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Kirchhoff's Laws

Table of Contents

1. 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 Maxwell's Equations.

1.1. KCL

Due to the 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}


\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 \(I_{n}\).

1.2. KVL

The Kirchhoff voltage law can be derived also from Maxwell's Equations, specifically the electrostatics equations that formulate the electric field as an electrostatic potential:

\begin{align} \label{} \vec{E} = -\vec{\nabla}V \end{align}

more specifically, the 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}
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