using 3 filter back to back

how to use 3 R-C LPFs back to back each with a cut-off frequency of 450 Hz and a Twin-Tee circuit with a notch positioned at 800 Hz. to achieve this :

H = tf([1], [R*C1,1])  % is a low pass filter

Answer 1

[Saurav]

Hey, Huijia Ma,

I understand that you want to achieve the filter response, using three RC low pass filters with a cut-off frequency of 450 Hz in a cascaded manner, along with a twin-tee notch filter centered at 800 Hz, as evident from the figure provided in the question.

Here is a possible solution:

• You will need to create transfer functions for each and cascade them together to achieve the overall response.
• To cascade three LPFs, you can raise the transfer function of one R-C LPF to the third power.

An example code for your reference is:

% Define the cutoff frequency for the LPF
fc = 450;  % in Hz

% Define the notch frequency for the Twin-Tee notch filter
fc_notch = 800;  % in Hz

% Choose a value for R, for example, 1 Ohm (for LPF), 
% Change the value of R according to your circuit
R = 1;  % in Ohms

% Calculate C based on chosen R and desired fc for LPF
C = 1 / (2 * pi * fc * R);  % in Farads

% Define the transfer function for one LPF
H_lpf = tf([1], [R * C, 1]);

% Cascading three identical LPFs
H_lpf_cascaded = H_lpf^3;

% Assuming values for the Twin-Tee notch filter components
% Convert notch frequency to radians per second
omega_notch = 2 * pi * fc_notch;

% Example values for the damping ratio and notch frequency (for simplicity)
zeta = 0.1;  % damping ratio (dimensionless)

% Define the transfer function for the Twin-Tee notch filter
H_notch = tf([1, 0, omega_notch^2], [1, 2 * zeta * omega_notch, omega_notch^2]);

% Cascading the LPF and notch filter
H_overall = H_lpf_cascaded * H_notch;

% Plot the PSD of the resultant signal
% Define the frequency range for plotting
f = logspace(0, 5, 10000);  % Frequency range from 1 Hz to 100,000 Hz (10^5 Hz)

% Calculate the frequency response of the total transfer function
[mag, ~, w] = bode(H_overall, 2 * pi * f);

% Convert magnitude response to power (PSD is proportional to magnitude squared)
psd = squeeze(mag).^2;

% Plot the PSD
figure;
semilogx(f, 10*log10(psd));
grid on;
xlabel('Frequency [Hz]');
ylabel('PSD [dB]');
title('PSD of the filter response');

% Highlight the notch frequency
hold on;
line([fc_notch fc_notch], ylim, 'Color', 'red', 'LineStyle', '--');
text(fc_notch, 10*log10(psd(round(end/2))), ' Notch Frequency', 'Color', 'red', 'HorizontalAlignment', 'left');
hold off;

You can refer to the following documentation for more information on the functions “tf” , “bode” and “squeeze” respectively:

1. https://www.mathworks.com/help/releases/R2022b/control/ref/tf.html
2. https://www.mathworks.com/help/releases/R2022b/control/ref/lti.bode.html
3. https://www.mathworks.com/help/releases/R2022b/matlab/ref/squeeze.html

I hope this helps!