Last month's article touched briefly on the two of the most important laws in power-quality analysis and seemed to have generated enough questions that a more in-depth discussion might be warranted. This month will focus on Ohm's Law, which is responsible for such memorable mnemonics as “Twinkle, twinkle, little star, P equals I squared R.”

Back in the early 19th century, Georg Simon Ohm developed a formula to describe the relationships among voltage, current and resistance. This has been extended from pure resistance to the term “impedance,” which accounts for inductance and capacitance as well as what is found in most circuits. He formulated that the current in a circuit is directly proportional to the applied voltage and inversely proportional to the impedance of the circuit (I = V / Z). Or, the voltage drop across two points is equal to the current times the impedance (V = I * Z)

Voltage is the difference in potential energy between those two points, while current is the pressure pushing the electrons between those two points. Impedance, as the word sounds, is trying to impede or resist that pressure. For the same voltage potential, a higher impedance means that less current is flowing between those two points and vice versa. Sounds pretty straightforward and simple.

A degree of complexity comes in when you look into what really makes up the impedance values in an electrical system. In some text books, they assume “ideal conditions,” which means that the wiring and equipment between the power source (the generator) and the power consumer (the load) have no impedance (hence no voltage losses along the way), that, basically, the generator has infinite capacity (that is, its output voltage and frequency do not change as a function of current being supplied), and that the load's impedance is a constant over the voltage range and sometimes also over the frequency range (since they assume only fundamental frequency components, such as 50 or 60Hz, make up the voltage and current waveforms).

For those of us who must operate in the real world, those conditions rarely exist in today's electrical environment. And if they did, there would be no reason for this column, since power quality phenomena would basically no longer exist. When a large motor started up and drew inrush currents six to 10 times the steady state condition, there would be no voltage drop through the transformers, wiring, breaker contacts and the like, so there would be no sags. With only fundamental frequency components present in the voltage and current waveforms, there would be no harmonic distortion. And no matter how much current is drawn out of the generator, the frequency would always be 60.000Hz. Ah, the “book life.”

While the math is relatively simple in the basic formula, it does get a bit tricky with impedance when there are capacitors and inductors to be considered besides just resistance. (Even resistance can get interesting, as there are components that have negative resistance-apply more current, and instead of the voltage drop getting larger, it gets smaller. But we'll leave those be for now.)

The impedance of an inductor increases with frequency; it might be 0.2 ohms at 60Hz, but it is 10 ohms at 3KHz (also known as the 50th harmonic of 60Hz). In comparison, a capacitor with a 0.2 ohm impedance at 60Hz would be only 0.004 ohms at 3KHz. This clearly shows why inductive devices, such as motors and transformers, do not like current harmonics to be present, since their internal voltage drops will increase, hence, the power. Internal power becomes heat, and heat leads to degradation in the device, forcing a derating of the operating point to keep the device from burning itself up. There are plenty of interesting sites on the Internet that have calculators that you can use to experiment with this concept, such as the site, www.cvs1.uklinux.net/calculators/.

When the impedance of the inductance in the circuit matches the impedances of the capacitance (usually from PF correction cap banks), a condition called resonance can occur. For example, 10mH of inductance and 28uF of capacitance at the 5th harmonic of 300Hz will form a tuned circuit that will effectively amplify the signals, causing overcurrent conditions that may damage other equipment. When the cap banks were switched in, the impedances resonated at the 7th harmonic current (which was present due to having numerous ASDs in the facility), and eventually led to failures in the cap banks.

In next month's article, we will apply Ohm's Law to various parts of a circuit, in conjunction with Kirchoff's Laws, and show how it is useful in analyzing voltage sags, swells, harmonics, flicker and even transient problems, to not only determine the source but to eliminate a PQ problem before it takes down a process in your facility. **EC**

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