Page 212 - Analog and Digital Filter Design

Bandstop Filters 209

Where X is the normalized lowpass series arm inductor value (15: in this case).
The capacitor value is not needed for this: it is however used to derive p. Higher-
order filters, with more than one series arm in the lowpass prototype, require
this process to be repeated for each series arm; Xis then L4. Lb, and so on.

Factor p is the squared resonant frequency of the parallel tuned circuit in the
normalized lowpass design. It can be derived from the series arm capacitor
and inductor values, C,,L,, and C,,L,,. The equations can be multiplied together:
CL,Lz, 1/wo2p, where woz = 4n'FL.FL.
=

Active Sandstop Filters

Active bandstop filters can be designed using pole and zero locations from the
frequency response's transfer function. Operational amplifiers (op-amps) are the
"active" part of the circuit. Op-amps have high input impedance and low output
impedance. They also buffer one filter stage from the next, which prevents inter-
action. Each stage can therefore be designed to provide the frequency response
of one pair of complex poles, Zeroes may also be required in the stopband, and
circuits that provide this function are usually more complex. Because stages are
buffered from one another, when all the stages are connected in series the overall
response should be that which is required.

Bandstop Poles and Zeroes

Using the normalized lowpass response pole and zero locations as a starting
point, frequency translation is required to find the normalized bandstop pole
and zero locations. Frequency translation in transfer functions and the S-plane
are found by replacing s with the following:

@O = and BW7is the bandwidth, w( - wL.

This is not particularly easy to evaluate. However, as in the bandpass case,
Williams and Taylor' have published equations for finding the Q and resonant
frequency, -fR, of each section of bandstop filters from a lowpass model. These
are all that are needed to design active bandstop filters. I have manipulated
Williams and Taylor's equations slightly, consistent with the bandpass filter
equations given in the previous chapter. To start with you must know the Q of
bandstop filter, Qss. and oand m, which are the real and imaginary parts of the
lowpass prototype pole location. The pole positions can be found from forrnu-