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Background Information

RC circuits can be used as low-pass filters. A low-pass filter allows low-frequency signals to pass while attenuating high-frequency signals. One practical application is audio systems, where low-frequency bass signals are directed to a woofer, while higher frequencies are suppressed. The frequency-dependent behavior of the filter is determined by the resistor and capacitor values. In the following experiment, the behavior of an RC circuit as a low-pass filter will be investigated.

The circuit shown can be interpreted as a frequency-dependent voltage divider. With increasing frequency, the capacitive reactance of capacitor $C_{\rm 1}$ decreases, which leads to a reduction in the amplitude of the output voltage $u_{\rm 2}$. The ratio of the output voltage to the input voltage as a function of frequency is called the transfer function: $$F(j\omega) = \frac {U_{\rm 2}}{U_{\rm 1}}$$

Applying the voltage divider rule yields the following expression for the transfer function:
$$F(j\omega) = \frac {U_{\rm 2}}{U_{\rm 1}} = \frac{\frac{1}{j\omega C_{\rm 1}}}{R_{\rm 1}+\frac{1}{j\omega C_{\rm 1}}} = \frac{1}{1+j\omega C_{\rm 1}R_{\rm 1}}$$
The transfer function depends on the angular frequency ($\omega = 2\pi \cdot f$) and can be expressed in terms of magnitude and phase.
The magnitude $|\frac{U_{\rm 2}}{U_{\rm 1}}| = \frac{1}{\sqrt {1+(\omega R_{\rm 1}C_{\rm 1})^2}}$ is referred to as the amplitude response.
The phase angle $\varphi$ between the input voltage $u_{\rm 1}$ and the output voltage $u_{\rm 2}$ is called the phase response. The following relation applies:

$\varphi_{\rm U_{\rm 2}}=−\arctan\omega R_{\rm 1}C_{\rm 1}$

The cutoff angular frequency $\omega_{\rm G}$ is defined as the frequency at which the following conditions apply to the amplitude ratio $\frac{U_{\rm 2}}{U_{\rm 1}}$ and the phase angle $\varphi$:

$\frac{U_{\rm 2}}{U_{\rm 1}}=\frac{1}{\sqrt2}, ~~~ \varphi=−45°$