+++ title = "Time Delay" author = ["Dehaeze Thomas"] draft = false +++ Tags : ## Phase induced by a time delay {#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 {#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](#table--tab:time-delay-phase-lag).
Table 1: Phase lag as a function of the frequency (relative to the sampling frequency )
| 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). ```matlab 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](#figure--fig:time-delay-induced-phase-lag). At the Nyquist frequency (5 kHz), the phase lag is 180 degrees. {{< figure src="/ox-hugo/time_delay_induced_phase_lag.png" caption="Figure 1: Phase lag induced by a time delay" >}}
## Bibliography {#bibliography}