Day 22 5/5/16: Series and Parallel RLC Circuits with Sources and Second order Op Amp Circuits

In class today we first learned a little about how wireless charging works; it really involves the induction of current in one coil by another. Inductive mode involved the transmitter working at a value slightly different than the resonant frequency, causing max power. It is achieved by having two inductors of equal diameter lined up closely together so that their separation is much smaller than the diameter. The other mode is resonance mode, in which the receiver works at resonant frequency. The seperation between these two coils is much larger.

We then reviewed step response of a series RLC circuit with a forced response (i.e. voltage source) connected. It turns out that the same exact concepts and formulas are used when just looking at a source free RLC circuit, with the addition of adding the forced response to the formulas. Next, we did a problem involving a voltage source in a series RLC circuit. We then reviewed step response of a parallel RLC circuit with a forced response (i.e current source). Again, it turns out that all the concepts and equations of a parallel source free RLC circuit hold true, with the exception that the forced response must be added to the equations. After, we did a problem involving a current source connected to a parallel RLC circuit. The two previously learned concepts that are helpful in the analysis of these circuits is that the capacitor resists voltage change and inductor resists current change. Therefore, at time zero the capacitor acts as a short circuit and the inductor as an open circuit. At time infinity (steady state) the capacitor is fully charged and acts like an open circuit; the inductor then acts like a short circuit.

After lecture, we did a lab titled "RLC Circuit Response". After the lab, we then learned about second order op amp circuits. As expected, nodal analysis is used in these circuits, and the op amps do not affect much the solution of the RLC circuit (i.e. they are solved very similarly to RLC circuits without op amps). The only time op amps are taken into account in these circuits is simply the initial values to find constants. In addition, the neper and natural frequencies of a second order op amp circuit are not the same as that of a series or parallel RLC circuit, and they are derived by using KCL and nodal analysis. The solutions to the differential equations though are the same as those for op-amp free RLC circuits.



LECTURE:

This problem involved a series RLC circuit with a voltage source. The objective was to find the current as a function of time in this circuit after the resistor and voltage source are short-circuited. This involved finding the neper and natural frequencies and from those values determining whether the circuit is underdamped, critically damped or overdamped. The rest of the problem is seen below.

The initial values for the circuit were then used to determine the constants of the current function.

This problem involved a current source connected to a parallel RLC circuit. Again, the objective was to find the current through the inductor as a function of time. This required to find the neper and natural frequencies and from there determining the damping of the circuit. The rest of the problem is found in the picture below.

The initial values for current across the inductor were then used to find the constants of the current function. 

LAB:

RLC Circuit Response:

Purpose:

The purpose of this experiment is to test the second order differential equation governing a series RLC circuit with a voltage source by finding theoretical values and comparing to measured values. The step response will also be tested and compared to the expected behavior, and data from the response will be compared to theoretical values.

Pre-lab:

 In the prelab of this experiment, the neper and natural frequencies of the second order RLC circuit being tested were obtained. Based on these values, it was found that the circuit is underdamped since the neper frequency is less than the natural. The damping ratio was then obtained, along with the values for the characteristic roots of the differential equation and the differential equation of this circuit itself.

Apparatus:

The equipment of this experiment consisted of the usual; an analog discovery, a breadboard, resistors, a capacitor, an inductor, a computer with Waveforms, wires, alligator clips and a DMM.

Procedure:

First, the circuit seen in the schematic in the prelab was constructed. In addition, the wavegen wire (yellow) was used to provide a square wave at 200 Hz with an amplitude of 1 V and an offset of 1 V. This allows providing to the circuit a voltage of 2 V while allowing it to enter its step response by turning off the applied voltage. Channel 1 was used to keep track of the input voltage in the oscilloscope window, and channel 2 was used to measure the voltage across the resistor in the RLC circuit (not the one used to impede the voltage source) and display it in the oscilloscope window.

Then, the square wave was applied to the circuit and the output voltage of the resistor was measured in the oscilloscope window. The window is shown in the data section. From the step response in the window, the experimental values of natural frequency, damping ratio, and characteristic roots were determined. This data is also shown in the data section of this blog.

Data:

As can be seen from the window above, which contains the input voltage and the voltage across the resistor as a function of time, the step response takes the form of an underdamped circuit, which is expected since the circuit was found to be underdamped in the prelab. N addition, based on the graph, the average voltage across the resistor was found to be about 40 mV when the 2 V is supplied, which is close to the expected value of 45.7 mV from using voltage divider. Using the step response, the natural frequency along with the damping ratio and other experimental values in this series RLC circuit were determined, and are shown below:

The percent error between the damping ratios was also calculated.

Data Analysis / Conclusion:

The experimental natural frequency from the obtained step response was found to be 1.28 x 10^4 rad/s, which is reasonably far from the theoretical value of 1.00 x 10^4 rad/s (a 28.0 % percent error). The discrepancy is most likely due to either the capacitance of the capacitor being off by the assumed value, or uncertainty in the inductance of the inductor. Because the inductance of the inductor cannot be measured and the resistance of the resistor was too small to be measured accurately, the same value of neper frequency was assumed, which in reality does skew our experimental values even further. The experimental damping ratio was then determined, which was found to be 0.0431, which is considerably off from the expected value of 0.0550. The percent error, as seen above, was calculated to be -21.7 %, which is a reasonably large value. Again, this is due to making many assumptions that are in reality not true; these assumptions were only made because the equipment needed to measure the real values was not available to us. These assumptions are the resistance of the resistor, the capacitance of the capacitor, the inductance of the inductor and therefore the same neper frequency. Because the neper frequency is dependent on the very small value of resistance used, any uncertainty in the resistance changes the true neper frequency drastically, and therefore changes the damping ratio even more.