R–L–C Series Resonant Circuit

Background information

“Resonance describes the oscillatory response of a system when it is excited by a frequency that matches or is close to the system’s natural frequency.”
The resonance frequency is given by: $$f_{\rm r} = \frac{1}{2\pi \sqrt{LC}}\text{, or} ~~~ \omega_{\rm r} = \frac{1}{\sqrt{LC}}$$
At resonance, the current is given by: $$I = \frac{U}{R}$$

Experimental Tasks

1. Set up the following measurement circuit:
   

   Fig. 1: R–L–C Series Resonant Circuit

   
   The corresponding phasor diagram is shown in figure 1.
   

2. The function generator shall be set to produce a sinusoidal signal, and the amplitude of the output voltage shall be 6 V. For the circuit shown above, the magnitude of the impedance is given by: $$Z = \sqrt{R^2 + (\omega L − \frac{1}{\omega C})^2}$$
   As can be seen from figure 1, the inductive and capacitive voltage components partially cancel each other. There exists a specific frequency at which these components cancel completely, causing the circuit to behave as a purely resistive circuit. This frequency is known as the resonance frequency.
   
3. Calculate the resonance frequency for the assembled measurement circuit.
   (The capacitance of the capacitor is determined using a measuring instrument)
   
   $f_{\rm r}~=~ ..........................................................$
   
   
4. Adjust the frequency of the function generator so that the ratio $\frac{f}{f_{\rm r}}$ corresponds to the values listed in table 1.
5. Measure the RMS voltage $U_{\rm R}$ using the oscilloscope and record the measured values in table 1.
   

   Tab. 1: R-L-C-circuit RMS voltage

   
   

6. Plot the current as a function of the ratio $\frac{f}{f_{\rm r}}$ on a logarithmic scale.

Test questions - Resonant Circuit