Butterworth Low Pass Filter

Butterworth Lowpass Filter

We can normalize the transfer function by dividing it by wc. H(sa) where sa = s/wc is the new normalized transfer function. It's gain is 1 and it's cutoff frequency is 1 rad/s.

Nomalized Low Pass Butterworth DenominatorTable

The table below shows the denominator needed to build a nth ordered low pass butterworth filter. Normalized means the wc = 1 rad/s

Steps to design Low Pass Butterworth Filters

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

2. Use the Sallen Key circuits 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.

Example1: Design a 1st order Low-Pass Butterworth Filter

Design a 1st order Low-Pass Butterworth Filter with unity gain and cutoff frequency 10 rad/s

1. This example is relatively easy. The table shows 1 pole. The number associated with that pole is 1. Unity gain means a gain of 1.

2. We can design this using a simple passive low pass filter. ( RC circuit with H(s) = 1/(s+1) ) We make sure it has one pole at -1. This means the filters cutoff frequency is 1 rad/s .

3. Perform Frequency Shifting to change the cutoff frequency to 10 rad/s

Example 2: 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. The sallen key provides this. (look at the low-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 low pass filter with cutoff frequency 1 rad/s.

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

Higher Order Low-Pass Butterworth Filters

• A 3rd order low-pass butterworth filter is just a 1st order in series with a 2nd order.

• A 4th order low-pass butterworth filter is just two 2nd orders in series

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