Aside: 200 reversals per second is 100 Hz, which is an unusual power frequency. Most of Europe runs at 50 Hz, and North America runs at 60 Hz, but that doesn’t matter here.
The mathy answer is that power is voltage times current. In fact, it is the average of the instantaneous voltage times the instantaneous current. The current may reverse 200 times a second, but so does the voltage! plus times plus is positive, and minus times minus is also positive.
A brief thought experiment. Connect a battery to a light bulb. The bulb glows, right? Now reverse the battery. The bulb still glows! It doesn’t recharge the battery. The bulb doesn’t care which way the battery is connected.
In the case with the battery the right way around, you might have a positive voltage and a positive current, result of multiplying them together gives a positive power delivered to the bulb.
With the battery backward, the current is negative, but so is the voltage. Multiply them and you still get a positive power.
Continuing our thought experiment, wire a reversing switch between the battery and the bulb, and flip it as fast as you can. No change! The bulb glows.
Hook it up to a motor so it reverses 200 times a second. No change!
Now you could have a poor switch that didn’t switch instantaneously, or you could have such long wires that the speed of light takes a role, but none of that applies here.
AC is used because it is conveniently convertible with transformers, which DC is not. It has nothing to do with power delivery. The lamps in your house don’t care which way the battery is connected, even if some lunatic is reversing it at insane speeds.
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