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Two-four-six-eight. We are going to Resonate.

The series RLC circuit.

The series RLC circuit.

In the last post “Electrical Noise part 2” I talked about tuned circuits.  Today’s post will be the first of several that talks about tuned circuits and resonance.  Today’s post will be very general and will talk the normal stuff.  Electricity is the same, no matter if you are talking micro-circuit electronics or massive power generation plant electricity.   However, different things become important in each situation.

Most of my professional experience has been in controls and a little less in power.  My hobby experience has been all over the map and included some playing with Amateur Radio (Ham Radio) many years ago.  In radio communications, tuned circuits are very important, both for the transmitter and for the receiver.

The values in the circuit in the first picture were chosen so XL and XC both are 1000 Ω at exactly 100 Hz.  At that point the lagging current in the inductor is exactly equal to the leading current in the capacitor and they cancel each other so the source only “sees” the resistor.  The resistor includes the resistance of the wire in the coil as well as the resistance of the wiring, so for the purposes of this post each component is ideal.  That is always the best starting point and then we can starting adding confusion as we understand things better and develop better tools.

The total current in the circuit for R = 1K

The total current in the circuit for R = 1K

In the first case I chose R to also be 1000 Ω.  The plot to the right shows how the current changes in the circuit depending upon the frequency of the source.  At exactly 100 Hz the current is 1 mA and this is exactly the same value it would be if the capacitor and inductor were not in the circuit.  As we deviate from 100 Hz the current drops off because at lower frequencies, the capacitor is blocking more current and at higher frequencies the inductor is blocking more current.

The phase of the total current in relationship to the source voltage.

The phase of the total current in relationship to the source voltage.

In this picture I show what happens to the phase of the total current.  In all of the phase plots we will be looking at zero phase shift means the phase is exactly the same as the phase of the voltage source.   As expected, when the capacitive reactance becomes greater (low frequencies) we get a 90 degree leading phase shift.   At high frequencies, the inductor becomes the blocking device and we have a lagging current.   At exactly resonance of 100 Hz, the current is in phase with the voltage.

Now we go into some strange things.  This circuit was analyzed using a procedure called sinusoidal analysis.   For each frequency the impedance of each device was calculated.  Total impedance (Z) equals the sum of the individual impedance values because this is a series circuit.  However, there is a little math trick which I have yet to explain where imaginary numbers were used.  That analysis is a real pain when done by pencil and paper, but with computers it is easy.   Once the total Z is known I = V/Z.  Now that I is known the voltage and phase of each component can be found by V=I*Z of the component.

The voltage across the capacitor.

The voltage across the capacitor in the series RLC circuit for various frequencies.

In the plot to the right I show the voltage across the capacitor. At 10 Hz it has all the voltage of the source across it.   At 1KHz it has next to no voltage across it as one would assume for higher frequencies across a capacitor.  The slope between 1 V at low frequencies and 0 V at high frequencies in the in-between range also is intuitive, because somehow it has to get high to low.  The thing that is not intuitive is the increase in voltage just below 100 Hz.

Voltage across the inductor in the series RLC circuitfor various frequencies.

Voltage across the inductor in the series RLC circuit for various frequencies.

The inductor Voltage plot looks very similar but is a mirror image.
So where does the “extra voltage come from?  Because the Capacitor and Inductor are storing energy in different parts of the cycle, the energy is being passed back and forth near resonance and this shows up as a higher voltage.


The phase of the voltage across the capacitor in the series RLC circuit.

The phase of the voltage across the capacitor in the series RLC circuit.

The next plot shows the phase of the voltage across the capacitor throughout the frequencies we are injecting into the circuit.   At low frequencies the capacitor is doing all the blocking so the complete voltage is impressed on the capacitor and there is no phase shift.  (However the current is being shifted… refer to the 3rd plot).   At high frequencies the inductor is doing the blocking and the shifting of the current.  This means the capacitor is creating an additional 90 degrees voltage shift on top of the 90 degrees current shift caused by the inductor so the total shift is 180 degrees from the source voltage.   (Is smoke coming out of your ears yet?)

The phase of the voltage across the inductor in the RLC series circuit.

The phase of the voltage across the inductor in the RLC series circuit.

The phase of the voltage across the inductor plot looks exactly the same as plot for the capacitor except it is shifted 180 degrees from the capacitor.  This means the voltage across the inductor leads the source voltage by 180 degrees at low frequencies.



This pretty much presents what is going on at resonance for a series RLC circuit, except we need to talk about what happens as the value of R changes.  Things get very very interesting.  I had read all the short answers in the books but I did not realize the true effects until I ran this model.   That will be the next post.  Then after that I probably need to explain the math and describe the program I created to do these models.   Once you are armed with that you will be able to model many versions of RLC circuits.   There are programs that will do all of that for you, but until you do it yourself you really don’t get the full understanding of what is going on.

Besides…. letting the smoke out of your head is a whole lot better than letting the smoke out of the circuits you build!


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