lecture14 - 14. Lecture 14 14.1 AC circuits: capacitors and...

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Unformatted text preview: 14. Lecture 14 14.1 AC circuits: capacitors and inductors 14.1.1 Capacitors Consider a circuit as in fig.78 which is called a low-pass filter for reasons we will see shortly. The AC generator determines the potential Va = V0 sin ω t (14.1) and we want to understand the behavior of the potential Vb at point (b). It is first convenient to see what happens if, instead of a sine we have a square wave as in fig.79. This is equivalent to putting a battery which periodically switches polarities as we see in the same figure. When the battery is of one polarity it charges the capacitor in time τ = RC as we saw before. After the capacitor charges nothing else happens and Vb remains equal to V0 . When the battery switches polarity the capacitor first discharges and then it charges with the opposite sign so Vb = −V0 and Vb continues to be −V0 until the battery switches back again. We see that, except for a small delay of τ = RC the potential Vb follows the value at Va . In that sense the capacitor, for time scales t ￿ RC behaves as an open circuit. This last point should be emphasized, we assumed that the period with which we switched the battery was much larger than τ = RC . If we switch the battery very fast then the capacitor has no time to charge and discharge and the potential across it is zero. Namely Vb = 0 and the capacitor is like a short-circuit, namely a cable. To summarize: ￿ 1 T ￿ τ; ω ￿ RC ; Capacitor → open circuit Capacitor (14.2) 1 T ￿ τ; ω ￿ RC ; Capacitor → short-circuit (cable) Using this information, if we go back to the circuit in fig.78 we conclude that ￿ 1 Va = V0 sin ω t if ω ￿ RC Vb = 1 0 if ω ￿ RC (14.3) For that reason it is called a low-pass filter, any high frequency signal does not appear at point (b). On the other hand if we have a circuit as in fig.80 we have, using the same rules: ￿ 1 0 if ω ￿ RC Vb = (14.4) 1 Va = V0 sin ω t if ω ￿ RC For that reason it is called a high-pass filter. Low frequencies, and in particular DC potentials are blocked by the capacitor. A typical application of the last circuit is in – 92 – what is called AC coupling. Suppose we have an audio signal mounted on a DC voltage. If we need to input that to an amplifier but we do not want to keep the DC voltage because it might affect the amplifier then we can use a high pass filter as in fig.81. The capacitor and resistors should be chosen so that they do not cut the frequencies we are interested in. Figure 78: AC generator connected to an RC circuit. Low-pass filter, allows only low frequencies to go through. 14.1.2 Inductors Inductors behave in the opposite way as capacitors. At low frequencies, including DC current, they behave as a short circuit, just like a cable. There is no voltage across its terminals. At high frequency however, there is no current, because any such current will vary very rapidly and would create a very large voltage across its terminals. What actually happens is that the voltage generated is enough to cancel any voltage applied and very little or no current circulates. So: ￿ L T ￿ τ = R; ω ￿ R ; Inductor → short-circuit (cable) L Inductor (14.5) L T ￿ τ = R; ω ￿ R ; Inductor → open circuit L Again, looking at the example of fig.82 we have ￿ 0 if Vb = Va = V0 sin ω t if – 93 – ω￿ ω￿ R L R L (14.6) Figure 79: Previous circuit connected to a square wave generator. It is equivalent to a battery that flips polarity. If the flips are at long intervals then the capacitor charges and discharges following the voltage. If they are very frequent then the capacitor has no time to charge and discharge and the voltage across it is very small. This is called a low-pass filter. In the case of fig.83 we have instead: ￿ Va = V0 sin ω t if ω ￿ R L Vb = 0 if ω ￿ R L (14.7) This is a high-pass filter. All these ideas are very nicely illustrated in the demo of fig.84 where changing the frequency of the generator we can see how the behavior of the capacitor and inductor changes according to the rules we saw before. It is interesting to find out the cutoff frequency for the capacitor. From the circuit one can see that C = 220µF . The light-bulb is a resistor. It is a 23W light-bulb when operated at 12V. From here we find: V2 P = 25W, V = 12V, P = , R = 6Ω (14.8) R Therefore 1 ω= = 800Hz (14.9) RC – 94 – Figure 80: AC generator connected to an RC circuit. High-pass filter, allows only high frequencies to go through. Figure 81: Example of AC coupling. If an AC signal is mounted on top of a DC voltage, only the AC part goes through (for adequate values of the capacitance and resistance). The frequency, as we saw before is f= ω 800 = Hz ￿ 130Hz 2π 2π (14.10) which can be easily verified in the demo by changing the frequency of the generator. – 95 – Figure 82: AC generator connected to an RL circuit. High-pass filter, allows only low frequencies to go through. Figure 83: AC generator connected to an RL circuit. Low-pass filter, allows only high frequencies to go through. – 96 – Figure 84: Demo: Variable frequency AC generator connected to an RL and RC circuit to demonstrate the AC properties of capacitors and inductors. – 97 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue University.

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