RLC Circuits

A resistor, an inductor and a capacitor in one loop behave exactly like a damped spring. Charge sloshes back and forth and fades away.

How it works

Second-order linear ODEs

In one sentence

The capacitor charge $q$ in a series RLC circuit satisfies $Lq'' + Rq' + q/C = E(t)$ ; with $E = 0$ and $R^2 < 4L/C$ it is a decaying oscillation at angular frequency $\sqrt{1/LC - R^2/4L^2}$ .

General formula

{\htmlData{sym=3}{L}}\,{\htmlData{sym=2}{q''}} + {\htmlData{sym=4}{R}}\,{\htmlData{sym=1}{q'}} + \frac{1}{\htmlData{sym=5}{{C}}}\,{\htmlData{sym=0}{q}} = {\htmlData{sym=6}{E(t)}}

Each term is a voltage drop: $Lq''$ across the inductor, $Rq'$ across the resistor, $q/C$ across the capacitor. Kirchhoff's voltage law says they add up to the source $E(t)$ .

What the symbols mean

$q$
Charge
Electric charge stored on the capacitor.
$q'$
Current
How fast charge flows around the circuit.
$q''$
Change in current
How fast the current itself is changing. The inductor pushes back against it.
$L$
Inductance
The inductor resists changes in current, the way mass resists changes in motion.
$R$
Resistance
Turns electrical energy into heat, like friction. It damps the oscillation.
$C$
Capacitance
How much charge the capacitor holds per volt. 1/C acts like a spring's stiffness.
$E(t)$
Applied voltage
The source driving the circuit, like an outside push on a spring.

Inductance $L$ plays the role of mass, resistance $R$ of friction and $1/C$ of spring stiffness.
When $R^2 < 4L/C$ the charge oscillates while it decays. That is the underdamped case shown here.
The natural frequency $\omega_0 = 1/\sqrt{LC}$ is what a radio tunes to a station.

Learn the math

Complex NumbersAlgebra · Lesson 7, Preliminaries Partial FractionsAlgebra · Lesson 38, Polynomial Functions Integrals Involving Trig FunctionsCalc II · Lesson 2, Integration Techniques Partial FractionsCalc II · Lesson 4, Integration Techniques Second Order Linear EquationsDiff Eq · Lesson 13, Second Order Differential Equations Complex RootsDiff Eq · Lesson 15, Second Order Differential Equations Mechanical VibrationsDiff Eq · Lesson 21, Second Order Differential Equations The Laplace TransformDiff Eq · Lesson 22, Laplace Transforms LC and RLC OscillationsE&M · Lesson 22, Electromagnetic Induction

Learn the math

Complex NumbersAlgebra · Lesson 7, Preliminaries

Partial FractionsAlgebra · Lesson 38, Polynomial Functions

Integrals Involving Trig FunctionsCalc II · Lesson 2, Integration Techniques

Partial FractionsCalc II · Lesson 4, Integration Techniques

Second Order Linear EquationsDiff Eq · Lesson 13, Second Order Differential Equations

Complex RootsDiff Eq · Lesson 15, Second Order Differential Equations

Mechanical VibrationsDiff Eq · Lesson 21, Second Order Differential Equations

The Laplace TransformDiff Eq · Lesson 22, Laplace Transforms

LC and RLC OscillationsE&M · Lesson 22, Electromagnetic Induction