Capacitors: Charge, Discharge, and RC Time Constants

What Is a Capacitor?

A capacitor is a passive component that stores electrical energy in an electric field between two conductive plates separated by an insulating material called the dielectric. The ability to store charge is measured in farads (F). One farad is a very large unit; practical capacitors are rated in microfarads (µF, 10⁻⁶ F), nanofarads (nF, 10⁻⁹ F), or picofarads (pF, 10⁻¹² F).

The relationship between charge, capacitance, and voltage is:

Q = C × V

where Q is charge in coulombs, C is capacitance in farads, and V is voltage in volts.

Dielectric Types and Capacitor Markings

The dielectric material determines a capacitor's characteristics, maximum voltage, and typical applications:

• Electrolytic (aluminum or tantalum): High capacitance in a small package; polarized — the positive lead (longer leg, marked +) must connect to the more positive voltage. Reverse polarity can rupture or explode the component.
• Film (polyester, polypropylene): Non-polarized; stable and low-leakage; common in audio and timing circuits.
• Ceramic: Very small values; non-polarized; used in high-frequency bypass applications.
• Mica and glass: Precision, low-loss; used in RF and precision circuits.

Reading capacitor markings: Large electrolytic capacitors print the value directly (e.g., 47 µF, 25 V). Smaller film and ceramic capacitors often use a three-digit code: the first two digits are the significant figures and the third is the multiplier (number of zeros to add), with the result in picofarads. For example, 104 means 10 × 10⁴ pF = 100,000 pF = 100 nF = 0.1 µF. A voltage rating is also marked; never exceed it.

Capacitor Charge and Discharge

When a DC voltage is applied to a capacitor through a resistor, the capacitor does not charge instantly. Current flows, charging the capacitor until the voltage across it equals the supply voltage. During charging, current is highest at the start (when the capacitor is empty) and drops exponentially as the capacitor fills. During discharging, the capacitor acts as a temporary voltage source, pushing current through the resistor until fully discharged.

This behavior follows an exponential curve, not a straight line. The voltage across a charging capacitor at any time t is:

Vₜ = V_s × (1 − e^(−t/τ))

And during discharge:

Vₜ = V₀ × e^(−t/τ)

where V_s is the supply voltage, V₀ is the initial voltage, e is Euler's number (≈ 2.718), t is time in seconds, and τ (tau) is the RC time constant.

The RC Time Constant

The time constant τ (Greek letter tau) defines how quickly an RC circuit charges or discharges:

τ = R × C

where R is in ohms and C is in farads; τ is in seconds.

After one time constant (1τ), the capacitor has charged to approximately 63.2% of the supply voltage (or discharged to about 36.8% of its starting voltage). The universal time-constant curve summarizes this:

| Time | Charge (% of V_s) | Discharge (% of V₀) |

|------|-------------------|----------------------|

| 1τ | 63.2% | 36.8% |

| 2τ | 86.5% | 13.5% |

| 3τ | 95.0% | 5.0% |

| 4τ | 98.2% | 1.8% |

| 5τ | 99.3% | 0.7% |

A capacitor is considered fully charged or discharged after 5τ for practical purposes.

Using the Oscilloscope to Measure τ

A digital oscilloscope displays voltage versus time. For this lab, a square-wave function generator provides repeating charge/discharge cycles — ideal for capturing the exponential curve. Key oscilloscope controls you will use:

• Time/div: Sets the horizontal scale. Choose a value so that at least one full charge or discharge curve is visible across the screen.
• Volts/div: Sets the vertical scale. Adjust so the waveform fills most of the screen without clipping.
• Trigger: Stabilizes the waveform. Use edge triggering on Channel 1 at the midpoint of the signal.
• Cursors: Two vertical (time) cursors allow you to mark two points on the waveform and read the time difference directly from the cursor readout.

Measurement method: Place cursor 1 at the point where the charging curve begins (the rising edge of the square wave). Place cursor 2 at the point where the voltage reaches approximately 63.2% of the final value. The time difference (Δt) displayed is your measured τ.

Safety Points

1. Electrolytic capacitor polarity: Always verify the positive terminal before applying power. A reversed electrolytic capacitor can vent or burst violently. Keep face and hands clear when first powering the circuit.

2. Capacitor voltage rating: Never apply a voltage greater than the capacitor's rated working voltage. Failure can be sudden and forceful.

3. Discharge before handling: Even after power is removed, a capacitor can hold a charge. Short the leads through a resistor (not a direct short) before touching terminals, especially on larger capacitors.

4. Bench grounding and ESD: Use an ESD wrist strap when handling sensitive components. Keep the work area dry.

5. Function generator output: Keep output amplitude within the voltage rating of your components. Confirm settings before connecting.