5th semester files
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\relax
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\@writefile{toc}{\contentsline {section}{\numberline {I}Constants}{1}{}\protected@file@percent }
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(rewritten before read)
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313
5th-Semester-Fall-2023/EEMAGS/EquationSheet/Equation-Sheet.fls
Normal file
313
5th-Semester-Fall-2023/EEMAGS/EquationSheet/Equation-Sheet.fls
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<inserted text>
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$
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l.114
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I've inserted a begin-math/end-math symbol since I think
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you left one out. Proceed, with fingers crossed.
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./Equation-Sheet.tex:114: Display math should end with $$.
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<to be read again>
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\tex_par:D
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l.114
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The `$' that I just saw supposedly matches a previous `$$'.
|
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So I shall assume that you typed `$$' both times.
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[2
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<./Transmission_line_element.png>] (./Equation-Sheet.aux) )
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@ -0,0 +1,35 @@
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# EEMAGS Equation Sheet
|
||||
|
||||
## Curl and Divergence Identities
|
||||
1. The Laplacian: can operate on a scalar or vector field
|
||||
$$\nabla \cdot (\nabla f) = \nabla^2 f$$
|
||||
$$\frac{\partial^2 f}{\partial x} + \frac{\partial^2 f}{\partial y} + \frac{\partial^2 f}{\partial z}$$
|
||||
2. The curl of a gradient is $0$
|
||||
$$\nabla \times (\nabla f) = 0$$
|
||||
3. The gradient of divergence is a scalar
|
||||
$$\nabla (\nabla \cdot \vec{f})$$
|
||||
4. The divergence of curl is $0$
|
||||
$$\nabla \cdot (\nabla \times \vec{v}) = 0$$
|
||||
5. Curl of curl
|
||||
$$\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$$
|
||||
|
||||
## Maxwell's Equations
|
||||
Gauss's Law for $\vec{E}$-fields:
|
||||
$$\Phi_E =\int \vec{E} \cdot d\vec{s} = \frac{Q_{enc}}{\varepsilon_0}$$
|
||||
$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$
|
||||
|
||||
Gauss's Law for $\vec{B}$-fields:
|
||||
$$\int \vec{B} \cdot d\vec{s} = 0 $$
|
||||
$$ \nabla \cdot \vec{B} = 0$$
|
||||
|
||||
Faraday's Law
|
||||
$$\int \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\vec{B} $$
|
||||
$$ \nabla \times \vec{E} = -\frac{d}{dt}\vec{B}$$
|
||||
|
||||
Ampere's Law
|
||||
$$\int \vec{B} \cdot d\vec{l} = \mu_0\left( \vec{J} + \varepsilon_0 \frac{d\vec{E}}{dt}\right) = \mu_0 I_{enc}$$
|
||||
$$ \nabla \times \vec{B} = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$$
|
||||
|
||||
## Work and Voltage
|
||||
$$\Delta V = -\int \vec{E} \cdot d\vec{l}$$
|
||||
$$-\nabla V = \vec{E}$$
|
||||
BIN
5th-Semester-Fall-2023/EEMAGS/EquationSheet/Equation-Sheet.pdf
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5th-Semester-Fall-2023/EEMAGS/EquationSheet/Equation-Sheet.pdf
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143
5th-Semester-Fall-2023/EEMAGS/EquationSheet/Equation-Sheet.tex
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5th-Semester-Fall-2023/EEMAGS/EquationSheet/Equation-Sheet.tex
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@ -0,0 +1,143 @@
|
||||
% DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
|
||||
% Version 2, December 2004
|
||||
%
|
||||
% Copyright (C) 2023 Aidan Sharpe <amsharpe102@pm.me>
|
||||
%
|
||||
% Everyone is permitted to copy and distribute verbatim or modified
|
||||
% copies of this license document, and changing it is allowed as long
|
||||
% as the name is changed.
|
||||
%
|
||||
% DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
|
||||
% TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
|
||||
%
|
||||
% 0. You just DO WHAT THE FUCK YOU WANT TO.
|
||||
|
||||
|
||||
|
||||
\documentclass[journal]{IEEEtran}
|
||||
|
||||
\usepackage{graphicx}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{esint}
|
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\usepackage{physics}
|
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|
||||
\DeclareMathOperator\arctanh{arctanh}
|
||||
|
||||
\setlength{\parindent}{0pt}
|
||||
|
||||
\title{EEMAGS Equation Sheet}
|
||||
|
||||
\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}}
|
||||
\newcommand{\veps}{\varepsilon}
|
||||
\newcommand{\vphi}{\varphi}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\section{Constants}
|
||||
$$\varepsilon_0 = 8.854 \times 10^{-12}$$
|
||||
$$\mu_0 = 1.257 \times 10^{-6}$$
|
||||
$$c = 3 \times 10^8$$
|
||||
$$\eta_0 = \sqrt{{\mu_0 \over \varepsilon_0}} = 377\Omega \approx 120\pi$$
|
||||
|
||||
\section{Fifth's Bullet Points}
|
||||
Find $\veps_r$ given $\lambda$ and $f$:
|
||||
$$\veps_r = {c^2 \over \mu_r \lambda^2 f^2}$$
|
||||
Find $v_p$ given $\delta$, $\alpha$, and $\beta$ given a good conductor:
|
||||
$$v_p = {\omega \over \beta}$$
|
||||
Find $P_\text{avg}$ given $\vec{H}(t)$ in air:
|
||||
|
||||
Find $\Gamma$ in polar and VWSR given $Z_L$ and $Z_0$:
|
||||
$$\Gamma = {Z_L - Z_0 \over Z_L + Z_0}$$
|
||||
$$\text{VWSR} = {1 + |\Gamma| \over 1 - |\Gamma|}$$
|
||||
Find $\vec{E}(z,t)$ given $\vec{H}(z,t)$ in a lossless medium of $\mu_r$ and $\veps_r$:
|
||||
|
||||
Find $\lambda$, $\veps_r$, and $\vec{H}$ given $\vec{E}$:
|
||||
$$\hat{H}_y = {\hat{E}_x \over \eta}$$
|
||||
$$\veps_r = {\beta^2 \over \omega^2 \mu_r \mu_0 \veps_0}$$
|
||||
$$\lambda = {v_p \over f}$$
|
||||
Find $\Gamma^2$ and $\langle P \rangle$ from $\veps$, lossless:
|
||||
|
||||
Find $\beta_\text{air}$, $\beta_\text{material}$, $\hat{\Gamma}$, $\hat{T}$ given $\vec{E}^i$, $\veps_r$, $\mu_r$, $\sigma_r$:
|
||||
|
||||
Find $v_p$, $L$, and $Z_0$ given $\veps_r$ and $C$, lossless:
|
||||
$$v_p = {1 \over \sqrt{\mu_0 \mu_r \veps_0 \veps_r}} = {\omega \over \beta}$$
|
||||
$$Z_0 = \sqrt{L \over C}$$
|
||||
$$L = {\mu_0 \mu_r \veps_0 \veps_r \over C}$$
|
||||
Find $C$ given $Z_0$, $R_L$, $f$, and VWSR:
|
||||
$$C = {L \over Z_0^2} - {G \over j\omega} + {R \over j\omega Z_0^2}$$
|
||||
$$\text{VWSR} = {1 + |\Gamma| \over 1 - |\Gamma|}$$
|
||||
$$\Gamma = {R_L - Z_0 \over R_L + Z_0}$$
|
||||
$$\omega = 2\pi f$$
|
||||
Find VWSR, $\Gamma$, $Z_\text{in}$ from $\lambda$, $Z_0$, $Z_L$, length:
|
||||
$$\Gamma = {Z_L - Z_0 \over Z_L + Z_0}$$
|
||||
$$\text{VWSR} = {1 + |\Gamma| \over 1 - |\Gamma|}$$
|
||||
$$\max[Z_\text{in}] = Z_0 \cdot \text{VWSR}$$
|
||||
$$\min[Z_\text{in}] = {Z_0 \over \text{VWSR}}$$
|
||||
Find $Z_0$, $\alpha$, $\beta$, $\gamma$, $\lambda$ given $Z_\text{in, sc}$ and $Z_\text{in, oc}$:
|
||||
$$Z_0 = \sqrt{Z_\text{in, sc} Z_\text{in, oc}}$$
|
||||
$$\gamma = {1\over l} \arctanh\left(\sqrt{Z_\text{in, sc} \over Z_\text{in, oc}}\right)$$
|
||||
|
||||
|
||||
\section{Boundary Conditions}
|
||||
Electric field boundary conditions:
|
||||
$$\vec{E}_{T1} = \vec{E}_{T2}$$
|
||||
$$\vec{D}_{N1} = \vec{D}_{N2}$$
|
||||
$$\vec{E} = \vec{E}_N + \vec{E}_T$$
|
||||
$$\vec{D} = \veps \vec{E}$$
|
||||
|
||||
Magnetic field boundary conditions:
|
||||
$$\vec{H}_{T1} = \vec{H}_{T2}$$
|
||||
$$\vec{B}_{N1} = \vec{B}_{N2}$$
|
||||
$$\vec{B} = \vec{B}_N + \vec{B}_T$$
|
||||
$$\vec{B} = \mu \vec{H}$$
|
||||
|
||||
Normal vectors:
|
||||
$$\vec{n} = \vec{E}_1 - \vec{E}_2$$
|
||||
$$\vec{n} = \vec{H}_1 - \vec{H}_2$$
|
||||
$$\hat{n} = {\vec{n} \over \|\vec{n}\|}$$
|
||||
|
||||
|
||||
|
||||
\section{Laplace and Poisson}
|
||||
The Laplacian:
|
||||
$$\vec{\nabla} \cdot (-\vec{\nabla}V) = -\vec{\nabla}^2 V = {\rho \over \veps_0}$$
|
||||
Laplace's Equation (charge free region):
|
||||
$$\vec{\nabla}^2 V = 0$$
|
||||
Poisson's Equation:
|
||||
$$\vec{\nabla}^2 V = -{\rho \over \veps_0}$$
|
||||
|
||||
\section{General Transmission Lines}
|
||||
\begin{center}
|
||||
\includegraphics[width=0.3\textwidth]{Transmission_line_element.png}
|
||||
\end{center}
|
||||
$$\begin{aligned}
|
||||
\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)
|
||||
\end{aligned}$$
|
||||
|
||||
The characteristic impedance:
|
||||
$$Z_0 = \sqrt{R + j\omega L \over G + j\omega C} = \sqrt{Z_\text{in, sc} Z_\text{in, oc}}$$
|
||||
$$Z_\text{in} = Z_0 {Z_L + Z_0 \tanh(\gamma l) \over Z_0 + Z_L \tanh(\gamma l)}$$
|
||||
|
||||
\section{Voltage Standing Wave Ratio}
|
||||
$$\Gamma = {V_\text{reflected} \over V_\text{incident}}\Big|_{z' = 0} = {Z_L - Z_0 \over Z_L + Z_0}$$
|
||||
$$\text{VWSR} = {|V_\text{max}| \over |V_\text{min}|} = {1 + |\Gamma| \over 1 - |\Gamma|}$$
|
||||
|
||||
$$v_p = {\omega \over \beta}$$
|
||||
|
||||
$$P_\text{avg} = {|V_0^+| \over 2 Z_0}(1 - |\Gamma_L|^2)
|
||||
|
||||
|
||||
\section{Transmission Line Design}
|
||||
Lossless lines:
|
||||
$$R = G = 0$$
|
||||
$$\alpha = 0$$
|
||||
$$\beta = \omega \sqrt{LC}$$
|
||||
|
||||
Distortionless lines:
|
||||
$${R \over L} = {G \over C}$$
|
||||
$$\alpha = R \sqrt{C \over L}$$
|
||||
$$\beta = \omega \sqrt{LC}$$
|
||||
|
||||
\end{document}
|
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