Resonance, harmonics and filters

Lesson 2 showed that inductive reactance rises with frequency while capacitive reactance falls. When the two are equal, a circuit is resonant, and currents or voltages can become far larger than the supply alone would suggest. On a pure 60 Hz system this would be a curiosity, but modern electrical systems are full of currents at other frequencies: the harmonics drawn by LED drivers, computer power supplies, variable frequency drives and battery chargers. Harmonics overheat neutrals and transformers, trip breakers, distort voltage, and can excite resonance with power factor capacitors, sometimes destroying them. This lesson explains series and parallel resonance, where harmonics come from and how they are measured, their effects on wiring and equipment, and the filters and other measures electricians use to control them.

Series resonance

In a series RLC circuit, the total reactance is \(X_L - X_C\). At one frequency, the resonant frequency, the two reactances are equal and cancel:

At resonance the impedance falls to the resistance alone, so the current is limited only by \(R\), and it is in phase with the voltage. The voltages across the inductor and the capacitor individually can be much larger than the supply voltage, because each equals the current times a reactance that may be many times \(R\); they are equal and opposite, so they cancel in the total. The ratio of the reactance to the resistance at resonance is the quality factor \(Q = X_L/R\), and the voltage across each element is \(Q\) times the supply voltage.

Parallel resonance

In a parallel LC circuit, the inductive and capacitive branch currents are 180° apart. At the resonant frequency (the same formula for ideal components) they are equal and cancel in the supply line, so the line current is very small and the impedance seen by the source is very high. Meanwhile a large current circulates between the inductor and the capacitor. Parallel resonance is the more common problem in power systems, because power factor capacitors are connected in parallel with the inductance of the supply transformer and utility system.

The resonant frequency of a capacitor bank with the system inductance can be estimated from the short-circuit capacity at the bus and the capacitor rating. Expressed as a harmonic number (the resonant frequency divided by 60 Hz):

where \(S_{sc}\) is the short-circuit capacity in kVA at the capacitor's location and \(Q_c\) the capacitor rating in kvar.

What harmonics are

A harmonic is a sinusoidal component at an integer multiple of the fundamental frequency: the 3rd harmonic of 60 Hz is 180 Hz, the 5th is 300 Hz, the 7th 420 Hz. Any repeating waveform, however distorted, can be built from the fundamental plus harmonics (Fourier's theorem). Lesson 1 introduced the idea; here we look at the sources and effects in detail.

Harmonic currents are produced by nonlinear loads, whose current is not proportional to the voltage. The most common is the rectifier with a capacitor at the input of electronic power supplies: it draws current only in short pulses near the peaks of the voltage wave. Typical sources:

The amount of distortion is measured by total harmonic distortion (THD), the RMS of all harmonic components divided by the fundamental:

Current THD can be large for individual loads; voltage THD, which affects every load on the bus, is usually much smaller and should normally stay below about 5% at the service. The total RMS current includes the harmonics: \(I_{\text{RMS}} = I_1\sqrt{1 + \mathrm{THD}^2}\), which is why true-RMS meters (Lesson 1) are essential: averaging meters can read 30% to 40% low on distorted currents.

Effects of harmonics

Neutral currents in three-phase systems

In a balanced three-phase four-wire system with linear loads, the phase currents cancel in the neutral (Lesson 6). Triplen harmonics (the 3rd, 9th, 15th and other odd multiples of 3) behave differently: their waves in the three phases are in phase with each other, so instead of canceling they add in the neutral. A neutral serving many single-phase electronic loads can carry more current than any phase conductor.

Code requirements reflect this: NEC 220.61(C) does not permit the neutral demand reduction for the portion of load consisting of nonlinear loads supplied from a 4-wire wye system, and 310.15(E)(3) requires the neutral to be counted as a current-carrying conductor when the major portion of the load is nonlinear.

Transformers

Harmonic currents increase eddy-current losses in transformer windings roughly with the square of the harmonic order, so a transformer serving nonlinear loads runs hotter than its load in kVA suggests. Options are derating the transformer, or using a K-rated transformer designed for harmonic heating (K-4, K-13 and K-20 ratings are common, with higher numbers for more severe harmonic content). In delta-wye transformers, triplen currents from the secondary circulate in the primary delta and do not reach the supply, but they still heat the transformer.

Other effects

• Voltage distortion: harmonic currents flowing through the source impedance distort the voltage, affecting sensitive equipment, causing flat-topped voltage waves and reducing the ride-through of power supplies.
• Capacitors: capacitive reactance falls with frequency, so capacitors absorb harmonic currents, overheating and failing; at resonance the effect is amplified.
• Breakers and fuses: thermal-magnetic breakers respond to true RMS heating, but some electronic trip units sampling peaks may trip unexpectedly; fuses may run warm.
• Motors: 5th harmonic voltage produces a negative-sequence rotating field that opposes rotation, adding heat and torque pulsations.
• Neutral-to-ground voltage at receptacles caused by triplen neutral currents, which can disturb sensitive electronics.

Controlling harmonics

Harmonic problems are solved at the source, in the distribution system, or at the point of common coupling with the utility:

• Line reactors (typically 3% or 5% impedance) and DC-link chokes on drives smooth the current pulses, cutting current THD from 80% to roughly 35% to 45%.
• Multi-pulse rectifiers (12- or 18-pulse) and phase-shifting transformers cancel the 5th and 7th harmonics between groups of drives.
• Active front-end drives and power-factor-corrected supplies draw nearly sinusoidal current.
• Passive tuned filters: a capacitor in series with a reactor tuned just below a harmonic (for example 4.7th for the 5th) provides a low-impedance path that absorbs that harmonic, while also supplying power factor correction at 60 Hz.
• Detuned capacitor banks: reactors in series with power factor capacitors tuned well below the lowest significant harmonic (typically 4.2nd or about 250 Hz, often specified as 7% reactors) prevent resonance without trying to absorb harmonics.
• Active harmonic filters: power electronic devices that measure the load's harmonic currents and inject equal and opposite currents, adapting to changing loads.
• Oversized neutrals, K-rated transformers and separate circuits for sensitive equipment manage the effects that remain.

Standards and measurement

IEEE 519 sets recommended limits for harmonic voltage distortion and for current distortion at the point of common coupling with the utility, scaled by the ratio of available short-circuit current to load current: larger customers on weak systems face tighter limits. Utilities may require harmonic studies before connecting large drive or rectifier loads. Measuring harmonics needs a power quality analyzer or harmonic-capable meter that reports THD and individual harmonic magnitudes on each phase and the neutral (Lesson 5). Measurements should be taken under typical and maximum load, since harmonic content varies with loading.