My January 2015 article, “It’s Just Math,” explained how various parameters used to evaluate power quality (PQ) are derived while avoiding formulas and other mathematical expressions. This article follows up on testing operational aspects of PQ monitors to verify that they meet your application’s needs. It is also useful to verify that they measure the various parameters in the same manner, or the results may differ significantly between monitors, leading to different conclusions.


The most basic mathematical algorithm used is the root-mean-square (rms) calculation. It is used for voltage, current, harmonic distortion and several other calculations. As the abbreviation states, the samples in the measurement window or period (often one cycle for voltage and current) are squared (v × v, or v2), summed up over the cycle and then divided by the number of samples in the interval (sum(v2)/n), which is the mean, and then the square root of that number is taken. This is the equivalent heating effect of a direct current (DC) value compared to the alternating current (AC) value. It works for all types of waveforms.


Vrms and Irms are standard on PQ monitors, but some older digital voltage meters (DVMs) use the averaging method to determine the readings they display. The numbers users get when measuring distorted waveforms, especially current, can be very different. 


For example, the typical waveform for a single-phase, rectified power supply input used in most laptops, printers and other similar equipment shows current drawn around the peaks of the sine waves only, rather than throughout the cycle. There can be errors greater than 40 percent from averaging versus rms calculations, depending on the waveforms.


Most PQ monitors compute the minimum, maximum and average values of Vrms and Irms over a time interval, usually 10 minutes. While you may think all instruments produce the same answers because the values are straightforward calculations (smallest, biggest and sum of values divided by number of values), it requires the rms calculations from the previous step to be computed the same. 


The most commonly referenced PQ standards, IEEE 1159 and IEC ­61000-4-30, specify computing the rms values over a one-cycle window (16.66 millisecond [msec] at 60 hertz [Hz]) but recompute it every half-cycle (8.33 msec). So in 1 second, 120 values would be computed, not just 60. If the voltage drops out for only a quarter of a cycle, an instrument that computes Vrms over a 1-second window will not compute the same minimum voltage as one that computes Vrms every cycle in half-cycle steps, 119 Vrms versus 83 Vrms, respectively.


Since “true” power factor is measured by the real power divided by the apparent power (watts/volt-amperes), we don’t have to worry about any trigonometric functions, such as cosine, to compute this. Using the cosine of the angle between the voltage and current waveforms as power factor is only accurate if there is no distortion, which is unlikely in today’s electrical systems. 


But where did the real power and apparent power come from? The latter is easy: Multiply Vrms × Irms every cycle. But we can’t compute real power from the values in today’s systems. Instead, we use instruments to compute the watts as the product of every voltage and current sample over the cycle and to compute the rms of those values. Using the 
Vrms × Irms × cosine (theta) will yield quite different answers, 25 percent or more off, depending on the waveform shape.


One more significant source of differences in the numbers can come from how the harmonic values are computed. Nearly all PQ monitors use a form of the discrete Fourier transform (DFT) to compute the basic magnitude and phase angles of the components used in harmonics, but this math is quite complex. 


How these are combined makes a big difference in some applications. IEEE 519 and IEC 61000-4-30 require a harmonic subgroup method. Just as one cycle window can be used for rms, the standards call out a 200-msec window or 12 cycles at 60 Hz, 10 cycles at 50 Hz. The result of this window size for the DFT results is a value every 5 Hz or 5-Hz bins. The figure above shows an example centered on the fifth harmonic in a 50-Hz system.


The 5-Hz bins that are adjacent to the actual harmonic bin (245, 250, 255 Hz in this example) often have values that are really part of the harmonic as the frequency or phase angles of the signals shift slightly in most systems. Those three adjacent bins are “rms-ed” together to produce each harmonic’s value. All of the other 5-Hz bins between each harmonic subgrouping are used to compute the interharmonic values.


While these numbers are often not very large, arc furnaces, welders, car crushers and very large motors in mining operations can produce meaningful interharmonic values. If these bins aren’t separated into the interharmonics, they get lumped into the harmonic values again, distorting the real numbers and possibly leading to incorrect conclusions.


While there is no need to break out your calculator and do square-root functions to calculate the PQ parameters, it is important to know how your instrument works, so you don’t get into disputes with users of different instruments over what the real values are.