2.1 The sine wave

On the kitchen outlet, the multimeter reads 120 V. The oscilloscope, on the same voltage brought down through a transformer (never on the outlet: its ground is tied to earth through the laptop, and the ground clip on the live wire is a dead short), draws a wave that climbs to a peak, drops as far and starts again sixty times a second; on the outlet itself, that peak would be 170 V. Both are right, and the whole RAS lives in the gap between those two numbers.

In 1882, Edison lit New York with direct current, with power stations a few streets from the lamps because the copper ate the rest. In 1886, William Stanley switched on for Westinghouse, at Great Barrington, the first alternating network with transformers in America, where the voltage climbs to travel and comes back down for the lamp; in 1888 Westinghouse licensed Tesla’s motor. The Edison company answered with a pamphlet on the danger of AC, and in 1890 the State of New York’s first electric chair ran on a Westinghouse generator. In 1893, Westinghouse lit the Chicago exposition and won Niagara Falls. The war of the currents was over.

An engineer’s question: how much is an alternating voltage worth? Its true average is zero, and yet it lights the lamp. The answer that held goes through heat. Heat a resistor with the AC, and look for the DC that heats it just as much; that DC is the effective value of the AC. Heat follows the square of the voltage, so the calculation squares every instant, averages them and takes the root. English names the recipe, root mean square, RMS; French names the effect, valeur efficace. For a sine wave, it is the peak divided by √2, 0.707 of the peak. The "120 V" mains is 120 V RMS; its peak is 120 × 1.414, 170 V, and 340 V from one peak to the other.

Behind the diode bridge of a power supply, the filter capacitor charges at the top of the wave and holds it; it sees 170 V, never 120. A capacitor rated 160 V in that job is dead on arrival; you take 200 V, usually 250. A wire’s insulation breaks down at the peak; the copper heats at the RMS.

Three numbers for the same wave. The peak, what the oscilloscope sees. The RMS, what the resistor feels, 0.707 of the peak. And the average of the rectified wave, what a needle measures, 0.637 of the peak, 108 V on the mains. Almost every cheap multimeter measures that one and multiplies it by 1.11 to display 120. On a square wave, where peak, RMS and average are the same number, the display reads 11 % too high. A "true RMS" multimeter does the calculation for real; fifteen to forty dollars more, and the diagnosis of the sixth target.

Phase remains. A bicycle’s brushless motor, turned by hand, gives on its three wires taken two at a time sine waves of the same frequency, offset by a third of a period, 120 degrees. Adding them took trigonometry at every line. The engineers of the 1880s were already drawing the sine wave as an arrow, as long as the amplitude, turned as far as the phase, and two signals add like two arrows laid end to end. Two 10 V signals 90 degrees apart make 14.1 V, not 20; 120 degrees apart, they make 10. In 1893, in Chicago, Charles Steinmetz, an engineer at General Electric, took the next step: the arrow becomes a complex number, and calculation replaces the drawing. That is the phasor.

The classic mistake: read the peak on the scope and call it the voltage. The peak heats nothing; the RMS heats. Between the two, the factor is 1.414 for a sine wave, 1 for a square wave, and unknown for anything that comes out of a chopper.