RC-Circuit as Lowpass Filter

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 $\varphi$:

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

Experimental Tasks

To analyze the behaviour of the RC lowpass filter, the following circuit is used:

1. For the following tasks, use the circuit built during the previous experiment. Remove all measurement leads and disconnect the alligator clips of the signal generator. Now extend the circuit by adding the level shifting circuit at the input.
   Explanation: The signal generator of the handheld oscilloscope provides a unipolar AC signal with \(V_{\mathrm{pp}} = 3\,\mathrm{V}\). The level shifting circuit converts this signal into a bipolar AC signal with \(\hat{u} = 3\,\mathrm{V}\).
   
   
2. Configure the function generator of the handheld oscilloscope to provide a sinusoidal signal. Use the following settings:
   
   
   • Amplitude: 3 V
   • Frequency: 2 000 Hz
   • Duty Cycle: 50%
     
     
3. Connect the alligator clips of the signal generator to the RC circuit to provide the input voltage $u_\mathrm{F}(t)$, and supply the MEXLE board with power via the Micro-USB connector. Then connect Channel 1 of the handheld oscilloscope to the input voltage $u_\mathrm{F}(t)$and Channel 2 to the capacitor voltage $u_\mathrm{C}(t)$. Sketch the waveforms displayed on the oscilloscope.
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
4. Measure the amplitude response and the time delay in milliseconds, then calculate the corresponding phase shift angle. Explain how the amplitude response and phase shift change as the frequency increases.