Update Content - 2023-10-13

content/zettels/discrete_transfer_functions.md

@@ -153,6 +153,61 @@ den = 1 + (2*Ts^2*wn^2 - 8) * z^-1 + (Ts^2*wn^2 - 4*Ts*wn*xi + 4) * z^-2
 And the transfer function is equal to `gain * num/den`.
 
 
+### Second Order Low Pass Filter {#second-order-low-pass-filter}
+
+Let's consider a second order low pass filter:
+
+\begin{equation}
+  G(s) = \frac{g}{ms^2 + cs + k}
+\end{equation}
+
+First the symbolic variables are declared (`Ts` is the sampling time, `s` the Laplace variable and `z` the "z-transform" variable).
+
+```matlab
+%% Declaration of the symbolic variables
+syms Ts g m c k s z
+```
+
+Then the bi-linear transformation is performed to go from continuous to discrete:
+
+```matlab
+%% Bilinear Transform
+s = 2/Ts*(z - 1)/(z + 1);
+```
+
+The symbolic formula of the notch filter is defined:
+
+```matlab
+%% Second Order Low Pass Filter - Symbolic representation
+Ga = g/(m*s^2 + c*s + k)
+```
+
+Finally, the numerator and denominator coefficients can be extracted:
+
+```matlab
+%% Get numerator and denominator
+[N,D] = numden(Ga);
+
+%% Extract coefficients (from z^0 to z^n)
+num = coeffs(N, z);
+den = coeffs(D, z);
+```
+
+```text
+gain = 1/(4*m + 2*Ts*c + Ts^2*k)
+```
+
+```text
+num = (Ts^2*g) + (2*Ts^2*g) * z^-1 + (Ts^2*g) * z^-2
+```
+
+```text
+den = 1 + (2*Ts^2*k - 8*m) * z^-1 + (4*m - 2*Ts*c + Ts^2*k) * z^-2
+```
+
+And the transfer function is equal to `gain * num/den`.
+
+
 ### Second Order High Pass Filter {#second-order-high-pass-filter}
 
 Let's consider a second order low pass filter: