Introduction
This experiment aims to examine the transient phase of a second order circuit. By definition, a second order circuit consists of circuit elements which are a resistor, an inductor and a capacitor connected either in series or parallel. The transient phase of the second order circuit used in this experiment was investigated by measuring the voltage across the capacitor and the resistor. The measured voltage will help determine whether the transient phase is; overdamped, critically damped or under-damped.
Equipment’s used in the Experiment
An oscilloscope, a function generator, resistors, an inductor and a capacitor.
Procedure
The voltage v(t) of the circuit 8.2 was observed using the method shown in the circuit 8.4 below.
Procedure 1
The DC source was replaced by the square wave generator that was used as a voltage source and was connected in series with the switch. The voltage v(t) was observed as the capacitor was charging up. The oscilloscope was connected and used in the signal analysis for R = 6.8 8kΩ. To achieve a better display of the waveform, the frequency of the square wave was adjusted after taking care of the DC offset.
The above procedure was repeated for resistor R=5608Ω, by observing the different waveforms for the signals from the oscilloscope display a determination was made on whether the signals were over, under or critically damped. For an under damped waveform the theoretical damping factor α and the damping frequency ωd were calculated from the equation shown below and the values compared to the experimental results. PSPICE simulation was conducted for all the three circuits described above, and a plot of v(t) vs. t was done.
Procedure 2
The procedure of Step 1 was repeated for the circuit below (circuit 8.5) thus the value of v(t) of circuit 8.3 was effectively measured. A completion of the PSPICE simulation for procedure two was done making the analysis of the theoretical, experimental and visual representations possible.
Procedure 2
Results and Observations:
The formulea used;
α= ωd2πlnV1V2 and ωd=2πTd
b2-4ac>0 representing an overdamped condition
b2-4ac<0 representing an under damped condition
b2-4ac=0 representing a critically damped condition
Formulas used:
1A: the circuit observed was as below;
Pspice simulation result showing critically damped for the 6.8k Ω resistor.
Oscilloscope showing critically damped for 6.8k Ω resistor.
1B: the observations of the circuit were as shown below;
Pspice simulation result showing under damped for the 560 Ω resistor.
Oscilloscope showing under damped for 560 Ω resistor.
2A: The observed results were as shown below:
Pspice simulation results showing under damped for the 6.8k Ω resistor.
Oscilloscope showing under damped for 6.8k Ω resistor.
2B: The observations of the circuit were as below:
Pspice showing critically damped for the 560 Ω resistor.
Oscilloscope showing critically damped for the 560 Ω resistor.
Conclusion:
As stated in the introduction of this report, the main aim was to gain better understanding of second order (RLC) circuits. An oscilloscope, a function generator, an RLC circuit were used in investigating whether a circuit is over, under or critically damped. By using specified inductors and capacitors on the RLC circuit, it was possible to determine the effects of different resistors on RLC circuits to become over, under or critically damped. The formula R2C2-4LC was used to calculate the value of the theoretical damping ratio. A positive result signifies an overdamped condition; a value of the damping ratio signifies an underdamped condition while a result of zero means that the system is critically damped. Validation of the calculated values was done by ELVIS and the results found to be correct. To reach a steady state of the output voltage v(t) on the oscilloscope, the frequency of the square wave was varied. A plot of the output voltage v(t) vs. t for each circuit was done using PSPICE and the results compared with the experimental values.
