Circuit Element

A circuit element is a basic unit that has certain rules governing the relationship of voltage across and current through that element beyond the simple conducting wire.

There are five ideal basic circuit elements:

• voltage source
• current source
• resistor
• capacitor
• inductor

There will be no proofs for the equations here. That's a physics thing that probably deserve their own page.

Ideal

This page will focus on ideal circuit elements. The real world is complicated; we make things convenient for ourselves by assuming certain properties despite being unrealistic. For example, an ideal voltage source outputs the same voltage regardless of current passing through it.

Resistor

The resistor dissipates energy depending on their resistance $R$. They follow Ohm's Law:

$v=ir$

The power disspated by a resistor can be calculated by

$p=iv$

You can substitute $i$ and $v$ such that it only takes in one of them and the resistance.

Capacitor

The capacitor is one of two basic energy storing circuit elements (the other being the inductor). They gather charge depending on voltage and capacitance $C$.

$Q=CV$

Taking time derivative on both sides, we have

$i=C{\frac {dv}{dt}}$

The energy stored within a capacitor is given by

$W={\frac {1}{2}}CV^{2}$

Inductor

The inductor is also an energy storing circuit element. They "resist" changes in current and stores energy based on inductance $L$.

$V=L{\frac {di}{dt}}$

The energy stored within a capacitor is given by

$W={\frac {1}{2}}LI^{2}$