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+++ title = "Time Delay" author = ["Dehaeze Thomas"] draft = false +++
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Phase induced by a time delay
Having some time delay can be modelled by a transfer function having constant amplitude but a phase lag increasing with frequency. Such phase lag is linearly proportional to the time delay and to the frequency:
\begin{equation} \phi(\omega) = -\omega \cdot T_s \end{equation}
with:
- \(\phi(\omega)\) the phase lag in rad
- \(\omega\) the frequency in rad/s
- \(T_s\) the time delay in s
Estimation of phase delay induced in sampled systems
Consider a feedback controller implemented numerically on a system with a sampling frequency \(F_s\).
The time delay associated with the limited sampling frequency \(F_s\) is:
\begin{equation} \phi(\omega) = -\frac{\omega}{F_s} \end{equation}
with:
- \(\phi(\omega)\) the phase lag in rad
- \(\omega\) the frequency in rad/s
- \(F_s\) the sampling frequency in Hz
Some values are summarized in Table 1.
| Frequency | Phase Delay [deg] |
|---|---|
| \(F_s/100\) | -3.6 |
| \(F_s/10\) | -36.0 |
| \(F_s/2\) | -180.0 |
This is the main reason to have a sampling frequency much higher than the wanted feedback bandwidth is to limit the phase delay at the crossover frequency induced by the time delay. Having a sampling frequency a 100 times larger than the crossover frequency is a good objective.
Take the example of a controller implemented with a sampling time of 0.1ms (10kHz sampling frequency).
t_delay = 1e-4; % Delay [s]
G_delay = exp(-t_delay*s);
The induced phase delay as a function of frequency is shown in Figure 1.
At the Nyquist frequency (5 kHz), the phase lag is 180 degrees.
{{< figure src="/ox-hugo/time_delay_induced_phase_lag.png" caption="<span class="figure-number">Figure 1: Phase lag induced by a time delay" >}}