Study of an Oscillating Electrical System Driven into Resonance
R. Thakrar
Laboratory Partner: F. Wahhab
Department of Physics and Astronomy
University College London
5th October 2009
Abstract:
Electrical resonance has been investigated on a series LCR circuit by measuring the change in amplitude and phase variation with frequency for different amounts of damping. The resonant frequencies seen in the experiment were found to be (28.5±0.06)kHz when R=500Ω, and (27.1±0.08)kHz when R=1500Ω. The theoretical natural angular frequency calculated was ω0=(0.18±0.01)x106 rads-1. The experimental values for the resonant angular frequency found were (0.178±0.03)x106rads-1, and (0.166±0.03)x106rads-1. These values were different to the expected value, and only one value was found to be in range of ω0. There is therefore reason to believe that there are sources of experimental error, such as extra “hidden” resistance in the circuit components, and systematic errors from the equipment. These are discussed further in the report.
Introduction[1]
The aim of this experiment was to study the response of an oscillating electrical series LCR circuit to a driving stimulus whose frequency can be varied and measured as a function of damping. The damping in this system can be measured by recording the changes in amplitude and phase variation of the voltage across components of the LCR circuit. This circuit is shown in Figure 1 below.
Figure 1: An LCR Series Circuit with Inductance (L), Capacitance (C), Resistance (R) and an Oscillator.
The oscillator drives the circuit in simple harmonic motion. When the oscillator drives the circuit close to its natural frequency, resonance occurs. At this point, the amplitude of oscillation dramatically increases. The amount it increases is solely dependent to the amount of resistance in the circuit. The more resistance present, the more a circuit is said to be “damped”.
The LCR circuit is driven by an alternating current (ac). The oscillator provides this current at amplitude, E0, and at a frequency, ω.
A theoretical value of the natural frequency at zero damping ω0, can be estimated by using equation 1 below:
Equation 1
If the potential is plotted as a function of ω, the maximum value in the curve may be found by differentiating equation 1 with respect to ω. This becomes:
Equation 2
The response of the circuit can be studied by varying the resistance of the circuit while measuring the frequency at which amplitude resonance occurs across a capacitor. This experiment determined the frequency at which amplitude resonance occurs across a capacitor. In this circuit, the voltage drop across the capacitor, Vc is measured by using an oscilloscope.
The values of resistance are predicted not to be fully accurate, due to a seemingly “hidden resistance”. In theoretical LCR circuits, the inductor stores magnetic energy, the capacitor stores electrical energy, and the resistor dissipate the energy that has been stored. In a real circuit, The oscillator provides energy to the circuit, and all of the components of the LCR circuit dissipate it. Real inductors and capacitors dissipate energy as they are made of resistive materials themselves, thus increasing the real value of R.
An alternative way to investigate the response on the LCR series circuit is by investigating the change in phase angle across capacitance relative to the circuit’s driving voltage. The oscilloscope can be set to a non-time based setting, where an ellipse is seen as in Figure 2 Below.
The X axis on the ellipse in figure 2 is the driving oscillation, while the Y axis is the driven oscillation. Values c and d represent the amplitude of Vc, whereas a and b represent the values where the ellipse crosses the Y axis (when E0=0).
The value of φ is determined by measuring distances ab and cd such that:
Equation 3
Figure 2: Diagram Showing the Combination of Two Time-Based Simple Harmonic Motions to Create an Ellipse.
Another part of this experiment involves measuring a “quality factor” (or Q-factor). A Q-factor describes how under-damped an oscillator is. The value will measure the sharpness of the Vc versus ω curves obtained in the investigation. The value of Q is such that:
Equation 4
Δω is the frequency difference between the two values of amplitude resonance at low and high damping.
As the value of Q is seen to depend on resistance, it is expected that Q will be greater in oscillations that are less damped. A higher value of Q indicates a lower rate of energy loss of the oscillator.[2]
Method
Experiment 1: Amplitude Resonance.
The circuit shown in Figure 1 is set up, with the values of capacitance, inductance and resistance being set as shown in Table 1.
Component / ValueInductance, L / ~15mH
Capacitance, C / ~2000pF
Resistance, R / ~500Ω, ~1500Ω
Table 1: Table Showing the value of components in the LCR Circuit.
A 12V peak-to-peak amplitude is set up on the oscilloscope around the capacitor, with a sinusoidal output. A theoretical calculation of the resonant frequency of free oscillations with zero damping is calculated using equation 1.
Next, the oscillator is used to measure the peak-to-peak amplitude across the capacitance as a function of oscillator frequency. The frequency is ranged over values above and below the theoretical resonant frequency. During the experiment, the central knob on the voltage scale is left in the “calibrated” position.
The experiment is repeated using the two different values of resistance shown in Table 1, for a comparison between low (~500Ω) and high (~1500 Ω) damping.
Experiment 2: Displaying Relative Phase Angle of the Voltage across the Capacitor.
This experiment is aimed at showing a qualitative study of a relative phase angle of the voltage from the oscillator compared to the voltage, Vc.
To set up this oscilloscope in this experiment, oscillator output, E0, is applied to channel I of the oscilloscope, and the voltage across the capacitance, Vc, is applied to channel II, using AC coupling on both. Automatic triggering is set on positive slope, and the DUAL button is turned on to set both signals on separate traces.
The oscillator is set to the resonant frequency that was verified in experiment 1, and the voltage ranges are set so that both channels fit on the oscilloscope screen.
The amplitude of Vc and E0 change as the frequency changes. This can be viewed on the oscilloscope and a qualitative conclusion can be obtained from viewing the relative amplitudes of the two as the frequency changes.
Experiment 3: Quantitative Study using Lissajous Figures
The relative phase, φ, of the voltage across the capacitor is measured by setting up the oscilloscope as described in Experiment 2. The time base of the oscilloscope is then turned off, so that the X input is given as E0, and the Y input is given as Vc. The oscillator output should look like an ellipse similar to that in Figure 2.
Values of ab and cd are recorded over a range of oscillator frequencies below and above that of the theoretical resonant frequency. The value of phi (φ ) is calculated using equation 3. Phi as a function of frequency may be plotted on a graph. Care must be taken to recognise the quadrant that phi is in. This is explained in Table 2 below.
Inclination of Ellipse
/Quadrant
/Value of φ
Right / First / 0 < | φ | < π/2Left / Second / π/2 <|φ| < π
Vertical or Horizontal / - / |φ|= π/2
Inclination of Line
/ /Value of φ
Right / - / |φ|= 0Left / - / |φ|= ±π
Table 2: The range of Values of φ due to the Orientation of the Lissajous Figure.
The experiment is repeated for low and high damping (at low and high resistances).
Results and Analysis
Graph 1 below shows the potential difference obtained from the circuit as a result of changing the frequency. These were the results obtained for experiment 1.
The graph shows a significant rise in voltage after 23kHz, with a peak at around (28.5±0.06)kHz when R=500Ω, and at (27.1±0.08)kHz when R=1500Ω. In the LCR circuit with R=1500Ω is seen to be heavily damped as the maximum amplitude is much lower than when R=500 Ω. In this way, the results agree with the theory as the degree of damping is dependent on R.
The theoretical value of the resonant frequency was calculated using equation 1 as ω0=(0.18±0.03)x105 rads-1. Using equation 2, we find that ω0≈ω, as values of R2/2L2 are very small when compared to ω02.
The aim of experiment 2 was to show the change in phase angle between the potential across the capacitor and across the oscillator. The amplitude of Vc, was seen to be much larger than that of E0 when the frequency approached values between 27kHz and 30kHz. This, notably, is similar to the values of the peaks seen in graph 1.
Graph two below shows the results of experiment three, where the relative phase angle was measured as the frequency was increased. The graph shows that at 28.5kHz the paths of the 2 lines cross. Theoretically, the two lines should cross at |φ|= π/2. But as this is not so, we can assume that there are sources of experimental error. The point where the line crosses also agrees with the values of amplitude resonance in graph 1.
The Q factor determined in this experiment was found to be Q500=(3.0±0.08)0x10-4, and Q1500=(3.6±0.09)x10-5. As Q500>Q1500, this agrees with theory is Q is dependent on 1/R2. The Q factor is expected to be more accurate for higher values of R, as the amount of “hidden resistance” becomes a smaller proportion of higher resistances.
The theoretical values Q1 and Q2 were found to be (3.2±0.04)x10-4 and Q2= (3.5±0.02)x10-5 respectively. These values correspond with the values of Q500 and Q1500 very well, showing that the data agrees qualitatively with theory.
Conclusion
In this experiment, three different qualities of an LCR series circuit were observed: the effect of resistance on the damping of the circuit, the effect of frequency on the amplitude, and the effect of changing frequency on relative phase angle.
It was verified that the dampening of amplitude was solely dependent on the amount of resistance present in the circuit.
The maximum amplitudes seen in graph 1 were seen between frequencies of 27kHz and 30kHz. This corresponded to the resonant angular frequency of the LCR series circuit, which was found to be ω1=(0.178±0.03)x106rads-1 and ω2=(0.166±0.03)x106rads-1.
These values were different to the expected value of ω, justifying that there were sources of experimental error in the experiment. These errors were both systematic and human, as reading oscilloscopes manually is less precise then a digital reader. At low resistance, some of the error can be accounted for with the “hidden resistance” in the circuit.
Though the results follow the trend that was expected, the accuracy of each result may be improved by selecting a narrower range of readings and taking more repeats at each frequency. The accuracy of the value of Δω could also be improved by using a computer application to work out the distance between the two peaks on graph 1, rather than manually.
If I was to further investigation the LCR series circuit, I would investigate the true values of the “hidden resistance” in the circuit.
References
[6]
[1] UCL Department of Physics and Astronomy Second Year Laboratory, “Study of an Oscillating Electrical System Driven into Resonance”, Experiment E7
[2] James H. Harlow (2004). “Electric power transformer engineering”. CRC Press. p.2–216. ISBN 9780849317040