Page 162 - Analog and Digital Filter Design

Highpass Filters 159

The relationship between the equations using highpass pole locations, and those
previously presented using lowpass pole locations, can be seen. Note that for
both resistors, the equations have a frequency-dependent factor in the denomi-
nator. Frequency scaling can therefore be achieved by dividing the normalized
highpass resistor values by 3@.

Operational Amplifier Requirements

Sallen and Key highpass filters are good if the requirements are not too demand-
ing, with section Q factors below 50. As with lowpass designs, the gain-
bandwidth product of the op-amps can limit the filter's cutoff frequency. The
lowpass cutoff frequency limit was given by the empirical expressions:

Gain -Bandwidth Product
Butterworth passband frequency limit =
(filter order)'
Gain - Bandwidth Product
Chebyshev (IdB) passband frequency limit =
(filter order)''

These equations can also be used for highpass filters. by letting the passband
frequency limit equal the highest frequency to be passed (Le., do not use the
-3dB cutoff frequency). Remember that if several amplifiers are cascaded. the
gain-bandwidth product of each one has to be higher than what is required
overall. This is because each one contributes to high frequency roll-off as the
gain-bandwidth frequency is approached.

The passband frequency limit for a given amplifier gain-bandwidth product is
for a maximum of 2dB amplitude error in the passband. A lower passband
frequency limit must be set if no amplitude error is acceptable. Although using
an amplifier having a greater gain-bandwidth product can raise the passband
frequency limit, it can lead to instability. Amplifiers that have a high gain-
bandwidth product are often unstable in a unity gain configuration.

Denormalizing Sallen and Key or First-Order Designs

In active filter designs the resistor values used should all be in the range 1 kS1 to
lOOkR where possible. If resistor values are lower than 1 kR there may be a