
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Find $P_\text{avg}$ given $\vec{H}(t)$ in air:
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Find $\Gamma$ in polar and VWSR given $Z_L$ and $Z_0$:
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$$\Gamma = {Z_L - Z_0 \over Z_L + Z_0}$$
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$$\text{VWSR} = {1 + |\Gamma| \over 1 - |\Gamma|}$$
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Find $\vec{E}(z,t)$ given $\vec{H}(z,t)$ in a lossless medium of $\mu_r$ and $\veps_r$:
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Find $\lambda$, $\veps_r$, and $\vec{H}$ given $\vec{E}$:
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$$\hat{H}_y = {\hat{E}_x \over \eta}$$
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$$\veps_r = {\beta^2 \over \omega^2 \mu_r \mu_0 \veps_0}$$
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$$\lambda = {v_p \over f}$$
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Find $\Gamma^2$ and $\langle P \rangle$ from $\veps$, lossless:
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Find $\beta_\text{air}$, $\beta_\text{material}$, $\hat{\Gamma}$, $\hat{T}$ given $\vec{E}^i$, $\veps_r$, $\mu_r$, $\sigma_r$:
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Find $v_p$, $L$, and $Z_0$ given $\veps_r$ and $C$, lossless:
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$$v_p = {1 \over \sqrt{\mu_0 \mu_r \veps_0 \veps_r}} = {\omega \over \beta}$$
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$$Z_0 = \sqrt{L \over C}$$
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$$L = {\mu_0 \mu_r \veps_0 \veps_r \over C}$$
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Find $C$ given $Z_0$, $R_L$, $f$, and VWSR:
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$$C = {L \over Z_0^2} - {G \over j\omega} + {R \over j\omega Z_0^2}$$
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$$\text{VWSR} = {1 + |\Gamma| \over 1 - |\Gamma|}$$
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$$\Gamma = {R_L - Z_0 \over R_L + Z_0}$$
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$$\omega = 2\pi f$$
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Find VWSR, $\Gamma$, $Z_\text{in}$ from $\lambda$, $Z_0$, $Z_L$, length:
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$$\Gamma = {Z_L - Z_0 \over Z_L + Z_0}$$
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$$\text{VWSR} = {1 + |\Gamma| \over 1 - |\Gamma|}$$
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$$\max[Z_\text{in}] = Z_0 \cdot \text{VWSR}$$
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$$\min[Z_\text{in}] = {Z_0 \over \text{VWSR}}$$
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Find $Z_0$, $\alpha$, $\beta$, $\gamma$, $\lambda$ given $Z_\text{in, sc}$ and $Z_\text{in, oc}$:
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$$Z_0 = \sqrt{Z_\text{in, sc} Z_\text{in, oc}}$$
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$$\gamma = {1\over l} \arctanh\left(\sqrt{Z_\text{in, sc} \over Z_\text{in, oc}}\right)$$
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\section{Boundary Conditions}
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Electric field boundary conditions:
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$$\vec{E}_{T1} = \vec{E}_{T2}$$
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$$\vec{D}_{N1} = \vec{D}_{N2}$$
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$$\vec{E} = \vec{E}_N + \vec{E}_T$$
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$$\vec{D} = \veps \vec{E}$$
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Magnetic field boundary conditions:
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$$\vec{H}_{T1} = \vec{H}_{T2}$$
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$$\vec{B}_{N1} = \vec{B}_{N2}$$
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$$\vec{B} = \vec{B}_N + \vec{B}_T$$
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$$\vec{B} = \mu \vec{H}$$
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Normal vectors:
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$$\vec{n} = \vec{E}_1 - \vec{E}_2$$
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$$\vec{n} = \vec{H}_1 - \vec{H}_2$$
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$$\hat{n} = {\vec{n} \over \|\vec{n}\|}$$
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\section{Laplace and Poisson}
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The Laplacian:
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$$\vec{\nabla} \cdot (-\vec{\nabla}V) = -\vec{\nabla}^2 V = {\rho \over \veps_0}$$
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Laplace's Equation (charge free region):
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$$\vec{\nabla}^2 V = 0$$
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Poisson's Equation:
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$$\vec{\nabla}^2 V = -{\rho \over \veps_0}$$
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\section{General Transmission Lines}
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\begin{center}
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\includegraphics[width=0.3\textwidth]{Transmission_line_element.png}
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\end{center}
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$$\begin{aligned}
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\gamma &= \alpha + j\beta = \left[(R + j\omega L)(G + j\omega C)\right]^{1/2} \\ & = {1\over l} \arctanh\left( \sqrt{Z_\text{in, sc} \over Z_\text{in, oc}}\right)
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\end{aligned}$$
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The characteristic impedance:
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$$Z_0 = \sqrt{R + j\omega L \over G + j\omega C} = \sqrt{Z_\text{in, sc} Z_\text{in, oc}}$$
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$$Z_\text{in} = Z_0 {Z_L + Z_0 \tanh(\gamma l) \over Z_0 + Z_L \tanh(\gamma l)}$$
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\section{Voltage Standing Wave Ratio}
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$$\Gamma = {V_\text{reflected} \over V_\text{incident}}\Big|_{z' = 0} = {Z_L - Z_0 \over Z_L + Z_0}$$
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$$\text{VWSR} = {|V_\text{max}| \over |V_\text{min}|} = {1 + |\Gamma| \over 1 - |\Gamma|}$$
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$$v_p = {\omega \over \beta}$$
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$$P_\text{avg} = {|V_0^+| \over 2 Z_0}(1 - |\Gamma_L|^2)
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\section{Transmission Line Design}
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Lossless lines:
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$$R = G = 0$$
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$$\alpha = 0$$
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$$\beta = \omega \sqrt{LC}$$
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Distortionless lines:
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$${R \over L} = {G \over C}$$
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$$\alpha = R \sqrt{C \over L}$$
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$$\beta = \omega \sqrt{LC}$$
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\end{document}
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