Day 5: More Capacitors

We left off with a capacitor integrator.
The next logical step is, of course, a differentiator!
Image from 310 lab manualPicture taken by HannahFollowing basically the same steps as the integrator, the only difference is that Vout is now across the resistor (instead of the capacitor).
In the following pictures, the yellow line indicates the input and the blue line indicates the differentiator output.


Note how the differentiator correctly shifts the sine wave! I think the most interesting is the ramp, as the derivative is the clearest.
Next, we discussed impedance a bit. I think it's easiest to think of as a form of resistance. When the frequency is 0, the impedance R-i/ωC approaches infinity. In contrast, when the frequency is infinity, the impedance approaches R. We also wanted to check what happens to the integrator circuit when the conditions of Vout<<Vin is violated and what happens to the differentiator circuit when 1/ωC<<R are violated. For the integrator circuit, the RC must be decreased; for the differentiator, the RC must be increased. We used a 1000 fold change to make the results very clear.The derivative of the square wave is clearly no longer correct!Then, we built a low pass filter with a 15 kΩ resistor and a 0.01 µF capacitor. This allows low frequencies through the circuit, but high frequencies are not permitted.Mathematically, we want ω3db = 1/RC, which should correspond to a Vout that is 70% of Vin. f_3dB = 1/(2πRC) = 1/(2π 15 kΩ 0.01 μF) = 1.1 kHz
The ultimate test is to see if experimentally we will get ~2.1 units of Vout when Vin is 3 units.

And wow! it is 1.2 kHz

The limiting phase shift for an integrator / low pass filter is in phase for low frequencies and 90 degrees out of phase for high frequencies. We will see later that this is the opposite for differentiator / high pass filter.

In phase

Out of phase

We also wanted to look at the 6 dB/octave attenuation (fancy word for reduction) above 3dB.


2 f3dB = 2.4 kHz -> reduced from 15 V to 6.6 V
4 f3dB = 4.8 kHz -> reduced from 15 V to 3.8 V

10 f3dB = 12 kHz -> reduced from 15 V to 1.5 V
20 f3dB = 24 kHz -> reduced from 15 V to 0.8 V

And the phase changes at various frequencies are also as predicted.
f << f3dB -> in phase
f3dB -> 45 degrees out of phase 
f >> f3dB -> 90 degrees out of phase

Why 45 degrees?
f3dB = 1/(ωC) 
Vout = Vin/(1+iωRC) 
1/sqrt(2)  tan^-1(1/sqrt(2)) = 45 degrees

Now onto the high pass filter! Basically the exact opposite of the low pass filter -- it uses a differentiator as the base, and permits only low frequencies. We just changed the positions of the capacitor and resistor.
f3dB is once again 1.2 kHz.



The limiting phase shift for a differentiator / high pass filter is in phase for high frequencies and 90 degrees out of phase for low frequencies, with Vin leading.  

Finally, we created a garbage detector. We wanted to measure the "garbage" in the 60 Hz AC power supply. We also did not want to die, so a transformer was used.

The setup

With a high pass filter, the high frequencies can be observed.

And as predicted, it is quite messy!