Download Investigation of LCR Resonance - Hong Kong University of Science

Science Day Camp
The Hong Kong University of Science and Technology
Investigation of LCR Resonance
Objective:
The main purposes of this project is to let you
1. Familiarize with the working principles of capacitor and inductor
2. Construct workable inductors and capacitors using simple components
3. Learn to tune the resonant frequency of LCR circuit
Pre-task (Before the Science Day Camp):
You are first required to construct one capacitor and one inductor by using household
devices or elementary electrical components before the day of experiment and bring
them to the Science Day Camp. The capacitor and inductor constructed are
recommended to reach the values of at least 50nF (for capacitor) and 1mH (for
inductor).
Task 1 (During the Science Day Camp):
You are asked to set up an appropriate LCR circuit using the capacitor and inductor
made in the “pre-task” together with other electronic components. You can then tune
and find out the resonant frequency by varying the frequency of signal generator.
After achieving the resonant frequency, you can proceed to the next task.
Task 2 (During the Science Day Camp):
In this task, you are required to try varying the capacitance or inductance in the
circuit so as to achieve different resonant frequency. For example, if you want to half
the resonant frequency, how can you tune to the “new” resonant frequency by varying
the capacitance or inductance?
Suggested apparatus:
Signal generator
Several resistors of different resistance
Cathode-Ray Oscilloscope (CRO)
Connecting wires
Multi-meter
Coil and aluminum foil
1
Science Day Camp
The Hong Kong University of Science and Technology
And any other materials you may find useful
Pre-knowledge:
Figure 1. A series LCR circuit
The capacitive reactance is
XC 
1
[1]
wC
and the inductive reactance is
X L  wL[2]
where w = 2f (f is frequency of a.c. supply)
L = inductance in henrys
C = capacitance in farads
Thus, the impedance of a LCR circuit depends on the frequency of a.c. supply.
The impedance of the circuit is Z  R 2  ( wL 
Current of a LCR circuit is I rm s 
1 2
) [3]
wC
E rm s
1 2
R  ( wL 
)
wC
[4] . The current would be
2
wL 
largest only when
f 
1
wC
1
2 LC
[5]
Erm s
[6]
R
By ohm’s law, the potential difference across resistor would be proportional to the
current V=IrmsR [7]. Therefore, the resonance frequency could be obtained when there
is maximum potential difference across the resistor. Most normal multimeters cannot
measure alternating current at frequencies higher than 1 kHz.
At this frequency, the current only depends on resistance of circuit. I rm s 
Inductor
Solenoid inductance is given roughly by L 
N B
[7] . N B  (nl )( BA) [8], where
I
2
Science Day Camp
The Hong Kong University of Science and Technology
n is the number of turns per unit length, B is the magnetic field inside the solenoid and
A is the cross-sectional area. Besides, magnetic field B of solenoid is given by
B  nI [9], where  permeability, I is current. The relative permeability is   k 0
where k is relative permeability, 0 is permeability constant equal to 4 x 10-7 H/m.
Finally, the inductance could be expressed as
L = n2 lA [10]
Capacitor
 0 A
[11] , where  is dielectric
d
constant, 0 is permittivity constant equal to 8.85 x 10-12 F/m, A is overlapping area of
two plates and d is separation between two plates.
Parallel plates capacitors have capacitance value C 
Questions:
1.
2.
3.
4.
5.
6.
7.
8.
What factors affect the capacitance?
What factors affect the inductance?
How capacitance and inductance affect the resonance frequency?
Can you measure the root-mean-square current instead of the potential difference
across the resister? Why or why not?
What are your expected resonance frequency and experimental result? Are they
consistent?
To what extent does your measured resonance frequency differs from the
theoretical one?
How could you achieve different resonance frequencies?
What are the sources of error? How can you minimize the error?
3