Part 2 - Passive filters
Data Persistence
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Capacitor charge and discharge
Exercise 2-1 - Next, we want to understand how capacitors store charge and resist changes in voltage. On your breadboard, build the following circuit.
Measure the voltage across the capacitor using the oscilloscope. What is it?
With your probes attached to the circuit, disconnect the lead from +5V to the circuit.
LED
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What happens to the LED?
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What happens to the voltage on the scope? Why? (Hint: there is a first and second-order answer to this question, a full explanation requires considering the LEDs I/V characteristics which you can look into if you want :)
Capacitors and filters:
The simplest form of filtering in electronics is by using a resistor and a capacitor. To get an intuition of how this can be achieved, you can think of capacitors as elements whose "resistance" changes depending on the frequency of the input. DC current can not pass through capacitors (frequency=0), so it has an infinite resistance against DC current. AC current, however, passes through the capacitor. The higher the frequency, the lower the "resistance" of the capacitor. More precisely the current through a capacitor is proportional to the rate of voltage changes (\(I\ = \ C\ dV/dt\)). Consider the 2 circuits below.
Question: Which one do you think is a high pass filter (allows higher frequencies to pass to the output) and which one is a low pass filter? (consider 2 extreme cases of very high and zero input frequencies)
Notice how easily combining resistors and capacitors in parallel or series makes a filter and modifies the frequency bandwidth of your circuit. This filtering is not always desirable. In practice, you sometimes filter signals unintentionally due to the resistive and capacitive properties of your recording system. For instance, voltage-clamp recordings are limited by the resistance and capacitance of your electrode. Although it is not always possible to avoid this problem, you should at least watch out for it.
Exercise 2-2: Assemble a highpass filter with the following values and use it to filter the function generator.
Note: If you have an electrolytic capacitor, the negative pin is the shorter one.
The frequency cutoff (defined as \(\sim 30\%\) reduction in voltage amplitude) of the filter is \(\frac{1}{2\pi R_{1}C_{1}}\).
To test the frequency response of this circuit connect your function generator to drive Vin and measure both this input signal and the output of the circuit using your scope inputs. From the "Gen" menu on the scope, adjust the output frequency of a sine wave, note the corresponding amplitudes of the input and output of your circuit, and fill out the following table:
| Frequency | Vin (V) | Vout (V) |
|---|---|---|
| 1 Hz | ||
| 5 Hz | ||
| 10 Hz | ||
| 50 Hz | ||
| 100 Hz | ||
| 500 Hz | ||
| 1 kHz | ||
| 5 kHz | ||
| 10 kHz | ||
| 50 kHz | ||
| 100 kHz |
- What happens to the amplitude of the output as the input frequency varies?
Exercise 2-3: Feed a 400 Hz sinusoidal signal to your circuit, and visualize the input and output of the high pass filter with 2 oscilloscope probes. Do you notice any difference between input and output signals other than the amplitude?
Filters, in addition to modulating the amplitude of signals, produce a lag, causing the phase of the output signal to be shifted from the input.
Change the input frequency to 1000 Hz, does the phase lag change?
Exercise 2-4: Assemble a low pass filter with the following values:
The frequency cutoff of the filter is \(\frac{1}{2\pi R_{1}C_{1}}\)
Again, try changing the frequency of the input sine wave to test the filter to fill out the following table:
| Frequency | Vin (V) | Vout (V) |
|---|---|---|
| 1 Hz | ||
| 5 Hz | ||
| 10 Hz | ||
| 50 Hz | ||
| 100 Hz | ||
| 500 Hz | ||
| 1 kHz | ||
| 5 kHz | ||
| 10 kHz | ||
| 50 kHz | ||
| 100 kHz |
What happens to the amplitude and phase of the output as the input frequency varies?
Generate a frequency sweep with the PicoScope
Connect the USB cable, start the software, and check that the software detects the hardware ("Picoscope 2204A). Select it and click on the "Gen" section on the top left and sweep, On, Up Down, and reasonable frequency.
Exercise 2-5: Connect the output of the PicoScope "AWG" to your oscilloscope and look at the raw signal as well as at the Fourier transform (FFT). What is the FFT doing? Connect the input to your filter circuit and look at the output. Does it all make sense?
Exercise 2-6: So far we have been recording sine waves. Square waves consist of a broad range of frequencies, with edges containing high frequencies. Produce a square wave with your Picoscope's function generator and then use the benchtop scope to measure the signal before and after the filter.
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What do you observe?
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Turn on the FFT function on each of your scopes' inputs (CONF button). What do you see? (Hint: if you don't see anything interesting, try changing the bandwidth of your measurement using the horizontal controls on your scope.)
You should be aware that, when your input signal contains a broad range of frequencies, filtering can affect each frequency component differently. The most obvious change is that each frequency's amplitude is changed (this is generally the purpose of filtering, after all). However, the relative phases of the signal can also change. This will manifest as distortions in the time domain (peaking and oscillations). There are many different types of filters that are designed to minimize the impact on certain aspects of the signal, but they always come with tradeoffs. Be careful when interpreting filtered signals, you need to understand exactly what effect they have across frequencies before comparing raw and filtered signals, or the results of different filter types. The phase response of a filter captures its effect on the phase of various frequency components. For the first order RC filter you have \(\text{phase shift}(f) = -\arctan(2\pi fRC)\)
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PicoScope could also be used for recording data, instead of the oscilloscope ... but they can be very fiddly (i.e. crash, bug, require to be restarted for things to work again) ... so do it at your own risk and keep the simpler oscilloscope at hand if you do. ↩