Inductance and capacitance

Resistance is not the only thing that opposes current in an AC circuit. Every coil, from a contactor coil to a motor winding or a transformer, has inductance, which opposes changes in current by storing energy in a magnetic field. Every pair of conductors separated by insulation, from a power factor capacitor to a long cable, has capacitance, which opposes changes in voltage by storing energy in an electric field. In AC circuits, where current and voltage change continuously, these effects produce reactance, an opposition to current that depends on frequency, and shift the phase of current relative to voltage. Understanding them explains why a contactor coil burns out on DC, why capacitors can be deadly after the power is off, why motors have poor power factor, and how filters and power factor correction work.

Inductance

A current in a conductor produces a magnetic field around it, and winding the conductor into a coil concentrates the field, especially with an iron core. When the current changes, the changing magnetic field induces a voltage in the coil itself (self-induction). By Lenz's law, the induced voltage opposes the change that causes it: it resists a rising current and tries to maintain a falling one. The property is inductance \(L\), measured in henries (H):

A coil has an inductance of 1 H if a current changing at 1 A/s induces 1 V. Inductance increases with the number of turns (as the square), the cross-sectional area of the coil, and the permeability of the core, which is why iron-cored coils have far more inductance than air-cored ones of the same size. Practical values range from microhenries in electronic circuits to several henries in the coils of large relays and contactors.

Inductive kick

Because an inductor resists changes in current, interrupting the current suddenly, by opening a switch or contact, produces a very large induced voltage as the magnetic field collapses. This inductive kick can reach thousands of volts across the opening contacts, causing arcing, pitting and damage to electronic switches. It is why relay and contactor coils driven by electronic outputs need suppression: a diode across DC coils, or an RC snubber or metal-oxide varistor across AC coils. The same principle is used deliberately in vehicle ignition coils and in electric fences.

Energy stored

An inductor carrying current \(I\) stores energy in its magnetic field, \(W = \tfrac12 LI^2\). This energy must go somewhere when the current is interrupted, which is the source of the inductive kick and of the arc when a large motor circuit is opened under load.

Inductive reactance

With AC applied, the current in an inductor changes continuously, so the inductor continuously opposes it. The opposition is the inductive reactance \(X_L\), in ohms:

Reactance is proportional to frequency: the faster the current changes, the more an inductor opposes it. At DC (\(f = 0\)), an ideal inductor has zero reactance and only its winding resistance limits current. In a purely inductive circuit, \(I = V/X_L\), and the current lags the voltage by exactly 90°, because the induced voltage is greatest when the current is changing fastest, which is as the current passes through zero.

Resistance and inductance in series

Real coils have both resistance and inductance, and many circuits, such as a motor or a fluorescent ballast with a lamp, behave as a resistance in series with an inductance. The same current flows through both; the voltage across the resistance is in phase with the current, while the voltage across the inductance leads the current by 90°. The source voltage is the phasor sum, so the voltages do not add arithmetically. The total opposition is the impedance \(Z\):

The relationship forms an impedance triangle with \(R\) along the base, \(X_L\) vertical and \(Z\) as the hypotenuse; the angle \(\theta\) is the angle by which the current lags the source voltage.

Capacitance

A capacitor consists of two conductive plates separated by an insulator, the dielectric. When a voltage is applied, charge accumulates on the plates, positive on one and negative on the other, creating an electric field in the dielectric that stores energy. Capacitance \(C\), in farads (F), is the charge stored per volt, \(C = Q/V\). It increases with plate area and the permittivity of the dielectric, and decreases with the separation between plates. A farad is very large; practical capacitors are rated in microfarads (µF), nanofarads (nF) and picofarads (pF), although supercapacitors reach thousands of farads.

The current into a capacitor is proportional to the rate of change of its voltage:

A capacitor therefore opposes changes in voltage: its voltage cannot change instantly, and a sudden change would require an infinite current. When connected to DC through a resistance, it charges along an exponential curve with time constant \(\tau = RC\), reaching 63% of the supply voltage in one time constant and practically full charge (over 99%) in five. Once charged, it blocks DC entirely.

Capacitor safety

A charged capacitor stores energy \(W = \tfrac12 CV^2\) and keeps it after the supply is disconnected. Power factor correction capacitors, variable frequency drive DC buses, UPS systems, motor run capacitors and even long cables can hold lethal charges for minutes. Large capacitors have built-in discharge resistors, and codes require capacitors to be discharged to a safe level within a specified time after disconnection, but resistors can fail open. Before working on such equipment, wait the time stated on the equipment label, then verify zero voltage with a meter at the capacitor terminals, and discharge with an approved discharge tool if required.

Capacitive reactance

With AC applied, a capacitor charges and discharges every half cycle, so alternating current flows in the circuit even though no charge passes through the dielectric. The opposition is the capacitive reactance:

Capacitive reactance falls as frequency rises: capacitors pass high frequencies easily and block DC. In a purely capacitive circuit the current leads the voltage by 90°, because the current is greatest when the voltage is changing fastest. A memory aid is ELI the ICE man: in an inductive circuit (L), voltage (E) comes before current (I); in a capacitive circuit (C), current comes before voltage.

Resistance and capacitance; series RLC circuits

In a series RC circuit, the impedance is \(Z = \sqrt{R^2 + X_C^2}\) and the current leads the voltage by \(\tan^{-1}(X_C/R)\). With resistance, inductance and capacitance in series, the inductive and capacitive voltages are 180° apart and partly cancel, so the net reactance is their difference:

If \(X_L > X_C\) the circuit is inductive and current lags; if \(X_C > X_L\) it is capacitive and current leads. When \(X_L = X_C\), the reactances cancel, the impedance equals \(R\), and the current is at its maximum; this is series resonance, at \(f_r = 1/(2\pi\sqrt{LC})\). At resonance the voltages across the inductor and capacitor individually can be many times the supply voltage. In power systems, resonance between power factor capacitors and the system inductance at a harmonic frequency can amplify harmonic voltages and currents and damage capacitors, which is why harmonic studies and detuned capacitor banks (capacitors with series reactors) are used where drives are common.

Parallel circuits

In parallel circuits the voltage is common and the branch currents add as phasors. A resistive branch draws current in phase with the voltage, an inductive branch draws current lagging by 90°, and a capacitive branch draws current leading by 90°. The total current is \(I = \sqrt{I_R^2 + (I_L - I_C)^2}\). This is the basis of power factor correction: a capacitor in parallel with an inductive load supplies current that cancels part of the load's lagging current, reducing the total current drawn from the supply (Lesson 4).

Lesson 3 shows what happens when inductive and capacitive reactances cancel at resonance, and how harmonics interact with the capacitors and inductances in a building.