Table of Contents
Related Reading
Hayes and Horowitz: Pages 78 - 84
Lawless: Chapters 10 - 17
Pasquale: Pages 83 - 100 (Note: this author covers operational amplifiers & diodes first)
Capacitors
We have a wide variety of capacitors in the lab, both in the form of ceramic capacitors (often small discs or rectangles) and electrolytic capacitors (large black cylinders). Both, at heart, consist of some pieces of metal separated by a dielectric material. Unfortunately, standards for labeling capacitors are not as uniform as those for resistors, and the manufacturing tolerance (how much their capacitance can deviate from the marked value) can often be 20%! Thus, if you need to know a precise value for a specific part, you'll need to test it yourself. Since you don't have a meter that can do that, you'll have to use your knowledge of circuits instead!
 Ceramic & film capacitors do not have a preferred polarity; it makes no difference which leg is where. They do tend to have relatively low capacitance, which means they are less useful in situations where lots of charge is needed (such as power supplies). |  Electrolytic capacitors have a preferred polarity! In our case, the side with the white arrow containing a “-” sign and short leg should always be at a lower voltage than the other side. |
 An illustration of the insides of a typical ceramic capacitor |  An illustration of electrolytic capacitor construction. |
 An annotated cross-section of a film capacitor. Source |  An annotated cross-section of an electrolytic capacitor. Source |
 The schematic symbol for a capacitor |  The schematic symbol for a polarized capacitor. |
Capacitors are defined by their capacitance, $C$ which is the ratio to the charge stored on a capacitor $q$ to the voltage across the capacitor $V$
| $C = \dfrac{q}{V}$ |
Since measuring stored charge isn't really a practical thing, we can instead consider the rate of change in the charge, which just so happens to be the current through the capacitor.
| $I = \dfrac{dq}{dt}$ |
We can the do some math to rearrange terms and express the capacitance in terms of just (change in) voltage and current, which we can readily measure
| Multiply both side by V | $CV = q$ |
| Differentiate | $\dfrac{d}{dt} (CV) = \dfrac{d}{dt}q$ |
| Pull C out because it is constant | $C\dfrac{dV}{dt} = I$ |
What's all this impedance stuff anyways?
Let's suppose that you want to express the relationship between the voltage across and the current through a capacitor in a way similar to what we do with resistors. This relationship is called impedance, which we'll represent with a capital $Z$
| $Z = \dfrac{V}{I}$ |
For a resistor, the impedance is just the resistance, as we're used to.
For a capacitor, the impedance is related to the angular frequency by:
| $Z_{cap} = \dfrac{1}{i\omega C} = \dfrac{-i}{\omega C}$ |
The $i$ term indicates that there is a phase shift between the current and voltage; if you assume that voltage is a sine wave then the current would be a cosine. The rest tells us that the impedance of the capacitor decreases as the frequency of the signal across it increases. As a rule of thumb,
| as $\omega \rightarrow \infty$, $|Z_{cap}| \rightarrow 0$ | $\therefore$ capacitors act like wires at very high frequencies |
| as $\omega \rightarrow 0$, $|Z_{cap}| \rightarrow \infty$ | $\therefore$ capacitors act like open circuits at very low frequencies |
Of course, these are rough approximations, but they're enough for you to be able to work out filter behavior. Integrators and differentiators act in an intermediate regime where these rules aren't helpful.
Get your imaginary numbers out of my physics!
We could do all of our circuit analysis just fine while sticking to the differential equation forms for capacitive (or inductive) behavior. However, you'd be stuck solving a (possibly gnarly) set of differential equations every single time you wanted to analyze what a circuit does. Since typically that'd be a huge amount of work, we instead use some assumptions about the behavior and properties of complex numbers as a shortcut.
It is likely that some students still haven't had much experience with complex numbers at this point, but that should be fine. Probably the most important thing to emphasize here is that the imaginary part has relevance in terms of phase (signals leading/lagging one another) and that we can only physically measure real numbers. Generally his means we'd have to multiply things by their complex conjugate and take a square root, but students aren't being asked to do that math here.
RC time constants
To start, you'll make an RC circuit, as shown in the figure below. Choose a capacitor in the 100 to 1000nF range and a resistor that's somewhere between 1,000 and 10,000 $\Omega$. (reminder: nano is $10^{-9}$)
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| An RC circuit for observing capacitor charging/discharging |
Note that the solid circles indicate electrical connections: Thus, the side of the capacitor connected to the resistor should be the point measured as $V_{out}$ , and the other side should be a common ground connection for the capacitor, the function generator, and the scope probe. To start, we'll drive the input with a square wave that goes between 0V and 1V.
- Start the function generator
- Press the Menu button
- Select the Square wave on the touchscreen
- Press the swap icon
next to the Ampl setting - Use the touchscreen or dial controls to set the high and low voltages to 1V and 0V, respectively.
The function generators we're using have a touchscreen, which you might not expect. This lets you select a number and type it in directly if you want to. You can also navigate by rotating / pushing the knob on the righthand side. The arrows under the knob select which digit you're changing.
On Measuring Signals
Many circuits we will build are designed to perform some operation on a signal or respond to an input. In such instances, it makes sense to observe both the signal that's put into the circuit and the signal coming out. Typically, the signal is a change in voltage but there are cases where a signal is a change in current. We'll still almost always be measuring a voltage anyways, as measuring current is a messier business. When you see a point labeled $V_{out}$, this is an indication that you'll be making a measurement there relative to ground. Eventually we'll chain together outputs and inputs, but that's for another lab.
Since we can measure two things at once on the oscilloscope, we should exploit this to compare our input and output signals simultaneously.
- Use the probe connected to channel one of the oscilloscope to measure the circuit's input
- This should be where your wire from the function generator and resistor meet
- The alligator clip should always connect to ground
- Make sure the attenuation is set to 10x
- Use the second probe to measure the circuit's output
- You can either insert a wire in the breadboard row where the resistor and capacitor connect or clip it to one of the legs
- Also make sure the attenuation is 10x
- If the second channel (blue) isn't displaying, toggle it with the blue
2button - Use the trigger menu to make sure that you're triggering on channel
1- Turn the knob clockwise a bit to set the trigger level to half a volt or so
- Check the left-hand side of the screen to see if the two channels' baseline is the same
- If you can only see a single arrow, you're probably fine.
 |
| The oscilloscope set up to observe both channels. Nothing is on currently, hence the flat lines. |
With the function generator off, your screen should look something like the picture above. Of particular note:
- The center left shows a single arrow pointed to the middle division of the scope
- The bottom left of the screen shows scale information for channel 1 and channel 2
- the bottom right indicates that we are triggering on a rising slope at 0V on channel 1
Note that, depending on the component values chosen, the starting frequency of 1kHz might not be enough time for you capacitor to charge, or it might be too much. Adjust your frequency until you get something similar to the middle image below.
 |  |  |
| Here the capacitor is charging and discharging too quickly to get a good measurement. | This is fairly optimal. the capacitor has just enough time to fully charge or discharge each period. | In this instance, the capacitor is staying charged or discharged for extended periods of time. Measuring the time constant at this scale will work, but is more prone to roundoff errors from digitization. |
Now, you'll measure the time constant $\tau = RC$ three different ways:
- Using the cursors, measuring the time it takes capacitor to discharge such that the output drops to $\frac{1}{e} = 37\%$ of its starting value.
- This will be easiest if you change the trigger to
fallinginstead ofrising
- Using the cursors, measuring the time it takes capacitor to charge such that the output rises to $1-\frac{1}{e} = 63\%$ of its final value.
- Changing the trigger to
risingagain helps here
- Using the scope's built in “rise time” ($t_r$) measurement, which measures the time it takes for the signal to change from 10% to 90% of its final value. After going through the math, the rise time turns out to be about 2.2 time the circuit's time constant (i.e. $t_r \approx 2.2\tau$ )
If students are choosing parts within the suggested range, then the time constant should be anywhere between $\tau = 100nF * 1000 \Omega = 10^{-5} s => 10 kHz$ and $\tau = 1000nF * 10000 \Omega = 10^{-2} s => 100 Hz$. If students are having trouble figuring out where they should be looking for a signal, the expected time constant gives you an idea of what horizontal scale should be appropriate.
Cursors
Press the CURSOR button on you oscilloscope, then use the softkeys (keys at the edge of the screen) to set the “Type” to Time and the “Source” to whichever channel you're using
Rotating the multipurpose knob (top center of the scope) will move the active cursor, while pressing the bottom two softkeys changes which cursor is selected
 |
| Cursor controls for a Tektronix scope. Your cursor button is in a slightly different location, but the others are identical. Click for larger image |
The time constant is the product of the resistance and capacitance, in SI units (i.e., $\tau = RC$ will have units of seconds when R is in Ohms and C is in Farads)
Other measurements
Using the measure button, you can select both the channel to be measured as well as the type of measurement using the softkeys and multipurpose knob. In this case, you'll use the softkey that's second from the top to select CH1, use the knob to highlight the measurement you want (Rise Time), and then press the knob to select it. The measurement will then appear on the bottom of the oscilloscope screen.
Differentiating & integrating signals
Instead of just characterizing a capacitor's capacitance, we can use the fact that the current through them is related to the rate of change of the voltage across them to perform calculus! Sort of. Part of your task will be to characterize the conditions needed to integrate or differentiate a signal.
These circuits will produce an output that is proportional to the derivative or integral of the input, but they will be scaled by an unknown factor. In practice, the output will be scaled down an order of magnitude or more.
Differentiator circuits
Build the differentiator circuit, shown below. Don't worry about finding the exact value of the capacitance for the $100 nF = 0.1 \mu F$, we're looking for large-scale trends in this portion of the lab
 |
| A basic differentiator. |
Starting with a triangle wave, observe the circuit's input and output simultaneously.
Determine (roughly) what range of frequencies result in an output voltage proportional to derivative of the input. The magnitude of the signal will be quite different.
The expected cutoff frequency is around $f = 1/2\pi RC = (2*\pi*4.7k\Omega * 100 nF)^{-1} \approx 330 Hz$. Thus, it will be below this frequency that you'll observe differentiating behavior. You basically want to be in a regime where the majority of the voltage in the circuit is dropped across the first element, and only a small bit is across $V_{out}$.
Try at least two different input waveforms, and document what the differentiated output looks like. You can do this by saving an image from the oscilloscope.
Does the circuit still differentiate in the same frequency range?
How do I save an image from the scope again?
Your computers are already set up so that you can copy a screenshot or data from them by using your lab computer.
- Open up the
Open Choice Desktopprogram from the desktop - Press the 'Select Instrument“ button and select whatever starts with
USB - Use the “Get Screen” button to capture a screenshot
- Use the “Waveform Data Capture” if you want to pull numerical information from the scope
If this doesn't work for some reason, you can plug a usb drive into the front of the scope and press the save button (located just beside the multipurpose knob). If all else fails, a photo taken with your phone is better than sketching things by hand.
Application note
Differentiators like this can be useful for detecting sudden changes in a system, so that something can be done with the information. For instance, you might want to detect sudden changes in volume from a microphone to identify unwanted loud noises.
Integrator circuits
Swap your resistor and capacitor, as shown in the figure below.
 |
| A simple, two-component integrator circuit |
Starting with a square wave this time, observe the circuit's input and output simultaneously.
Determine (roughly) what range of frequencies result in an output proportional to the input
Again, try a couple different inputs and record the results.
The expected cutoff frequency is the same: $f = 1/2\pi RC = (2*\pi*4.7k\omega * 100 nF)^{-1} \approx 330 Hz$. This time, it will be above this frequency that you'll observe differentiating behavior.
One particular thing to look out for: students will often say that the integral of a triangle wave looks like a sine wave. But, the integral of a constant (what the triangle wave is, piecewise) is not sine. What they're actually seeing is a set of joined quadratic curves, as $\int x \propto x^2$. If someone really wants to see the difference, they can output a sine wave on the second function generator channel and scale it to overlay the integrator output.
Keep your circuit built for the next part.
Filtering signals
As you noticed with the previous cases, these circuits have behavior that changes with the frequency of the input signal. We can exploit this to create circuits that allow sine waves of certain frequencies to pass through unattenuated, while decreasing the amplitude of other frequencies.
Low-pass filters
Using a sine wave input, observe how $V_{out}$ changes relative to $V_{in}$ as a function of the signal frequency. After you find ranges where the output goes from passing (i.e., not substantially changing) the input to attenuating the input, you'll be ready to make quantitative measurements.
Experimentally determine thecutoff frequency $f_{cutof\!f}$
Cutoff Frequency
The Cutoff Frequency of a filter is the threshold used to define when when the output voltage changes significantly from the input. Frequencies above/below (depending on filter type) are significantly attenuated and thus 'cut off' from the signal.
(also called the 3 decibel frequency $f_{3dB}$), which is the frequency where $V_{out} = \dfrac{V_{in}}{\sqrt{2}}$
Compare your observed value to the expected value of $f_{3dB} = \dfrac{1}{2\pi RC}$ , using the nominal values of your components (i.e. what we wrote in the diagram). Is your observed value reasonably close (within 10%)?
Characterize the filter's gain at high frequencies by measuring the $V_{out}$ for a spread of frequencies higher than $f_{cutof\!f}$
The students will have seen the circuit's behavior before, but now we're asking them to quantify things a bit. As noted earlier, capacitors have notoriously poor tolerance, so it isn't worrying if the expected and measured cutoff frequencies are off by quite a bit.
As for the gain at high frequencies, the expectation is that the output will fall off at a rate of 20 dB power per decade, which translates to the voltage being cut by a factor of 10 every time the frequency increases by a factor of 10. From the math side, you can relate this to the fact that the capacitor's impedance is directly proportional to $1/f$.
Angular frequencies
In lecture or other resource, you may have seen the frequency-domain behavior characterized in terms of angular frequency, $\omega$. This helps keep the math compact by eliminating the need to write factors of $2\pi$ everywhere and is a useful descriptor when talking about behavior in the complex plane. However, when it comes down to time to physically measure frequencies, we'll be measuring $f = \dfrac{\omega}{2\pi}$. If you're ever working with capacitors or inductors and find something off by a factor of about 6, it may very well be a frequency vs angular frequency conversion issue.
Circuit Gain
The gain $g$ of a circuit is the ratio of the amplitude of the output to the input, i.e. $g = \dfrac{V_{out}}{V_{in}}$. This term will come up regularly throughout the course, particularly when working with amplifiers whose purpose is to increase the magnitude of a signal while keeping its shape.
High-pass filters
Swap the resistor and capacitor to make a high-pass filter, shown in the figure below.
| |
| Our suspiciously familiar high-pass filter |
Experimentally determine the cutoff frequency $f_{cutof\!f}$ (also called the 3 decibel frequency $f_{3dB}$ ), which is the frequency where $V_{out} = \dfrac{V_{in}}{\sqrt{2}}$
Compare your observed value to the expected value of $f_{3dB} = \dfrac{1}{2\pi RC}$, using the nominal values of your components. Is your observed value reasonably close (within 10%)?
Observe what happens to the phase of the signals both above and below the cutoff frequency. When are the signals most in phase? When are they out of phase the most?
Here you expect the gain at low frequencies to instead decrease proportionally to $f$ when below the cutoff frequency.
Above the cutoff frequency, the signals will both become increasingly aligned in phase (i.e. if you overlay them there will be less of a time shift between the signals). Below the cutoff frequency, the output will start to consistently lead the input, to a maximum of a $90^\circ$ shift.
To think about this physically, at high frequencies the capacitor will be nearly un-charged at all times. It'll alternate between charging and discharging so rapidly that there's little time for its time dependent behavior to be relevant, so the voltage across the resistor will be in phase with the input.
At lower frequencies, we need to consider the capacitor's charging behavior. From the relation $C\dfrac{dV}{dt} = I$, the capacitor will have the largest current through it when the voltage is changing the quickest. For a sine wave, this corresponds to where $V_{in}$ is 0V (think about the slope of a sine curve). Since there's one current in the circuit from $V{in}$ to the capacitor to the resistor to ground, then $V_{out}$ as measured across a resistor will be largest for the largest current. There's a lot of nuance to the details here, but limiting cases are the most interesting for now.