Update Content - 2023-10-13

content/zettels/time_delay.md

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+title = "Time Delay"
+author = ["Dehaeze Thomas"]
+draft = false
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+
+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).
+
+<a id="table--tab:time-delay-phase-lag"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:time-delay-phase-lag">Table 1</a>:</span>
+  Phase lag as a function of the frequency (relative to the sampling frequency )
+</div>
+
+| 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.
+
+<div class="exampl">
+
+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.
+
+<a id="figure--fig:time-delay-induced-phase-lag"></a>
+
+{{< figure src="/ox-hugo/time_delay_induced_phase_lag.png" caption="<span class=\"figure-number\">Figure 1: </span>Phase lag induced by a time delay" >}}
+
+</div>
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>