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#+title: Kirchhoff's Laws
#+author: Preston Pan
#+description: basic laws of circuit analysis
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* Introduction
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
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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}^{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
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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}