Capacitor Theory
Part of Electrical Theory
The physics of how capacitors store charge and energy, and how this behavior enables filtering, timing, and energy storage applications.
Why This Matters
Capacitors are one of the three fundamental passive circuit elements (alongside resistors and inductors). They appear in virtually every functional circuit: power supply filters smooth voltage fluctuations, timing circuits use RC time constants, coupling capacitors pass AC signals while blocking DC, and motor-start capacitors create the phase shift needed to start single-phase motors.
Understanding capacitor theory—not just the formulas but the physical picture of what is actually happening—enables confident circuit design and troubleshooting. Why does a power supply voltage drop when a heavy load is suddenly applied? Because the capacitor’s stored charge is depleted faster than the source can replenish it. Why does a coupling capacitor block DC? Because at DC, current stops flowing once the capacitor is fully charged and the voltage gradient across the dielectric collapses. These answers come from understanding the physics.
The Physical Picture
A capacitor is two conductors (plates) separated by an insulator (dielectric). When connected to a voltage source:
- The positive terminal of the source pulls electrons away from the near plate, leaving it positively charged
- The negative terminal pushes electrons onto the other plate, leaving it negatively charged
- An electric field exists between the two plates, pointing from positive to negative
- Energy is stored in this electric field
The charge cannot flow through the insulator—but it accumulates on the plates. As more charge accumulates, the electric field grows stronger and opposes further charge accumulation. Eventually the capacitor voltage equals the source voltage and current stops.
The Capacitance Formula
Q = C × V
Where:
- Q = charge stored (coulombs)
- C = capacitance (farads)
- V = voltage across the capacitor
The farad is an enormous unit—a 1 farad capacitor charged to 1 volt holds 1 coulomb of charge (6.24 × 10¹⁸ electrons). Practical capacitors are measured in microfarads (µF = 10⁻⁶ F), nanofarads (nF = 10⁻⁹ F), or picofarads (pF = 10⁻¹² F).
Physical factors determining capacitance: C = ε₀ × ε_r × A / d
Where:
- ε₀ = permittivity of free space (8.85 × 10⁻¹² F/m)
- ε_r = relative permittivity of the dielectric material (dimensionless)
- A = area of the plates (m²)
- d = separation between plates (m)
Design implications:
- Larger plate area → more capacitance (proportional)
- Closer plate spacing → more capacitance (inversely proportional)
- Better dielectric (higher ε_r) → more capacitance
This is why commercial capacitors use very thin dielectrics, and why high-value capacitors are physically large.
Dielectric Materials
The dielectric is the insulating material between the plates. Its permittivity determines how much capacitance a given plate area and separation provides.
| Dielectric | ε_r | Notes |
|---|---|---|
| Vacuum | 1.0 | Reference |
| Air | 1.0006 | Effectively same as vacuum |
| Paper | 2–4 | Traditional, inexpensive |
| Waxed paper | 2.5–3.5 | Waterproofed paper |
| Mica | 5–8 | Stable, precise |
| Glass | 4–10 | Available, stable |
| Beeswax | 2.4 | Locally producible |
| Ceramic | 10–10,000 | Very high values in small size |
| Electrolytic (aluminum oxide) | ~10 but very thin | Allows very large capacitance |
Dielectric strength: The maximum electric field the dielectric can withstand before it breaks down (conducts). Breakdown permanently damages most dielectrics.
Air: ~3 MV/m (1mm air gap withstands ~3kV) Paper: ~15 MV/m Mica: ~100 MV/m Glass: ~10–30 MV/m
The working voltage of a capacitor must stay below the breakdown voltage of its dielectric, with a safety margin of at least 50%.
Energy Storage
The energy stored in a charged capacitor:
E = ½ × C × V²
Note the voltage squared: doubling the voltage quadruples the stored energy. This is why high-voltage capacitors can store dangerous amounts of energy even when the capacitance is small.
Example: A 100µF capacitor charged to 400V: E = ½ × 100×10⁻⁶ × 400² = ½ × 100×10⁻⁶ × 160,000 = 8 joules
8 joules delivered in a millisecond (typical discharge time) is 8,000 watts—potentially lethal. High-voltage capacitors must be treated with respect even when the power source is disconnected.
Power delivery: A capacitor can release its stored energy very quickly—limited mainly by the resistance in the discharge path. This makes capacitors excellent for supplying brief high-current pulses: camera flashes, ignition systems, and spot welders all use capacitor discharge.
Charging and Discharging Through Resistance
When charging a capacitor through a resistor from a voltage source V:
V_C(t) = V × (1 - e^(-t/RC))
The current during charging: I(t) = (V/R) × e^(-t/RC)
When discharging through a resistor: V_C(t) = V₀ × e^(-t/RC)
The time constant τ = RC determines how quickly the charging or discharging occurs.
| Time elapsed | Charge (fraction of final) | Discharge (fraction remaining) |
|---|---|---|
| 1τ | 63.2% | 36.8% |
| 2τ | 86.5% | 13.5% |
| 3τ | 95.0% | 5.0% |
| 5τ | 99.3% | 0.7% |
Practical implications:
RC timing: A 1µF capacitor and 1MΩ resistor gives τ = 1 second—a convenient timing interval. Vary R or C to set the time constant for any application.
Power supply filtering: A large capacitor on a power supply output reduces voltage ripple. The time constant RC (where R is the load resistance) must be much larger than the period of the ripple to smooth it effectively.
Decoupling: Capacitors placed close to circuit loads absorb momentary current demands, preventing the supply voltage from drooping. The capacitor supplies current during the brief interval before the source responds.
Capacitors in Series and Parallel
Parallel capacitors: Total capacitance = C₁ + C₂ + C₃ + … All share the same voltage. Capacitances add.
Series capacitors: 1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + … The series combination has less capacitance than the smallest individual capacitor, but it can withstand higher voltage (each capacitor withstands only a fraction of the total voltage).
Mismatched capacitors in series do not share voltage equally—the capacitor with less capacitance gets more voltage across it. For high-voltage series strings, equalizing resistors are added in parallel with each capacitor to force equal voltage sharing.
Current-Voltage Relationship in AC
At AC, the capacitor current leads the voltage by 90°: the current is maximum when the voltage is changing fastest (at zero crossing), and zero when the voltage is at its peak (rate of change = 0).
This 90° phase lead is a key diagnostic: measuring the phase angle between current and voltage in an AC circuit identifies whether the dominant reactive element is capacitive (current leads) or inductive (current lags).
For a purely capacitive circuit:
- Real power (watts) = 0 (no energy dissipated)
- Reactive power (vars) = V × I
- Power factor = 0
In practice, real capacitors have some series resistance (ESR) and leakage current through the dielectric. Good capacitors have ESR of milliohms and leakage of microamps—negligible for most purposes. Degraded or improperly made capacitors show higher ESR (causing heat) and higher leakage (reducing effective capacitance and causing charge to leak away over time).