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.

 
 
(1)
 
 
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.
 
 
 

 

(12)
 
(13)
 
(1)
 
 
 
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: