RLC Circuit Differential Equations

The RLC circuit is a spring-mass-damper in disguise

Kirchhoff's voltage law around a series RLC loop (with no source) says the voltage drops across inductor, resistor, and capacitor sum to zero: $L \frac{dI}{dt} + R \, I + \frac{Q}{C} = 0$

Using $I = dQ/dt$, this becomes a second-order ODE for the charge $Q(t)$: $\boxed{\, L \, Q'' + R \, Q' + \frac{1}{C} Q = 0 \,}$

Compare term by term with the damped mass-on-spring (#202) $m x'' + c x' + k x = 0$:

• $L \leftrightarrow m$ — inductance plays the role of inertia.
• $R \leftrightarrow c$ — resistance dissipates energy like a dashpot.
• $1/C \leftrightarrow k$ — the "stiffness" of the capacitor (resistance to charge accumulation).
• $Q \leftrightarrow x$ — charge plays the role of position.
• $I = Q' \leftrightarrow x'$ — current plays the role of velocity.

It's literally the same ODE with re-labelled variables. This is one of the most beautiful unifications in physics — mechanical oscillation and electrical oscillation share the exact same mathematical skeleton.

The given system

• $L = 1$ H, $R = 10$ Ω, $C = 0.01$ F.

Plug in: $1 \cdot Q'' + 10 \, Q' + \frac{1}{0.01} Q = 0$ $Q'' + 10 \, Q' + 100 \, Q = 0$

Step-by-step

Step 1 — Characteristic equation. $r^{2} + 10 r + 100 = 0$

Discriminant: $\Delta = 100 - 400 = -300 < 0$ → complex conjugate roots. $r = \frac{-10 \pm \sqrt{-300}}{2} = -5 \pm 5\sqrt{3}\, i.$

So $\alpha = -5$ (decay rate), $\beta = 5\sqrt{3} \approx 8.66$ (oscillation frequency).

Step 2 — Classify the regime.

Critical resistance for this L, C: $R_{\text{crit}} = 2\sqrt{L/C} = 2\sqrt{1/0.01} = 2 \cdot 10 = 20 \text{ Ω}.$

Our $R = 10 < R_{\text{crit}}$ → under-damped. The circuit rings.

Step 3 — General solution. $\boxed{\, Q(t) = e^{-5 t}\bigl(C_1 \cos(5\sqrt{3}\, t) + C_2 \sin(5\sqrt{3}\, t)\bigr) \,}$

Natural (undamped) frequency: $\omega_0 = 1/\sqrt{LC} = 1/\sqrt{0.01} = 10$ rad/s. Damped frequency: $\omega_d = \sqrt{\omega_0^2 - \gamma^2} = \sqrt{100 - 25} = 5\sqrt{3}$ rad/s. Decay rate: $\gamma = R/(2L) = 5$ s⁻¹. Envelope time constant: $1/\gamma = 0.2$ s.

Step 4 — Current.

$I(t) = Q'(t)$. Differentiate: $I(t) = e^{-5 t}\bigl[(-5 C_1 + 5\sqrt{3} C_2) \cos(5\sqrt{3}\, t) + (-5 C_2 - 5\sqrt{3} C_1) \sin(5\sqrt{3}\, t)\bigr].$

Energy sloshes between the inductor (magnetic field, $\tfrac{1}{2} L I^2$) and the capacitor (electric field, $\tfrac{1}{2} Q^2 / C$), with the resistor steadily converting it to heat.

Verification (structural)

$e^{-5t} \cos(5\sqrt{3}\, t)$ and $e^{-5t} \sin(5\sqrt{3}\, t)$ solve $Q'' + 10 Q' + 100 Q = 0$ by construction — plug in $r = -5 \pm 5\sqrt{3} i$ to $r^2 + 10 r + 100$: $(-5)^2 \pm 2 \cdot (-5)(5\sqrt{3}) i + (5\sqrt{3}i)^2 + 10(-5 \pm 5\sqrt{3} i) + 100$. Real and imaginary parts both cancel.

The RLC-mass-spring dictionary

| RLC | Spring-mass | |---|---| | $L$ (inductance, H) | $m$ (mass, kg) | | $R$ (resistance, Ω) | $c$ (damping, N·s/m) | | $1/C$ (inverse capacitance, 1/F) | $k$ (stiffness, N/m) | | $Q$ (charge, C) | $x$ (position, m) | | $I = Q'$ (current, A) | $v = x'$ (velocity, m/s) | | $\tfrac{1}{2} L I^2$ (magnetic PE) | $\tfrac{1}{2} m v^2$ (kinetic) | | $\tfrac{1}{2} Q^2/C$ (electric PE) | $\tfrac{1}{2} k x^2$ (elastic PE) |

Bullets instead of table rows in case of narrow viewports:

• $L \equiv m$, $R \equiv c$, $1/C \equiv k$, $Q \equiv x$, $I \equiv v$.

Resonance with a driving voltage

Add a source $E(t)$ and the ODE becomes non-homogeneous: $L Q'' + R Q' + Q/C = E(t).$

If $E(t) = E_0 \cos(\omega t)$, resonance occurs at $\omega = \omega_0 = 1/\sqrt{LC}$ — driving at the natural frequency pumps energy in most efficiently. This is the principle behind every tuned radio receiver: adjust $C$ (or $L$) until $\omega_0$ matches your desired station.

The three regimes (same as spring-mass)

• $R < 2\sqrt{L/C}$: under-damped — oscillates. (Our case.)
• $R = 2\sqrt{L/C}$: critically damped — fastest non-oscillatory settle.
• $R > 2\sqrt{L/C}$: over-damped — sluggish return.

Quality factor $Q_{\text{factor}} = \omega_0 L / R = \sqrt{L/C} / R$. Our $Q_{\text{factor}} = 10 / 10 = 1$ — moderately damped.

Where this ODE actually shows up

• Radios and TVs: every tuner has an LC resonant circuit.
• Power-supply filters: smooth DC rails with LC low-pass behaviour.
• Antenna matching: broadcast and reception depend on the circuit's $\omega_0, Q$.
• MRI coils: rotating magnetic fields in tuned LC tanks.
• Ring-down spectroscopy: measure $R, L, C$ by observing the decay of an excited oscillation.
• Quantum LC analogue: the quantum harmonic oscillator has the same math; superconducting qubits are engineered LC circuits.

Common mistakes

• Using $C$ instead of $1/C$ for stiffness. Capacitance $C$ is like compliance — bigger $C$ means softer. The stiffness term in the ODE is $1/C$.
• Sign of the voltage drops. With the convention $I = dQ/dt$, KVL gives all terms on one side: $L Q'' + R Q' + Q/C = V_{\text{source}}$.
• Confusing $Q$ and $I$. In an RLC problem, "solve for Q" and "solve for I" give different ODEs (the one for $I$ is the derivative of the $Q$ equation, but structurally the same second-order form).
• Wrong critical resistance. It's $2\sqrt{L/C}$, not $\sqrt{L/C}$ or $\sqrt{LC}$. Double-check the factor of 2.

Try it in the visualization

Draw the R, L, C components in a loop with the current direction shown. Animate the charge/voltage on the capacitor as the under-damped oscillation unfolds, with the $Q(t)$ curve plotted alongside. Slide $R$ from 0 to 40 Ω — watch the oscillation morph from pure ring (R = 0, no damping, a perpetual-motion LC tank) through under-damped, critical, and over-damped.