Flick on a flashlight and the bulb seems to light instantly — but zoom in on time and nothing in electronics is truly instant. Think of filling an empty bucket through a thin straw: water rushes in at first, then trickles slower and slower as the bucket nears full. Charging a capacitor through a resistor behaves exactly the same way. The resistor is the narrow straw that limits the flow; the capacitor is the bucket that fills up. That gradual "fill-up," and how fast it happens, is what every circuit on this page is really about.
Name the pieces
Three parts do all the work. A resistor $R$ slows the flow of charge. A capacitor $C$ stores charge on its plates. An inductor $L$ stores energy in a magnetic field and resists any sudden change in current. When you close the switch in the charging circuit, the capacitor voltage climbs along $V_C = V_0\,(1 - e^{-t/\tau})$, where the single number $\tau = RC$ — the time constant — sets the pace. Try everyday values: $R = 100\,\Omega$ and $C = 100\,\mu\text{F}$ give $\tau = (100)(100\times10^{-6}) = 0.01\,\text{s}$, or 10 ms. After one $\tau$ the capacitor reaches about 63% of full voltage; after $5\tau$ (here 0.05 s) it is past 99% — effectively done.
The precise picture
Each circuit is just Kirchhoff's voltage law written as a differential equation. RC and RL circuits give first-order decays, each ruled by one time constant — $\tau = RC$ for the capacitor, $\tau_L = L/R$ for the inductor. Put an inductor and a capacitor in the same loop and the equation becomes second-order, $L\ddot{Q} + R\dot{Q} + Q/C = 0$ — the exact electrical twin of a mass on a damped spring. Its behaviour is decided by comparing the damping $\gamma = R/2L$ with the natural frequency $\omega_0 = 1/\sqrt{LC}$: when $R < 2\sqrt{L/C}$ the charge rings as a decaying cosine at $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$; push $R$ past that critical value and the energy drains away before even one full swing. The $R$, $C$, and $L$ sliders move you across exactly these regimes.
Try this in the sim above
In RC Charging, watch the τ readout, then double $R$ — the charging curve stretches out and $\tau$ doubles right along with it. Switch to RLC Underdamped and slowly raise $R$: the oscillation both slows and shrinks, until near $R = 2\sqrt{L/C}$ it stops ringing altogether — that boundary is critical damping. Finally, open the Phase tab and compare the underdamped case (an inward spiral) with the overdamped case (a single smooth sweep straight to the origin).
Section 03
Equations & Derivation
Transient analysis describes how voltage, current, and charge in a circuit evolve from one steady state to another — for example, when a switch closes and a capacitor begins charging. The behavior is governed by first-order (RC, RL) or second-order (RLC) linear differential equations derived from Kirchhoff's voltage law.
1. RC Circuit — Charging
Apply Kirchhoff's voltage law around the loop:
RC Charging Equation
$$V_0 = IR + \frac{Q}{C}, \qquad I = \frac{dQ}{dt}$$
$$R\frac{dQ}{dt} + \frac{Q}{C} = V_0$$
Define natural frequency $\omega_0 = 1/\sqrt{LC}$ and damping coefficient $\gamma = R/(2L)$. The characteristic equation $s^2 + 2\gamma s + \omega_0^2 = 0$ has roots:
$$s = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}$$
Three regimes appear depending on the sign of $\gamma^2 - \omega_0^2$:
Regime
Condition
Behavior
Underdamped
$R < 2\sqrt{L/C}$
Oscillates while decaying
Critically damped
$R = 2\sqrt{L/C}$
Fastest non-oscillating decay
Overdamped
$R > 2\sqrt{L/C}$
Slow exponential decay
For the underdamped case, with damped frequency $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$:
Underdamped RLC Solution
$$Q(t) = Q_0\,e^{-\gamma t}\cos(\omega_d t + \phi)$$
The R, C, L, and V₀ sliders set the corresponding circuit elements directly. The simulation integrates the ODEs numerically using a 4th-order Runge–Kutta scheme so that all curves remain accurate even for stiff parameter combinations. The phase plot shows $I$ vs $V_C$ — for underdamped oscillation it traces an inward spiral; for overdamped decay, a smooth curve heading to the origin.
Reference: Halliday, Resnick & Walker — Fundamentals of Physics, 10th Ed., Ch. 27.9 'RC Circuits' & Ch. 31.5 'Damped Oscillations in an RLC Circuit'; Serway & Jewett — Physics for Scientists and Engineers, 8th Ed., Ch. 32.2 'RL Circuits' & Ch. 32.5 'Oscillations in an LC Circuit'.
Section 04
Frequently Asked Questions
The canvas shows a live schematic of the active circuit (RC, RL, or RLC), with a moving needle on the source, current arrows whose thickness scales with |I(t)|, and a glowing capacitor whose intensity scales with |V_C|. The graph below plots V, I, Q, U, or a phase-space trajectory in real time as the ODE is integrated by RK4. Adjusting any slider while the simulation runs immediately changes the system parameters, but the integrator continues from the current state, so you can see how a change of, say, R affects an in-progress oscillation.
RC filters set the cutoff frequency in audio circuits and camera flash timing (the flash capacitor charges through R, then dumps energy through the bulb). RL circuits govern the current build-up in electric motors, relays, and ignition coils. RLC circuits are the heart of every radio tuner, oscillator, and the magnetic-resonance pulse circuits in MRI machines. Defibrillators are essentially heavily damped RLC discharge devices delivering ~360 J in milliseconds.
Dimensional analysis settles it: [R]·[C] = (V/A)·(C/V) = C/A = s. Physically, τ is the time for the capacitor to charge to (1−1/e) ≈ 63.2% of V₀, or equivalently to discharge to 1/e ≈ 36.8%. After 5τ the system is at >99% of steady state, which is why engineers treat 5τ as 'effectively done'.
Critical damping is the boundary case R = 2√(L/C) where the system reaches equilibrium without overshooting in the shortest possible time. Overdamped systems also don't oscillate but approach equilibrium more slowly because the larger resistance dissipates energy before the inductor can move charge. Door-closers, car suspensions, and analog meter movements are deliberately tuned to be slightly overdamped or critical — never underdamped, or the door would slam back and forth.
A capacitor stores energy in its electric field: U = ½Q²/C. That field persists as long as charge separation is maintained, regardless of whether current is flowing. A resistor dissipates energy as heat: P = I²R. Heat radiated to the environment cannot be recovered, so a resistor has no memory. This asymmetry is what gives RC circuits their characteristic exponential transient — the capacitor's stored energy drives the decaying current.
Energy oscillates between the capacitor's electric field (U_C = ½Q²/C) and the inductor's magnetic field (U_L = ½LI²). Total energy U_C + U_L would be conserved if R = 0, but the resistor steadily drains energy as heat at a rate I²R. So the oscillation amplitude decays as e^(−γt) and the total energy as e^(−2γt), falling to zero asymptotically.
It is the damped frequency ω_d = √(ω₀² − γ²) that you observe, not ω₀. Increasing R increases γ = R/(2L), which subtracts from ω₀² inside the square root. As R approaches the critical value 2√(L/C), ω_d → 0 — there is no oscillation at all. This is a non-obvious but important effect: heavy damping does not just reduce amplitude, it slows the oscillation.
Resources: HyperPhysics — RC, RL, RLC Circuit pages (hyperphysics.phy-astr.gsu.edu); MIT OpenCourseWare 8.02 'Electricity & Magnetism' Lecture 19; Khan Academy — 'Capacitors and Capacitance', 'Inductance'; Paul's Physics Notes — Transient Circuits.
Section 05
Common Misconceptions
❌ Misconception: 'After one time constant τ, the capacitor is fully charged.'
✅ Correction: After τ the capacitor has reached only 63.2% of V₀. It takes about 5τ to reach >99% — and mathematically, full charge V₀ is approached only as t → ∞. The 'effectively done at 5τ' rule is a useful engineering shortcut, not the actual physics.
📖 Reference: Halliday/Resnick/Walker, Fundamentals of Physics, 10th Ed., §27-9 'RC Circuits', Example 27-13
❌ Misconception: 'Current cannot flow through a capacitor.'
✅ Correction: While DC steady-state current cannot pass an ideal capacitor, transient current absolutely flows during charging and discharging — that is exactly what powers the capacitor's stored energy. In AC circuits, displacement current $\varepsilon_0\,dE/dt$ in the dielectric region completes the loop continuously, so AC current 'passes through' a capacitor every cycle.
📖 Reference: Griffiths, Introduction to Electrodynamics, 4th Ed., §7.3.2 'Maxwell's Correction to Ampère's Law'
❌ Misconception: 'In an RL circuit, increasing L makes the current rise faster.'
✅ Correction: Exactly the opposite. The inductor opposes change in current — larger L means the inductor resists the build-up more strongly, so the time constant τ_L = L/R is longer and the current rises more slowly. To make I(t) rise faster, you must either decrease L or increase R.
📖 Reference: Serway & Jewett, Physics for Scientists and Engineers, 8th Ed., §32.2 'RL Circuits', Equation 32.7
❌ Misconception: 'A critically damped RLC circuit returns to equilibrium fastest because it is right at the boundary.'
✅ Correction: Among non-oscillating responses (no overshoot allowed), critical damping is fastest. But if you simply ask which case reaches equilibrium fastest, an underdamped circuit's first zero-crossing occurs sooner than the critical-damping settling time. Engineers choose critical damping when overshoot would damage the system (galvanometers, door closers), not because it is the fastest possible.
📖 Reference: Kleppner & Kolenkow, An Introduction to Mechanics, 2nd Ed., §11.3 'Damped Harmonic Oscillator' (mathematics is identical)
❌ Misconception: 'In RLC, energy is conserved as long as R is small.'
✅ Correction: Energy is only conserved when R = 0 exactly. Any non-zero resistance, however small, dissipates energy at rate I²R every cycle. The total stored energy decays as e^(−2γt), where γ = R/(2L). Small R means slow decay, never zero decay. This is a critical point: damping is qualitative, not quantitative — even the cleanest LC tank circuit eventually rings down.
📖 Reference: Halliday/Resnick/Walker, Fundamentals of Physics, 10th Ed., §31-5 'Damped Oscillations in an RLC Circuit', Equation 31-30
❌ Misconception: 'The inductor's stored energy is in its current, just like the capacitor's is in its charge.'
✅ Correction: Energy in an inductor is stored in the magnetic field in the surrounding space (U = ½LI²), not 'in the current' itself. If you suddenly cut the wire, the field collapses and induces a huge voltage spike (this is why turning off an inductor without a flyback diode can destroy circuits). Charge can sit on a capacitor plate indefinitely, but a current is not a 'thing' that can be stored — only the field it creates is.
📖 Reference: Griffiths, Introduction to Electrodynamics, 4th Ed., §7.2.4 'Energy in Magnetic Fields'
Misconception research: Thacker, B. A. et al. — 'Comparing problem solving performance of physics students in inquiry-based and traditional introductory physics courses', Am. J. Phys. 62, 627 (1994); Sokoloff & Thornton — 'Tools for Scientific Thinking: Microcomputer-Based Laboratories', Phys. Teach. 28, 162 (1990); McDermott & Shaffer — 'Research as a guide for curriculum development: An example from introductory electricity', Am. J. Phys. 60, 994 (1992); Arnold Arons — A Guide to Introductory Physics Teaching, Wiley, 1990, Ch. 9.