The Physics
Opus in profectus

Alternating Current

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All of Australia and Europe run on 50 Hz alternating current. Africa, Asia, and Oceania use mostly 50 Hz. North and South America use mostly 60 Hz. Japan is the only country with two standards — the northeast uses 60 Hz and the southwest uses 50 Hz. The choice of frequencies was somewhat arbitrary as neither has a technical advantage over the other.

These standards were settled on because they worked well for general purpose lighting in the 20th century. Incandescent light bulbs (good old fashioned, Thomas Edison inspired, bulbs of glass with a glowing hot filament) flicker at twice the rate of the current — once when it flows in the positive direction and again when it flows in the negative direction. Above the flicker fusion threshold, a light source will appear to have constant brightness. Like all things physiological this varies from human to human, but somewhere between 15 and 60 Hz is about right. Doubling 50 and 60 Hz gives 100 and 120 Hz, which are well above these values. Modern LED lighting uses direct current (even when it screwed into an AC socket) so is not affected by this phenomenon.

When large scale commercial alternating current systems were first being designed at the turn of the 20th century, they tended to favor lower frequencies. The first large scale commercial electric power plant to feature alternating current was built in Niagara Falls, New York in the 1890s. The original frequency chosen was 25 Hz because it worked better for low speed, high power applications like industrial machinery — its primary use at the time. This became the de factor standard for the state as a result. When New York City opened its first subway line in 1905, the electrical system included a single central generating station producing 3 phase, 25 hertz alternating current at 11,000 volts. This was transformed to 600 volt direct current for use by the trains through the third rail. This was done using rotary converter — a hybrid of an AC motor and a DC generator. I can't find an exact date, but I think most of this equipment was removed sometime in the 1990s. The 21st century New York City subway system is fed with US standard 60 Hz AC.


The average value of a sinusoidally varying quantity is zero, since half the time it's positive and half the time it's negative. This is true for the voltage and current in an alternating system. Both have an average value of zero. Power, however, is different. It's average value is always greater than zero in any electrical system.

Take a look at this equation…

P = VI

Voltage and current have the same sign since the former causes the latter. A positive voltage produces a positive current. Multiply the two and you get a positive power. A negative voltage produces a negative current. Multiply those values and you still get a positive power.

As long as a device has a resistance that does not depend on the direction of the current, the power it consumes is also given by one of these equations…

P = I2R =  V2

What these equations say is that electric power is proportional to the square of voltage or the square of current. So when we talk about an alternating current, an average of the square of these quantities would be closer to what we want. Better still would be the square root of those averages, since we could use them in many of the equations we derived for direct current. Another name for an average is a mean, so what we need is the root of the mean of the square.


The root mean square (rms) value of a quantity is a measure of its typical magnitude disregarding the mathematical sign determined by taking the positive square root (r) of the mean (m) of the square (s) of the quantity.

The rms of a sinusoidally varying quantity is its peak value divided by the square root of two. You need integral calculus to prove this mathematical relationship.

Given a quantity x that varies sinusoidally in time t with period T

x = xpeak sin



Square it…

x2 = x2peak sin2



Take its integral over one cycle.

T   T            

x2 dt = 

x2peak sin2


0   0            

This is one of my favorite definite integrals since you don't need to know fancy calculus to solve it. Visualize the sine squared curve traced out over one cycle. Note how it divides the rectangle bounding it into equal halves.

Graph of the sine squared function dividing one cycle into two equal halves

The height of this rectangle is the peak value squared and its width is one period. Multiply height by width to get the area of the bounding rectangle. Then divide that by two. Integral completed.


x2 dt =  x2peakT

Divide the quantity above by period to get the mean of the square. Then take the root of that to get the root of the mean of the square.

xrms =  xpeak

As I said earlier, the rms of a sinusoidally varying quantity is its peak value divided by the square root of two. Thus…

Vrms =  Vpeak


Irms =  Ipeak

Interestingly, the relation between peak and rms for power does not work the same way.

Prms = VrmsIrms  
Prms =  Vpeak   Ipeak
√2 √2
Prms =  Ppeak
Prms = Pave  

Since power is always positive, its rms value is the same as its average value.


In the US


According to the Merck Manual of Medical Information, Second Home Edition

Alternating current… is more dangerous than direct current. Direct current tends to cause a single muscle contraction often strong enough to force the person away from the current's source. Alternating current causes a continuing muscle contraction, often preventing people from releasing their grip on the current's source. As a result, exposure may be prolonged. Even a small amount of alternating current — barely enough to be felt as a mild shock — may cause a person's grip to freeze. Slightly more alternating current can cause the chest muscles to contract, making breathing impossible. Still more current can cause deadly heart rhythms.

Merck Manual of Medical Information, 2004