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Letters to the Editor

Differentiating Circuits Dr. G. W. Beeler's letter on differentiating circuits (Circ Res 28:503-505, 1971) is timely but somewhat incomplete and possibly misleading. The passive RC circuit does not provide "a satisfactory derivative" for all signal frequencies below f0 = 1/(2TTRC). The amplitude error is 30% at f0. One should set f0 at least 10 times the highest frequency of interest, i.e., for cardiovascular work, the RC time constant of passive circuits should be 1 msec or less to provide reasonable derivatives (0.5% amplitude and 6° phase error) to 16 Hz. To disregard the calibration factor obtainable from the time constant of the circuit or to rely on "relative measurements" is to weaken confidence in the results and ease of data reduction. Quality components are available that make predictable circuits easily attainable. We should, of course, check the system by calibration under experimental conditions. With the excellent and simple active differentiator shown in Figure 2 of Dr. Beeler's letter, there are four important time constants. In addition to the three that he mentions, Rf Cf progressively attenuates very high frequencies, such as chopper hash, tape recorder carrier frequencies, and RF noise. For an effective system with rapid cutoff of high frequency noise, and yet one which meets a 1% amplitude error criterion for the highest frequency of interest, RfC should equal RCf. Beeler's approach of having f2 > fi provides less high frequency attenuation with about the same range of accurate differentiation than setting fi = f.. The values of R and C should be selected to match the gain characteristics of the associated in-line signal processors, and the values Rf and Cf should be chosen to give a critical frequency, fc (the frequency of maximum gain) of at least 10 times the highest frequency of interest. For example, for data with frequencies from 0 to 16 Hz, R,C and RCf should be 0.001 sec for a maximum gain at 159 Hz [1/(277X0.001)]. With this amplitude error will be < 1% and phase error 11° at 16 Hz. There is, in our opinion, no unambiguous description of practical differentiators other than either the circuit diagram or a list of amplitude and phase responses of the system used. There are many types of low pass filters; a single RC network is but one. At the 3-db point the 30% error is large, especially in view of the fact that the frequency components of the isovolumetric phase of ventricular contraction are in the range of 10 to 20 Hz! All who use differentiators should understand that serious differentiation error (up to 30%) occurs at frequencies below those represented by the characteristic frequency, f0, for the passive circuit and fi, f2 or fe for the active circuit. "The Applications Manual for Operational Amplifiers" (Philbrick/Nexus Research, p. 49), has some useful information for those wanting more detail, as does an article by B. J. Gersh et al. (Cardiovasc Res 5:32-40, 1971).