Control Engineering Basics
Which of the following correctly describes the frequency characteristics of a time-delay element e^(−Ls) with delay time L, and its effect on a control system whose loop transfer function contains it?
- AIt only delays the appearance of the response and does not change the frequency characteristics, so it does not affect the stability of the control system
- BIts gain falls at −20 dB/dec and its phase is a constant −90°, so like an integrator it brings the steady-state error to zero
- CIts gain is 1 (0 dB) at every frequency, but its phase lags by ωL [rad], growing in proportion to frequency, so it reduces the phase margin
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Answer: C. Its gain is 1 (0 dB) at every frequency, but its phase lags by ωL [rad], growing in proportion to frequency, so it reduces the phase margin
A time delay is an element whose output is the input delayed by a time L; e^(−jωL) always has a magnitude of 1 and a phase of −ωL [rad]. The phase lag grows without limit as the frequency rises, so it reduces the phase margin at the gain crossover frequency. For example, with L = 10 ms the phase lags by as much as about 57° at 100 rad/s. Transport delay through piping and the computation and output delays of digital control are examples, and with a large time delay the gain often has to be lowered at the cost of a slower response.
- A. The phase of the frequency characteristics changes, which strongly affects stability.
- B. This describes an integrator. The gain of a time delay is constant regardless of frequency.
- C. Correct. Only the phase lags, in proportion to frequency.