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Diffstat (limited to 'mindmap/Kirchhoff's Laws.org')
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1 files changed, 29 insertions, 2 deletions
diff --git a/mindmap/Kirchhoff's Laws.org b/mindmap/Kirchhoff's Laws.org index be20a41..fc88086 100644 --- a/mindmap/Kirchhoff's Laws.org +++ b/mindmap/Kirchhoff's Laws.org @@ -3,13 +3,14 @@ :END: #+title: Kirchhoff's Laws #+author: Preston Pan +#+description: basic laws of circuit analysis #+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 +Kirchhoff's Laws, along with [[id:3cdce475-7644-4529-a447-6e790ad4055f][Ohm's Law]], create the axioms of [[id:a7d6d6e9-9f7a-446f-b6af-255c802f86b1][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: @@ -29,7 +30,33 @@ If the total amount of charge in the wires are conserved: Therefore: \begin{align} \label{} -\sum_{n}I_{n} = 0 +\sum_{n}^{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 +:PROPERTIES: +:ID: 92c952ee-f1f3-4782-b9e2-6fecb56caac6 +:END: +The Kirchhoff voltage law can be derived also from [[id:fde2f257-fa2e-469a-bc20-4d11714a515e][Maxwell's Equations]], specifically the [[id:63713308-0ff7-433f-8103-8b64ba9bdbe1][electrostatics]] equations +that formulate the electric field as an [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][electrostatic potential]]: +\begin{align} +\label{} +\vec{E} = -\vec{\nabla}V +\end{align} +more specifically, the [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][potential difference]] across a circuit element can be defined by +$\int \vec{E} \cdot d\vec{l} = V(b) - V(a)$, where $a$ and $b$ correspond to the positions before and after the circuit element. +We know from electrostatics that: +\begin{align} +\label{} +\oint \vec{E} \cdot d\vec{l} = 0 +\end{align} +and from the superposition principle we know: +\begin{align} +\label{} +V_{tot} = \sum V_{i} +\end{align} +so the total voltage drop, or potential difference around the entire circuit must be zero: +\begin{align} +\label{} +\sum_{n=0}^{N}V_{n} = 0 +\end{align} |