In the past, when power flowed as a sine wave, the square root of three was used to convert between line-to-line voltages and line-to-neutral voltages. One phase could be measured and the results multiplied by square root of three times the square root of three (also known as “3”), to get the three-phase total power. These formulas were based on sinusuoidal voltage and currents under balanced load conditions, if you read the fine-print footnotes. The primary load in the past was three-phase electric motors, which are balanced and linear loads if they are properly functioning.

Rectified-input power supplies and other non-linear loads have been estimated to have increased from 25 percent of the total load in the United States in 1985, to 65 percent in 2000. These loads draw current during only part of the waveform, resulting in current distortion, and depending on the harmonic impedances, voltage distortion as well. This has resulted in increased distortion and imbalance at the equipment load, the point-of-common-coupling, and even some substation levels. Harmonic distortion and imbalance can have a significant effect on some equipment, especially transformers and motors. Measuring and monitoring these parameters accurately is important when trying to maximize asset usage, prolong the life of the equipment and prevent catastrophic failures.

Another vestige of the past that is no longer a good rule of thumb is the power triangle, shown in Figure 1. The horizontal axis is the real power, measured in Watts, and abbreviated by P. This is the power that does “real work,” such as the current through a resistor times the voltage across the resistor. The vertical axis is the reactive power, measured in VARs, and abbreviated Q. The current is 90 degrees out of phase from the voltage, either leading (for capacitive loads) or lagging (for inductive loads). In power systems, these VARs are often said to be “doing no useful work,” just flowing back and forth through the wires without contributing to the work of the load. It’s kind of a gross oversimplification, but it has a bit of merit to the idea. The actual power that the utility company or generator must provide is the vectorial sum of P and Q, referred to as the apparent power, and is abbreviated by S. In a purely resistive load, P will equal S.

The power factor is the ratio of the real power to the apparent power, or P/S. This is often referred to as the “true” power factor. This does not imply that the other method, called displacement power factor (DPF), is a lie. Rather, displacement power factor is a measure of the angle between the voltage and current fundamental frequency sinusoids. But remember the opening to this saga—voltage and current are no longer made up of just fundamental frequency sinusoids, but a plethora of harmonic (and interharmonic) frequencies as well. So the fundamental phase angle between the voltage and current can be zero degrees which would imply a power factor of 1, yet due to the distortion, the power factor would be significantly less than one. The generator still has to produce all of the power needed to provide for the harmonic losses, eddy current losses, VARs, etc.

The power triangle doesn’t take into account that the VARs are frequency dependent. The impedance of a capacitor or inductor changes, depending on the frequency, whereas the impedance of a resistor is frequency independent. For capacitors, the impedance decreases as the frequency goes up; for inductors, it increases. So depending on the harmonic signals levels at any point in time, the equivalent power calculation in the VAR will vary based on voltage, current and frequency. There was some discussion on a three-dimensional power triangle, which add another vector in the Z axis, representing the distortion power, D.

However, a more rigorous set of mathematical methods have been developed in the *IEEE Std-1459-2000* “Trial Use Standard Definitions for the Measurement of Electric Power Quantities under Sinusoidal, Nonsinusoidal, Balanced or Unbalanced Condition.“

It provides values that more adequately characterize the power flow, such as effective apparent power, fundamental effective apparent power, fundamental positive-sequence apparent power, fundamental unbalanced power, effective harmonic distortion power, along with parameters based on vector and arithmetic sums. As Figure 2 shows, the true power factor can be a significantly different value, based on the calculation method, changing from –0.78 to –0.93.

And then there is the DPF. There have been customers who have PF rate structures in their electric bills who have complained that their bills went up when a “new” wattmeter was installed at their facility. In the past, they were charged for low power factors based on the DPF, which weren’t too severe, as they were above 0.90. When the meter was replaced with one that measured true power factor, their bills increased, as now the power factor was close to 0.8. Their complaint was that their loads haven’t changed, so why should they pay more? The utility did put the old meter back in, but in reality, they had been underpaying for years, as the Watts consumed versus Volt-Amperes provided by the utility was what the new meter said.

Lastly, there has been the controversy over whether the electromagnetic wattmeters accurately measure the watt-hours under distorted and unbalanced conditions. There has been significant research showing that there is a discrepancy, however, the discrepancy has been found to result in both under- and overstated values. So at this time, it is correct to say, “It is different, but that who that difference favors is still under consideration.” Just remember that our distorted and unbalanced world has even affected who has the power. **EC**

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