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Unformatted text preview: Physics 54 AC Circuits If we don't succeed, we run the risk of failure. — Dan Quayle AC generators and rms values It is relatively easy to devise a source (a “generator”) which produces a sinusoidally varying emf. A rotating coil in a magnetic Feld gives an important example; another is the sinusoidally varying potential across the inductor in an L-C oscillator, used in many situations to provide an emf for another circuit. This kind of oscillating source is called an AC (alternating current) generator. Its output is described by E = E max sin ω t . Here E max is the maximum emf during a cycle of the oscillation. But meters that measure the strength of sinusoidally varying voltages and currents do not usually measure the maximum value. Instead they measure the rms (root-mean-square) value. This is a statistical measure, deFned for any varying quantity by: RMS values The rms value of a quantity f that varies over a distribution is given by f rms = ( f 2 ) av , where the average is taken over the distribution. That is: square the quantity, take the average of that, then take the square root. In the case of sinusoidal time variation, the average is taken over the time for one cycle. Since the average of sin 2 ω t over a cycle is 1/2, we Fnd a simple rule: E rms = E max / 2 . This is what an AC voltmeter, placed across the generator’s terminals, will read. ¡or example, the ordinary household outlets in the USA deliver an AC potential difference with maximum about 170 V, oscillating at frequency 60 Hz. What a voltmeter will read, placed across the terminals, is about 120 V. This is the rms value. PHY 54 1 AC Circuits Series LCR circuit We will analyze a circuit containing a capacitor, an inductor, a resistor and an AC generator, all in series as shown. We apply the loop rule at an instant when the current is running clockwise, putting positive charge on the lower capacitor plate. Then we fnd E − L dI dt − Q / C − IR = . Using I = dQ / dt and rearranging, we have d 2 Q dt 2 + R L dQ dt + 1 LC Q = 1 L E max sin ω t . This diFFerential equation has the same mathematical Form as the equation describing a driven oscillator with damping, so its solutions have the same Form. The "steady-state" solution (describing the situation where the energy dissipated per cycle is exactly...
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This note was uploaded on 10/19/2011 for the course PHYSICS 54L taught by Professor Thomas during the Summer '09 term at Duke.
- Summer '09