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diff --git a/mindmap/Kirchhoff's Laws.org b/mindmap/Kirchhoff's Laws.org new file mode 100644 index 0000000..be20a41 --- /dev/null +++ b/mindmap/Kirchhoff's Laws.org @@ -0,0 +1,35 @@ +:PROPERTIES: +:ID: 1d586d6b-bd97-4c59-ad57-8894ae4ac8ba +:END: +#+title: Kirchhoff's Laws +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +Kirchhoff's Laws, along with Ohm's law, create the axioms of circuit analysis. The two laws are the Kirchhoff Voltage Law +(KVL) and Kirchhoff's Current Law (KCL). They can be derived from an approximation of [[id:fde2f257-fa2e-469a-bc20-4d11714a515e][Maxwell's Equations]]. +** KCL +:PROPERTIES: +:ID: 9f7e61fa-a6ed-4d9b-8cdf-7f4ffdd80f06 +:END: +Due to the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] for electrodynamics, current is always conserved locally. In an ideal one-dimensional +wire, the surface integral can be reduced to a simple line integral, given the current only moves in one direction +(which we will assume for circuits). +\begin{align} +\int I \cdot d\vec{l} = -\frac{\partial Q_{enc}}{\partial t} +\end{align} +If the total amount of charge in the wires are conserved: +\begin{align} +\label{} +\int \vec{I} \cdot d\vec{l} = 0 +\end{align} +Therefore: +\begin{align} +\label{} +\sum_{n}I_{n} = 0 +\end{align} +where the total current $\vec{I}$ can be decomposed into many currents of each branched path $I_{n}$. +** KVL |