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    Butterworth had a reputation for solving "impossible" mathematical problems. At the time filterdesign required an amount of designer experience due to limitations of the theory then in use.The filter was not in common use for over 30 years after its publication. Butterworth stated that:

    "An ideal electrical filter should not only completely reject the unwanted frequencies but shouldalso have uniform sensitivity for the wanted frequencies".

    Such an ideal filter cannot be achieved but Butterworth showed that successively closerapproximations were obtained with increasing numbers of filter elements of the right values. Atthe time, filters generated substantial ripple in the passband, and the choice of component valueswas highly interactive. Butterworth showed that a low pass filtercould be designed whose cutofffrequency was normalized to 1 radian per second and 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. If = 1, the amplitude response of this type of filter in the passband is1/2 0.707, which is half power or 3 dB. Butterworth only dealt with filters with an evennumber of poles in his paper. He may have been unaware that such filters could be designed withan odd number of poles. He built his higher order filters from 2-pole filters separated by vacuumtube amplifiers. 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 his original graph.

    Butterworth solved the equations for two- and four-pole filters, showing how the latter could becascaded when separated byvacuum tubeamplifiers and so enabling the construction of higher-order filters despite inductorlosses. In 1930 low-loss core materials such as molypermalloy hadnot been discovered and air-cored audio inductors were rather lossy. Butterworth discovered thatit was possible to adjust the component values of the filter to compensate for the windingresistance of the inductors.

    He used coil forms of 1.25 diameter and 3 length with plug in terminals. Associated capacitorsand resistors were contained inside the wound coil form. The coil formed part of the plate loadresistor. Two poles were used per vacuum tube and RC coupling was used to the grid of thefollowing tube.

    Butterworth also showed that his basic low-pass filter could be modified to give low-pass,high-pass,band-pass andband-stop functionality.

    [ed it] Overview

    http://en.wikipedia.org/wiki/Filter_designhttp://en.wikipedia.org/wiki/Filter_designhttp://en.wikipedia.org/wiki/Image_parameter_filterhttp://en.wikipedia.org/wiki/Low_pass_filterhttp://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Inductionhttp://en.wikipedia.org/wiki/Inductionhttp://en.wikipedia.org/wiki/MPPhttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/High-pass_filterhttp://en.wikipedia.org/wiki/High-pass_filterhttp://en.wikipedia.org/wiki/High-pass_filterhttp://en.wikipedia.org/wiki/Band-pass_filterhttp://en.wikipedia.org/wiki/Band-stop_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=2http://en.wikipedia.org/wiki/Image_parameter_filterhttp://en.wikipedia.org/wiki/Low_pass_filterhttp://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Angular_frequencyhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Vacuum_tubehttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Inductionhttp://en.wikipedia.org/wiki/MPPhttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/High-pass_filterhttp://en.wikipedia.org/wiki/High-pass_filterhttp://en.wikipedia.org/wiki/Band-pass_filterhttp://en.wikipedia.org/wiki/Band-stop_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=2http://en.wikipedia.org/wiki/Filter_designhttp://en.wikipedia.org/wiki/Filter_design
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    The Bode plot of a first-order Butterworth low-pass filter

    The frequency response of the Butterworth filter is maximally flat (has no ripples) in thepassband and rolls off towards zero in the stopband.[2] When viewed on a logarithmic Bode plotthe response slopes off linearly towards negative infinity. A first-order filter's response rolls offat 6dB peroctave (20 dB perdecade) (all first-order lowpass filters have the same normalizedfrequency response). A second-order filter decreases at 12 dB per octave, a third-order at 18dB 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 aslowerroll-off, and thus will require a higher order to implement a particularstopbandspecification, but Butterworth filters have a more linear phase response in the pass-band thanChebyshev Type I/Type II and elliptic filters can achieve.

    [edit] Example

    A third-order low-pass filter (Cauer topology). The filter becomes a Butterworth filter withcutoff frequency c=1 when (for example) C2=4/3 farad, R4=1 ohm, L1=3/2 henry and L3=1/2henry.

    A simple example of a Butterworth filter is the third-order low-pass design shown in the figureon the right, with C2 = 4/3 F,R4 = 1 ,L1 = 3/2 H, andL3 = 1/2 H. Taking the impedanceof thecapacitors Cto be 1/Cs and the impedance of the inductorsL to beLs, wheres = +j is thecomplex frequency, the circuit equations yield the transfer function for this device:

    http://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/Ripple_(filters)http://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Octave_(electronics)http://en.wikipedia.org/wiki/Decade_(log_scale)http://en.wikipedia.org/wiki/Chebyshev_filterhttp://en.wikipedia.org/wiki/Elliptic_filterhttp://en.wikipedia.org/wiki/Roll-offhttp://en.wikipedia.org/wiki/Stopbandhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=3http://en.wikipedia.org/wiki/Electrical_impedancehttp://en.wikipedia.org/wiki/Electrical_impedancehttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/File:LowPass3poleICauer.svghttp://en.wikipedia.org/wiki/File:LowPass3poleICauer.svghttp://en.wikipedia.org/wiki/File:Butterworth_filter_bode_plot.pnghttp://en.wikipedia.org/wiki/File:Butterworth_filter_bode_plot.pnghttp://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/Ripple_(filters)http://en.wikipedia.org/wiki/Bode_plothttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Octave_(electronics)http://en.wikipedia.org/wiki/Decade_(log_scale)http://en.wikipedia.org/wiki/Chebyshev_filterhttp://en.wikipedia.org/wiki/Elliptic_filterhttp://en.wikipedia.org/wiki/Roll-offhttp://en.wikipedia.org/wiki/Stopbandhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=3http://en.wikipedia.org/wiki/Electrical_impedancehttp://en.wikipedia.org/wiki/Transfer_function
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    The magnitude of the frequency response (gain) G() is given by

    and thephaseis 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 isa measure of the distortion in the signal introduced by phase differences for differentfrequencies. The gain and the delay for this filter are plotted in the graph on the left. It can beseen 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 functionH(s) is plotted in complex frequency spacein the second graph on the right. The function is defined by the three poles in the left half of thecomplex frequency plane.

    http://en.wikipedia.org/wiki/Phase_(waves)http://en.wikipedia.org/wiki/Phase_(waves)http://en.wikipedia.org/wiki/Group_delayhttp://en.wikipedia.org/wiki/File:Butterworth_Filter_s-Plane_Response_(3rd_Order).svghttp://en.wikipedia.org/wiki/File:Butterworth_Filter_s-Plane_Response_(3rd_Order).svghttp://en.wikipedia.org/wiki/File:Butterworth3_GainDelay.pnghttp://en.wikipedia.org/wiki/File:Butterworth3_GainDelay.pnghttp://en.wikipedia.org/wiki/Phase_(waves)http://en.wikipedia.org/wiki/Group_delay
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    Log density plot of the transfer function H(s) in complex frequency space for the third-orderButterworth filter with c=1. The three poles lie on a circle of unit radius in the left half-plane.

    These are arranged on a circle of radius unity, symmetrical about the reals axis. The gainfunction 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, ahigh-pass

    Butterworth filter is obtained.Aband-pass Butterworth filter is obtained by placing a capacitor in series with each inductor andan inductor in parallel with each capacitor to form resonant circuits. The value of each newcomponent must be selected to resonate with the old component at the frequency of interest.

    Aband-stop Butterworth filter is obtained by placing a capacitor in parallel with each inductorand an inductor in series with each capacitor to form resonant circuits. The value of each newcomponent must be selected to resonate with the old component at the frequency to be rejected.

    [edit] Transfer function

    Plot of the gain of Butterworth low-pass filters of orders 1 through 5, with cutoff frequency 0 =1. Note that the slope is 20n dB/decade where n is the filter order.

    Like all filters, the typicalprototypeis the low-pass filter, which can be modified into a high-pass filter, or placed in series with others to formband-passandband-stop filters, and higherorder versions of these.

    The gain G() of an n-order Butterworth low pass filter is given in terms of the transfer functionH(s) as

    where

    n = order of filter

    c = cutoff frequency (approximately the -3dB frequency)

    http://en.wikipedia.org/wiki/Complex_frequency_spacehttp://en.wikipedia.org/wiki/High-pass_filterhttp://en.wikipedia.org/wiki/High-pass_filterhttp://en.wikipedia.org/wiki/Band-pass_filterhttp://en.wikipedia.org/wiki/Band-stop_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=4http://en.wikipedia.org/wiki/Prototype_filterhttp://en.wikipedia.org/wiki/Prototype_filterhttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Band-passhttp://en.wikipedia.org/wiki/Band-passhttp://en.wikipedia.org/wiki/Band-stophttp://en.wikipedia.org/wiki/Band-stophttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Cutoff_frequencyhttp://en.wikipedia.org/wiki/File:Butterworth_Filter_Orders.svghttp://en.wikipedia.org/wiki/File:Butterworth_Filter_Orders.svghttp://en.wikipedia.org/wiki/Complex_frequency_spacehttp://en.wikipedia.org/wiki/High-pass_filterhttp://en.wikipedia.org/wiki/Band-pass_filterhttp://en.wikipedia.org/wiki/Band-stop_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=4http://en.wikipedia.org/wiki/Prototype_filterhttp://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Band-passhttp://en.wikipedia.org/wiki/Band-stophttp://en.wikipedia.org/wiki/Transfer_functionhttp://en.wikipedia.org/wiki/Cutoff_frequency
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    G0 is the DC gain (gain at zero frequency)

    It can be seen that as n approaches infinity, the gain becomes a rectangle function andfrequencies below c will be passed with gain G0, while frequencies above c will be suppressed.For smaller values ofn, the cutoff will be less sharp.

    We wish to determine the transfer functionH(s) wheres = +j (from Laplace transform).

    SinceH(s)H(-s) evaluated ats = 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 transferfunction itself will be specified by just the poles in the negative real half-plane ofs. The k-th poleis specified by

    and hence;

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

    The denominator is a Butterworth polynomial ins.[edit] Normalized Butterworth polynomials

    The Butterworth polynomials may be written in complex form as above, but are usually writtenwith real coefficients by multiplying pole pairs which are complex conjugates, such ass1 andsn.The polynomials are normalized by setting c = 1. The normalized Butterworth polynomials thenhave the general form

    for n even

    for n odd

    To four decimal places, they are

    n Factors of Polynomial Bn(s)

    1 (s + 1)

    http://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=5http://en.wikipedia.org/wiki/Laplace_transformhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=5
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    2 s2 + 1.4142s + 13 (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 anylow-pass filtercut-off frequency c, as follows

    , where

    Transformation to other bandforms are also possible, seeprototype filter.

    [edit] Maximal flatness

    Assuming c = 1 and G0 = 1, the derivative of the gain with respect to frequency can be shown tobe

    which ismonotonically decreasing for all since the gain G is always positive. The gainfunction of the Butterworth filter therefore has no ripple. Furthermore, the series expansion ofthe 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 passbandonly and ripples are allowed in the stopband, then it is possible to design a filter of the sameorder, such as the inverse Chebyshev filter, that is flatter in the passband than the "maximallyflat" Butterworth.

    [edit] 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-offis therefore 20n dB/decade, or 6n dB/octave (the factor of20 is used because the power is proportional to the square of the voltage gain; see20 log rule.)

    [edit] Filter design

    http://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Prototype_filterhttp://en.wikipedia.org/wiki/Prototype_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=6http://en.wikipedia.org/wiki/Monotonichttp://en.wikipedia.org/wiki/Monotonichttp://en.wikipedia.org/wiki/Inverse_Chebyshev_filterhttp://en.wikipedia.org/wiki/Inverse_Chebyshev_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=7http://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Roll-offhttp://en.wikipedia.org/wiki/20_log_rulehttp://en.wikipedia.org/wiki/20_log_rulehttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=8http://en.wikipedia.org/wiki/Low-pass_filterhttp://en.wikipedia.org/wiki/Prototype_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=6http://en.wikipedia.org/wiki/Monotonichttp://en.wikipedia.org/wiki/Inverse_Chebyshev_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=7http://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Roll-offhttp://en.wikipedia.org/wiki/20_log_rulehttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=8
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    There are a number of different filter topologiesavailable to implement a linear analogue filter.The most often used topology for a passive realisation is Cauer topology and the most often usedtopology for an active realisation is SallenKey topology.

    [edit] Cauer topology

    The Cauer topology uses passive components (shunt capacitors and series inductors) toimplement a linear analog filter. The Butterworth filter having a given transfer function can berealised 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 theLk are k odd and the Ck arek even.

    [edit] SallenKey topology

    The SallenKey topologyuses active and passive components (noninverting buffers, usually opamps, resistors, and capacitors) to implement a linear analog filter. Each SallenKey stage

    implements a conjugate pair of poles; the overall filter is implemented by cascading all stages inseries. 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 SallenKey circuit shown to the right the transfer function is given by;

    http://en.wikipedia.org/wiki/Electronic_filter_topologyhttp://en.wikipedia.org/wiki/Electronic_filter_topologyhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=9http://en.wikipedia.org/wiki/Cauer_topology_(electronics)http://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=10http://en.wikipedia.org/wiki/Sallen%E2%80%93Key_topologyhttp://en.wikipedia.org/wiki/Sallen%E2%80%93Key_topologyhttp://en.wikipedia.org/wiki/Op_amphttp://en.wikipedia.org/wiki/Op_amphttp://en.wikipedia.org/wiki/RC_circuithttp://en.wikipedia.org/wiki/File:Sallen-Key.svghttp://en.wikipedia.org/wiki/File:Cauer_lowpass.svghttp://en.wikipedia.org/wiki/Electronic_filter_topologyhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=9http://en.wikipedia.org/wiki/Cauer_topology_(electronics)http://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=10http://en.wikipedia.org/wiki/Sallen%E2%80%93Key_topologyhttp://en.wikipedia.org/wiki/Op_amphttp://en.wikipedia.org/wiki/Op_amphttp://en.wikipedia.org/wiki/RC_circuit
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    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 undefined component values that may be chosen at will.

    [edit] Digital implementation

    Digital implementations of Butterworth and other filters are often based on thebilinear transformmethod or the matched Z-transform method, two different methods to discretize an analog filterdesign. In the case of all-pole filters such as the Butterworth, the matched Z-transform method isequivalent to the impulse invariance method. For higher orders, digital filters are sensitive to

    quantization errors, so they are often calculated as cascadedbiquad sections, plus one first-orderor third-order section for odd orders.

    [edit] Comparison with other linear filtersHere is an image showing the gain of a discrete-time Butterworth filter next to other commonfilter types. All of these filters are fifth-order.

    http://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=11http://en.wikipedia.org/wiki/Bilinear_transformhttp://en.wikipedia.org/wiki/Bilinear_transformhttp://en.wikipedia.org/wiki/Matched_Z-transform_methodhttp://en.wikipedia.org/wiki/Matched_Z-transform_methodhttp://en.wikipedia.org/wiki/Impulse_invariancehttp://en.wikipedia.org/wiki/Digital_biquad_filterhttp://en.wikipedia.org/wiki/Digital_biquad_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=12http://en.wikipedia.org/wiki/File:Electronic_linear_filters.svghttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=11http://en.wikipedia.org/wiki/Bilinear_transformhttp://en.wikipedia.org/wiki/Matched_Z-transform_methodhttp://en.wikipedia.org/wiki/Impulse_invariancehttp://en.wikipedia.org/wiki/Digital_biquad_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=12
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    The Butterworth filter rolls off more slowly around the cutoff frequency than the Chebyshevfilteror the Elliptic filter, but without ripple.

    [edit] References

    http://en.wikipedia.org/wiki/Chebyshev_filterhttp://en.wikipedia.org/wiki/Chebyshev_filterhttp://en.wikipedia.org/wiki/Elliptic_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=13http://en.wikipedia.org/wiki/Chebyshev_filterhttp://en.wikipedia.org/wiki/Chebyshev_filterhttp://en.wikipedia.org/wiki/Elliptic_filterhttp://en.wikipedia.org/w/index.php?title=Butterworth_filter&action=edit&section=13