E84 Lab: Active Filters

• Derive the frequency response functions of the second order Sallen-Key low-pass and high-pass filters discussed here.

• Derive the frequency response function of active twin-T.

  Here $R_4$ and $R_5$ are the top and bottom parts of a potentiameter which controls the amount of feedback from the output to the twin-T filter:

  $$V_{feedback}=\frac{R_5}{R_4+R_5}V_{out}=\alpha V_{out},\;\;\;\;\;\; \alpha=R_5/(R_4+R_5)$$ (1)

  The middle point between $R_4$ and $R_5$ is the wiper position of the potentiameter that can move from top ($R_4=0$, $\alpha=1$) to bottem ( $R_5=\alpha=0$).

  Express the frequency response function (FRF) of the circuit in the canonical 2nd order form and determine $\omega_n$ and $Q$ in terms of R, C, and $\alpha$.

• Select component values for your twin-T filter circuit so that the 60 Hz powerline noise is attenuated by 20 dB or more, while frequencies below 50 and above 70 Hz should be attenuated by less than 3 dB. The width of the twin-T notch filter is configurable using the potentiometer. Use $C=270\,pF$. Make a Bode plot and show your design meets these specifications.

• Select component values for the Sallen-Key filter so that it can be used as an antialias low-pass filter that have a passband of 0-140 Hz and a stopband of 1 KHz and above to allow for 2 KHz sampling. The attenuation should be less than 3 dB in the passband but >30 dB in the stopband. Make a Bode plot to show your design meets these specifications.

  Hints: As these Sallen-Key filters are used as pure low-pass filters, a peak at $\omega_n$ needs to be avoided, i.e, $\zeta\ge 1/\sqrt{2}$ (see here).

• Construct the filter circuit that combines the twin-T filter and the Sallen-Key filter you designed above. Test and verify the circuit to show that its FRF satisfies the specifications.

  

• Test your EKG circuit with and without the filter, and show that the filter eliminates 60 Hz (which may or may not be detectable, depending on the specific location in the lab) without affecting the EKG signal.

What to turn in:

1. Hand analysis of your design of both filters in terms of the component value selection.
2. Multisim or Matlab Bode plots showing that your filters agree with the hand analysis and meets the specifications.
3. Measurement results of the circuit showing that its FRF matches the simulation and the specifications. Pay special attention to frequencies of interest such as the shape of the notch and the transition between pass and stop band.
4. Test results of the filter circuit with the EKG signals.