Cascade Active Filters

We’ve seen that odd-order filters may be built by cascading second-order stages or sections (with appropriately chosen poles, zeros and gain) and one first-order section, and even order filters can be built by cascading second order sections. For example: n=1 n=2

Cascade Active Filters The transfer function of a second-order section may be written as: And that of a first order stage can be written as: The values of  and  are given in Cartinhour’s tables

Cascade Active Filters Active first- and second-order filter sections may be realized as active networks. One opamp is sufficient for a first-order section, second order sections may be built using one, two, three or four opamps. We’ll use the single-amplifier realization. A first-order section is shown in the next slide:

Cascade Active Filters R C - + RaRa RfRf X(s) Y(s)

Cascade Active Filters Here’s a Sallen and Key second-order section: R C - + RaRa RfRf X(s) Y(s) R C

Cascade Active Filters This filter has the following transfer function: And we want to set it’s coefficients equal to:

Cascade Active Filters So we’ll pick resistors and capacitors such that

Cascade Active Filters Note that Ak is determined by  k and  k,, so we can’t freely determine the section gain – it’s determined by the parameters of the quadratic equation in the denominator of the transfer function. We do need to be able to set the overall filter gain, which can be done by selecting the gain for the first- order section (if the filter order is odd) or by an additional gain stage (if the filter order is even.

Cascade Active Filters There are other procedures for selecting the component values for the Sallen and Key filter, some of which may allow more freedom in selction of the section gain. Some procedures are optimized for a particular quality, such as insensitivity to variations in component value. There are also other second-order filter topologies (circuits) using from 1 to 4 opamps, each of which has its own advantages and disadvantages. This isn’t an active filter course, so we won’t consider the others.

Cascade Active Filters One more thing: Cartinhour states on p. 95 that active filters are not suitable for frequencies above the audio range, because of opamp frequency response limitations. This is no longer true, because opamps have been vastly improved over the last years in every respect, and some are now suitable for use in the RF range. At high frequencies, selection of the component type, and circuit board layout become critical, however.

Highpass Filters We can take a normalized lowpass filter (  c = 1, in Cartinhour’s nomenclature) and transform it to a denormalized highpass filter. Recall that to denormalize a lowpass prototype, we substituted s/  c for s in H(s). To denormalize and transform to highpass, we substitute  c /s for s.

Highpass Filters If we take a second order section of the form: Now substitute  c /s for s:

Highpass Filters Multiplying by s 2 /s 2 : and then dividing the numerator and denominator by  k 2 : where

Highpass Filters This way of deriving the transformation is somewhat different than Cartinhour’s, but the results are equivalent. Let’s try a second order Butterworth highpass filter, with f c = 10 Hz. From the tables:

Highpass Filters Using Cartinhour’s procedure,

Highpass Filters Using my procedure, So the results are the same. Ya pays your dollar, ya takes yer choice.

Highpass Filters Either procedure may be used to transform a normalized Butterworth or Chebyshev lowpass prototype to a denormalized highpass filter of the same type. As Cartinhour points out, the amplitude response of an N th order Butterworth filter is:

Highpass Filters And that of a Chebyshev highpass filter is: These are similar to the lowpass amplitude responses, but (f/f c ) has been replaced by (f c /f). Examining the plots on pages 97 and 98 of the book, the amplitude response of the highpass filter is simply that of the lowpass prototype, but reversed in frequency.

Active Highpass Filters These are similar to active lowpass filters, and may be constructed by cascading first- and second-order sections. Here’s a first-order section: R C - + RaRa RfRf X(s) Y(s)

Active Highpass Filters R - + RaRa RfRf X(s) Y(s) R CC Here’s a Sallen and Key highpass section:

Active Highpass Filters And we want to set it’s coefficients equal to: This filter has the transfer function:

Required Filter Order Typically, we will have a requirement for a filter with a particular passband, stopband, minimum/maximum passband gain, and minimum stopband attenuation. The first thing we need to do is choose the filter type and order. We’ll assume the type has already been chosen, so we need to determine the order.

Required Filter Order From Cartinhour for a Butterworth lowpass filter:

Required Filter Order For other filter types, refer to Cartinhour, filter design software, or a book of filter tables. Problems: