In digital control with a sampling period Ts = 1 ms, the zero-order hold of the D/A conversion acts as a time delay of about Ts/2. With a gain crossover frequency of 100 Hz, which of the following is closest to the phase lag caused by this delay?

Answer: C. About 18°

The phase lag due to a time delay is ωL, so it is 2π × 100 × 0.0005 ≈ 0.314 rad, about 18° in degrees. A zero-order hold keeps each sample value until the next sample, so on average the signal is delayed by half a period, and its phase lag is exactly ωTs/2. In practice delays such as computation time add to this, and a delay of one full sample (Ts) gives about 36°. To avoid losing much phase margin, the sampling frequency is chosen well above the gain crossover frequency (for example, at least ten times higher).

• A. This reads 0.314 rad directly as degrees, without converting it.
• B. This uses the frequency f = 100 Hz instead of the angular frequency ω = 2πf.
• C. Correct. When the sampling frequency is only ten times the gain crossover frequency, the phase margin is reduced considerably.