28-Apr-2015: Series RLC Circuit Step Response

PURPOSE

The purpose of this lab was to implement what we learned about an RLC circuit in series in a real-life experiment.

PRE-LAB

Before setting up our circuit, we set up a second order differential equation to relate Vout and Vin. We also did some calculations to estimate the damping ratio (ζ), the natural frequency (ωo), and the damped natural frequency (ωd). In order to do so, we had to first find the actual values of the circuit elements. We found the resistances of the resistor and the inductor to be 1.4 Ω and 1.7 Ω, respectively. Furthermore, we found the inductance of the inductor to be 0.999 mH and the capacitance of the capacitor to be 0.437 µF. From these values, we found the necessary variables as shown in Figure 1.

PROCEDURES

Figure 2

To begin our experiment, we constructed the RLC circuit as shown in Figure 2. It consisted of a resistor, an inductor, and a capacitor, all connected in series. Then, we applied a 2 V step input at 1 Hz to the circuit and plotted Vout and Vin on the oscilloscope. The resulting graphs are shown in Figures 3 and 4.

Figure 3: Under-damped circuit (1)

Figure 4: Under-damped circuit (2)

For an under-damped circuit, the response oscillates at the damped natural frequency. In order to find the damped natural frequency, we multiplied the inverse of the period (which we found by analyzing the graphs in Figures 3 and 4) by 2π. We found this value to be 52359.9 rad/s. This experimental value was somewhat close to the theoretical value of 47835.3 rad/s. In fact, the percent difference between the two values was 9.459 percent. Although this number was slightly bigger than what we would have preferred, we believed that the results were acceptable since there were many possible sources of error in this experiment such as the actual capacitance of the capacitor.

Figure 5

For the second part of the lab, we estimated the resistor value that would cause the circuit to be critically damped (Figure 5). As it can be seen from the image, we found this value to be 93.93 Ω. However, since we did not have this resistor value available for use, we chose a 100 Ω resistor instead. The measured value of the resistor (97.8 Ω) is shown in the schematic of the revised circuit in Figure 5.

Figure 6: Critically damped circuit

Figure 6 shows the response of the revised circuit to a 2 V step input oscillating at 1 Hz.