An AC Circuit's Steady State

Drive a circuit with an alternating voltage and, once the startup transient dies away, the charge sloshes at exactly the source's frequency, but delayed and resized. Guess a sine-plus-cosine and solve for the two coefficients.

How it works

Undetermined coefficients

In one sentence

For $ay'' + by' + cy = g(t)$ with $g$ a sine, cosine, exponential or polynomial, a particular solution has the same form as $g$ (times $t$ if that form already solves the homogeneous equation), with coefficients found by substitution.

General formula

{\htmlData{sym=1}{L}}{\htmlData{sym=0}{q}}'' + {\htmlData{sym=2}{R}}{\htmlData{sym=0}{q}}' + \frac{\htmlData{sym=0}{{q}}}{\htmlData{sym=3}{{C}}} = {\htmlData{sym=4}{E_0}}\cos{\htmlData{sym=5}{\omega}} t \;\Rightarrow\; {\htmlData{sym=0}{q}}_p = A\cos{\htmlData{sym=5}{\omega}} t + B\sin{\htmlData{sym=5}{\omega}} t, \quad A = \frac{{\htmlData{sym=4}{E_0}}\left(\frac1{\htmlData{sym=3}{C}} - {\htmlData{sym=1}{L}}{\htmlData{sym=5}{\omega}}^2\right)}{\Delta}, \;\; B = \frac{{\htmlData{sym=4}{E_0}}{\htmlData{sym=2}{R}}{\htmlData{sym=5}{\omega}}}{\Delta}

Sines in, sines out: derivatives of $\cos$ and $\sin$ are $\cos$ and $\sin$ again, so a combination of them is the only shape that can balance the equation.

What the symbols mean

$q$
Charge
Charge on the capacitor: the unknown function.
$L$
Inductance
The circuit's electrical inertia.
$R$
Resistance
What turns electrical energy into heat.
$C$
Capacitance
Charge stored per volt.
$E_0$
Source amplitude
The AC source's peak voltage (1 V in the graph).
$\omega$
Source frequency
How fast the source alternates, in radians per second.

An AC source $E_0\cos\omega t$ drives a series RLC circuit. After the transient dies out, the charge oscillates at the source's frequency.
Undetermined coefficients: guess $q_p = A\cos\omega t + B\sin\omega t$ , substitute, and match the cosine and sine terms.
That gives two linear equations for $A$ and $B$ , with $\Delta = \left(\frac1C - L\omega^2\right)^2 + (R\omega)^2$ . The amplitude is $E_0/\sqrt{\Delta}$ .
The response lags the source, and it is largest when $\omega$ is near $1/\sqrt{LC}$ .

Learn the math

Undetermined CoefficientsDiff Eq · Lesson 19, Second Order Differential Equations Variation of ParametersDiff Eq · Lesson 20, Second Order Differential Equations Phasors and ImpedanceE&M · Lesson 23, AC Circuits Resonance and FiltersE&M · Lesson 25, AC Circuits