We then analyzed source free RLC circuits with all components in series. KVL was used to determine the second order differential equation of this circuit. One possible solution to this differential was used (i = Ae^st) to determine its validity, but it turned out that it didn't quite possibly explain all aspects of the circuit. It was still plugged back into the equation to find the two natural frequencies "s", or the roots of the obtained quadratic. Using these frequencies the neper frequency alpha (R/2L) and the undamped frequency omega (1/(LC)^-1/2) were found. Because there were two values of s obtained and the initial predicted solution did not explain all of the aspects, it was found that the solution contains two components A1 and A2. It was then determined that when alpha is greater than omega, the circuit is overdamped; when they are equal, it is critically damped; and when alpha is less than omega, it is underdamped. We then learned about the different current solutions used for those three cases and their respective graphs. We then did a problem involving calculating the roots and figuring out if the circuit is overdamped, underdamped, or critically damped. A lab titled "Series RLC Circuit Step Response" was then performed.
We then went over parallel RLC circuits that were source free. These circuits are very similar, with the exception that KCL was used to find the second order differential equation. In addition the DEQ of a parallel RLC circuit relates voltages whereas the DEQ of a series circuit related currents. In addition, alpha of a parallel circuit is now 1/2RC and not R/2L. Everything else stays the same, including equations for different damped cases and the other formulas. We then did a problem applying these concepts to a parallel RLC circuit.
LECTURE:
In this problem, the circuit involved an inductor in series with a voltage source and a capacitor parallel to the 2 ohm resistor. At DC conditions the inductor acts as a wire and the capacitor acts as an open switch. Then, using KVL the current was obtained in the above simplification of the circuit and the voltage across the 2 ohm resistor was calculated.
In this problem, the objective is to determine the second order differential equation governing the series RLC circuit. It was found that the differential equation can be dependent on charge or current. di/dt at time zero was found using the equation for voltage of an inductor. Then, using the value for di/dt at time zero and the second order DEQ the initial voltage was determined.
In this derivation the second order DEQ of the series RLC circuit with respect to current was determined. Then, a plausible solution for current involving an exponential function was plugged into the DEQ and simplified in terms of the solution.
The previous derivation was continued even further, and the solutions for the value s were obtained using a quadratic equation since the obtained formula for the plausible solution was a quadratic. Then, the solutions were written in terms of alpha (the neper frequency) and the angular frequency (omega). The neper frequency for the series RLC circuit is R/2L and the angular frequency is (LC)^(-1/2).
The objective of the problem seen above is to find the two constants "s" of the above RLC circuit and whether the circuit is underdamped, critically damped or overdamped. It was found that the circuit is overdamped since its neper frequency is greater than the angular frequency. Then, knowing that the solutions to the values of s is -alpha +/- (alpha^2-omega^2)^(1/2), the values were calculated.
In this problem, the equation for output voltage as a function of input voltage was obtained for the series RLC circuit. Then, the circuit was determined to be overdamped because the neper frequency is larger than the angular frequency. The damping ratio was then obtained, which is just alpha over omega and termed by the Greek letter zeta. As expected, zeta hould be higher than one since the circuit is overdamped. Finally, the two values of the constant s was calculated.
In this problem, a parallel RLC circuit was analyzed, and the damped status and voltage functions were determined, as seen above. The two values of the constant s were also determined. It was found to be overdamped, and the constants in the voltage function were found by using v(0) and dv(0)/dt. The rest of the problem is seen below.
LAB:
Series RLC Circuit Step Response:
Purpose:
The purpose of this experiment was to model and test a series RLC circuit. The step response was determined and tested by comparing the measured values of natural and neper frequencies to the expected. Another purpose of the experiment was to design a critically damped RLC circuit without changing the natural frequency. The step response was then remeasured and compared t theoretical values.
Prelab:
In this prelab, the second order differential equation of the series circuit was used to find the output voltage (voltage across the capacitor) in terms of the input voltage. Then, the natural frequency, damping ratio and damping frequency were determined, along with the type of damping the circuit is. Then, the two values of the constant s were determined.
Apparatus:
The equipment of the experiment consisted of the usual equipment, such as an analog discovery toolkit, a breadboard, a laptop with Waveforms, resistors, an inductor, a capacitor, a DMM, alligator clips and wires.
Procedure:
First, the circuit schematic seen in the prelab was built. A 1 ohm resistor, a 1 mH inductor and a 4.7 uF capacitor were used. Then, a 1 V square step input voltage with an offset of 1 V (alternates between 2 V and 0 V) at a frequency of 100 Hz was applied using the wave generator function. Channel 1 was used to measure the input voltage and Channel 2 was used to measure the voltage across the capacitor. The voltages were then measured using the oscilloscope window. The graph f the voltages is pasted in the data section.
Then, the natural frequencies, damping ratio and period were estimated from the oscilloscope window.
Next, the circuit was modified to obtain a critically damped circuit without changing the natural frequency of the circuit. This was done by changing the resistance and keeping the inductance and capacitance the same, since the angular frequency is dependent on inductance and capacitance. The new circuit consisted of a 29.5 ohm resistor and the same inductance (1 mH) and capacitance (4.7 uF), all in series, exactly like the underdamped circuit. Then, the same procedure was used for the underdamped circuit, as seen above. The oscilloscope window containing the voltage graphs is pasted in the data section.
Data:
The above picture of the oscilloscope window is the graphs of the input voltage and output voltage for the underdamped circuit. The step response is the quickly decaying oscillation seen at the changes in voltage. This step response is shown closer in the diagram below.
The calculated natural frequency and damping frequency are posted in the data analysis section. The above step response is expected because it indicates an underdamped system, which is the type of circuit used in this experiment.
The above picture is the oscilloscope window / step response for the critically damped circuit. Practically the circuit is slightly overdamped since it was difficult to accurately obtain the needed resistance of 29.2 ohms. This step response is also expected because it correlates for the most part with a critically damped system, which is what we have. The two voltages never meet, which is correlating to a critically damped circuit.
Data Analysis / Conclusion:
The above picture is the calculated data for the underdamped circuit from the step response of the oscilloscope window. The angular frequency was found to be 1.5 x 10^4 rad/s, which is very close to our theoretical value of 1.46 x 10^4 rad/s. In addition, the damped ratio was found to be 0.0533, which is slightly off from 0.0343. This is mostly due to the resistance actually being 1.6 ohms and not 1 ohm, changing the neper frequency significantly. Because of this and due to being the value being super small, any small change causes a large difference.
The neper frequency is found to be 800 rad/s from the data, which is also far off from the expected of 500 due to, again, the large uncertainty in the resistance. Therefore, the damped frequency was found to be 14978 rad/s, which is close to the theoretical value of 14578 rad/s. This is because the value is so large, so any change will not affect it much.
Lastly, the values of s were calculated from the data to be -800 +/- 14978i, where the theoretical vaues were -500 +/- 14578i, which the difference again is due to alpha being large because of the large uncertainty in the resistance.
No comments:
Post a Comment