Series-Resonant LCR Circuit EM05
(1/11/09)
The object of this experiment was to plot resonance curves for LCR circuits with different resistances and calculate the circuit magnification factor for each different circuit. My results revealed that as the resistance increased from zero, the peaks on the graphs got lower and lower. I discovered that the steepness and height of the peaks was directly affected by the circuit magnification factor. I found that the higher the circuit magnification factor, the lower the resistance. Therefore the higher the circuit magnification factor, the higher and steeper the peak.
Introduction
In this experiment my aim was to plot resonance curves and determine the circuit magnification factor for various resistances. To do this I set up a series circuit containing a capacitor, an inductor and a resistor. These instruments were connected to a signal generator and the circuit was used to plot resonance curves at various resistances. I also aimed to find the resonant frequency, which should’vebeen the same at all resistances. I also compared the value of the resonant frequency to that of a calculated theoretical value in order to view the error involved in the calculation. I used the various values obtained from the circuit to plot the resonance curves. I then used different methods to calculate the circuit magnification factor of the circuit.
Theory
Resonance occurs in an LCR circuit when the impedance of the Inductor and Capacitor, becomes equal to zero i.e. . [1]
The current across the circuit is given by the equation;
Current across the capacitor is given by the equation;
Therefore a graph of against frequency, f would be equivalent to plotting a graph of current against frequency.
The magnification factor, Q, which determines the sharpness and height of the peak of a resonance curve, is determined by using each of the following methods;
1)
2)
where is the angular resonant frequency
3)
where and are points on the graph at which the value of has fallen to of its maximum value,
Experimental
1) Instrumentation
i) Signal generator
ii) Capacitor
iii) Inductor
iv) Resistor
v) Digital voltmeter x2 (DVM)
2) Diagram
Plan of Measurement
1)Set up the instrumentation as shown.
2)Set the resistance to 200Ω and set the output amplitude of the signal generator to a suitable voltage.
3)Carry out a preliminary experiment to discover the resonant frequency.
4)Remove the resistor.
5)Determine how varies with frequency by noting down values for and the frequency.
6)Calculate values for Cand
7)Plot a graph of C against
8)Repeat steps 5), 6) and 7) for resistances of 200Ω and 500Ω.
9)Determine values for Q using the shown methods.
Safety Procedures
1)It was made sure that the capacitor was discharged before touching the wires connecting it to the circuit.
General Lab safety procedures applied.
Sources of error
The errors in this experiment were both random and systematic.Reading errors occurred when reading the frequency values off the signal generator. Error values were also estimated when considering the number of significant figures stated on the digital voltmeter. An error value was also calculated for the generator output amplitude by observing its variation as the resonant frequency was traversed.
Results
Graph 1: Graph to show variation of Angular Frequency, with C when there is no resistor connected to the circuit.
Graph 3: Graph to show variation of Angular Frequency, with C when there is a 200Ω resistor connected to the circuit.
Errors
Error in angular frequency,
Error in =+/-0.05
Error in C:
Circuit Magnification factor, Q
For circuit with no resistor
1)
2)
3)
=46.9(+/-0.09)
For circuit with 200Ω resistor
1)
=16.78(+/-0.06)
2)
=26.00(+/-0.05)
3)
=17.28(+/-0.09)
For circuit with 500Ω resistor
1)
=8.81(+/-0.09)
2)
=11.01(+/-0.05)
3)
=10.10(+/-0.09)
Discussion
I believe on the whole this experiment was successful in revealing the variation of current with angular frequency in an LCR circuit at different resistances. As expected the resonant frequency was the same for each experiment. Therefore the resonant frequency is not related to the amount of resistance in the circuit. However the graphs show that as the resistance in the circuit is increased, the current () decreases. Sources of error were taken into account when considering the graph. In the end I calculated an overall error for of +/-0.09. I believe this error value could be decreased by using a signal generator and digital voltmeters with more accurate scales. The circuit magnification factor was calculated in a number of different ways with differing results. Despite the odd anomaly these calculations generally revealed that at lower resistances the circuit magnification factor is higher. Potential errors were also taken into account in these calculations and once again this value could have been improved by using more accurate scales.
Conclusion
I believe that the resonant frequency does not change because at resonance therefore as the inductance and capacitance never change, the resonant frequency does not change. The graphs also show that as the resistance in the circuit increases, the current decreases. I believe this is due to this relationship;
as the resistance in the circuit increases the value for also increases which directly affects the value for the current. The circuit magnification factor calculations revealed that the lower the resistance, the sharper and higher the peak of a resonance curve is.
References
[1] Young H.D. and Freedman R.A. (2007), University Physics (with Modern Physics) (12thedition), Addison-Wesley, ISBN 0-321-50131-4 (UL: 530 YOU)