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Butterworth filter   by jekky

Original Paper Butterworth had a reputation for solving mathematical problems that were thought to be impossible to solve. His paper was far ahead of its time and the filter was not in common use for over 30 years after its publication. At the time filters were largely designed by trial and error because of their mathematical complexity. Butterworth stated the goal of his paper thusly: “An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies.” At the time the frequency response of filters contained substantial amounts of ripple in the passband and the choice of component values was highly interactive. Butterworth showed that low pass filters could be designed whose frequency response (gain) was , where is the angular frequency in radians per second and n is the number of reactive elements (poles) in the filter. Butterworth only dealt with filters with an even number of poles in his paper. His plot of the frequency response of 2, 4, 6, 8, and 10 pole filters is shown as A, B, C, D, and E in this original graph. He made small errors in the plots for 6, 8 and 10 poles as can be seen by comparison to a modern computer graph. Frequency response plot from the original paper. Modern computer plot showing errors in the original graph Butterworth solved the equations for two and four pole filters and showed how the latter could be cascaded when separated by vacuum tube amplifiers. This made possible the construction of higher order filters in spite of inductor losses. In 1930, low loss core materials such as molypermalloy had not been discovered and air core audio inductors were rather lossy. Butterworth discovered that it was possible to adjust the component values of the filter to compensate for the winding resistance of the inductors. Butterworth also showed that his basic low pass filter could be modified to give low-pass, high-pass, band-pass, and band-stop versions. He may have been unaware that this type of filter could be designed with an odd number of poles. The Bode plot of a first-order Butterworth low-pass filter 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 lowpass filters have the same normalized frequency response). For a second-order lowpass filter, the response ultimately decreases at 12 dB per octave, a third-order at 18 dB, and so on. Butterworth filters have a monotonically changing magnitude function with , unlike other filter types that have non-monotonic ripple in the passband and/or the stopband. 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 filters have a more linear phase response in the passband than the Chebyshev Type I/Type II and elliptic filters. A simple example A third-order low-pass filter (Cauer topology). The filter becomes a Butterworth filter with cutoff frequency c=1 when (for example) C2=4/3 farad, R4=1 ohm, L1=3/2 henry and L3=1/2 henry. Log density plot of the transfer function H(s) in complex frequency space for the third-order Butterworth filter with c=1. Note the three poles which lie on a circle of unit radius in the left half plane. A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with C2 = 4 / 3 farad, R4 = 1 ohm, L1 = 3 / 2 and L3 = 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 = + j is the complex frequency, the circuit equations yield the transfer function for this device: The magnitude of the frequency response (gain) G() is given by: and the phase is given by: Gain and group delay of the third-order Butterworth filter with c=1 The group delay is defined as the derivative of the phase with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different frequencies. The gain and the delay for this filter are plotted in the graph on the left. It can be seen that there are no ripples in the gain curve in either the passband or the stop band. The log of the absolute value of the transfer function H(s) is plotted in complex frequency space in the second graph on the right. The function is defined by the three poles in the left half of the complex frequency plane. These are arranged on a circle of radius unity, symmetrical about the real s axis. The gain function will have three more poles on the right half plane to complete the circle. By replacing each inductor with a capacitor and each capacitor with an inductor, a high-pass Butterworth filter is obtained. A band-pass Butterworth filter is obtained by placing a capacitor in series with each inductor and an inductor in parallel with each capacitor to form resonant circuits. The value of each new component must be selected to resonate with the old component at the frequency of interest. A band-stop Butterworth filter is obtained by placing a capacitor in parallel with each inductor and an inductor in series with each capacitor to form resonant circuits. The value of each new component must be selected to resonate with the old component at the frequency to be rejected. The transfer function Plot of the gain of Butterworth low-pass filters of orders 1 through 5. Note that the slope is 20n dB/decade where n is the filter order. Like all filters, the typical prototype is the low-pass filter, which can be modified into a high-pass filter, or placed in series with others to form band-pass and band-stop filters, and higher order versions of these. The gain G() of an n-order Butterworth low pass filter is given in terms of the transfer function H(s) as: where n = order of filter c = cutoff frequency (approximately the -3dB frequency) G0 is the DC gain (gain at zero frequency) It can be seen that as n approaches infinity, the gain becomes a rectangle function and frequencies below c will be passed with gain G0, while frequencies above c will be suppressed. For smaller values of n, the cutoff will be less sharp. We wish to determine the transfer function H(s) where s = + j. Since H(s)H(-s) evaluated at s = j is simply equal to |H(j)|2, it follows that: The poles of this expression occur on a circle of radius c at equally spaced points. The transfer function itself will be specified by just the poles in the negative real half-plane of s. The k-th pole is specified by: and hence, The transfer function may be written in terms of these poles as: 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 s1 and sn. The polynomials are normalized by setting c = 1. The normalized Butterworth polynomials then have the general form: for n even for n odd To four decimal places, they are: n Factors of Polynomial Bn(s) 1 (s + 1) 2 s2 + 1.4142s + 1 3 (s + 1)(s2 + s + 1) 4 (s2 + 0.7654s + 1)(s2 + 1.8478s + 1) 5 (s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1) 6 (s2 + 0.5176s + 1)(s2 + 1.4142s + 1)(s2 + 1.9319s + 1) 7 (s + 1)(s2 + 0.4450s + 1)(s2 + 1.2470s + 1)(s2 + 1.8019s + 1) 8 (s2 + 0.3902s + 1)(s2 + 1.1111s + 1)(s2 + 1.6629s + 1)(s2 + 1.9616s + 1) The normalized Butterworth polynomials can be used to determine the transfer function for any low-pass filter cut-off frequency c, as follows , where Transformation to other bandforms are also possible, see prototype filter. Maximal flatness Assuming c = 1 and G0 = 1, the derivative of the gain with respect to frequency can be shown to be: which is monotonically decreasing for all since the gain G is always positive. The gain function of the Butterworth filter therefore has no ripple. Furthermore, the series expansion of the gain is given by: In other words, all derivatives of the gain up to but not including the 2n-th derivative are zero, resulting in “maximal flatness”. If the requirement to be monotonic is limited to the passband only and ripples are allowed in the stopband, then it is possible to design a filter of the same order that is flatter in the passband than the “maximally flat” Butterworth. Such a filter is the inverse Chebyshev filter. High-frequency roll-off Again assuming c = 1, the slope of the log of the gain for large is: In decibels, the high-frequency roll-off is therefore 20n dB/decade, or 6n dB/octave (The factor of 20 is used because the power is proportional to the square of the voltage gain; see 20 log rule.) Filter design There are a number of different filter topologies available to implement a linear analogue filter. The most often used topology for a passive realisation is Cauer topology and the the most often used topology for an active realisation is Sallen-Key topology. Cauer topology The Cauer topology uses passive components (shunt capacitors and series inductors) to implement a linear analog filter. The Butterworth filter having a given transfer function can be realised using a Cauer 1-form. The kth element is given by: ; k = odd ; k = even The filter may start with a series inductor if desired, in which case the Lk are k odd and the Ck are k even. Sallen-Key topology The Sallen-Key topology uses active and passive components (noninverting buffers, usually op amps, resistors, and capacitors) to implement a linear analog filter. Each Sallen-Key stage implements a conjugate pair of poles; the overall filter is implemented by cascading all stages in series. If there is a real pole (in the case where n is odd), this must be implemented separately, usually as an RC circuit, and cascaded with the active stages. For the second order Sallen-Key circuit shown to the right the transfer function is given by We wish the denominator to be one of the quadratic terms in a Butterworth polynomial. Assuming that c = 1, this will mean that and This leaves two component values undefined, which may be chosen at will. Digital implementation Digital implementations of Butterworth filters often use bilinear transform or matched z-transform to discretize an analog filter. For higher orders, they are sensitive to quantization errors. For this reason, they are often calculated as cascaded biquad sections and a cascaded first order filter, for odd orders. Comparison with other linear filters Here is an image showing the gain of a discrete-time Butterworth filter next to other common filter types. All of these filters are fifth-order. The Butterworth filter rolls off more slowly around the cutoff frequency than the others (Chebyshev filter and Elliptic filter), but shows no ripples. References ^ “On the Theory of Filter Amplifiers”-S. Butterworth ^ Giovanni Bianchi and Roberto Sorrentino (2007). Electronic filter simulation & design. McGraw-Hill Professional. p. 1720. ISBN 9780071494670. http://books.google.com/books?id=5S3LCIxnYCcC&pg=PT32&dq=Butterworth-approximation+maximally-flat&lr=&as_brr=3&ei=SiyWSt_yH5jGM5TdidcH#v=onepage&q=Butterworth-approximation%20maximally-flat&f=false. Categories: Linear filters | Network synthesis filters | Electronic design

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