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 | $C\dfrac{dV}{dt} = I$ |

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.

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.

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 100 and 10,000 $\Omega$. (reminder: nano is $10^{-9}$)

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 - Use the touchscreen or dial controls to set the high and low voltages to 1V and 0V, respectively.

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.

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.

- Using the cursors, measuring the time it takes capacitor to charge such that the output drops to $1-\frac{1}{e} = 63\%$ of its final value.

- 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. $t_r \approx 2.2\tau$ )

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

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)

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.

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.

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

Starting with a triangle wave, observe the circuit's input and output simultaneously.

Determine (roughly) what range of frequencies result in an output proportional to the input.

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.

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 Desktop`

program 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.

Does the circuit still differentiate in the same frequency range?

**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.

Swap your resistor and capacitor, as shown in the figure below.

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.

Keep your circuit built for the next part.

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.

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 $f_{cutof\!f}$

**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.

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.

Swap the resistor and capacitor to make a low-pass filter, shown in the figure below.

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?