Butterworth Filter

The Butterworth filter is one type of Electronic Filter design. It is designed to have a Frequency Response which is as flat as mathematically possible in the Passband . Another name for them is 'maximally flat magnitude' filters.

The Butterworth type filter was first described by the British Engineer Stephen Butterworth in his paper "On the Theory of Filter Amplifiers", ''Wireless Engineer'' (also called ''Experimental Wireless and the Radio Engineer''), vol. 7, 1930, pp. 536-541.

OVERVIEW

The frequency response of the Butterworth filter is maximally flat (has no ripples) in the passband, and rolls off towards zero in the stopband. When viewed on a logarithmic Bode Plot , the response slopes off linearly towards negative infinity. For a first-order filter, the response rolls off at −6 DB per Octave (−20 dB per Decade ) (All first-order filters, regardless of name, have the same normalized frequency response). For a second-order Butterworth filter, the response decreases at −12 dB per octave, a third-order at −18 dB, and so on. Butterworth filters have a monotonically changing magnitude function with ω.
The Butterworth is the only filter that maintains this same shape for higher orders (but with a steeper decline in the stopband) whereas other varieties of filters ( Bessel , Chebyshev , Elliptic ) have different shapes at higher orders.

Compared with a Chebyshev Type I/Type II filter or an Elliptic Filter , the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular Stopband specification. However, Butterworth filter will have a more linear phase response in the passband than the Chebyshev Type I/Type II and elliptic filters.

A SIMPLE EXAMPLE

A simple example of a Butterworth filter is the 3rd order Low-pass design shown in the figure on the right, with

C_2=4/3

farad,

R_4=1

ohm,

L_1=3/2

and

L_3=1/2

henry. Taking the Impedance of the capacitors ''C'' to be ''1/Cs'' and the impedance of the inductors ''L'' to be ''Ls'', where

s=\sigma+j\omega

is the complex frequency, the circuit equations yields the Transfer Function for this device:

:

H(s)=rac{V_o(s)}{V_i(s)}=rac{1}{1+2s+2s^2+s^3}

The magnitude of the frequency response (gain)

G(\omega)

is given by:

We Wish To Determine The Transfer Function ''H(s)'' Where <math>s \sigma+j\omega</math> Since ''H(s)H(-s)'' evaluated at ''s = jω'' is simply equal to H(&omega)<sup>2</sup>, it follows that:

\qquad\mathrm{k = 1,2,3, \ldots, n}

and hence,

:

s_k = \omega_c e^{rac{j(2k+n-1)\pi}{2n}}\qquad\mathrm{k = 1,2,3, \ldots, n}

The transfer function may be written in terms of these poles as:

:

H(s)=rac{G_0}{\prod_{k=1}^n (s-s_k)/\omega_c}

The denominator is a Butterworth polynomial in ''s''.

Normalized Butterworth polynomials

The Butterworth polynomials may be written in complex form as above, but are usually written with real coefficients by multiplying pole pairs which are complex conjugates, such as

s_1

and

s_n

. The polynomials are normalized by setting

\omega_c=1

. The normalized Butterworth polynomials then have the general form:

:

B_n(s)=\prod_{k=1}^{rac{n}{2}} \left[s^2-2s\cos\left(rac{2k+n-1}{2n}\,\pi
ight)+1
ight]

for n even
:

B_n(s)=(s+1)\prod_{k=1}^{rac{n-1}{2}} \left[s^2-2s\cos\left(rac{2k+n-1}{2n}\,\pi
ight)+1
ight]

for n odd

To four decimal places, they are:

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Butterworth Filter