## 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 sine-wave test signal Vin = 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 VR = i*R has two parts: a sine part with magnitude A that is in-phase with the input signal, and a cosine part with magnitude B that is out-of-phase 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 Vin = 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 wR of the circuit.

 (12)   (13)   (1)

• Note the term (1/(Cw) - Lw) in the denominator of the expressions for the coefficients A and B. When this term is zero, M (the amplitude of our output voltage) attains its largest value. Thus, the resonant frequency of our chosen system is given by
• or

so that , which agrees with the above plot. Hurray!

• Remember that this circuit we are analyzing is just one channel of an equalizer. Thus this channel of our graphic equalizer controls the output voltage (volume) of frequencies in the range of about 1 to 1000 hertz (see the above plot). This should be beginning to make sense: each channel of the equalizer will have a particular resonant frequency (frequency around which the Vout is large) for the chosen L and C in the circuit. If the L and C of each channel is carefully chosen, we will have a continuous spectrum of frequencies that we control the volume of by turning the knob (potentiometer) of each channel.

• 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.