Butterworth High Pass Filter

Butterworth High Pass Filter

Transfer function H_hp(s) of a Butterworth highpass filter of order n and cutoff frequency ωc has the same poles as that of H_lp(s), the transfer function of a Butterworth lowpass filter of the same order and with the same cutoff frequency, but in addition, H_hp (s), has n zeros, all located at s = 0, whereas H_lp(s) has none. This means we can use the same table used for the low pass

Nomalized Low Pass Butterworth DenominatorTable

Steps to design High Butterworth Filters

1. Figure out which order circuit you need. (Higher order means steeper transition)

2. Use the High Pass Sallen Key circuit to design even ordered filters. For odd ordered filters it will be a combination of a first order filter and Sallen Key circuits.

3. Calculate the element values needed to realize the filter

4. Perform frequency scaling to change the cutoff frequency

5. Perform magnitude scaling to get more realistic values for the circuit elements.

Example : Design a 2nd Order Low-Pass Butterworth Filter

Design a 2nd order Low-Pass Butterworth Filter with unity gain and cutoff frequency 300 rad/s.

1. This example is a second order. We need a circuit that gives us two poles in denominator and we also need 2 zeros. The high pass sallen key provides this. (look at the high-pass sallen key transfer function)

2. Pick values for the sallen key that will give you the correct denominator. Again, for simplicity we can choose the resistors to be 1 ohm. Just pick The capacitor values to realize the correct denominator. Now you have a high pass filter with cutoff frequency 1 rad/s.

3. Perform frequency shifting to change the cutoff frequency to the correct value.

Higher Order High-Pass Butterworth Filters

• This is the same as the low pass filter. We just cascade first order high pass filters with 2nd order high pass filters to form higher order filters.