Capacitors and inductors in DC circuits

Resistors turn electrical energy into heat. Capacitors and inductors do something different: they store energy, a capacitor in an electric field and an inductor in a magnetic field, and give it back later. This makes them behave in time-dependent ways that resistors do not: a capacitor resists sudden changes in voltage, and an inductor resists sudden changes in current. In DC circuits, these effects show up when a circuit is switched on or off. They explain why drive and power supply capacitors stay dangerously charged after power is removed, why relay coils produce voltage spikes that destroy electronics, how timers and delays work, and why motor starting currents and contact arcing occur. This lesson introduces both components, their time constants and stored energy, and the safety practices they require. ELEC-210 extends them to AC.

Capacitors

A capacitor consists of two conducting plates separated by an insulator called the dielectric. When a voltage is applied, charge flows onto one plate and off the other until the voltage across the capacitor equals the supply. The amount of charge stored per volt is the capacitance, measured in farads (F):

One farad is very large; practical capacitors are rated in microfarads (µF, millionths), nanofarads (nF) and picofarads (pF), although supercapacitors reach thousands of farads at low voltages. Capacitance increases with plate area and dielectric quality and decreases with plate spacing. Every capacitor also has a voltage rating; exceeding it can puncture the dielectric and cause failure, sometimes violently.

Types include ceramic capacitors (small values, electronics), film capacitors (motor run capacitors, power factor correction, drive DC links), electrolytic capacitors (large values in small packages, power supplies and drives, polarized so they must be connected the right way round), and supercapacitors (energy storage for short bursts). Motor start and run capacitors used with single-phase motors are AC capacitors (ELEC-210).

Charging and discharging: the RC time constant

When a capacitor charges through a resistor, the current is largest at first (the empty capacitor offers no opposing voltage) and falls as the capacitor voltage rises. The voltage rises along an exponential curve. The time constant is:

In one time constant, the capacitor charges to about 63% of the supply voltage; after five time constants it is over 99% charged and considered fully charged. Discharging follows the same curve in reverse: after one time constant, 37% of the voltage remains; after five, under 1%.

Capacitors in series and parallel

Capacitors combine opposite to resistors. In parallel, the plate areas effectively add, so capacitances add: \(C_T = C_1 + C_2 + \cdots\). In series, the effective spacing increases, so capacitance falls: \(1/C_T = 1/C_1 + 1/C_2 + \cdots\). Capacitors are connected in series to handle higher voltages (with balancing resistors so the voltage divides evenly), as in drive DC links where two banks rated 450 V are connected in series across a 680 V bus.

Inductors

An inductor is a coil of wire, often wound on an iron or ferrite core. When current through it changes, the changing magnetic field induces a voltage that opposes the change (Lesson 9, Lenz's law). Inductance, measured in henries (H), sets how much voltage is induced for a given rate of change of current:

An inductor therefore resists sudden changes in current. With steady DC, the current is constant, no voltage is induced, and the inductor behaves like an ordinary conductor with only its winding resistance. Inductors are found as chokes in drives and power supplies, in relay, contactor and solenoid coils, in motor windings, in ballasts and in filters. Inductance increases with the number of turns squared and with an iron core.

An inductor stores energy in its magnetic field:

The RL time constant

When a DC voltage is applied to an inductor in series with a resistance, the current cannot jump instantly; it rises exponentially toward its final value \(V/R\) with time constant:

After one time constant the current reaches 63% of its final value, and after five it is essentially complete. This delay is why contactors take a few milliseconds to pull in and why large DC motor fields take time to build up.

Inductive kick: switching off a coil

Interrupting the current in an inductor is the dangerous part. The current tries to keep flowing, and because \(v = L\, \Delta i/\Delta t\), a very fast interruption induces a very high voltage, often hundreds or thousands of volts from a 24 V coil. This "inductive kick" causes arcing across switch contacts, erodes relay contacts, destroys transistors and PLC outputs, and can give a nasty shock when disconnecting a coil or a motor field.

The standard remedies give the coil current somewhere to go when it is switched off:

• Flyback (freewheeling) diode across DC coils, connected so it does not conduct normally but carries the coil's current when the switch opens, limiting the spike to about 1 V. It slows the coil's release slightly.
• Resistor-capacitor (RC) snubbers or metal oxide varistors across AC or DC coils.
• Transient voltage suppression diodes for faster release with limited spike.

Many relays and contactors are available with suppression modules built in, and PLC manufacturers specify suppression for inductive loads on their outputs (ELEC-330).

Capacitors and inductors compared

Safe work with stored energy

Stored electrical energy is a recognized hazard in electrical safety standards. Before working on equipment with capacitors, such as drives, UPS systems, power supplies, power factor correction banks and medium-voltage equipment:

• Isolate and lock out the supply.
• Wait the time marked by the manufacturer.
• Verify the absence of voltage with a properly rated meter, measuring across the DC bus or capacitor terminals as well as the input terminals.
• Discharge capacitors through an appropriate resistor tool if required; never short them with a screwdriver.
• For high-voltage capacitor banks, apply grounding and shorting devices before contact.

NFPA 70E requires stored energy, including capacitors, to be considered in the risk assessment and in establishing an electrically safe work condition.