TUTORIALS 727

Abstract: This application note explains a method that uses integrator blocks and some simple mathematical manipulation to produce filter responses of any order. The method is precise, easy to apply, and an alternative to a \"standard\" set of filter networks when a nonstandard filter response is needed.

The transfer functions of the integrator in Figure 1 and its symbolic representation are shown in the expression in

A sequence of mathematical steps is then applied to the transfer function to obtain an expression of the form:

VNote that all frequency-dependent terms (occurrences of S) in the resulting expression must appear in the denominator terms. This is because the final circuit will be composed of integrators, i.e., functions of 1/S. Three basic mathematical steps are required. The expression obtained at each stage is given below._{OUT}= ƒ(V_{IN}, V_{OUT}, 1/S)

S² × V_{OUT}+ ω_{0}/q × S × V_{OUT}+ ω_{0}² × V_{OUT}= ω_{0}² × V_{IN}

V_{OUT}+ [(ω_{0}/q) × V_{OUT}]/S + (ω_{0}² × V_{OUT})/S² = ω_{0}² × V_{IN}/S²

VThe equation produced in Step 3 is now the defining equation for a network of integrator blocks that will perform the required filtering function._{OUT}= V_{IN}× (ω_{0}²/S²) - V_{OUT}× (ω_{0}²/S²) - V_{OUT}× [(ω_{0}/q)/S]

The remaining two design steps are somewhat intuitive, but the rules are simple. First, an integrator network drawing is produced. This uses a number of integrators and summing nodes to produce a network described by the defining equation in Step 3. To produce this network, start by considering the form of the defining equation. This expresses the output voltage (V

Considering these terms from left to right, the first term is a function of V

To complete the network, the correct sign, inverting or noninverting, must be allocated to each summing-junction input. The transfer function of the integrator shown in Figure 1 is of the form -1/ST, so the integrator blocks will have a signal inversion built in. Signs are allocated to the summing junctions working from the output back toward the input. The third term in the equation of Step 3 shows that the feedback path from V

The integrator time constants, T

-ωSimilarly, from the first and second terms of the defining equation we find:_{0}/qS = -1/ST_{2}→ T_{2}= q/ω_{0}

ωValues for ω_{0}²/S² = 1/S²T_{1}T_{2}→ T_{1}= 1/ω_{0}²T_{2}= 1/ω_{0}q

The final step is to translate the integrator network of Figure 4 into an operational amplifier/resistor/capacitor circuit. A standard, inverting, op-amp integrator block, comprising an op amp, a feedback capacitor, and an input resistor, is equivalent to a single (noninverting) summing node followed by an (inverting) integrator. Multiple input summing nodes are then accommodated by the addition of more input resistors to the op-amp integrator block.

The circuit in

Note the trick with the feedback to IC

The above example can be implemented using a simple, dual-op-amp IC and a handful of passive components. Where higher order systems are being considered, the overall design task can be simplified considerably by the use of multistage filter ICs. Two examples of this type of component are the MAX274/MAX275. These devices provide, respectively, fourth- and eighth-order continuous-time filtering based on a series of integrator blocks. The filter time constants for these devices are defined by external resistor values only, as the feedback capacitor for each integrator stage is provided on-chip.

If the designer wishes a higher degree of programmability for the filter design, then a switched-capacitor-filter approach may well be suitable. There are switched-capacitor building-block ICs available that can be adjusted with a programmable clock or resistors. Some parts are also available with microprocessor-interface capability. The MAX260 and MAX268 families of switched-capacitor-filter building-block ICs provide a full range of control methods for anyone looking for programmable filtering functions.

The design process described here is powerful and versatile. It can be applied to virtually any active filtering requirement and to functions of any order. In addition, the resulting implementation of simple integrator blocks eases the selection and the tolerance issues for components. Some active-filter implementations exacerbate the effects of basic component tolerances, whereas the integrator approach produces the same basic tolerance susceptibility as that furnished by a passive LCR filter circuit. Further, the effects of op-amp-bandwidth variation are relatively simple to calculate, because the (desired) operating unity-gain bandwidth of each integrator block is simply given by 1/T rad/s = 1/2πRC Hz.