Download resonance experiment

RESONANCE EXPERIMENT
ABSTRACT
We looked at a resonant circuit that was made up of an inductor a capacitor and some
resistors. We calculate its frequency and measured it. The phase was measured and
1
graphs plotted. Resonance is defined as
, which was shown to be true. The
2π LC
resonant frequency of the circuit was 73412.7 Hz. The phase shift varied between 1.5
and –1.5. The quality factor was 10.8 for R1 33 ohms and 2.3 for R1 330 ohms. The
circuit behaved as expected.
INTRODUCTION
1
Resonance is defined as
2π LC
. The voltage across the capacitor and the voltage
across the inductor are equal, so the circuit looks like a resistor. No energy is
dissipated at resonance. We looked at a typical resonant LCR circuit as shown in the
diagram below. The values of the capacitor, inductor and resistor were
4700Picofarads, 1MH and either 33ohms or 330ohms depending on which one was
connected. We calculated the resonance frequency using the previous equation then
measured the phase and the Q factor.
f =
1
2π LC
=
1
−3
2 π 1×10 × 4700× 10−12
= 73412.7
THEORY
In a resonant circuit the voltage in the inductor and the voltage in the capacitor are
equal. So:
VL = VC
⇒ IX L = IX C
1
ωC
1
⇒ ω2 =
LC
⇒ ωL =
We can therefore define the resonant frequency as:
fres =
1
2π LC
And the Q factor as:
ω0 L
1
=
R
ω0 CR
|I|
Q=
ω1 ω2
Figure 1: A resonant circuit
ω
EXPERIMENT
We were given the circuit
2
50Ω
I
330Ω
30Ω
S/C
3
Vs
1mH
~
4
4700pF
10Ω
1
5
V0
The resonant frequency was measured with the 33ohm resistor in place. We did this
by looking at the phase.
The phase was plotted.
The Q factor was measured for both resistors.
RESULTS
The resonant frequency was measured as 74503.54 Hz.
Hz
Voltage
20000 12.704
22500 12.085
25000 10.174
27500 7.4178
30000 4.7597
32500 3.2137
35000 3.4141
37500 5.3649
40000 8.5132
42500 12.111
45000 15.692
47500 19.419
50000 24.133
52500 31.040
55000 41.184
57500 54.910
60000 71.575
62500 89.624
65000 107.01
67500 121.78
70000 132.56
72500 138.83
75000 140.82
77500 139.15
80000 134.48
82500 127.26
85000 117.75
87500 106.22
90000 93.262
92500 79.917
95000 67.586
97500 57.683
100000 51.188
1.5 102
Voltage
1 102
50
0
0.00
20000.00
40000.00
60000.00
80000.00
100000.00
120000.00
Frequency
The phase was measured:
1.5
1
0.5
B
0
-0.5
-1
-1.5
-2
40000.00 50000.00 60000.00 70000.00 80000.00 90000.00 100000.00 110000.00
A
And again:
1.5
Phase difference (msec)
1
0.5
0
-0.5
-1
-1.5
-2
40000.00 50000.00 60000.00 70000.00 80000.00 90000.00 100000.00 110000.00
Frequency
And the Q-factor for two different resistors:
1.2 102
1 102
millivolts
80
60
40
20
0
-20
40000.00 50000.00 60000.00 70000.00 80000.00 90000.00 100000.00 110000.00
Frequency
Big resistor: circles. Smaller resistor squares (330 ohm and 33 ohm).
And again:
1.5 102
millivolts
1 102
50
0
-50
-1 102
40000.00 50000.00 60000.00 70000.00 80000.00 90000.00 100000.00 110000.00
Frequency
DISCUSSION
The plot of voltage against frequency has a maximum at the resonant frequency,
which we found to be 74kHz.
The plot of phase against frequency goes from negative to positive with the phase
difference at Fres being zero.
The resistor changed the Q factor. The bigger resistor gave the lowest Q factor.
The results were a bit noisy (see the last graph). I could have done better with more
accurate equipment.
CONCLUSIONS
We looked at a resonant circuit. It behaved how we expected from what we have
learned in the lectures.
REFERENCES
1. Horowitz and Hill, The Art of Electronics
2. Lecture notes
3. http://www.ee.ucl.ac.uk/teachlab.html