Capacitor Charging Curve

How a capacitor charges

A capacitor is a pair of conductors separated by an insulator. Push current onto one plate and an equal-and-opposite charge accumulates on the other. The accumulated charge sets up a voltage across the dielectric: V = Q/C. The instant you connect a discharged capacitor to a voltage source through a resistor, current rushes in. As charge piles up on the plates, it pushes back against the source. The harder it pushes back, the smaller the net driving voltage across the resistor, and the slower charge can keep flowing.

That feedback is what makes the curve exponential. Apply Kirchhoff's voltage law to the loop:

V₀ = i·R + V_C
   = R·(dQ/dt) + Q/C

Differentiate, separate, integrate, and you get the standard charging solution:

V_C(t) = V₀ · (1 − e^(-t/τ))     where τ = RC
i(t)   = (V₀ / R) · e^(-t/τ)

Voltage rises from zero asymptotically toward V₀; current starts at V₀/R and decays toward zero. Both share the same time constant τ. The product RC has units of seconds: ohms × farads = (V/A)·(A·s/V) = s.

Worked example: 1 µF through 1 kΩ

Take the canonical lab circuit. R = 1 kΩ. C = 1 µF. Source V₀ = 5 V. The time constant is

τ = R·C = 1000 Ω × 1×10⁻⁶ F = 1×10⁻³ s = 1 ms

Plug t = 1 ms into the charging law:

V(1 ms) = 5 · (1 − e⁻¹) = 5 · (1 − 0.368) = 5 · 0.632 = 3.16 V

That 63.2% number is the universal signature of the exponential — it's where every RC charging curve sits at one time constant, regardless of values. Continuing:

t = 2τ:  V = 5·(1 − e⁻²) = 5·0.865 = 4.32 V   (86.5%)
t = 3τ:  V = 5·(1 − e⁻³) = 5·0.950 = 4.75 V   (95.0%)
t = 5τ:  V = 5·(1 − e⁻⁵) = 5·0.993 = 4.97 V   (99.3%)
t = 7τ:  V = 5·(1 − e⁻⁷) = 5·0.9991 = 4.995 V (99.9%)

The energy finally stored on the capacitor is

E = ½ · C · V² = 0.5 · 1×10⁻⁶ · 5² = 12.5 µJ

An equal amount of energy is dissipated as heat in R during charging. That 50% efficiency cap is why charging a discharged capacitor from a voltage source through a resistor is intrinsically lossy — power supplies recover this with switching converters that don't dump energy into a series resistor.

Charging curve, drawn

The shape is universal — the only knobs are τ (which stretches the time axis) and V₀ (which scales the voltage axis):

V/V₀
 1.0 ┤                          ╶─────── 100%
 0.99┤                    ╶─────              ← 5τ
     │              ╶─────
 0.86┤        ╶─────                          ← 2τ
     │     ╶─
 0.63┤   ╶─                                   ← 1τ
     │  ╱
 0.0 ┤─╱
     └──────────────────────────────── t/τ
     0    1    2    3    4    5    6

Discharging is the mirror — replace the source with a short and you get V(t) = V₀·e^(-t/τ), falling from full to 36.8% in one τ.

Capacitor families compared

Family	Cap density	ESR	Voltage	Tolerance	Typical role
Aluminum electrolytic	High (1 µF–47 mF)	0.05–1 Ω	10–500 V	±20%	Bulk supply, ripple smoothing
Tantalum electrolytic	High (1–680 µF)	0.1–0.5 Ω	2.5–50 V	±10%	Compact bulk, low-ripple analog
Ceramic class 1 (C0G/NP0)	Low (pF–nF)	<0.01 Ω	10–500 V	±1–5%	Timing, RF, precision filters
Ceramic class 2 (X7R/X5R MLCC)	Medium (nF–100 µF)	0.005–0.05 Ω	4–100 V	±10–20%	Bypass on every IC pin
Film (polypropylene/polyester)	Low (nF–10 µF)	<0.05 Ω	50 V–10 kV	±1–10%	Audio, snubbers, AC line
Supercapacitor (EDLC)	Very high (0.1 F–5000 F)	0.5 mΩ–1 Ω	2.5–3.0 V/cell	±20%	Memory back-up, regen brakes

The choice is driven by the τ you need, the ripple current you must handle, and the frequency band where ESR/ESL matter. Electrolytics dominate where you need millifarads in cubic centimeters; MLCCs dominate everywhere noise must be flat above 1 MHz; supercaps dominate where you need to dump or absorb tens of joules in a second.

Real-world numbers

• 0.1 µF MLCC bypass cap placed within 3 mm of every IC supply pin. With a 0.5 Ω parasitic loop, τ ≈ 50 ns — fast enough to source the current spike of a CMOS gate switching in 2 ns.
• 10 mF aluminum electrolytic on a 12 V regulator output. Charged through a 0.1 Ω source, τ = 1 ms — power-on settles in 5 ms.
• Maxwell BCAP3000 supercapacitor: 3000 F at 2.7 V, holding ½·3000·2.7² = 10,935 J ≈ 11 kJ per cell. Bus regen-brake banks chain dozens of these to reclaim 30–40% of braking energy.
• Camera flash capacitor: 200 µF at 330 V, charged through a 5 V boost converter. Energy ½·200µ·330² = 10.9 J. Dumped through a xenon tube in ~1 ms — that's 10 kW peak.
• Tantalum on a smartphone PMIC rail: 47 µF, ESR 0.1 Ω, volume of a grain of rice. Replaces three aluminum electrolytics that would not fit.

Variants

• Charging vs discharging. Same τ, mirrored curve. V_charge = V₀(1−e^(-t/τ)); V_discharge = V₀·e^(-t/τ). The 36.8% number for discharge is the natural counterpart of the 63.2% charge number.
• Tantalum vs aluminum electrolytic. Tantalum is smaller, has flatter ESR vs frequency, and tolerates higher temperature — but fails as a short circuit when overstressed (sometimes catastrophically, with audible pops). Aluminum is bigger, dries out over decades, and fails open or as ESR creep.
• Constant-current charging. Dump a fixed current i into the cap and the voltage rises linearly: V(t) = (i/C)·t. Used in flash strobes and photodiode readouts; the integrating cap on a CMOS image sensor pixel works exactly this way.
• Two-stage RC. Cascading two RC sections gives an S-shaped step response — the second section can't begin to charge until the first develops voltage. A useful trick to soft-start motors and avoid inrush.
• Constant-current discharge. Replace the resistor with a current source and the cap voltage falls linearly. This is how supercapacitor energy is metered: linear droop means E = ½C(V₁² − V₂²) is easy to bookkeep.

Failure modes

• ESR-driven heating. Ripple current i flowing through ESR dissipates i²·ESR. For a 1 A RMS ripple in a 0.5 Ω ESR electrolytic that's 0.5 W — enough to dry out the electrolyte over years and accelerate failure. Switching-supply caps are the #1 reason consumer electronics die at the 5–10 year mark.
• Voltage derating ignored. Class 2 ceramics lose 50–80% of their nominal capacitance at rated voltage due to dielectric saturation. A 10 µF X7R rated at 10 V might be only 2 µF at 9 V. Designers who don't read the DC bias curves blow stability margins.
• Inrush current. An empty large cap looks like a short circuit at t=0. A 4700 µF cap on a 24 V rail through 0.05 Ω draws 480 A peak — enough to weld switch contacts and trip breakers. Mitigated with NTC thermistors, soft-start FETs, or pre-charge resistors.
• Reverse polarity on electrolytics. Aluminum and tantalum electrolytics rely on a chemically formed oxide layer that only insulates one direction. Reverse them and the layer breaks down, the electrolyte boils, and the can vents (or, for tantalum, ignites). Always check the silkscreen polarity stripe.
• Mechanical cracking on ceramics. MLCCs are brittle and small; PCB flex during depanelization or in-field shock cracks them silently. Cracks short or open intermittently. Place chip caps perpendicular to PCB stress lines and avoid edge placement.
• Dielectric absorption. A discharged cap can spontaneously redevelop voltage as molecules in the dielectric relax — high in electrolytics, near-zero in C0G ceramics. Hazardous in high-voltage systems where shorted-out caps still bite hours later.

Where the curve shows up

• Power-on settling on every regulator output.
• RC oscillators (555 timer, neon-lamp relaxation oscillators).
• Sample-and-hold amplifiers in ADCs — the hold cap must charge to within ½ LSB before the converter samples.
• Camera flashes, defibrillators, rail-gun pulsers.
• Touchscreen capacitance sensing — a finger changes C, which shifts τ, which an MCU reads as a count.
• Soft-start circuits that ramp gate voltage on a power FET to avoid inrush.