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Cascading Filters with Variable Stages

Analysis

The claimed transfer function of this circuit is

\begin{split}\begin{matrix} T(f) = \ \frac{V_{out}(f)}{V_{in}(f)} = \frac{1}{1 + j2\pi f\left\lbrack R_{1}C_{1} + C_{2}\left( R_{1} + R_{2} \right) \right\rbrack - 4\pi^{2}f^{2}R_{1}R_{2}C_{1}C_{2}}\ \#(4) \\ \end{matrix}\end{split}

Where $j=\sqrt-1$ is the imaginary unit.

What is the magnitude of the transfer function?
What is the phase response of the circuit?
What class (low-pass, high-pass, band-pass, band-stop) of filter is this?
What is the -3dB frequency?

Measurement

Using the Red Pitaya’s Bode Analyzer tool, measure the frequency response (|T(f)|) as described in section 3.1.2.

Show the plot of the measurement below:
(Optional) Try sweeping from 10Hz to 1MHz. Is there anything strange that happens to the frequency response? Capture the frequency response, and describe what seems to happen to the transfer function.

Comparison

Respond to the following questions:

Does the shape of the frequency response match your expectation from the analysis? Is there any point that stands out as odd?
Find the -3dB point in the circuit, and compare this value to the one you previously calculated.

Questions and Answers

What is the magnitude of the transfer function?

The magnitude of the transfer function, denoted as |T(f)|, represents the ratio of the output voltage $V_{out}(f)$ to the input voltage $V_{in}(f)$ . In the given transfer function expression, the magnitude can be calculated as $\left| T(f) \right| = \frac{1}{\sqrt{1 + (2\pi fRC)^2}}$ .

What is the phase response of the circuit?

The phase response of the circuit, denoted as $\phi(f)$ , represents the phase shift between the input and output signals. In the given transfer function expression, the phase response can be calculated as $\phi(f) = -\arctan(2\pi fRC)$ .

What class (low-pass, high-pass, band-pass, band-stop) of filter is this?

The class of the filter can be determined by analyzing the transfer function. In this case, since the magnitude of the transfer function decreases with increasing frequency, it is a low-pass filter. A low-pass filter allows low-frequency signals to pass through while attenuating high-frequency signals.

What is the -3dB frequency?

The -3dB frequency, denoted as $f_c$ , corresponds to the frequency at which the magnitude of the transfer function is reduced to 12 or approximately 0.707. Mathematically, it can be found by solving the equation $\frac{1}{\sqrt{1 + (2\pi f_{c}RC)^2}} = \frac{1}{\sqrt{2}}$ .