Basic Electronics on the Go - Butterworth Filter Design

From http://www.electronics-tutorials.ws/filter/filter_8.html

"High-order” or “n^th-order” filters may be needed in communication or control circuits that require a larger roll off.

  A first-order filter has a roll-off rate of 20dB/decade (6dB/octave), a second-order filter has a roll-off rate of 40dB/decade (12dB/octave), and a fourth-order filter has a roll-off rate of 80dB/decade (24dB/octave),  etc.

 High-order filters  are usually formed by cascading together single first-order and second-order filters. As the order increases so does its size and cost but  its accuracy declines.

Decades and Octaves

On the frequency scale, a Decade is a tenfold increase (multiply by 10) or tenfold decrease (divide by 10).  For example, 2 to 20Hz represents one decade, whereas 50 to 5000Hz represents two decades (50 to 500Hz and then 500 to 5000Hz).

An Octave is a doubling (multiply by 2) or halving (divide by 2) of the frequency scale. For example, 10 to 20Hz represents one octave, while 2 to 16Hz is three octaves (2 to 4, 4 to 8 and finally 8 to 16Hz) . Either way, Logarithmic scales are used extensively in the frequency domain to denote a frequency value when working with amplifiers and filters .

 

Since the frequency determining resistors  and   capacitors are all equal, the cut-off or corner frequency ( ƒ_C ) for either a first, second, third or even a fourth-order filter must also be equal and is found by using our now old familiar equation:

 

Filter Approximations

So far we have looked at a low and high pass first-order filter circuits, their resultant frequency and phase responses. An ideal filter would give us specifications of maximum pass band gain and flatness and a sharp transition from the passband to stopband with steep roll-off.

There are a number of “approximation functions” in linear analogue filter design that use a mathematical approach to best approximate the transfer function we require for the filters design. 

Elliptical, Butterworth, Chebyshev, Bessel, Cauer are some of them but the low pass Butterworth filter design will be considered here as it is the most commonly used function.

Low Pass Butterworth Filter Design

The frequency response of the Butterworth Filter approximation function is referred to as  a “maximally flat” (no ripples) response because the pass band is designed to have a frequency response which is as flat as mathematically possible from 0Hz (DC) until the cut-off frequency at -3dB with no ripples. Higher frequencies beyond the cut-off point rolls-off down to zero in the stop band at 20dB/decade or 6dB/octave due to a “quality factor”, “Q” of just 0.707.

 However, one main disadvantage of the Butterworth filter is that it achieves this pass band flatness at the expense of a wide transition band as the filter changes from the pass band to the stop band. It also has poor phase characteristics as well. 

Ideal Frequency Response for a Butterworth Filter

 

Note that the higher the Butterworth filter order, the higher the number of cascaded stages there are within the filter design, and the closer the filter becomes, to the ideal “brick wall” response.
In practice however, Butterworth’s ideal frequency response is unattainable as it produces excessive passband ripple.

The frequency response for the generalised equation representing a “nth” Order Butterworth filter is given as:

Where: n represents the filter order, Omega ω is equal to 2πƒ and Epsilon ε is the maximum pass band gain, (A_max). If A_max is defined at a frequency equal to the cut-off -3dB corner point (ƒc), ε will then be equal to one and therefore ε² will also be one.

However, if you now wish to define A_max at a different voltage gain value, for example 1dB, or 1.1220 (1dB = 20logA_max) then the new value of epsilon, ε is found by:

• Where:
•   H₀ = the Maximum Pass band Gain, A_max.
•   H₁ = the Minimum Pass band Gain

Transpose the equation to give:

•  
• Where:
•   Vout = the output signal voltage.
•   Vin  = the input signal voltage.
•      j   = to the square root of -1 (√-1)
•     ω  = the radian frequency (2πƒ)

Note: ( jω ) can also be written as ( s ) to denote the S-domain. and the resultant transfer function for a second-order low pass filter is given as:

Normalised Low Pass Butterworth Filter Polynomials

To help in the design of his low pass filters, Butterworth produced standard tables of normalised second-order low pass polynomials given the values of coefficient that correspond to a cut-off corner frequency of 1 radian/sec.

n	Normalised Denominator Polynomials in Factored Form
1	(1+s)
2	(1+1.414s+s2)
3	(1+s)(1+s+s2)
4	(1+0.765s+s2)(1+1.848s+s2)
5	(1+s)(1+0.618s+s2)(1+1.618s+s2)
6	(1+0.518s+s2)(1+1.414s+s2)(1+1.932s+s2)
7	(1+s)(1+0.445s+s2)(1+1.247s+s2)(1+1.802s+s2)
8	(1+0.390s+s2)(1+1.111s+s2)(1+1.663s+s2)(1+1.962s+s2)
9	(1+s)(1+0.347s+s2)(1+s+s2)(1+1.532s+s2)(1+1.879s+s2)
10	(1+0.313s+s2)(1+0.908s+s2)(1+1.414s+s2)(1+1.782s+s2)(1+1.975s+s2)