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Capacitors and Alternating Current.

deltav sine

Rate of Change in a Sine wave.

One of the important concepts I hope I have driven home in the two previous posts about capacitors is the current in and out of a capacitor is related to how much difference there is between the source and voltage stored in the capacitor. We are going to make a lot of use of that concept tonight.  The question we have to deal with now is:  “What happens if the voltage from the source is continually changing?”

A series RC circuit with an AC source.

A series RC circuit with an AC source.

If the frequency is high enough the answer is that the capacitor capacitor is never given enough time to charge in one direction before it is requested to charge in the opposite direction.   At this point the capacitor conducts almost as well as a wire.  There is another interesting fact that happens within a cycle of sine wave.   The first picture shows that the fastest change within a cycle occurs near the zero crossing point while the slowest rate of change occurs near the peaks.   The rate of change of a sinusoid is also a sinusoid but shifted 90 degrees ahead.  In other words, if the wave is equal to Sine(ωt,)  then the rate of change is Sine(ωt + 90 degrees).   ω represents the frequency in radians per second or 2*pi*frequency  (2Πf).  You will be seeing 2Πf a lot.

Because we have to worry about this phase shift we cannot call the measure of how much a capacitor blocks current from flowing resistance.   The special word for this is Reactance and is given the value Xc.   The formula for calculating Xc is:

Xc

frequency and Xc

frequency and Xc

Notice several things: First our new found friend 2Πf appears in the denominator. So the higher the frequency the less reactance (blocking). Also, the bigger the capacitor the less the reactance. The relationship of frequency is shown very clearly in this table for a 10μF capacitor at various frequencies.
Now we will move on to determining the actual current flowing in a circuit with a resistor and capacitor (RC) in series.

Because the capacitor causes a 90 deg phase shift in current we are back to our old friend the right angle and right triangle when we have to calculate the total impedance.  Impedance takes both the Reactive elements, (capacitors for now, soon to include inductance), and the resistive elements.  The symbol for impedance is Z and in a series circuit Z = √ (R2 + Xc2).  This is our old friend the Pythagorean theorem.

Waveforms of a 100 Hz signal on a RC circuit (see text)

Waveforms of a 100 Hz signal on a RC circuit (see text)

In the circuit diagram shown earlier, assume the source is set at 100 Hz and the resistor has been replaced with a  159.15Ω resistor. Z in this circuit would be √ (159.152 + 159.152) or 225.07Ω.
Since I set the Resistance and Reactance to the same value this is a right triangle with the other two angles equal to 45 degrees.

As shown in the diagram to the right the current through the resistor would be 45 degrees ahead of the voltage in (the black line).  Ohms law still works for the resistor so the voltage across it would also be 45 degrees leading. (red line).  Kirchhoff’s voltage law still applies so for each point on the graph Vin – Vr = Vc and the voltage on the capacitor is 45 degrees behind the voltage in and 90 degrees behind the current.  This is show in the blue line on the graph.

Although I have not typed a lot of words in this post, my guess is if you are not already familiar with all of this I probably have your head swimming with all these concepts so I think I will end the post here at this point.  Please read and reread if necessary to make these concepts you own.   At this point the concepts are more important that learning the equations.  Very soon I will be arming you with a math concept that will give you more powerful tools.  I decided to do this in this order so you will see a need to learn the math.  The next electrical post will be about some common uses for capacitors before I throw the math at you.  The math is not that bad but until you see a use for it why learn it?

Gary


 

 

 

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