vision electricity: Prototype filter

Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionaliseddesign 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 passband that the filter possesses. The usual bandforms are lowpass, highpass, bandpass and bandstop but others are possible. In particular it is possible for a filter to have multiple passbands. In fact, in some treatments, the bandstop filter 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.

Low-pass prototype

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 ω_c'= 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 filter prototype 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.

Frequency scaling

The prototype filter is scaled to the frequency required with the following transformation;

$i \omega \to \left( \frac{\omega_c'}{\omega_c}\right) i \omega$

where ω_c' is the value of the frequency parameter (eg cut-off frequency) for the prototype and ω_c is the desired value. So if ω_c' = 1 then the transfer function of the filter is transformed as;

$A(i\omega) \to A\left( i\frac{\omega}{\omega_c}\right)$

It can readily be seen that to achieve this the non-resistive components of the filter must be transformed by;

$L \to \frac{\omega_c'}{\omega_c}\,L$ and, $C \to \frac{\omega_c'}{\omega_c}\,C$

Impedance scaling

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;

$Z \to \frac{R}{R'}\,Z$

It may be more convenient on some elements to scale the admittance instead;

$Y \to \frac{R'}{R} \,Y$

The prototype filter above transformed to a 600Ω, 16kHz lowpass filter

It can readily be seen that to achieve this the non-resistive components of the filter must be scaled as;

$L \to \frac{R}{R'} \,L$ and, $C \to \frac{R'}{R} \,C$

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;^[1]

$L \to \,\frac{\omega_c'}{\omega_c}\,\frac{R}{R'} \,L$ and, $C \to \,\frac{\omega_c'}{\omega_c}\,\frac{R'}{R} \,C$

Bandform transformation

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;^[2]

$\frac{i\omega}{\omega_c'} \to \frac {\omega_c}{i\omega}$

where ω_c is the point on the highpass filter corresponding to ω_c' on the prototype. The transfer function then transforms as;

$A(i\omega) \to A\left( \frac{\omega_c \, \omega_c'}{i\omega} \right)$

Inductors are transformed into capacitors according to,

$L' \to C= \frac{1}{\omega_c \,\omega_c'\,L'}$

and capacitors are transformed into inductors,

$C' \to L = \frac{1}{\omega_c \,\omega_c'\,C'}$

the primed quantities being the component value in the prototype.

Lowpass to bandpass

In this case the required frequency transformation is;^[3]

$\frac{i\omega}{\omega_c'} \to Q \left( \frac {i\omega}{\omega_0}+\frac {\omega_0}{i\omega} \right)$

where Q is the Q-factor and is equal to the inverse of the fractional bandwidth;^[4]

$Q=\frac{\omega_0}{\Delta\omega}$

If ω₁ and ω₂ are respectively, the lower and upper frequency points of the bandpass response corresponding to ω_c' of the prototype then,

$\Delta\omega=\omega_2-\omega_1\,$ and $\omega_0=\sqrt{\omega_1\omega_2}$

Δω 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;

$A(i\omega) \to A\left( \omega_c' Q \left[ \frac {i\omega}{\omega_0}+\frac {\omega_0}{i\omega} \right] \right)$

The prototype filter above transformed to a 50Ω, 6MHz bandpass filter with 100kHz bandwidth

Inductors are transformed into series resonators,

$L' \to L= \frac{\omega_c' Q}{\omega_0}L' \,,\,C= \frac{1}{\omega_0 \omega_c' Q}\frac{1}{L'}$

and capacitors are transformed into parallel resonators,

$C' \to C= \frac{\omega_c' Q}{\omega_0}C' \, \lVert \,L= \frac{1}{\omega_0 \omega_c' Q}\frac{1}{C'}$

Lowpass to bandstop

The required frequency transformation for lowpass to bandstop is;^[5]

$\frac{\omega_c'}{i\omega} \to Q \left( \frac {i\omega}{\omega_0}+\dfrac {\omega_0}{i\omega} \right)$

Inductors are transformed into parallel resonators,

$L' \to L= \frac{\omega_c'}{\omega_0 Q}L' \,\lVert \,C= \frac{Q}{\omega_0 \omega_c'}\frac{1}{L'}$

and capacitors are transformed into series resonators,

$C' \to C= \frac{\omega_c'}{\omega_0 Q}C' \, , \,L= \frac{1}{\omega_0 Q\omega_c'}\frac{1}{C'}$

Lowpass to multi-band

Filters with multiple passbands may be obtained by applying the general transformation;

$\frac{\omega_c'}{i\omega} \to \dfrac{1}{Q_1 \left( \dfrac {i\omega}{\omega_{01}}+\dfrac {\omega_{01}}{i\omega} \right)}+ \dfrac{1}{Q_2 \left( \dfrac {i\omega}{\omega_{02}}+\dfrac {\omega_{02}}{i\omega} \right)}+ \cdots$

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.

The same response can equivalently be obtained, sometimes with a more convenient component topology, by transforming to multiple stopbands instead of multiple passbands. The required transformation in those cases is;

$\frac{i\omega}{\omega_c'} \to \dfrac{1}{Q_1 \left( \dfrac {i\omega}{\omega_{01}}+\dfrac {\omega_{01}}{i\omega} \right)}+ \dfrac{1}{Q_2 \left( \dfrac {i\omega}{\omega_{02}}+\dfrac {\omega_{02}}{i\omega} \right)}+ \cdots$

Alternative prototype

In his treatment of image filters, Zobel provides an alternative basis for constructing a prototype which is not based in the frequency domain.^[6] 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 network rather 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 admittance Y of a filter half-section. See the article Image impedance for a description of half-sections. This quantity is nondimensional, adding to the prototypes generality. Generally, ZY is a complex quantity,

$ZY = U + iV\,\!$ and as U and V are both, in general, functions of ω we should properly write,

$ZY = U(\omega) + iV(\omega)\,\!$

With image filters, it is possible to obtain filters of different classes from the constant k filter prototype 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,

$ZY = U_k(\omega) + iV_k(\omega)\,\!$

In the case of dissipationless networks, ie no resistors, the quantity V(ω) is zero and only U(ω) need be considered. U_k(ω) ranges from 0 at the centre of the passband to -1 at the cut-off frequency and then continues to increase negatively into the stopband regardless 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;