An Oscillator's Phase Portrait

In an LC circuit, energy swings back and forth between the capacitor's charge and the inductor's current. Plot charge against current and the state goes around in loops. Add resistance and every loop is a little smaller: a spiral into rest.

Click or drag anywhere to start a new trajectory there.

How it works

Systems with complex eigenvalues

In one sentence

If $A$ has eigenvalues $\alpha \pm i\beta$ with $\beta \ne 0$ , solutions of $\mathbf x' = A\mathbf x$ are $e^{\alpha t}$ times combinations of $\cos\beta t$ and $\sin\beta t$ ; the origin is a spiral (stable if $\alpha < 0$ ) or a center if $\alpha = 0$ .

General formula

{\htmlData{sym=0}{q}}' = {\htmlData{sym=1}{i}}, \qquad {\htmlData{sym=1}{i}}' = -\frac{1}{{\htmlData{sym=2}{L}}{\htmlData{sym=3}{C}}}\,{\htmlData{sym=0}{q}} - \frac{\htmlData{sym=4}{{R}}}{\htmlData{sym=2}{{L}}}\,{\htmlData{sym=1}{i}}

Charge drives current and current changes charge, a constant chase around the origin. The resistance term pulls every loop slightly inward.

What the symbols mean

$q$
Charge
Charge on the capacitor.
$i$
Current
Current through the inductor, the rate of change of charge.
$L$
Inductance
Electrical inertia: it keeps current flowing.
$C$
Capacitance
How much charge the capacitor holds per volt.
$R$
Resistance
What drains energy and makes the spiral wind in.

Write the second-order circuit equation as a system in charge $q$ and current $i = q'$ .
Its matrix $\begin{bmatrix}0 & 1\\ -1/LC & -R/L\end{bmatrix}$ has complex eigenvalues $\alpha \pm i\beta$ when $R^2 < 4L/C$ .
The imaginary part $\beta$ makes trajectories rotate: energy swaps between the capacitor and the inductor.
The real part $\alpha = -R/2L$ shrinks each loop: resistance burns energy, so the spiral winds into the origin.

Learn the math

The Phase PlaneDiff Eq · Lesson 32, Systems of Differential Equations Complex EigenvaluesDiff Eq · Lesson 34, Systems of Differential Equations LC and RLC OscillationsE&M · Lesson 22, Electromagnetic Induction