It is my sincere hope that my last post “Two, Four, Six, Eight We are going to Resonate” resonated well with you. (Sorry about the pun, sometimes I can’t help myself.) In this blog post I am going to build upon the last one and expand it just a little.

In the mechanical world resonant systems are all around us. Musical instruments depend upon resonance, and anyone who has ever swung in a swing knows the swing is going to dictate the timing of the pushes to get the swing to move higher. The suspensions of automobiles is also a resonance system and the car would bounce for a long time if it was not for the shock absorbers. These are called dampeners in UK English and I hate to admit it, but that is really a better name for the function they provide. In our series RLC circuit the resistor provides the dampening function.

There are two numbers used to describe the dampening action. The first one is called the Q or Quality factor. This primarily used in radio circuits because as we are going to find the bigger this number is the narrower the bandpass of the filter. Q is defined as:

Q= 2*π * energy stored per cycle / energy dissipated per cycle.

In our circuit Q = X/R where X is the reactance of either the capacitor or the inductor at resonance.

In control systems resonance is usually not a good thing, and the term used is the damping ratio. The damping ratio is given the Greek symbol ζ and ζ = 1/(2*Q).

In this post I am going to show what happens when the R value is changed. I will leave the capacitor and inductor values the same so the resonance frequency will remain at 100 Hz and each of those will have a reactance value of 1000 Ω at the resonance frequency. In all cases the source voltage will be 1 V. We will assume this is rms, although for most of the discussion it does not matter as long as all values are measured the same way.

Phase changes will be exactly as shown in last night’s post, except the changes will occur over varying bandwidths depending upon the Q. I will only show the total current plots and the voltage across the capacitor for each R value.

The graph to the right shows the total current flowing in the circuit for an R value of 1KΩ. This is the same plot used last night and the Q for the circuit is 1.

The plot to the left is the same circuit except R = 10K. The Q for this circuit would be 0.1. Notice how much wider the bandpass is compared to the previous plot. Because we are holding the voltage constant in all examples the scale on the Y axis changes to reflect the lower current.

Now lets see what happens if we go down by a factor of 10 from the original value for the R value.

The plot to the right shows the same circuit, except R = 100Ω and Q = 10. Notice how much “tighter” the band pass is in this case.

Now we will start getting into the really interesting things. The plot to the left is the voltage across the capacitor with R = 1000Ω or Q = 1. If you remember last night we talked about why the voltage across the capacitor is greater than the source voltage.

Increasing the resistance makes a less interesting case. This plot is with R=10K and Q = 0.1.

Things get a whole lot more interesting if we decrease the resistance. The plot to the left shows the voltage across the capacitor with R = 100Ω and Q = 10. Notice now we are getting 10 Volts across the capacitor with only a one volt source. Wow Man… Free energy! No! remember the energy is just being passed back and forth from the capacitor to the inductor and very little is being transferred to the resistor.

Being that was kind of interesting….. What will happen if we use a very small resistor; one very close to zero. (I can’t use zero because the computer program will simply die from a divide by zero error.)

The plot to the right is the voltage across the capacitor with R = 0.000001 ohms or Q = 1,000,000,000. For one volt in we get 1,000,000,000 Volts across the capacitor. That will fry you like a fritter! Besides the fact that this would be almost impossible to build, the 1 V input would have to supply 1,000,000 A at resonance into that small resistor. This would require a 1 MW power source. (I^{2}R).

However, what if we used a little smaller power source, but used a resonance circuit and then coupled it to a another coil with more windings forming a step-up transformer? We would end up with a Tesla coil. There are probably a whole bunch of construction details we have to worry about but we are getting the idea. Not the least of these details is how do we adjust everything into resonance without frying ourselves? Oh well, we can enjoy the feats of others. Especially when they are playing the National Anthem. (The National Anthem if you live in Alabama).

It will probably be next week before I can model the parallel RCL circuit. It is very similar, but duality of course will make things slightly different. For the rest of this week I will try to introduce the math I used to make this model. There really is only a couple of small concepts beyond what you already know.

Gary

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