RC time constant

When a resistor and capacitor are in series with a DC source, current is limited by R, and capacitor voltage is limited by how much charge has already piled up. Early on, little opposing voltage exists on C, so current is larger and voltage rises quickly. Later, C’s voltage nearly cancels the source, current shrinks, and the last few volts crawl in slowly.

That feedback creates the familiar exponential shape.

τ = R × C

• τ (tau) in seconds
• R in ohms
• C in farads

During each time constant on a charge toward a final voltage Vf, the capacitor voltage changes by about 63.2% of the remaining gap.

Example: charge toward 100 V from 0 V:

Discharge from 100 V toward 0 V falls by 63.2% of the remaining voltage each τ: 36.8 V after 1τ, 13.6 V after 2τ, and so on, near zero after 5τ.

Designers often treat 5τ as “fully charged” or “fully discharged.”

Example: C = 50 µF = 50 × 10⁻⁶ F, R = 100 kΩ = 100,000 Ω:

τ = 100,000 × 50×10⁻⁶ = 5 s

Total ≈ 5 × 5 = 25 s

If you need a total charge time of 0.2 s with C = 100 pF, first find one τ = 0.2 / 5 = 0.04 s, then:

R = τ / C = 0.04 / (100×10⁻¹²) = 400 MΩ

(Those numbers show how tiny capacitors need huge resistors for slow timing — timing circuits choose R and C together.)

Always convert µF, nF, pF to farads before multiplying. Mixing 50 “µF” as if it were 50 F is a classic exam and field calculation error.

• Soft-start and delay circuits
• Debounce and noise filtering (short τ)
• Camera flash charge indicators (large energy, visible seconds)
• Power-supply hold-up after AC loss
• Safety: knowing roughly how long a DC bus capacitor stays dangerous after shutdown

A 1000 µF bus capacitor with a high-value bleed resistor can stay charged for a long time. A low-value bleed resistor dumps energy faster but wastes heat continuously — designers choose the tradeoff.

The same τ governs both directions if R and C are the same. What changes is the path: charging through one resistor, discharging through another (or a short) changes the effective R and therefore τ.

If you discharge through a screwdriver (do not), R is tiny → τ is tiny → current spike is huge. That is why you discharge through a rated resistor, not a metal tool.

R=47 kΩ, C=100 µF → τ=4.7 s → ~23.5 s to practical full charge. R=1 MΩ, C=0.01 µF → τ=0.01 s → ~0.05 s full.

Bleed resistor example: 1000 µF bus with 100 kΩ bleed → τ=100 s → many minutes to be “safe enough.” Always meter before you trust the math alone.

Situation. A control board uses an RC delay so a contactor picks up about 2 seconds after a call for cooling. After a board repair, the delay is almost instant and the compressor short-cycles on pressure.

What happened. The tech replaced a 10 µF timing capacitor with 1 µF “because it looked the same.” With R unchanged, τ dropped by 10×. Five time constants became a blink.

Applied lesson. Timing is R × C. Changing either part changes the delay. Match capacitance and voltage rating; if R was also changed, recalculate τ.