Download Em05: Series-Resonant LCR Circuit

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’ve been 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
1
 0 . [1]
Capacitor, Z LC becomes equal to zero i.e. Z LC  L 
C
The current across the circuit is given by the equation;
Vˆ
Vˆ0
I 0 
2
Z LCR
1 

2
R   L 

C 

Current across the capacitor is given by the equation;
Vˆ
Vˆ
I  C  C  CVˆC
1
ZC
C
Nomenclature
I=Current
V̂0 =Generator output amplitude
Z LCR =impedance of Capacitor, Inductor
and Resistor in series.
R=Total circuit resistance
L=Inductance
C=Capacitance
Angular frequency
VˆC =amplitude of voltage across
capacitor
Therefore a graph of CVˆC 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;

(VC ) max
1) Q 
Vˆ0
(0 L)
R
where 0 is the angular resonant frequency
2) Q 
3) Q 
0
2  1

where 1 and 2 are points on the graph at which the value of CVC has fallen to
1 / 2 of its maximum value,
Experimental
1) Instrumentation
i) Signal generator
ii) Capacitor
iii) Inductor
iv) Resistor
v) Digital voltmeter x2 (DVM)
2) Diagram
DVM
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 VC varies with frequency by noting down values for VC and
the frequency.

6) Calculate values for C VC and 

7) Plot a graph of C VC 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
V̂0 by observing its variation as the resonant frequency was traversed.
Results

Graph 1: Graph to show variation of Angular Frequency, with C VC when there is
no resistor connected to the circuit.
No Resistor
0.000009
0.000008
0.000007
0.000006
CV
0.000005
…
..
0.000004
0.000003
0.000002
0.000001
0
30.00
40.00
50.00
60.00
70.00
80.00
90.00
Angular Frequency, 

Graph 2: Graph to show variation of Angular Frequency, with C VC when there is
a 200Ω resistor connected to the circuit.
Resistance 200
0.000009
0.000008
0.000007
0.000006
CV
0.000005
0.000004
0.000003
0.000002
0.000001
0
20.00
30.00
40.00
50.00
60.00
Angular Frequency, 
70.00
80.00
90.00

Graph 3: Graph to show variation of Angular Frequency, with C VC when there is
a 200Ω resistor connected to the circuit.
Resistance 500
0.000009
0.000008
0.000007
0.000006
 CV
0.000005
0.000004
0.000003
0.000002
0.000001
0
20.00
30.00
40.00
50.00
60.00
Angular Frequency
Errors
Error in angular frequency, 

Error in VC =+/-0.05

Error in C VC :
(CVˆC )
2
 0.05  (0.05) 2   /  0.07
CVˆ
C
70.00
80.00
90.00
Circuit Magnification factor, Q
For circuit with no resistor

(VC ) max
1) Q 
Vˆ0
56.50( /  0.05)
Q
 39.51( /  0.06)
1.43( /  0.03)
(0 L)
R
(65660( /  0.05) * 0.0872)
Q
 283.44( /  0.05)
20.2
2) Q 
3) Q 
0
2  1
65660( /  0.05)
=46.9(+/-0.09)
Q
66800( /  0.05)  65400( /  0.05)
For circuit with 200Ω resistor

(VC ) max
1) Q 
Vˆ0
24.00( /  0.05)
=16.78(+/-0.06)
Q
1.43( /  0.03)
2) Q 
Q
(0 L)
R
65660( /  0.05) * 0.0872)
=26.00(+/-0.05)
220.2
3) Q 
0
2  1
65660( /  0.05)
=17.28(+/-0.09)
Q
67800( /  0.05)  64000( /  0.05)
For circuit with 500Ω resistor

(VC ) max
1) Q 
Vˆ
0
Q
12.6( /  0.05)
=8.81(+/-0.09)
1.43( /  0.03)
(0 L)
R
65660( /  0.05) * 0.0872)
Q
=11.01(+/-0.05)
520.2
2) Q 
3) Q 
0
2  1
65660( /  0.05)
=10.10(+/-0.09)
Q
69000( /  0.05)  62500( /  0.05)
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 ( CVC ) decreases.
Sources of error were taken into account when considering the graph. In the end I

calculated an overall error for CVC 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
1
C
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
I believe that the resonant frequency does not change because at resonance L 
current decreases. I believe this is due to this relationship;
Vˆ
Vˆ0
I 0 
2
Z LCR
1 

2
R   L 

C 

as the resistance in the circuit increases the value for Z LCR 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) (12th edition), Addison-Wesley, ISBN 0-321-50131-4 (UL: 530 YOU)