Published on *EC Mag* (http://www.ecmag.com)

Several times recently, someone has asked why two different instruments reported the same values for watts (W) and volt-amperes (VA), but they don’t show the same values for volt-ampere-reactive (VAR). A quick Internet search will turn up a lot of explanations why some instruments are reporting the value incorrectly, but the sites have incorrect information about how to calculate VARs. One of the favorite stops for a quick definition, Wikipedia, had so much misinformation that it is no wonder the concepts are confusing to users of test equipment.

The false information is based on information valid back when the 60 Hz (or

50 Hz in some parts of the world) fundamental power frequency was the only significant component. Now, voltage distortion is often 3–5 percent of the total signal, and current distortion can be many times that. The power triangle doesn’t work anymore, yet some instruments still try to use it.

Back to the question at hand. How can two legs of the power triangle be correct and match another instrument but the VAR be wrong (generally too large)? We need to go back to why there are VARs in the first place. The impedance of electrical loads and wiring is made up of three basic components: resistance, capacitance and inductance. Some loads have mostly one, such as a toaster oven being almost entirely resistive, while some have two primary components, such as a motor being inductive and resistive. Wires that bring the electricity from the generator through the transmission and distribution systems to the building wiring and load have all three elements.

The equivalent impedance value of a resistor is not dependent on the frequency of the signals. If one ampere (A) of a 60-Hz signal flows through a resistor, it will produce the same voltage, hence the same power consumption, as if it were 1A of a 180-Hz signal (the third harmonic of 60 Hz). Such is not true for inductors and capacitors. The equivalent impedance of an inductor is proportional to the frequency of the signal—it goes up as the frequency goes up. A capacitor is the opposite—it goes down as the frequency goes up. In addition, the voltage and current signals will be the same in time or in-phase for a resistive element, but an inductor has the current lagging the voltage (due to the physics of an inductor, which we will skip over). The voltage will lag the current for a capacitor. That is why capacitor banks are added to electrical systems to offset the effects of motors, trying to get the voltage and current in phase with each other and make the net VARs equal to zero. This reduces the “useless” power flow of VARs on the power system and will make the real or effective power (W) equal to the apparent power produced by the generator (minus all the resistive losses in the system).

The majority of today’s instruments convert the voltage and current waveforms to digital values or samples using an analog-to-digital-converter. The waveforms are sampled typically between 32 and 512 times each cycle, which, at 60 Hz, is 16.66 milliseconds long. If the voltage and current samples are taken at approximately the same time, those two samples can be multiplied together, and the sum of them over the cycle can be used to determine the watts. Computing the root-mean-sum-of-squares or rms value of voltage and current also uses all of the samples within one cycle, but the voltage and current samples are used separately. Vrms is then multiplied by Irms to give the VA value. Based on the accuracy of the hardware and the sampling rate, these values might differ slightly from instrument to instrument, but they will all be relatively close since they use the same methods.

Computing the VAR properly requires a complex mathematical process to determine the fundamental frequency waveforms of the voltage and current from the signals that have lots of harmonics mixed in. This requires continual computing or processing for every cycle. Instead, the Pythagorean theorem is invoked, and the W squared is subtracted from the VA squared, then the square root of that value is taken and used as the VARs. That value has the harmonic distortion power mixed in with it and isn’t the fundamental frequency VAR value only. One could calculate the VAR value for each harmonic present, but for power system purposes, we usually only care about the fundamental frequency VAR when adding in capacitor banks to get that power factor

**BINGHAM**, a contributing editor for power quality, can be reached at 732.287.3680.