RLC Circuit Response
In this lab lab we wire the RLC circuit shown in the top left diagram (however the 49.4 Ω and 1 Ω resistors are actually supposed to be switched). We analyze the circuit step theoretically as it would look when 2 V are applied for a long enough time and then taken away using a step voltage. The two diagrams representing the two states are shown next to the original diagram. Next, the characteristic second order differential equation is shown, in terms of the neper frequency and undamped natural frequecny, ɑ and ω0, respectively. The values for these variables are then calculated below and plugged back in to the differential equation. Next we calculate the natural frequencies s1 and s2 and the damping ratio ζ.
Here we see our circuit wired up with the yellow and black wires providing step voltage and ground. The orange wires measure the voltage input on the oscilloscope, while the blue wires measure the resistor output voltage. We now apply a 2 V step voltage and obtain the following graphs on the oscilloscope.
Here we see the input voltage (orange) and the output voltage (blue). We can see that the output voltage has a quick oscillation behavior that is dampened and dies out until the next step.
This graph shows a close up view of the previous graph. Here we have a better view of the oscillations. We can see that the voltage steps down from 40 mV and overshoots to about -14 V which is 35 % and had a rise time of 265 μs. This graph almost looks like it is barely underdamped and almost critically damped. The oscillations seem to die out quickly but not enough to be critically damped. The damping ratio is almost 1 which agrees with this hypothesis.
However, this lab has been troublesome to analyze because our initial diagrams may have not been completely correct. We assumed that when the voltage is zero, the voltage source acts like an open circuit (seen in the top right diagram of the first picture), which would then mean that the resistor R1 could be ignored and the remaining circuit could be treated like a series RLC circuit. This is not true however, because when the voltage is 0 V, the source actually acts like a wire. Therefore the R1 resistor must be taken into account. This means that the circuit is not a series RLC circuit. If we examine the circuit closer, we can see that it is not a parallel circuit either. For these reasons, we may not be able to calculate the theoretical values by the traditional 'plug-and-chug-the-values' into a formula method. Therefore, we only make qualitative observations in this laboratory rather than quantitative ones. A deeper analysis is required to find the theoretical values needed to compare to our experimental values.
However, this lab has been troublesome to analyze because our initial diagrams may have not been completely correct. We assumed that when the voltage is zero, the voltage source acts like an open circuit (seen in the top right diagram of the first picture), which would then mean that the resistor R1 could be ignored and the remaining circuit could be treated like a series RLC circuit. This is not true however, because when the voltage is 0 V, the source actually acts like a wire. Therefore the R1 resistor must be taken into account. This means that the circuit is not a series RLC circuit. If we examine the circuit closer, we can see that it is not a parallel circuit either. For these reasons, we may not be able to calculate the theoretical values by the traditional 'plug-and-chug-the-values' into a formula method. Therefore, we only make qualitative observations in this laboratory rather than quantitative ones. A deeper analysis is required to find the theoretical values needed to compare to our experimental values.
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