About this video
This circuits lesson derives and plots the natural response of an RC circuit, where a charged capacitor discharges through a resistor with no source. Using 1,000 ohms and 1 microfarad, it shows that the time constant RC is one millisecond and that the voltage decays exponentially. The same exponential shape governs current, and the instructor notes it appears in almost every circuit.
You'll be able to
- compute the time constant RC for a given resistor and capacitor.
- describe the exponential decay of voltage in a discharging RC circuit.
- use the natural response equation to find voltage at a given time.
Key words
- Natural response
- The behavior of a circuit with no independent source, driven only by stored energy.
- Time constant
- The product RC, which sets how quickly an RC circuit's voltage or current decays.
- Capacitor
- A component that stores charge and creates a voltage across its plates.
- Exponential decay
- A curve that drops by a constant fraction over each equal interval of time.
Transcript
- [Voiceover] We just derived what the current is and the voltage. These are both the natural response of the RC. Now what I did is I went ahead and I plotted out this using a computer, just using Excel, to plot out what these two expressions look like, and let me show you that.
So let's do a real quick example here with real values just to see how this equation works. We'll say that this is 1,000 ohms. That's our resistor. And we'll say C is one microfarad. And what we wanna work out is R times C equals 10 to the third times 10 to the minus six and that equals 10 to the minus three seconds or one millisecond.
That's the product of R and C. I forgot V-naught. Let's say we put two volts on this capacitor to start with, like that. And now we can say V of t equals V-naught two volts times e to the minus t over one millisecond. And that's our natural response for this particular circuit.
Now let me show you what that looks like. This is V of t on this side. Equals two e to the minus t over one millisecond. And you can see it starts at two volts and then sags down as we predicted, and that's an exponential curve. And then over here on this side, i, as we said before, starts out at zero in the capacitor, the current in the capacitor is zero, and as soon as we throw open that switch, the charge charges over through the resistor and this is the equation here, i of t equals two volts over 1,000 e to the minus t over RC or t over one millisecond.
So this is what we call the natural response of an RC circuit, and you'll run into this in almost every circuit you ever build.
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