- Prototype filter
Prototype filters are
electronic filterdesigns that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed. They are most often seen with regard to electronic filters and most especially linear analogue passive filters. However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.
Filters are required to operate at many different frequencies, impedances and bandwidths. The utility of a prototype filter comes from the property that all these other filters can be derived from it by applying a scaling factor to the components of the prototype. The filter design need thus only be carried out once in full, other filters being obtained by simply applying a scaling factor.
Especially useful is the ability to transform from one bandform to another. In this case the transform is more than a simple scale factor. By bandform is meant the category of
passbandthat the filter possesses. The usual bandforms are lowpass, highpass, bandpassand bandstopbut others are possible. In particular it is possible for a filter to have multiple passbands. In fact, in some treatments, the bandstopfilter is considered to be a type of multiple passband filter having two passbands. Most commonly, the prototype filter is expressed as a lowpass filter but other techniques are possible.
The prototype is most usually given as a low-pass filter with a cut-off frequency (image filters) or 3dB bandwidth frequency (
network synthesis filters) which has an angular frequency of "ωc '" = 1 rad/s. Occasionally, frequency "f" '' = 1 Hz is used instead. In principle, any non-zero frequency point on the filter response could be used as a reference for the prototype design.
Likewise, the nominal or characteristic impedance of the filter is set to "R"
'= 1 Ω.
The prototype filter can only be used to produce other filters of the same class and order. For instance, a fifth order
Bessel filterprototype can be converted into any other fifth order Bessel filter but it cannot be transformed into a third order Bessel filter or a fifth order Tchebyscheff filter.
The prototype filter is scaled to the frequency required with the following transformation;
'" is the value of the frequency parameter (eg cut-off frequency) for the prototype and "ωc" is the desired value. So if "ωc '" = 1 then the transfer function of the filter is transformed as;
It can readily be seen that to achieve this the non-resistive components of the filter must be transformed by;
Impedance scaling is invariably a scaling to a fixed resistance. This is because the terminations of the filter, at least nominally, are taken to be a fixed resistance. To carry out this scaling to a nominal impedance "R", each impedance element of the filter is transformed by;
It may be more convenient on some elements to scale the admittance instead;
It can readily be seen that to achieve this the non-resistive components of the filter must be scaled as;
Impedance scaling by itself has no effect on the transfer function of the filter (always provided that the terminating impedances have the same scaling applied to them). However, it is usual to combine the frequency and impedance scaling into a single step; [ Matthaei et al, pp96-97]
In general, the bandform of a filter is transformed by replacing "iω" where it occurs in the transfer function with a function of "iω". This in turn leads to the transformation of the impedance components of the filter into some other component(s). The frequency scaling above is a trivial case of bandform transformation corresponding to a lowpass to lowpass transformation.
Lowpass to highpass
The frequency transformation required in this case is; [Matthaei et al, pp412-413.]
where "ωc" is the point on the highpass filter corresponding to "ωc
'" on the prototype. The transfer function then transforms as;
Inductors are transformed into capacitors according to,
and capacitors are transformed into inductors,
the primed quantities being the component value in the prototype.
Lowpass to bandpass
In this case the required frequency transformation is; [Matthaei et al, pp438-440.]
where "Q" is the Q-factor and is equal to the inverse of the fractional bandwidth; [Farago, p69]
If "ω1" and "ω2" are respectively, the lower and upper frequency points of the bandpass response corresponding to "ωc
'" of the prototype then,
Δ"ω" is the absolute bandwidth and "ω0" is the resonant frequency of the resonators in the filter. Note that frequency scaling the prototype prior to lowpass to bandpass transformation does not affect the resonant frequency, but instead affects the final bandwidth of the filter.
The transfer function of the filter is transformed according to;
Inductors are transformed into series
and capacitors are transformed into parallel resonators,
Lowpass to bandstop
The required frequency transformation for lowpass to bandstop is; [Matthaei et al, pp727-729.]
Inductors are transformed into parallel resonators,
and capacitors are transformed into series resonators,
Lowpass to multi-band
Filters with multiple passbands may be obtained by applying the general transformation;
The number of resonators in the expression corresponds to the number of passbands required. Lowpass and highpass filters can be viewed as special cases of the resonator expression with one or the other of the terms going to zero as appropriate. Bandstop filters can be regarded as a combination of a lowpass and a highpass filter. Multiple bandstop filters can always be expressed in terms of a multiple bandpass filter. In this way it can be seen that this transformation represents the general case for any bandform and all the other transformations are to be viewed as special cases of it.
In his treatment of image filters, Zobel provides an alternative basis for constructing a prototype which is not based in the
frequency domain. [Zobel, 1930, p3] The Zobel prototypes do not, therefore, correspond to any particular bandform, but they can be transformed into any of them. Not giving special significance to any one bandform makes the method more mathematically pleasing but it is not in common use.
The Zobel prototype considers filter sections, rather than components. That is, the transformation is carried out on a
two-port networkrather than a two-terminal inductor or capacitor. The transfer function is expressed in terms of the product of the series impedance, Z, and the shunt admittanceY of a filter half-section. See the article Image impedancefor a description of half-sections. This quantity is nondimensional, adding to the prototypes generality. Generally, ZY is a complex quantity,
and as "U" and "V" are both, in general, functions of "ω" we should properly write,
With image filters, it is possible to obtain filters of different classes from the
constant k filterprototype by means of a different kind of transformation (see composite image filter). Constant k being those filters for which Z/Y is a constant. For this reason, filters of all classes are given in terms of "U(ω)" for a constant k, which is notated as,
In the case of dissipationless networks, ie no resistors, the quantity "V(ω)" is zero and only "U(ω)" need be considered. "Uk(ω)" ranges from 0 at the centre of the
passbandto -1 at the cut-off frequencyand then continues to increase negatively into the stopbandregardless of the bandform of the filter being designed. To obtain the required bandform, the following transforms are used,
For a lowpass constant k prototype that is scaled;
the independent variable of the response plot is,
The bandform transformations from this prototype are,
and for bandpass,
Electronic filter topology
Composite image filter
*Zobel, O J,"Theory and Design of Uniform and Composite Electric Wave Filters", Bell Systems Technical Journal, Vol. 2 (1923), pp. 1-46.
*Zobel, O J, "Electrical wave filters", US patent 1 850 146, filed 25 Nov 1930, issued 22 Mar 1932. Gives many useful formulae and a non-frequency domain basis for defining prototypes.
* Matthaei, Young, Jones "Microwave Filters, Impedance-Matching Networks, and Coupling Structures" McGraw-Hill 1964.
*Farago, P S, "An Introduction to Linear Network Analysis", English Universities Press, 1961.
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