Different kinds of filter designs
INTRODUCTION.
There are several varieties of active filter. Some of them, also available in passive form, are:
High-pass filters – attenuation of frequencies below their cut-off points.
Low-pass filters – attenuation of frequencies above their cut-off points.
Band-pass filters – attenuation of frequencies both above and below those they allow to pass.
Notch filters – attenuation of certain frequencies while allowing all others to pass.
To design filters, different types are available to set the component value based on mathematical properties (which define "shape" of the frequency bands)this include
Chebyshev filter
Butterworth filter
Bessel filter
Elliptic filter
Chebyshev filters.
Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized filter characteristic and the actual over the range of the filter, but with ripples in the passband. This type of filter is named in honor of Pafnuty Chebyshev because their mathematical characteristics are derived from Chebyshev polynomials.
Because of the passband ripple inherent in Chebyshev filters, filters which have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications.
You must select four parameters to design a Chebyshev filter:
(1) a high-pass or low-pass response,
(2) the cutoff frequency,
(3) the percent ripple in the passband,
(4) the number of poles. Just what is a pole? Here are two answers.
Elliptic filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer) is an electronic filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not). Alternatively, one may give up the ability to independently adjust the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.
As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a Butterworth filter.
The gain of a lowpass elliptic filter as a function of angular frequency ω is given by:
where Rn is the nth-order elliptic rational function (sometimes known as a Chebyshev rational function) and
ω0 is the cutoff frequency :ε is the ripple factor :ξ is the selectivity factor
The value of the ripple factor specifies the passband ripple, while the combination of the ripple factor and the selectivity factor specify the stopband ripple.
Butterworth filter
Operation
The Butterworth filter is composed of a series of "branches" which are alternately connected in series or shunt with the source-to-load path. In low-pass and high-pass filters, each branch is either a capacitor or an inductor. In band-pass and band-stop filters, each branch is either a series or parallel resonant circuit composed of a capacitor and an inductor. (see figures 1 and 2). The first branch of any filter may be selected as either a series or a shunt branch.
Upon running the program, the user must first specify if a low-pass/high-pass or a band-pass/band-stop filter is desired. If the low-pass high-pass design is chosen, the user must then specify two frequencies in Hz, each followed by the respective attenuation desired (this may be a very small value but never exactly zero). The source resistance must then be specified. The program will then calculate and display the minimum number of branches required for the specifications, and request if the first branch is to be series or shunt. The program will then calculate all required component values rounded to 3 significant digits and displayed in the most common units. The rounding feature may be eliminated by changing statement 830 to "LET X1=X". The program will then provide a table of attenuation vs. frequency for any specified range and increment of frequency.
If the band-pass/band-stop design is selected, the process is similar except that first the center frequency of the filter must be specified, and then two frequency bandwidths with their respective attenuations. These filters are geometrically symmetrical, i.e. the center frequency is not exactly the arithmetic average of the limit frequencies of any bandwidth. The relationship is really fc=SQRT(f1xf2).
The given examples demonstrate the use of this program. The first filter designed is a band-stop filter for 11 meters, providing 20 dB attenuation over 26.965 to 27.405 MHz (bandwidth of 440 kHz) and only 1 dB attenuation over 26.37 to 28.0 MHz (bandwidth of 1.63 MHz). This type of filter may be designed for the input of a linear amplifier to prevent operation on the 11 meter band while permitting operation on the 10 meter band. This filter requires only 3 branches (total of 6 components), and its response over 26 to 28 MHz is shown in the printout.
The second example shows design of a low-pass filter, such as used for TVI suppression.. An attenuation of 30 dB is specified at 54 MHz, while a loss of 1 dB is specified at 28 MHz. This filter would require 7 elements, and its response over 5 to 55 MHz is shown in the table.
This program offers the radio amateur the ability to synthesize modern Butterworth filter circuits with all calculations performed by a computer. All previously published articles in amateur magazines on this subject present tables of normalized filter values or only a few "typical" values for certain common frequencies. This program offers the full flexibility possible by using the original formulas for synthesis of Butterworth filters. It is possible to write such programs for Chebyshev and other modern filters, and these will require slightly more complicated formula derivations. This is one of the author's future projects for programing on the Timex Sinclair 1000.
DESIGN.
BANDSTOP FILTER DESIGN
CENTER FREQUENCY 27.185 MHZ
BANDWIDTHFREQUENCY
ATTENUATION (DB) 440000 201630000
1RESISTANCE (OHMS): 50
NO BRANCHES=3
1ST EL. (SERIES=1, SHUNT=-1) -1
BANDSTOP FILTER COMPONENTS
BRANCH 1 SHUNT L 1 = 6.12 UH IN SERIES WITH C 1 = 5.6 PF
BRANCH 2 SERIES C 2 = .00122 UF IN PARALLEL WITH L 2 = .028 UH BRANCH 3 SHUNT L 3 = 6.12 UH IN SERIES WITH C 3 = 5.6 PF
LOWPASS FILTER DESIGn
FREQMHZ
ATTENUATION (DB)28 154 30
RESISTANCE = 50 OHMS
NO. BRANCHES=71ST EL. (SERIES=1, SHUNT=-1) -1
LOWPASS FILTER COMPONENTS
BRANCH 1 SHUNT C 1 = 45.9 PF
BRANCH 2 SERIES L 2 = 0.322 UH
BRANCH 3 SHUNT C 3 = 186 PF
BRANCH 4 SERIES L 4 = 0.516 UH
BRANCH 5 SHUNT C 5 = 186 PF
BRANCH 6 SERIES L 6 = 0.322 UH
BRANCH 7 SHUNT C 7 = 45.9 PF
Comparison with other linear filters
Here is an image showing the elliptic filter next to other common kind of filters obtained with the same number of coefficients:
Reference
http://www.qsl.net/kp4md/butrwrth.htm
