Resonance
Remember this circuit is one channel in our graphic equalizer for our
stereo system. In the last section, we found the equations of motion, guessed
a class of solutions, and solved for the parameters A, B, D, E which created
a specific solution to satisfies the diffeqs of motion for this circuit.
Remember that the input, which we decided should be a sinewave test
signal V_{in} = sin(wt), is under our
control; we can impose any input frequency w
that we wish. The values of R, L, and C are constants, but they are up
to us too, when we build the circuit.

Note that the output V_{R} = i*R has two parts: a sine part with
magnitude A that is inphase with the input signal, and a cosine
part with magnitude B that is outofphase with the input signal.
Both of these parts oscillate with frequency w.
A good measure of the total magnitude of the output is

(1)


Suppose L = 12 mH, C = 3000 mF, R = 100 W.
Let the input frequency be f=0.1 Hz (w=0.62).
Here is a plot of the input and output voltages:
In this figure the amplitude of the input voltage is unity
(since V_{in} = sin(wt) ). The magnitude
of the output voltage is given by (1) and for the values chosen, M=0.185,
as you can tell from the amplitude of the output voltage on the figure.
Now suppose we change the frequency of our input, keeping the resistor
and inductor the same. Let's consider f=10Hz  now M is approximately
unity!:
What about if f increases further, say to f=10000  then M=0.132:
So we can see that our circuit is weighting different input
frequencies differently  for some frequencies, my output voltage is nearly
my input voltage, while for other frequencies my output voltage will be
essentially zero.

A better way to visualize this is to plot M versus as a function
of w. This is the frequency response curve
for the circuit.

Locate the peak of the magnitude curve. This w is
called the resonant frequency w_{R
}of the circuit.

Finally, notice that R does not affect the resonant frequency, however
it clearly impacts the output voltage signal via its presence in A and
B. How does R affect the height of the resonant peak, and its width?
You can see clearly that larger R values lead to a broader
range of resonant frequencies. And it looks like a small value of R might
be affecting the height of the resonant peak  but be careful. Remember
these plots are generated numerically  even though I am using the analytical
solution I still have programmed it into a short code and set up a frequency
array during which to calculate the amplitude values for plotting. Do you
think the reduction of the M value from 1 at the resonant frequency in
the above plot is real for R=1? Answer.
What Frequencies Can We Hear?
Now that we have a grasp of what the graphic equalizer is doing for us,
and you will get more practice in the homework, let's connect the numbers
to our biology a little bit. Here are some interesting factoids:

Humans hear frequencies from 20 to 20,000 Hz. But as a person ages, the
high frequencies are the first to go. Your parents and professors probably
hear about 25 to 14,000 Hz.

Middle A (above middle C) is 440 Hz.

Musical instruments differ in their sound, even though they are playing
the same note because of overtones/harmonics (multiples or integer fractions
of the fundamental frequency). This is why a flute sounds different from
a trumpet playing the same note.

Pitch is frequency (high pitch = high freq.).

One octave is a doubling of frequency (A=440 Hz, high A = 880 Hz).

Male speech is 300 to 600 Hz; Female speech is about 1 octave higher.

Hum of light fixtures is at the line frequency of 60 Hz. In Europe, the
hum is at 50 Hz.

The human body chest cavity resonates at O(1) Hz. At a rock concert the
pressure that you feel on your chest is this frequency.

A woofer is a speaker that operates at 20 to 1000 Hz. A tweeter operates
at 1000 to 17,000 Hz.