# It's Just Math

By | Jan 15, 2015

In attempting to explain to a colleague why we care about negative sequence components, I tried to follow the guidance that Bob Lawrie, another member of the NFPA 70B Electrical Equipment Maintenance committee, offered to me many years ago: never use a formula in an article. While engineers may take joy in formulas and math, the general public typically does not. Personally, I still refer to an invaluable 3-inch stack of note cards with thousands of formulas that I used for review during my undergraduate courses and to take the engineer-in-training exam. I set out to describe negative sequence and why it has evolved as a valuable tool for some applications in the power quality.

For a single-phase circuit with a purely resistive load, we can start to describe what goes on by just the root-mean-square (rms) value of voltage and current. Of course, the rms value is actually a formula where the number of samples are first squared, then added and divided by the number of samples, and reduced to the square root. But since we are not mentioning formulas, let’s just push on.

In this example, power is the voltage times the current. Power factor, the measure of how effectively the power is being transferred from the source to the load, is always 1. If the load has a capacitive or inductive component but no distortion, the phase angle between the voltage and current is no longer zero, so the power factor is now the voltage multiplied by the current multiplied by the cosine of the angle between the voltage and current (often called displacement power factor).

Adding trigonometry to calculate the cosine into the mix isn’t really necessary, nor is it appropriate if the load is nonlinear and produces harmonics. The phase angle is only for a single frequency, so each frequency—from the fundamental to the nth harmonic—has its own phase-angle displacement of the voltage and current components at that particular frequency. Determining the values in magnitude and phase angle of each of the harmonic components requires really geek math, known as the fast (or discrete) Fourier transform. A more appropriate measure of the power factor (often called “true power factor”) is the power in watts divided by the apparent power, which is the voltage multiplied by the current—some more simple math.

Expanding the process into three-phase systems requires more ways of looking at the various signals. In a balanced system, the magnitude of each of three voltage signals should be the same, as should the angle between them: 120 degrees. We often represent this on a polar graph called a phasor diagram, where we can easily see the relationship in terms of magnitude and the angle, especially useful when we add the current. These lines are called “phasors” or “vectors.” In the phasor diagram shown here, the voltages are balanced in both magnitude and phase, but the currents aren’t as well balanced in magnitude. The offset of the voltages from the currents is fairly consistent, being about 20 degrees lagging, which is typical of an industrial facility with moderate motor loading.

While a picture may be worth a thousand words, sometimes just one word or two characterizes some aspect of the system. Just like power factor is a single number to represent how effectively the power is being used by a load, the sequence components represent how balanced the three-phase system is. The process to derive the positive-, negative- and zero-sequence values is another of those geeky mathematical transforms that involves vectorial rotation and addition, so let’s invoke the Bob Lawrie method again, and skip over those details.

In physical terms, an unbalanced three-phase system can be divided into three sets of balanced phasors. The positive sequence phasors have the same rotation as the original system, with equal magnitudes and phase angles (120 degrees) between them. Now add a set of negative-sequence phasors, each having equal magnitudes and phase angles but rotating in the opposite direction as the positive sequence. Zero sequence are three phasors without any phase angle between them but of equal magnitude. If one were to vectorially add each of the phasors for the same phase, the original set of unbalanced phasors would be re-established.

So, back to the original question, why do we care about these three numbers? Next month, I will cover this and some other single-number characteristics of three-phase systems.

## Richard P. Bingham

BINGHAM, a contributing editor for power quality, can be reached at 908.499.5321.

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