Introduction 5
Background Information 5
RC circuit 6
RL Circuit. 6
Objectives 8
Experimental Procedure. 8
Experiment 1: The Capacitive Circuit (RC Circuit) 9
Step response. 9
Sinusoidal Response. 11
Experiment 2: The RLC circuit 12
Step response 14
Sinusoidal Response 16
Conclusion 18
Bibliography 19
This is an experiment up that shows the electrical response of different types of built circuits. Three different circuits consisting of different components which are capacitors, inductors and resistors are made. All the components are connected in series. The two types of circuits under investigation are listed below.
Resistor-capacitor circuit.
Resistor-capacitor-inductor circuit.
Background Information
The lab manual gives has an overview of three different combinations of circuits but only two of them are considered in this experiment. Each of the circuits should have either two or three components but one must possess all of them. Figure 1 below shows a diagram of the RCL circuit that contains all the three components.
Figure 1: RCL circuit connected in series
A detailed information concerning RC and RL circuits is discussed below (Gao, pg. 45)
RC circuit
Figure 2: RC circuit connected in series.
Using Kirchhoff’s current law,
Voltage overtime is given by,
RCdvcdt + Vc = V
Voltage across the capacitor Vc is given by,
Vc = V (1-e-tRC)
Current is given by,
i = VR [1-e-RLt
RL Circuit.
Figure 3 : RL circuit connected in series.
Similarly, using Kirchhoff’s Voltage law,
Ldidt+ iR = V
Making i the subject in the equation above yields:
i = VR [1-e-RLt
Time constant = L/R.
A plot of equation above is shown in figure 4 below
Figure 4: Step Response of a series L-R Circuit - Current build-up
(Alizadeh, Barzegari & Alizadeh, 2015)
Figure 5: Figure 5 L-C-R Circuit: Step Response –Dependence on ζ
Objectives
Experimental Procedure.
Two experiments involving a RC circuit and a RCL circuit are performed. The circuit’s response to a step voltage and a sinusoidal voltage are observed in the experiment.
The procedures, investigations and analysis of each experiment performed are outlined below.
Experiment 1: The Capacitive Circuit (RC Circuit)
Procedure
The circuit is set up as shown in figure 6 with a function generator connected to a capacitor and resistor in series. The function generator and the capacitor are both connected separately to an oscilloscope to show the difference in waveforms. Below is a diagram of the circuit set up.
Figure 6: Series R-C Circuit –Experimental Set-up
Step response.
The function generator is turned on and the value of the capacitor is set to 0.1µF and the resistor to 1KΩ. A 10 V peak to peak wave is applied at a frequency of 500Hz. The oscilloscope is adjusted to enable the viewing of the waveforms for the input and output voltages. Below is an image of the results obtained from the oscilloscope.
Figure 7: step response of the capacitive circuit.
The square waveform at the top of the screen in yellow lines the voltage output from the function generator and the one below in the blue line is the voltage across the capacitor.
The time constant to be obtained is visible experimentally by zooming in the oscilloscope. A time constant of value of 10E-4s was observed.
The theoretical value of the time constant is found using the formula T=RC.
Where, R and C are the values for the resistance and the capacitance respectively
Therefore, T=1000Ω x 0.1c x 10E-6
=1 x 10E-4s.
In the experiment, it was noticed the theoretical and the experimental values were similar. This can be attributed to the approximated result but in reality, it should be slightly higher than the theoretical.
T = L/R
:. 1 x 10E-4s = L/1000
L = 0.1H.
Changing the value of the resistance will increase the time constant and lowers the time of charging the capacitor.
Sinusoidal Response.
Using the same circuit in figure 6, the waveform from the function generator is switched to a sinusoidal. The same 10 V peak to peak sinusoidal voltage is used but with a reduced frequency of 100Hz. The readings of the voltage across the capacitor are taken from the oscilloscope. The experiment is repeated using different frequencies of 200, 500, 1000, 2000, 5000, and 10000Hz. A graph of the voltage across the capacitor against the frequencies is plotted.
Using the same circuit as in experiment 1, the circuit is altered a little by adding an inductor to it in series. The function generator and the capacitor are connected to the oscilloscope. The cicuit is shown in figure 9 below.
Figure 9: Series L-C-R Circuit- Experimental Set-up
The values of the components remained the same. The values of resistance = 1KΩ, capacitance = 0.1µF, and the inductor is set to 200mH. The function generator produces the same voltage of 10 V peak to peak with a square wave having a frequency of 200Hz. The snap shots of the waveforms were taken because the experiment with different frequencies, 200, 3000 and 6000 ohms followed. The following images shows the input and output voltages of the RCL circuit with different frequencies.
R = 1k Ω.
R = 200Ω
R = 3kΩ
R=6KΩ
Step response
The damped and undamped natural frequencies are calculated for each resistance. The formula below is used for calculating the damping ratio,ζ, and is given by,
ζ=RC2L
The damped ratio is for different resistances is also calculated
R = 1 kΩ
ζ=1030.1*10-620.2
ζ = 0.354
R = 200 Ω,
ζ=200*0.1*10-620.2
ζ= 0.071
R = 3 kΩ
ζ=3000*0.1*10-620.2
ζ =1.06
R = 6 kΩ
ζ=6000*0.1*10-620.2
ζ =2.12
An example of mechanical system that behaves and provides a similar response as a RCL circuit is a spring system in a car.
Sinusoidal Response
The same circuit set up is retained with the function generator providing a 10V peak to peak with a frequency of 100Hz. A resistance of 200 ohms was used different frequencies i.e. 200, 500, 1000, 2000, 5000, 7000, and 10000Hz. The readings of the voltage across the capacitor for the difference frequencies were taken. A graph of the voltage across the capacitor against the sinusoidal voltage frequencies was then plotted. The theoretical values of Vc/V were determined and compared to the experimental values obtained.
The experiment that involve the RL and RC circuits were done carefully and the results obtained were discussed above. From the discussion, it is clear that the values have a slight variation for each of the two experiment. The results has a slight variation or totally agree with theoretical values. This shows a high degree of experimental execution. The same can be improve through more trials on the same, using more improvised equipment and proper taking of readings. In summary, the experiment was done successfully within the expected limit.
Bibliography
Alizadeh, T., Barzegari, S., & Alizadeh, A. (2015). Reverse engineering of RLC circuits using Matlab: An experiment for electrical circuits course. International Journal of Electrical Engineering Education. http://dx.doi.org/10.1177/0020720915622472
Gao, Z. (2007). Modeling and simulation of the coupled mechanical electrical response of dielectric elastomers.
