If you have looked at other website articles on RC circuits, you would have noticed that you quickly descend into some serious mathematics and calculus. You could be excused for asking, “what has this got to do with anything electronic I’m doing?”
Good question! And the answer is probably almost nothing. Most of these discussions start with a charge and discharge circuit and the constants’ calculus derivation, but where would you use this?\
Here’s a confession: I worked in the electronics industry for 40 years without getting to grips with the math and the reason for it. As this is a beginners article, we will not go there. Instead, we will look at the practical side of RC circuits.
But if you want the math, read a great article here.
What are they used for
First off, RC circuit is a circuit with both a resistor (R) and a capacitor (C). It is a frequent element in electronic devices and plays an important role in the transmission of electrical signals.
There are some good reasons for knowing your way around RC circuits using some basic arithmetic to get a grip of how they work. In my line of work (audio and RF), I constantly had to match impedances from one amplification stage to another, prevent high or low-frequency roll-off, or actually force a particular roll-off. For example, seeing an audio amplifier does not have gain above the audible range, thus preventing oscillation.
Any amplifying stage will have points where the high-frequency end gets less (roll-off) with any increase in frequency and likewise at the lower end frequency. These points are conveniently called the 3dB points and are when the power has fallen to half or 3dB. Expressed as a voltage, this is 0.707 of the main level. This may sound like a lot but is barely perceptible.
If you are a musician or Hi-Fi person, you may have found certain connecting cables have caused a loss of high-frequency response, also due to RC (the C is inside the cable).
How They Work
Just to recall, a capacitor can store energy and a resistor placed in series with it will control the rate at which it charges or discharges. This produces a characteristic of time dependence and a crucial parameter that describes the time dependence is the “time constant” R C.
In the above circuit, assuming the C1 is discharged, when the switch is closed, the current flows into the capacitor, which looks pretty much like a short at this point. So a large current starts to rush in and fill the cap. As the voltage on the cap rises, the difference in voltage between the cap and the battery gets less and less, so the charging current tapers off. As it reaches the voltage of the battery, the charging slows noticeably.
As an experiment, and to prove the theory, I used a stopwatch (very crude) to time how long it took to reach the battery voltage using the above values. This took about 18 seconds. From the curve below, this was at the 4t point, so t must have been about 18/4 = 4.5sec.
The theory says time constant is simply R*C. So RC = 100k*47uF = 4.7sec which checks nicely.
The time constant can also be defined as the time it takes to reach 0.63 of the final value, or even the time between 10% and 90% of charge values.
If the battery is replaced with a short circuit, the cap will now discharge as in the curve below. In this case, t is reached when the value has dropped to 0.36 of V.
Applications of RC Circuits
We mentioned power bandwidth and frequency response earlier. They are related to each other.
In the circuit above, the output of amplifier Q1 is coupled to a high impedance load R2 of 100k via a coupling capacitor C1. Note that because the source impedance R1 on the collector is much lower than R2, we ignore it. Now the frequency at which the 3dB point occurs is fc=1/(2πRC),
so, Fc = 1/2*π* 100k*0.05uF = 28Hz.
I measured the actual components to be more exact and measured the 3dB points with a signal generator and an oscilloscope and got 28Hz as well. So the theory is good! The time constant is RC = 100k*0.05uF = 0.005.
The time constant is related to 3dB bandwith by t=0.35/BW(3db).
Beware, though, if the source impedance is high, say R1 was 50k, we have to consider the total impedance as though they were in series, i.e., now 150k. We might say this circuit is a high pass filter. If C1 and R2 were swapped, we would have a low pass filter.
Above is another amplifier, and we are going to use our RC skill to tailor the lower roll-off point.
Imagine for a moment that C2 was removed. The circuit’s gain would be roughly R1/(R2+R3) = 22/(2.2+5.6) or about 2.8.
If C2 is back in R3 and is now bypassed to lower AC, the gain jumps to 22/2.2 = 10, but at a well-defined point where XC2=R3, i.e., 28Hz.
We have seen some uses for RC circuits and how they might also give us unexpected results. Like given the length of a co-axial cable in a guitar cord or a microphone cable with high capacitance (often as high as 200pF for a 5m cable), this will cause significant high frequency losses due to the RC effect.