Robotics · Electronics basics · Grade 7

Capacitor: charging and the RC time

Why does a toy’s light fade slowly after it is switched off? Where does τ = RC come from?

The situation

A small toy car has a headlight. If you switch it off, the light does not go out at once — for a second or two it fades slowly. Open it up and next to the LED there is a small blue “barrel” — a capacitor. How does it store charge, why does it give it back slowly, what is “RC time”? On a breadboard, charge a capacitor through a resistor, watch it with the multimeter and the graph, make an LED “fade out slowly” — and then connect one the wrong way round on purpose and see what happens.

Topic

Capacitor: charging and the RC time. The lamp stays lit for a moment after switching off: capacitor charge and τ = RC

Key ideas

Condenser
τ = RC
Energy CU²/2

Formulas

q = C \cdot U

Charge is proportional to the capacitance and the voltage: 1 F = 1 C / 1 V. Ordinary capacitors are measured in microfarads (1 µF = 10⁻⁶ F).

τ = R \cdot C

Time constant: when charging, the voltage reaches 63 % of the source in time $τ$ , when discharging it falls to 37 %; after about 5τ the process is practically over.

u_{C} (t) = U (1 - e^{- t / τ}), u_{C} (t) = U_{0} e^{- t / τ}

The charging (left) and discharging (right) curves: the current is large at first and falls as the capacitor fills.

W = \frac{C U^{2}}{2}

Energy: a capacitor stores it in the electric field; 1000 µF, 9 V → 40 mJ.

C_{par} = C_{1} + C_{2}, \frac{1}{C_{ser}} = \frac{1}{C_{1}} + \frac{1}{C_{2}}

In a parallel connection the capacitances add; in a series connection the total capacitance falls and the voltage divides in inverse proportion to the capacitance: $U_{1} / U_{2} = C_{2} / C_{1}$ .

Theory

What a capacitor is

Two metal plates with a thin dielectric (an insulator) between them. Connected to a battery, one plate collects positive charge and the other negative: $q = C U$ . The capacitance $C$ is how much charge fits per volt; the unit is the farad, in practice the microfarad (µF) and the nanofarad (nF). There are two kinds: electrolytic — large capacitance (10 µF to thousands of µF), a blue or black “barrel”, polarised: the white striped band on its side marks the minus leg, and the body shows the capacitance and the voltage rating (6.3 V, 16 V, 25 V); ceramic — a small flat disc, small capacitance (100 nF — “104”), non-polarised, fitted either way round.

Charging and τ = RC

When the switch is connected to the battery, current flows through the resistor into the capacitor. At first the capacitor is empty — the current is greatest: $I_{0} = U / R$ . As the capacitor fills, its voltage rises, less is left across the resistor, and the current falls — so the curve is steep at first and then flat. The time constant $τ = R C$ : 10 kΩ · 100 µF = $10^{4} \cdot 10^{- 4}$ = 1 s. After $τ$ the voltage reaches 63 %, after $3 τ$ 95 %, after 5τ 99 % — we call the capacitor full. Make the resistor 10 times larger and it is 10 times slower; the same with the capacitance. The multimeter (V⎓, 10 MΩ) is connected in parallel with the capacitor and hardly affects the circuit.

Discharging and the toy’s light

If the switch is moved to the position that connects the resistor to ground, the capacitor discharges through that resistor: in time $τ$ the voltage falls to 37 %. In the toy, in place of the resistor there is an LED with its resistor: after switching off, the LED gets its current not from the battery but from the capacitor — and fades slowly. But the LED has a threshold: when the voltage falls to $U_{L E D}$ ≈ 1.9 V the current stops, it does not “wait” until the end. The fading time is roughly $t \approx R C \cdot \ln \frac{U_{0} - U_{L E D}}{I_{\min} R}$ : 470 Ω, 1000 µF, 4.5 V give ≈ 1.1 s; a larger capacitance or resistor — longer.

Energy

The energy in a capacitor is $W = C U^{2} / 2$ : 100 µF, 4.5 V — only 1 mJ; 1000 µF, 9 V — 40 mJ. This energy keeps the toy’s LED lit for a second or two. In a camera flash the capacitor is charged to about 300 V and discharges in an instant — several joules, which is why it is dangerous. If the voltage doubles, the energy grows four times.

Polarity, the voltage rating and series connection

If you connect an electrolytic capacitor the wrong way round (the stripe on the plus side), the electrolyte inside heats up, gas is released, the can swells and bursts — within a few seconds. The same happens above the voltage rating printed on the body: a 6.3 V capacitor must not be connected to 9 V; the rule is a rating at least 1.5 times the working voltage (9 V → 16 V). In a series connection the same charge passes through both capacitors, $U = q / C$ — the voltage falls more across the smaller capacitance: 100 µF + 470 µF, 9 V → 7.4 V and 1.6 V. Two 6.3 V capacitors in series “survive” 9 V only if their capacitances are equal. In a parallel connection the capacitances add and the voltage is the same.

About the model

The circuit is solved by the nodal method in small time steps (at least 1/40 of τ). The capacitor is ideal: leakage current and internal resistance are ignored (a real electrolytic leaks a few µA, and a charge left for hours slowly disappears). Swelling is a simple model: ≈ 2.5 s when reversed, proportional to the excess when the rating is exceeded; real capacitors differ, but this is the order. A burst capacitor is treated as open (a break). The LED is a piecewise model (1.9 V, then 2 Ω), and it is “out” when the current is below 0.5 mA. The voltmeter is 10 MΩ.

Did you know?

The first capacitor — the Leyden jar — was made in 1745–1746 independently by two people: Ewald von Kleist in Germany and Pieter van Musschenbroek in Leiden in the Netherlands; this glass vessel was coated with metal inside and outside.
The unit of capacitance, the farad, is named after Michael Faraday; 1 F is very large: ordinary electrolytic capacitors are measured in microfarads. Modern supercapacitors (ionistors), however, have capacitances of thousands of farads: in trams and buses they store braking energy and return it when pulling away.

Inside the experiment

A real board in a hand-inked isometric drawing: turn it, place the parts, wire the circuit, write and upload the code — current and voltage are computed at once, and a wrong connection burns a part. Inside are missions, theory, a quiz and a guided tour.

More experiments in this section

The electric circuit: a torch — Grades 5–6
Breadboard: rows — Grades 5–6
LED and resistor: Ohm’s law — Grade 7
Resistors: the colour code — Grade 7
Voltage divider and potentiometer — Grade 7
Diode and rectifier — Grade 8
The transistor — a switch — Grade 8
Measuring with a multimeter — Grade 8