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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 lowpass 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 shortcircuit,
namely a cable.
To summarize:
1
T τ;
ω RC ; Capacitor → open circuit
Capacitor
(14.2)
1
T τ;
ω RC ; Capacitor → shortcircuit (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 lowpass 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 highpass 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. Lowpass 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 → shortcircuit (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 lowpass 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 highpass 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
lightbulb is a resistor. It is a 23W lightbulb 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. Highpass 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. Highpass ﬁlter, allows only low
frequencies to go through. Figure 83: AC generator connected to an RL circuit. Lowpass 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.
 Fall '11
 NA

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