# Yet Another Use: The Letter of the Laws

Though Ohm’s Law is a generalized construct that is not as encompassing as Maxwell’s Equations, it is far more understood and used by those in the electricity business. We have used Ohm’s and Kirchhoff’s Laws to understand how sags and swells occur and how to help pinpoint their origins. We also used Ohm’s Law for doing power analysis. It can be expressed as R=V/I, where R is resistance, V is voltage and I is current. Another version is useful for approximating impedance, including source and load impedance. It involves taking the difference (delta) between two voltages and their respective currents at an instance in time and dividing the delta V by delta I.

With this, we can understand and hopefully prevent a serious overvoltage condition (resonance) caused when there are significant harmonic current components close in frequency to the harmonic impedance resonance. Think of it as an amplification of the voltage caused by the effect of current at a particular harmonic frequency. Figure 1 shows an example of such an effect caused when a power factor capacitor is switched into a circuit that causes a resonance to occur near the 7th harmonic.

When power factor capacitors typically are switched in, there is a relatively short oscillatory transient that lasts less than a quarter of a cycle. In this resonance case, the frequency of the oscillation is around 420 hertz, the 7th harmonic, and lasts for cycles. This occurred because the circuit’s impedance—when the capacitor’s impedance is included—causes the harmonic impedance to increase around the 7th harmonic over six times compared to the other harmonic impedances, as shown in Figure 2.

This data is derived from the harmonic voltage and current values at two points in time. The Vhn1 (nth harmonic voltage at time T1) is subtracted from the Vhn2 at time T2. I did this for n=2 through 15 in this example. Then the Ihn1 value (nth harmonic current at T1) is subtracted from the value at T2. The subtracted voltages are divided by the subtracted currents, yielding an approximate Zhn (nth harmonic impedance). The graph plots these values over the range of n. Note that there is also an impedance increase around the 15th harmonic, though not as large as the 7th.

By expressing Ohm’s Law as V=I×Z, it is clear that a large increase in Z for the same current levels will cause a large increase in voltage. This is only part of what occurs in Figure 1. In addition, due to the increased currents from charging up the capacitors that were previously at 0 volts, there is even more current, which results in even more voltage. Hence, the result is a large voltage distortion around the switching as well as the ongoing distortion.

At this particular site, the excessive voltage condition was even more severe at times and resulted in the catastrophic capacitor destruction at an expensive rate. The root cause of this was determined with the help of the PQ monitoring data and a bit of the “what changed” analysis. The facility had gone through a very significant energy reduction program, changing out always-on-full induction motors with adjustable speed drive systems for moving product on conveyors. Lighting was changed to high efficiency lighting. The overall wattage was significantly reduced, and the power factor (PF) changed significantly from the heavily inductive motor lagging PF to actually a leading PF.

No one had paid attention to the fact that many of the energy-savings measures use power supplies that were no longer linear but had rectified input switching power supplies that are prime sources of harmonic currents. For three-phase equipment, the most dominant contributors are the 5th and 7th harmonic. The change in the resulting network impedance coupled with the PF capacitors made a resonance point (calculated by a formula not included here), right near the 7th harmonic.

Once again, this pointed to the fact that changes should not be made blindly. Not monitoring the quality of the supply before and after, as well as not considering the full spectrum of the electrical impact from the changes, can be costly. In this case, the money that they saved on their energy bill was needed by pay for replacing blown cap banks.