Introduction
This experiment is about the testing, simulation, and analysis of a second-order RLC circuit. The basic resistor-inductor-capacitor series circuit is analyzed for state variable response. The output is measured as the voltage of the capacitor. The experimental, simulated, and theoretical results are all compared and analyzed.
Purpose of the Experiment
The purpose of the experiment is to analyze a second-order RLC circuit for a state variable response.
Equipment and Materials Used
The equipment used in the experiment is listed below:
Function generator
Oscilloscope
Triple Output Power Supply
R = 100Ω, L = 25 mH, C = 0.1 uF
MULTISIM
MATLAB
The first three are used in the experimental setup; the last two are used in the simulation part of the analysis. The values of the resistor, inductor, and capacitor are listed as such.
Procedure
The procedure of the experiment is as follows:
1. Set up the circuit as shown below:
The input is taken from a function generator. The circuit is configured in series as resistor-inductor-capacitor. The output is taken from the capacitor.
2. Use the function generator to apply a square wave input of 10 Vpk-pk at a frequency of 100 Hz. The input will swing from -5 V up to 5 V and will have a period of 10 ms.
3. Measure the instantaneous voltage at the peaks of your response. Note down the value of the voltage and time readings. The voltage values are considered peaks if they are significantly greater than the corresponding peak of the square wave input. The voltage value is taken from the y-axis, and the corresponding time reading is taken from the x-axis.
4. Simulate the circuit on MULTISIM. Note that you should use PULSE Voltage as your source. Set TD, TF, TR, DC, AC, V1 to zero and V2 to 5V. Set PW to 5ms and PER to 10ms {MULTISIM constraints}. The PULSE Voltage input in MULTISIM is set in such a way that it will produce a square wave signal that would swing from 0 V to 5 V, no DC voltage component, negligible rise time and fall time, pulse width of 5 ms, and period of 10 ms.
5. The simulation for the output should resemble the figure shown below:
The red signal signifies the square wave input; the green signal signifies the capacitor voltage output. Notice that there are peaks that are significantly greater than the 5 V square wave peak.
6. Fill out the following table and extend the number of columns according to the number of peaks displayed on your oscilloscope. The voltage peaks and timings are captured accordingly from simulation using zooming options of the graph display. Only the positive peaks are captured because the negative counterparts are simply mirror images.
7. Solve for an expression for the state variable vC(t) and iL(t). Be sure to add all your handwritten work in the appendix of your lab report. The state variable equations are solved using differential equation methods.
8. Add a third row to the table and input theoretical values at those instantaneous points from your worked out solution from step 7. In this case, since the output signals contained 6 significant peaks, the table is designed in such a way that the columns are for simulated, experimental, and theoretical, while the rows are for the voltage and time readings for the peak voltages.
9. Plot the MATLAB response {code given} and comment on the damping effect as seen by your results and compare both your simulated responses with your hardware readings as well as your theoretical expectations. A background of the principles of damping is presented in the calculation and discussion section.
10. Why was a square wave input used instead of a switch?
11. Solve for the output graph on MATLAB.
The succeeding sections present the results of the experiment and simulation, then the theoretical calculations and results. The three sets of circuit values (experimental, simulated, and theoretical) are tabulated in a single data table.
Results
The results of the simulated and experimental results are shown in the following table:
An image of the MULTISIM simulation output is shown below:
An image of the MATLAB output is shown below:
Calculation and Discussion
Solving the series RLC circuit, the following equation is derived from KVL:
Rditdt+Ld2itdt2+itC=0
The solution is of the form: it=Aest
The damping factor is:
α=R2L=100225m=2000
The resonant frequency is:
ω0=1LC=125m0.1μ=20000rads
In this case, ∝<ω0, the system is under-damped. The form of the equation follows:
it=e-αtB1cosωdt+B2sinωdt
Where ωd=ω02-α2 is the damped natural frequency, which is equal to:
ωd=200002-20002=19900rads
The solution is:
iLt=0.0101e-2000tsin19900t
vCt=5+-5cos19900t-0.5025sin19900t
The updated table of results with theoretical values is:
It is just natural that the experimental voltage values are greater than the others because the voltage settings are greater. The setting is at 10 volts peak-to-peak, while the others are at 5 volts peak-to-peak. However, the time values are almost the same for all the peaks measured.
There are 6 significant voltage peaks observed in every setup. After the six peaks, the ripples are very small and can be considered at steady-state.
For the experimental setup, the square wave is used as an input so that the displayed output is assured to include the transient response and not just the steady-state response.
The theoretical computations determined that the system is under-damped. The under-damped response is verified both experimentally and in simulation. The output waveforms contained sinusoidal ripples that vanish in steady-state.
The simulated and theoretical values are almost identical in terms of both voltages and time readings. All the three sets of values are almost identical in terms of the time readings. It is interesting to note that although the experimental setup has a higher input voltage, the peak timings and the decay towards steady-state is the same as the two others.
