Physics 204: Spring 2011

Oscilloscopes and Capacitors

The idea of charge Q on a capacitor C is rather abstract, but with an oscilloscope, we can actually “see” the charge, and even see it moving on and off the capacitor. What we actually see is the voltage V across the capacitor, but as you know, that is just proportional to Q, because V=Q/C and C is constant. In this lab we will watch the charge Q on a capacitor C in two different (but closely related) situations.

First we have to get acquainted with the oscilloscope. An oscilloscope is really a fancy voltmeter. What it shows is the voltage between its two inputs. It is conventional to ground one input. Then what the oscilloscope displays is the potential of the other one, represented as the deflection of an electron beam, deflected up for a positive voltage and down for a negative one. The fancy part is that while the vertical deflection is happening, the electron beam is also being deflected rapidly in the horizontal direction, from left to right. Thus, assuming the up/down deflection changes while the beam is sweeping horizontally, the beam draws an interesting graph on the screen of voltage vs. time. It is very clever!

The program O3 that we used in the Audio Spectra lab was a simulation of an oscilloscope, with many of the features of a real one. It had a “vertical gain” which was essentially a way of changing the unit on the vertical scale. Oscilloscopes have that too, a knob that allows you to choose the vertical unit, as so-and-so many volts/division, where “division” refers to the coordinate system ruled on the screen. A typical unit might be 0.1 V/division, if you are measuring voltages a little less than 1 volt. O3 allowed you to change the sampling rate. In the oscilloscope this corresponds to changing the horizontal sweep speed, a knob that tells you the horizontal unit as so-and-so many seconds/division. A typical unit might be 50 microseconds/division, meaning you can easily see things that happen on a very short timescale. O3 also had a “trigger”, meaning it wouldn’t begin a sweep until a certain amplitude was exceeded. Oscilloscopes have this too, and it is useful for synchronizing successive sweeps to a periodic signal, so that they all begin at the same place, and superpose perfectly – the result looks stationary, even though it is actually happening again and again (but always the same way).

Other useful knobs on the oscilloscope are vertical and horizontal positioning controls. These just apply a constant voltage to the vertical or horizontal deflecting plates to move the whole graph up or down, left or right. This can be useful for reading the relative coordinates of two points on a trace. Just move the whole trace to a convenient location, with the first point, for example, at the origin of coordinates, then note where the second point is. The triggering level is also an interesting knob to adjust – this determines when the sweep starts, so for some signals you can move the trace horizontally by changing its starting point.

I. Looking at a signal generator

The signal generator can produce sinusoidally oscillating voltage (AC) at an amplitude and frequency that you select. With two wires, connect the outputs of the signal generator to the inputs of the oscilloscope (connect ground to ground – the black side – and red to red), and choose 1 kHz for the frequency, with the amplitude control somewhere in the middle of its range. Fiddle with the oscilloscope until you see the sinusoidal voltage displayed on the oscilloscope screen. You may have to adjust all the controls mentioned above, including the vertical unit, and the sweep speed. The signal generator sends out a trigger pulse, so there should be a wire from “trigger out” on the signal generator to “trigger in” on the oscilloscope, with the oscilloscope set to expect an “external” trigger.

Once you can see the signal, try interpreting the screen. Using the horizontal unit, can you estimate the frequency just from what you see there? Suppose you increase the frequency from the signal generator. What happens on the screen? Try changing the amplitude control. Is this what you expect?

Practice changing the settings of the oscilloscope to get a good picture of signals generated at widely different frequencies and amplitudes.

The signal generator can also produce “square waves” – have a look at a square wave on the oscilloscope, and see how it changes when you alter frequency and amplitude. A square wave is very useful, because it is essentially like throwing a switch, connecting to a battery, but it can be done repeatedly, at high frequency, by the signal generator.

II. The RC time constant

Now let the signal generator drive the circuit below, with a resistor R and a capacitor C, and connect the oscilloscope so that it displays the voltage across C (which as we know is Q/C, where Q is the charge on C). Use a square wave.

The square wave in effect suddenly connects the capacitor C to a battery that should charge it up, then just as suddenly switches the connection, so that now the battery tries to charge it up the other way, and so on. You probably see that the square wave succeeds in doing this: the positive charge does appear on the capacitor, then the negative charge appears, and so on. The full voltage V appears across the capacitor, just as if you had connected to the signal generator directly.

If you do this at a high enough frequency, though, you begin to see something interesting. The capacitor C does not charge up instantaneously. Instead you see how the charge rises to its final value, taking a noticeable time to do it. The oscilloscope trace will look something like

You can even measure how long the charging takes. The process is exponential, which means the charge on the capacitor is actually a little logarithmic clock, just like radioactive decay of carbon 14 or the draining of a tank by Poisseuille flow. The difference is that this clock measures very short times! The decay time, or characteristic time, for a logarithmic clock is just the time it takes to fall to 1/e of its old value, which is about 1/3 of its old value. If you arrange the trace as above, falling 3 screen divisions in all, then the decay time is the time it takes to fall 2 screen divisions, approximately. That looks like a little less than 0.6 of a horizontal division in the figure, so that the decay time would be perhaps 0.55 of the time per division (whatever that is – you would have to check the oscilloscope settings). To get a nice picture of the process, you should feel free to change the amplitude of the signal generator, because we are not interested in the vertical scale here, only the horizontal scale (time). And of course you should adjust the frequency of the square wave so that there is enough time for the capacitor to charge up.

Theoretically the decay time t (in seconds) is just RC, where R is in Ohms and C is in Farads. Fix R at a convenient value and measure the decay time t for several values of C. Is it true that t is proportional to C? Try plotting t vs C. What is the slope of the line?

III. High frequency AC response of a capacitor

If you give a capacitor less time than RC to charge, it won’t finish charging. That means the voltage across it will be less than the voltage that was driving charge onto it. It will have to turn around and discharge before it was done. This shows up in the AC response of the capacitor at high frequency, where “high” means greater than 1/RC. The charge and the voltage don’t reach the full value that you find at low frequency, because there isn’t time.

Check this out qualitatively. Switch the signal generator to a sinusoidal signal (not a square wave), and look at the voltage across the capacitor as you increase the frequency. If you can get to high enough frequency, you will see that it goes down. It would be good to choose R and C so that RC is about 1 millisecond, and look at frequencies from about 2 kHz up to perhaps 10 kHz. These are moderate values where the signal generator can perform well.

Now try it quantitatively. Theory says that at high frequency , where the voltages are peak-to-peak, i.e., measured from the minimum to the maximum of the sinusoidal trace, and is the angular frequency. Since the time the capacitor has to charge in this case is proportional to 1/f, and the time it needs is RC, the theoretical expression has a simple meaning. It says that the amount of charge that gets onto the capacitor is proportional to the charging time, measured in the natural unit RC. This only works at high frequency because at low frequency (long charging time) there is plenty of time for the capacitor to saturate, with the full charge corresponding to the voltage V. Measure the voltage across the capacitor as a function of frequency, in the appropriate range, and see if it is proportional to 1/f. It would also be interesting to interpret the slope of this proportionality relationship, assuming that the relationship holds at all.