8.3 RC Time Constants, Energy Storage & Testing

8.3 RC Time Constants, Energy Storage & Testing

Quick Answer: τ = R × C sets the speed of charging and discharging. Voltage changes exponentially — about 63% of the way to the final value in one τ, and essentially settled by ~5τ. Energy stored is W = ½CV². Test capacitors with proper instruments after safe resistive discharge; never rely on a casual short across large charged parts.

Static C = Q/V tells you how much charge a capacitor holds at a voltage. Aircraft and avionics circuits also care how fast that charge moves when R and C interact, how much energy is banked, and how to prove a part is healthy without getting shocked.

The Time Constant τ = RC

In a series RC circuit:

τ = R × C

Worked example 1 — find τ. R = 10 kΩ, C = 47 µF.

τ = 10 000 × 47 × 10⁻⁶ = 0.47 s.

Worked example 2 — find C from τ. A designer wants τ = 2.0 ms with R = 4.7 kΩ.

C = τ / R = 0.002 / 4700 ≈ 4.26 × 10⁻⁷ F ≈ 0.43 µF (nearest preferred value in practice).

Larger R or larger C → larger τ → slower exponential.

Exponential Charge (DC Series RC)

Assume an uncharged capacitor, series R, and a DC step of voltage V_S applied at t = 0.

• Capacitor voltage: v_C(t) = V_S (1 − e^(−t/τ))
• Charging current starts at I₀ = V_S / R and falls toward zero: i(t) = (V_S / R) e^(−t/τ)

Key percentages Module 3 expects:

Worked example 3 — voltage at 1τ. V_S = 28 V, τ = 0.47 s. At t = τ:

v_C ≈ 0.63 × 28 ≈ 17.6 V.

Worked example 4 — nearly charged. Same circuit: after 5τ ≈ 2.35 s, v_C is within about 1% of 28 V — call it charged for exam “fully charged” language unless the stem demands the exponential formula.

Exponential Discharge

A capacitor initially charged to V₀, then connected across R alone (source removed):

v_C(t) = V₀ e^(−t/τ)

Worked example 5 — discharge. V₀ = 50 V, R = 100 kΩ, C = 10 µF → τ = 1.0 s. After 1 s, v_C ≈ 0.37 × 50 ≈ 18.5 V. After 5 s, only about 0.3–0.5 V remains in the ideal model.

Exam trap: “63%” on charge means up toward final; on discharge, voltage falls by about 63% in one τ (remaining ≈ 37%). Read the stem carefully.

Energy Storage: W = ½ C V²

The energy stored in the electric field of a capacitor is:

W = ½ C V²

Also W = ½ Q V = Q² / (2C) with Q = CV.

Worked example 6 — energy. C = 100 µF charged to 50 V.

W = 0.5 × 100 × 10⁻⁶ × 50² = 0.5 × 100 × 10⁻⁶ × 2500 = 0.125 J.

Worked example 7 — voltage effect. Same 100 µF at 100 V: W = 0.5 × 100 × 10⁻⁶ × 10 000 = 0.5 J — four times the energy at twice the voltage, because energy scales with V².

That V² dependence is why high-voltage filter capacitors are hazardous even at modest microfarad values: energy — and shock/arc risk — climb quickly with voltage.

Testing Capacitors

Typical Module 3 / workshop checks:

1. Visual inspection — bulging electrolytic cans, leaked electrolyte, cracked ceramics, burnt smell, damaged leads.
2. Discharge first — before connecting meters or handling terminals (see safety below).
3. Capacitance measurement — LCR meter or capacitance function; compare to marked value within tolerance.
4. Resistance / insulation / leakage — a good capacitor (discharged) does not look like a steady low-resistance short on a simple ohmmeter after the brief charging kick; electrolytics show more complex behaviour. Excessive DC leakage is a fail.
5. ESR (equivalent series resistance) — elevated ESR on electrolytics often means “drying out” or degradation even if capacitance is still near nominal.
6. Substitution / in-circuit caution — parallel paths can fool meters; lift a lead when required by procedure.

Ohmmeter behaviour (idealised teaching point): On a discharged good capacitor, an analogue ohmmeter may kick as the meter charges C, then the needle drifts toward high resistance. A shorted capacitor reads low ohms continuously. An open capacitor may show no charging kick and infinite resistance. Confirm with a proper capacitance instrument when marks matter.

Safety Discharge

Capacitors can retain dangerous charge long after power is removed — especially large electrolytics on power supplies and high-voltage assemblies.

Safe practice:

• Assume charged until proven otherwise.
• Discharge through a resistor of appropriate value and wattage (and insulated probes), not by casually shorting terminals with a screwdriver (arc, pitted terminals, tool damage, and hazard).
• For large energy, use a documented discharge tool/procedure; verify with a voltmeter that voltage is near zero.
• Repeat verification — dielectric absorption can let voltage “reappear” slightly on some capacitors after an initial discharge.
• Wear required PPE; follow aircraft/equipment maintenance manuals for high-voltage assemblies.

Worked safety estimate. A 470 µF capacitor at 300 V stores W = 0.5 × 470 × 10⁻⁶ × 90 000 ≈ 21 J — enough for a painful shock and a substantial arc. Treat it with respect.

Bleed resistors across high-voltage filter capacitors in equipment provide intentional slow discharge when power is removed; still verify with a meter before touching.

Putting τ, Energy, and Testing Together

Master τ = RC, the 63%/37%/5τ rules, W = ½CV², and discharge-before-test discipline. That closes CAAS Module 3 topic 3.9 Capacitance/Capacitor and prepares you for later AC capacitive reactance work in R, L, C circuit topics.