It seems like every decade or so, elementary school educators employ a new way of teaching math. When parents try to help, they are greeted with “That’s not how we do it.” Yet the fundamental laws of mathematics haven’t changed since the time of the ancient Greek and Persian mathematicians.


The same is true with the rules of electrical engineering, notably the famous eight original equations of electromagnetism. When a group of mostly nontechnical people wanted to know why I can’t just put a power quality monitor at a breaker panel and determine where the harmonics originated, my first response was that it’s new math. The old rules don’t apply anymore. Those rules were the quadrants of power flow, which are based on the figure below. The four quadrants combined negative or positive directivity and inductive or capacitive (lagging or leading). 


The problem occurs because it is based on the elements in the example triangle, which illustrates the relationship between real, reactive and apparent power and the power factor being equal to cosine of the angle between voltage and current. These equations are mathematically correct only when there is a single frequency. 


In today’s nonlinear-load electrical systems, those assumptions don’t hold. Watts and volt-amperes reactive (VAR) usually don’t vectorially add up to volt-amperes (VA) in today’s electrical systems. There is a distortion vector D that extends from the fundamental frequency VARs to the end of the VA vector and is based on the combined harmonic power from a multitude of harmonics with different amplitudes and phase angles.


After further pondering about harmonic power directivity, I concluded that the math isn’t really new; it is just that the shortcuts used in the past don’t apply in today’s systems. At a given junction point or node, harmonic currents are usually changing in both amplitude and phase angle relative to the fundamental voltage and each other. The harmonic levels at a breaker panel will be different than at the point-of-common coupling (PCC) or at individual loads. Two nth-order harmonic currents may not be in phase and may even cancel each other out if 180 degrees out of phase.


How can we answer that question, then, particularly considering the development of standards that limit the harmonic contribution from a facility, such as IEEE 519 2014, or for individual loads, such as IEC-61000-3-2? There have been several attempts to develop algorithms that can accurately and repeatedly determine it, though none seem to have gained universal acceptance in the power quality (PQ) community.


One could assume the generators or power sources produce only pure sine waves so all distortion comes from loads. Loads, downstream from the source, contribute harmonic currents that alter the voltage that is provided, based on the harmonic impedances of the wiring, transformers and other parts of the electrical distribution system. That distorted voltage is then passed along further down the distribution system to be distorted even more by the next facility with harmonic currents.


Even a facility with purely resistive loads (though highly unlikely these days) will have harmonic currents measured at the PCC because harmonic voltages were present in the voltage provided by the utility. Harmonic voltages interact with harmonic impedances to produce harmonic currents, just like Ohm’s Law (and Maxwell’s equations) predict. It seems that the most accurate way to determine the contribution is to measure with all of the loads downstream turned off, then measure again after they are turned on, and hope that everything is steady state, including the voltage being provided and all of the loads downstream. This is impractical, since business is often conducted 24/7 and wouldn’t agree to a complete shutdown.


It seems silly to develop enforceable standards without having a proven method for actively monitoring possible polluters to determine their contribution. But that’s the PQ world we live in right now, no matter if your math is old or new.