If an AC emf given by 
is used to drive current through a resistor, a capacitor, and an inductor
connected in series, then the current through each element must be the same.
The voltages across the various elements obey the rules given above, and the
sum of these voltages must, by the loop theorem, be equal to the applied emf.
This sum must be taken at a particular
instant of time,
which is complicated because each voltage difference will be at a different part
of its cycle.
Solving this complicated problem gives the following solution for the
current I(t) that flows in the circuit:
where
and where
Note that the amplitude of I is given by 
, which looks about
like the DC relation I = V / R.
The quantity Z is called the impedance,
it has units of ohms,
and it plays the same role in AC circuits as resistance does in
DC circuits.
Unlike the resistance, however, it depends on the driving frequency,
so the current that flows in the circuit depends sensitively on
the driving frequency.
Be careful, however; this formula for the impedance only applies to the
series RLC circuit.
Each different circuit has its own impedance formula.
Resonance: Consider the series RLC circuit discussed above.
The formula for the current makes it easy to see how things
should be adjusted to get as much current as possible from a
given driving emf 
: simply make Z as small as possible.
If the circuit value of R is fixed (as is usually the case)
then the only way to get more current is by fiddling with 
and 
.
And it is clear that the smallest value of Z will be obtained
when 
, which a little algebra shows is equivalent to
.
But this simply says that things should be adjusted so that
the driving frequency is equal to the natural frequency of
the circuit (without the correction due to resistance).
So, if the driving frequency is near the natural frequency, very large
currents can result.
When a circuit is driven near its natural frequency, we say that it is being
driven at resonance.
And the formula for Z shows that
the smaller the resistance of the circuit, the larger the response at resonance
will be.
This is what makes the radio tuner work.
The antenna of the radio picks up radio signals from every station in the
area,
but only the station whose frequency matches the natural frequency
of the tuning circuit will cause large currents to flow in the circuit.
These currents, when amplified, are the ones that produce the sound you hear.
If the circuit is not properly tuned, then it may pick up two stations
equally well, an effect you have probably heard many times.