Capacitance and Inductance
Part of Electrical Theory
How capacitors and inductors store energy and oppose changes in voltage and current—the reactive elements that make AC circuits fundamentally different from DC.
Why This Matters
Every real electrical circuit contains capacitance and inductance, whether intended or not. Wires running parallel to each other have capacitance between them. Any coil of wire has inductance. Every motor winding is an inductor; every length of cable is a combination of resistance, inductance, and capacitance.
Understanding these reactive elements transforms AC circuit analysis from mystery to prediction. Why does a motor draw more current at startup than at full speed? Inductive behavior. Why does a power supply voltage fluctuate less when a capacitor is added? Because the capacitor stores charge and releases it to fill valleys. Why does a simple coil and capacitor circuit “ring”? Because energy sloshes back and forth between the magnetic field of the inductor and the electric field of the capacitor.
These phenomena have direct practical consequences for building power supplies, motor controls, radio circuits, and filtering systems. Capacitance and inductance are not complications to avoid—they are tools to use deliberately.
Capacitance: Storing Electric Fields
A capacitor stores energy in an electric field between two conductors separated by an insulator. The stored energy is:
E = ½ × C × V²
Where C is capacitance in farads and V is voltage.
The defining equation: I = C × dV/dt
In words: the current into a capacitor equals the capacitance times the rate of change of voltage. This has profound implications:
- At DC (constant voltage, dV/dt = 0): no current flows. A capacitor is an open circuit to DC.
- At AC (continuously changing voltage): current flows continuously. The higher the frequency, the larger the dV/dt, and the more current flows for a given voltage.
Capacitive reactance: X_C = 1/(2πfC)
Capacitive reactance decreases with increasing frequency. A capacitor passes high-frequency signals more readily than low-frequency ones—the basis of high-pass filters.
Time constant in DC circuits: When a capacitor charges through a resistor, it does not charge instantly. The voltage follows: V(t) = V_supply × (1 - e^(-t/RC))
The time constant τ = RC (seconds = ohms × farads). After one time constant, the capacitor has charged to 63% of the final voltage. After 5τ, it is essentially fully charged (>99%).
This RC timing is used in delay circuits, oscillators, and debouncing switches (a capacitor absorbs the multiple contact bounces of a mechanical switch, presenting only the first clean transition to subsequent circuitry).
Inductance: Storing Magnetic Fields
An inductor stores energy in a magnetic field around a current-carrying coil. The stored energy is:
E = ½ × L × I²
Where L is inductance in henries and I is current.
The defining equation: V = L × dI/dt
In words: the voltage across an inductor equals the inductance times the rate of change of current. Implications:
- At DC (constant current, dI/dt = 0): no voltage develops. An ideal inductor is a short circuit to DC.
- When current is suddenly interrupted, dI/dt becomes very large, and the voltage spike is enormous. This is why switching off an inductive load (motor, relay coil) without protection causes damaging voltage spikes.
- An inductor resists changes in current—it is a “current stabilizer.”
Inductive reactance: X_L = 2πfL
Inductive reactance increases with frequency. An inductor passes low-frequency signals readily but opposes high frequencies—the basis of low-pass filters.
Time constant in DC circuits: When current builds through an inductor and resistor in series: I(t) = (V/R) × (1 - e^(-t × R/L))
The time constant τ = L/R (seconds = henries / ohms). After one time constant, current has reached 63% of its final value V/R. After 5τ, current is essentially at steady state.
Energy Comparison
| Storage element | Energy stored | Responding to… |
|---|---|---|
| Capacitor | ½CV² (electric field) | Voltage changes |
| Inductor | ½LI² (magnetic field) | Current changes |
The duality is exact: every capacitor formula has an inductor analog with C↔L and V↔I swapped.
The LC Oscillator: Energy Sloshing
Connect a capacitor and inductor with no resistance. Charge the capacitor; then connect it to the inductor. What happens?
- Charged capacitor begins to discharge through inductor
- Current builds in the inductor, creating a magnetic field; capacitor loses charge
- Capacitor fully discharged; inductor current at maximum; energy stored as magnetic field
- Magnetic field collapses; drives current into capacitor in opposite direction; capacitor charges with reversed polarity
- Inductor current falls to zero; capacitor fully recharged in reverse
- Process reverses; cycle repeats
This is electrical resonance. The frequency of oscillation: f = 1/(2π√(LC))
In a real LC circuit, resistance dissipates energy each cycle, and oscillations gradually decay. With a source of energy to compensate (a transistor, a mechanical interrupter, or feedback from the oscillation itself), sustained oscillations can be maintained—this is the basis of all electronic oscillators and radio transmitters.
Variable tuning: Adjusting either C or L changes the resonant frequency. A variable capacitor (adjustable plate spacing or area) is the traditional radio tuning control—turning it adjusts L/(2π√(LC)) to match a desired station’s frequency.
Real-World Capacitance and Inductance
Every electrical structure has some capacitance and inductance whether desired or not:
Parasitic capacitance:
- Between adjacent wires in a cable: approximately 50–100 pF per meter for typical wire spacing
- Between a wound coil and its core: degrades high-frequency performance
- Across the contacts of a switch: causes the switch to pass high-frequency signals even when “open”
Parasitic inductance:
- In any wire, especially long straight runs: typically 1–2 µH per meter
- Inside resistors (wire-wound resistors are particularly problematic at RF)
- In solder joints and connection leads
At power frequencies (50/60 Hz), parasitic effects are negligible for practical wiring. At radio frequencies (>1 MHz), parasitic effects dominate and must be considered in any design.
Practical Filters
Capacitance and inductance create frequency-selective circuits:
Low-pass filter (series inductor + shunt capacitor): Passes low frequencies, blocks high frequencies. Used to remove switching noise from power supplies, protect audio amplifiers from RF interference.
High-pass filter (series capacitor + shunt inductor): Passes high frequencies, blocks low. Used to remove DC and low-frequency hum from signal circuits.
Band-pass filter (LC resonant circuit): Passes a narrow band of frequencies around the resonant frequency. The fundamental circuit of all radio receivers.
Band-stop (notch) filter: Rejects a narrow frequency band while passing all others. Used to eliminate a specific interference source.
These filter functions are the reason that AC circuits with inductors and capacitors can do things that purely resistive DC circuits cannot—they process signals differently at different frequencies.