ABSTRACT
With the realization of second-order filters discussed in the previous chapters of this section, we will now treat methods for practical filter implementations of order higher than two. Specifically, we will investigate how to realize efficiently, with low sensitivities to component tolerances, the input-to-output voltage transfer function
(15.1)
where n ≥ m and n > 2. The sensitivity behavior of high-order filter realizations shows that, in general, it is not advisable to realize the transfer function H(s) in the so-called direct form [5, ch. 3] (see also Chapter 73 in this book). By direct form we mean an implementation of (15.1) that uses only one or maybe two active devices, such as operational amplifiers (op amps) or operational transconductance amplifiers (OTAs), embedded in a high-order passive RC network. Although it is possible in principle to realize (15.1) in direct form, the resulting circuits are normally so sensitive to component tolerances as to be impractical. Since the direct form for the realization of high-order functions is ruled out, in this section we present those methods that result in designs of practical manufacturable active filters with acceptably low sensitivity, the cascade approach, the multiple-loop feedback topology, and ladder simulations. Both cascade and multiple-loop feedback techniques are modular, with active biquads used as the fundamental building blocks. The ladder simulation method seeks active realizations that inherit the low
In the cascade approach, a high-order function H(s) is factored into low-(first-or second-) order subnetworks, which are realized as discussed in the previous chapters of this section and connected in cascade such that their product implements the prescribed function H(s). The method is widely employed in industry; it is well understood, very easy to design, and efficient in its use of active devices. It uses a modular approach and results in filters that, for the most part, show satisfactory performance in practice. The main advantage of cascade filters is their generality, i.e., any arbitrary stable transfer function can be realized as a cascade circuit, and tuning is very easy because each biquad is responsible for the realization of only one pole pair (and zero pair): the realizations of the individual critical frequencies of
H s V
V
N s
D s
a s a s a s a
s b s b s b out
n( ) = = ( )( ) = + + + +
+ + + + −
passband sensitivity properties of passive doubly terminated LC ladder filters (see Chapter 9).
