Capacitance and transients
Predict first-order charge and discharge behavior to choose a timing or filtering capacitor without mistaking a transient model for an ideal delay.
- Calculate an RC time constant, estimate transient voltages, and diagnose a capacitor polarity, leakage, or connection fault from a measured curve.
Understand
The working idea
A capacitor stores separated charge and therefore energy in an electric field. Its defining relation is Q = CV: at a given voltage V, a larger capacitance C stores more charge Q. Current describes how quickly charge changes, so i = C dV/dt. A capacitor's voltage cannot jump instantaneously in the ideal model unless infinite current is available. This makes capacitors useful for smoothing noise, holding a reference briefly, and shaping timing.
In a resistor-capacitor circuit driven by a DC step from 0 to V_s, capacitor voltage is V_C(t) = V_s(1 − e^(−t/RC)). The time constant τ = RC has units Ω·F = s. After one τ, voltage reaches about 63% of its final value; after five τ, it is above 99% in the ideal model. During discharge from V_0 through R, V_C(t) = V_0e^(−t/RC). There is no sharp “finished” instant.
Suppose a sensor input chatters from short glitches. A 10 kΩ resistor with 10 µF gives τ = 0.10 s, which may suppress very brief events but also delays genuine changes. That latency is a build tradeoff, not free filtering. Real capacitors have tolerance, leakage, equivalent series resistance, and voltage limits. Polarized electrolytics must be oriented correctly.
A common misconception is that capacitance blocks all DC immediately. During charging, current flows; after settling in the ideal DC case, it approaches zero. Another failure is reading 100 µF as 100 F or assuming the nominal time constant is an exact measured delay. Use the curve, define a threshold, and account for component tolerance and source/load resistance. The associated RC simulation is a bounded ideal first-order model: it does not include every parasitic or instrument-loading effect.
Follow the reasoning
Worked example
A 47 kΩ resistor charges a 10 µF capacitor from a 5 V step. Find τ and ideal capacitor voltage after one τ and two τ.
- 1
Convert values: 47 kΩ = 47,000 Ω and 10 µF = 10×10^-6 F.
- 2
τ = RC = 47,000 × 10×10^-6 s = 0.47 s.
- 3
At t = τ, V_C = 5(1 − e^-1) ≈ 3.16 V.
- 4
At t = 2τ, V_C = 5(1 − e^-2) ≈ 4.32 V.
The time constant is 0.47 s; ideal voltage is about 3.16 V at 0.47 s and 4.32 V at 0.94 s.
Sanity checkBoth voltages lie between 0 and 5 V, increase with time, and approach rather than exceed the supply.
Try it yourself
Independent check
Move to the bench
Compare a measured low-voltage RC charge curve with the ideal time-constant prediction.
- 3×AA battery holder
- 47 kΩ resistor
- 10 µF electrolytic capacitor rated above 5 V
- multimeter
- breadboard
- timer
- jumper wires
Experiment
- With power absent, confirm capacitor polarity and build the series resistor-capacitor circuit; calculate τ from measured or nominal values.
- Discharge the capacitor through the resistor, then apply the battery and record actual capacitor voltage at 0, 0.5τ, 1τ, 2τ, and 5τ.
- Compare readings with ideal expected values and note meter timing and component tolerance.
- Remove power, discharge safely, then deliberately move the resistor so the capacitor has no charging path; repower and diagnose the flat or erratic trace as an open-path fault.
- Restore the path and repeat one checkpoint.
Observe
- Expected ideal values are about 39%, 63%, 86%, and 99% of final voltage at 0.5τ, 1τ, 2τ, and 5τ; actual readings must be kept distinct.
- Expected fault clue: an open charging path prevents the smooth rise and may leave a residual voltage determined by leakage and the meter.
Choose R or C to target a 50 ms time constant for suppressing a brief button glitch, then state the response delay that choice introduces.
- Use only the low-voltage battery holder and orient the electrolytic capacitor correctly.
- Remove power and discharge through the resistor before rewiring; never short a charged capacitor directly.
Keep evidence
Bench note
Record what you expected, what you actually measured or observed, and one difference worth investigating. This is self-reported evidence, not a certification.