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# Sketching exponentials

Voltages and currents often have an exponential shape. We look at some properties of exponential curves and learn how to rapidly sketch accurate waveforms. Created by Willy McAllister.

## Want to join the conversation?

• Hello!
In transient time energy stored in an element is going to decay twice as current or voltage finally reaches zero...
With in half time energy stored in that element becomes zero...then how the current flow with zero energy in the remaining half time....
• I really wish there were a lot more practice problems for circuits. That would be extremely helpful.
• Why is the vo drawn on the 2nd quadrant of the graph?
• The second quadrant of the graph is to the left of the y-axis (the voltage axis). It represents what is happening prior to time = 0. I defined the voltage source to have a special behavior. It's voltage is Vo for t < 0, and it switches its voltage to 0V right at t=0.
(1 vote)
• At , how could a tangent line exist at that point in the graph? Can't tangent line only exist in points where there is a smooth curve? And not in point of the graph where the line abruptly and suddenly changes directions?
(1 vote)
• You align the tangent to the exponential part of the curve, while ignoring the flat initial voltage before t = 0. Pretend the exponential part extends to the left of the vertical axis.
• You should probably specify that when we divide by v0 that it is not equal to 0 , since diving by 0 is problematic
(1 vote)
• when the voltage source is equal to vo , the voltage across the capacitor is not equal vo at instant , the capacitor takes time to charge through R until the voltage across it equals to vo, I am right?
(1 vote)
• The example in the video sketches the natural RC response where the starting state is some voltage V=V0, and lets the RC sag to zero. I think you are describing a step response, where the voltage starts at 0 and steps up to V0. If that's the case, the sketching instructions work, you just have to adjust the starting and ending voltages. For step response the voltage curves upward from 0 to Vstep. The curve has exactly the same curviness, just flipped upside down.
(1 vote)
• The slope of the curve changes with time (decreasing and eventually becoming zero as time approaches infinity) then what is the significance of calculating slope at just one point, extending the line, finding the equation of line and calculating where it cuts the time axis?
(1 vote)
• What type of voltage/current graphs that do not have an exponential function?
(1 vote)
• Exponential signal shapes happen in response to sudden changes of the input (a step function, or a switching event). Exponential shapes happen all the time in fast digital systems where you rapidly switch between high voltage and low voltage.

In systems where the input changes smoothly, like music or talking, the input signal resembles a smoothly changing curve, and we often model it as a sine wave. In this case, all the resulting signals in the system also look like sine waves of different size and delay.
(1 vote)

## Video transcript

- [Voiceover] Now I want to show you a really useful manual skill you could use when you have voltages that look like exponentials. And we're gonna talk about this exponential curve here that's generated as part of the natural response of this RC circuit. And we worked out that the voltage across here, this voltage V of T on the capacitor, its natural response is equal to V zero, V knot times e to the minus t over RC. This value of V knot is the starting voltage that our input source provides and then it immediately steps down to zero and this circuit then does its natural response. We got a current coming out of this capacitor flowing around in a circle like this. And that's the natural response of this RC circuit. I wanna look at the properties of this function right here, it's got some interesting properties. I'll write it right here. V knot, e to the minus t over RC. So first thing we can look at is this V knot value. That's this value here. This is V knot. In this particular chart, V knot was equal to one volt. That's what it originally charged up to. And now we wanna look at this point right here. Right when it goes through time equals zero and the curve starts to drop. We know how high it is, it's V knot volts high. What I wanna look at now is what's the slope right at this point? The slope of a curve is the derivative of a curve evaluated at this time, at time equals zero. Let's take a derivative of our function right here so its d, dt, V knot, e to the minus t over RC. And that equals V knot and the RC comes down, minus one over RC comes down. Minus one over RC, e to the minus t over RC. And that's the derivative of this exponential for all time. So now we evaluate at t equals zero and we get V knot over RC and there's a minus sign. And it's e to the, I plug in zero for t, it's e to the zero and e to the zero is one. So that equals the slope. At time equals zero, that equals the slope at time equals zero. So that's the value right here. That's what that slope looks like. That tangent line to the curve. Okay, now the next thing I wanna do is actually take this line and extend it all the way down till it crosses this axis, till it crosses the time axis. And the next question we're gonna ask is what is this point right here? What's that point? Let me move up a little bit. So now we have a line, we've defined a line. That means we have an equation of a line. So the equation of a line is y equals slope times x plus b, b is the intercept on the voltage axis, m is the slope. I can plug in my values here for our chart, y is the voltage axis. We know what the slope is, it's sitting right here. It's minus V knot over RC and we multiply by time and we add the y intercept or the voltage intercept which we know is V knot, plus V knot. Alright and now what I wanna do is find out at what time, what time does voltage equal zero for this orange line? So we're gonna plug in zero for volts and work out what time is. Zero equals V knot times one over RC and don't lose the minus sign times t plus one. If I divide both sides by V knot, I get zero on this side and this term on this side. So I can say zero equals minus one over RC times t plus one and we wanna isolate t so we'll take the one on the other side and multiply by RC so one more step. Minus one equals minus one over RC times t. And in the end, we come up with t equals RC. T equals RC is that time right there. That's how many seconds after the step that this line hits the time axis. And notice something here, there's no mention of V knot in here. There's no V knot. There's no V knot, it's not here. And it divided out in this step back here. It's one of our steps, it disappeared going from this step to this step here. So independent of how high this thing starts, it can be high or it can be low, the slope of this line always goes right through the time equals RC. And one more thing we wanna work out is at time equals RC, what is the value of the exponential? What is this voltage here when time equals RC? So we can use our equation again. We can plug into our equation and find out how to do that. If I go back to V equals V knot, e to the minus t over RC, and this time we're gonna plug in RC right here for this value here and find out what V is. So V equals V knot, e to the minus RC over RC or equals V knot, e to the minus one. Now the value of e is roughly equal to 2.7 and the value of one over e is roughly equal to 0.37 or another way I can say that is 37 percent. So in the end, this voltage right here is about 37 percent of V knot. That's that value right there. So that's two little things we're gonna tuck away in our head. The time for that line to hit the time axis is RC. If I wanna know the value of where the exponential actually is at time equals RC, it's roughly 37 percent of where it started from. Okay that's the basic idea and in the next video, I'll show you how to use these ideas to sketch exponentials really quickly.