which is the impedance of an
RLC series AC circuit. For circuits without a resistor, take
; for those without an inductor, take
; and for those without a capacitor, take
.
Calculating impedance and current
An
RLC series circuit has a
resistor, a 3.00 mH inductor, and a
capacitor. (a) Find the circuit’s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for
and
are the same as in
[link] and
[link] . (b) If the voltage source has
, what is
at each frequency?
Strategy
For each frequency, we use
to find the impedance and then Ohm’s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.
Solution for (a)
At 60.0 Hz, the values of the reactances were found in
[link] to be
and in
[link] to be
. Entering these and the given
for resistance into
yields
Similarly, at 10.0 kHz,
and
, so that
Discussion for (a)
In both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that
dominates at high frequency and
dominates at low frequency.
Solution for (b)
The current
can be found using the AC version of Ohm’s law in Equation
:
at 60.0 Hz
Finally, at 10.0 kHz, we find
at 10.0 kHz
Discussion for (a)
The current at 60.0 Hz is the same (to three digits) as found for the capacitor alone in
[link] . The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in
[link] . The inductor dominates at high frequency.
How does an
RLC circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law,
, and the expression for impedance
from
gives
The reactances vary with frequency, with
large at high frequencies and
large at low frequencies, as we have seen in three previous examples. At some intermediate frequency
, the reactances will be equal and cancel, giving
—this is a minimum value for impedance, and a maximum value for
results. We can get an expression for
by taking
Substituting the definitions of
and
,
Solving this expression for
yields
where
is the
resonant frequency of an
RLC series circuit. This is also the
natural frequency at which the circuit would oscillate if not driven by the voltage source. At
, the effects of the inductor and capacitor cancel, so that
, and
is a maximum.
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