A representation of transfer functions for digital simulation programs

Abstract

Efficient representation of an arbitrary filter in a digitai simulation of a control system has long been a problem. A technique often used [ l ] is an approximation that replaces terms of the form (TS+l) in the denominator of the transfer function by TS. This severely limits the types of transfer functions that can be simulated. An exact representation of the effect of a filter on a signal is given by either a convolution of the signal and the impulse response in the time domain or a product of the transforms of the signal and filter in the frequency domain. In many 'real time" simulations, time is incremented within a "DO" loop. With either a convolution or a spectral technique, an integration must be performed within each iteration of the loop. During recent investigations of a phase-locked loop (31, an alternative amroach was d e v e l o d . The tech-.. nique uses properties of the inverse Laplace transform in the frequency domain to represent the convolution integral in the time domain as an integral in real time. The advantage of this approach is that it eliminates the need for a separate integration within each iteration of the loop. The technique applies to any transfer function with poles of order one. In the particular program for the second-order phase-locked loop the techfunction of the form: FINAL BEAR:N.: munox nique reduced running time. Given a transferFig. 3. Single ball-bearing phase-measurement technique.Fig. 2.We achieved this by depositing the bearings into the helix one at a time and turning the helix once between each deposition. Since ball bearings are spherical and helix tolerances very small, each bearing could be positioned identically in each turn.We then reasoned that, since the bearings are small, the reflections caused by each bearing would reach the input port in phase at some frequency. This would occur a t the frequency at which each bearing was separated by some multiple of 2 H phase shift. Thus, at that frequency, the reflection coefficient R would be a maximum. A reflectometer was used to measure R. Ltlhen we vibrated the helix to set the bearings in motion, and then allowed the vibration to subside, the value of R did not change. A maximum in R was noted near the correct frequency. But, the technique had two errors.Reflections of the bearings furthest from the input port must pass a number of other bearings. Each bearing involved has a slight transmission phase shift. Since many small bearings must be used to produce a measurable reflection and a sharp peak in R, the phase error for the furthest reflections can be appreciable. This happens even though the bearings are small.A second error enters because each bearing produces a frequency sensitive reflection which may maximize a t frequencies associated with the bearing diameter. Even with these drawbacks, the technique is sufficiently accurate to predict the broadside frequency to within 30 Mc/s. It also results in a chse estimate with only a single reflectometer measurement.For a more accurate measurement technique, we used...