Physics 226 Lab
Lab
12
Electromagnetic Resonance: Building A Radio
What You Need To Know:
The Physics Well, here you’re going to apply what you learned in the LCR circuit
lab. In fact we will go further and describe “forced” oscillation rather than just natural
oscillation at resonance or ringing. In the previous lab 11, what you did was
analogous to striking a bell and listening to the ringing.
Quick recap of LCR-circuit…
LCR circuit:
This is a perfect example of a damped driven harmonic oscillator which you will find
throughout your studies in physics in mechanics as well as optics and electronics.
Look at the voltage drops around the LCR circuit.
ε = IR + L
dq
dI q
divide this by L and using I =
we get …
+
dt C
dt
− Rt
d 2 q R dq
ε
q
where … q(t ) = qo e 2 L cos(ωt + φ )
+
+
=
2
L dt LC L
dt
with the resonance frequency being … ωo = 2π f o =
1
LC
From which we find the voltage across the capacitor as
tan φ =
X L − XC
R
2
 R 
2
 − ωo where ωo =
 2L 
and ω = 
1
LC
VC (t ) =
q (t )
where
C
is the natural or resonance
frequency of the oscillator . We can write the supply max voltage ε o = I o Z .
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Electromagnetic Resonance: Building A Radio
The reactance’s are defined as, X C =
Physics 226 Lab
1
and X L = ω L .
ωC
The impedance Z is a complex quantity, Z = R + i( X L − X C ) you can find the
magnitude and phase in the usual way.
 X − XC 
2
Z = R 2 + ( X L − X C ) and Z = Z eiφ where φ = tan −1  L

R


Also ω = 2πf where f is the frequency in Hz or per second. The frequency f =
where T is the period of the oscillation.
1
T
The impedance is a minimum when
X L = X C . Then the natural frequency is given by ω = ωo =
1
. When the two
LC
reactance’s cancel out the circuit is purely resistive, that’s when you get the natural
frequency or resonance condition.
So now the new stuff…
Forced Oscillation:
If we try to drive the circuit at a frequency other than the natural frequency, with an ac
supply say, then the circuit will respond only reluctantly so long as the driving freq is
“close to” the natural freq.
You may well ask how “close” will work. The frequency at which the amplitude of
the charge on the capacitor reaches its maximum value is called the resonance freq (or
natural freq). If you drive the circuit off resonance then the capacitor will never reach
full charge. We need to choose driving frequencies that allow at least ½ of the
maximum energy to be stored in the capacitor. In some circuits these two freq. may be
far removed from the resonance freq. This would imply that there is a broad band of
freq.’s that would excite the circuit. In other circuits you must have a freq very close
to the resonance freq to get any excitation at all. This range of frequency is related to
the quality factor of the circuit. See first diagram above. The sharpness of the
resonance curve, is measured by the quality Factor Q. This is defined by Q = f0/Δf
where f0 is the resonance frequency and Δf is the band width or width of the
resonance curve between the “half power” points.
Since the energy stored in a capacitor is given by U = q2/(2C) , the half power points
occur at q0/21/2 . (Maximum q is q0 ). The quality factor is a dimensionless quantity.
Large Q’s correspond to sharply peaked curves while small Q’s correspond to broad
curves. Note that if f0 is very large, then the Δf can be quite large and still be related
to a sharply peaked resonance curve.
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Electromagnetic Resonance: Building A Radio
Physics 226 Lab
The band width, Δf , can be shown to be related to the decay constant τ =
2L
by
R
Δf = 1/(πτ) = R/(2πL).
These expressions tell you that circuits which lose energy slowly (have large τ) will
be very sharply peaked and can only be excited at freq’s close to resonance. By
contrast, those circuits that lose energy rapidly, have small τ, have very broad peaks
and can be excited by a wide range of frequencies. For example, a bell made of brass,
has a very definite pitch, a very narrow band width and will ring for a long time so
has large τ. A wooden “chime” will not have a definite pitch, can be excited by a
wide range of frequencies and has a sound that will die out quickly, small τ. Below
are a few examples of resonance curves, all of which have the same resonance
frequency ω0 = 2π f0 = (LC)-1/2 , but different quality factors, Q. Note that L and C
determine the resonant freq. but R and L determine the band width, Δf = 1/(πτ) =
R/(2πL).
What You Need To Do:
Experiment 1
In this experiment you will investigate the properties of an LCR circuit when
subjected to a continuous sinusoidal applied voltage. In lab 11, you “whacked” the
LCR circuit with bursts of energy via mutual inductance, using a primary coil hooked
up to a square wave pulse generator. We watched the energy slosh back and forth
between the capacitor and the inductor at a rate known as the resonance or natural
frequency. Now we are going to drive the circuit continuously at a frequency which is
NOT the natural oscillation freq of the circuit.
Hook up the circuit as shown on the next page.
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Electromagnetic Resonance: Building A Radio
Physics 226 Lab
We are trying to minimize the resistance because that will damp out the oscillation.
A) Set the signal generator to sinusoidal output.
B) Using the dual trace mode of the oscilloscope, you make the necessary connection
so that you can simultaneously measure the voltage supplied by the signal generator
to the energy injector coil (primary coil of 400 turns) and the voltage across the
capacitor.
You are to determine the response of the circuit to signals of varying frequency. It
would be best to hold the voltage from the signal generator constant... or as nearly
constant as possible. The idea is that we want to isolate parameters and not change
freq and voltage at the same time.
(Do a quick check to see how the voltage changes with different frequencies from the
signal generator.)
Complete the table on the next page and graph your results of Voltage vs freq.
The first curve is for just the inductor resistance and a second curve on the same graph
paper for the inductor resistance plus added resistance of 100 Ω. You should get two
curves the first sharper than the second. It should look like the diagram on page 3.
You will be asked for the resonant frequency, the band width and the Q-factor, please
fill those in on the sheet provided.
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Name:_____________________________
Phys 226 Lab 12
Frequency (Hz)
Voltage (V)
R = RCoil
Voltage (V)
R = RCoil + 100 Ω
500
1,000
1,400
1,800
2,000
2,200
2,400
2,800
3,200
3,600
4,000
4,500
5,000
6,000
7,000
8,000
10,000
1) The resonance frequency, voltage output maximum across LC, f0 =____________
The band width, Δf , is the width of the curve between power at ½ max points.
2) Find the band width Δf = ______________
3) The Q-factor = f0 / Δf = ______________
4) Compare the two curves you plotted, the second at the higher resistance. What can
you say about the band width and the Q-factor for the second curve?
Electromagnetic Resonance: Building A Radio
Physics 226 Lab
Experiment 2: Building a simple Radio
Now you get to build a simple radio receiver. Once you have the receiver working
you will be able to tune into an AM broadcast we are transmitting. Our broadcast
contains super secret messages… if we told you what they were we’d have to kill you.
In the previous experiment you probably noticed that when the freq of the source
(signal generator) was close to the resonant freq the amplitude of the voltage
measured across the LC was increased. The frequency of the source can be made to
match the resonant freq of the circuit. The reverse is also true. We can tune a circuit
so that it matches the freq of the source… that’s what you’re doing every time you
tune your car radio to your favorite station!
It is possible to tune (change) the natural frequency of an LCR circuit by adjusting the
values of the component inductor and capacitor. Once the natural frequency of the
circuit is brought close to the frequency of the radio source, the circuit will begin to
oscillate in sympathy with the source and the voltage within the circuit will become
large. We just need to build a circuit whose natural frequency can be adjusted within
the radio freq band.
The radio band has a large range. We will be aiming at the medium frequency range,
AM signals. AM is in the middle of the MF range, FM is in the upper half of the
VHF range. Signals below 20 kHz are audio frequency or AF. (The audio band is 20
Hz to 20 KHz)
RF Spectrum Ranges:
Name
Abbrv.
Very low freq
Low freq
Medium freq
High freq
Very high freq
Ultra high freq
Super high freq
Extremely high freq
VLF
LF
MF
HF
VHF
UHF
SHF
EHF
Range
3 kHz - 30 kHz
30 kHz- 300 kHz
300 kHz - 3 MHz
3 MHz - 30 MHz
30 MHz- 300 MHz
300 MHz - 3 GHz
3 GHz - 30 GHz
30 GHz - 300 GHz
So let’s say you have your circuit how do we get the electromagnetic energy “in the
air” from the radio station into the circuit? Well, we need an antenna. Instead of
connecting the primary coil to a signal generator we connect one end to a ground and
the other to a wire antenna hooked up to the ceiling. We need the ground as a charge
source.
You can dump a lot of charge into the ground and pull a lot of charge from the ground
and the ground voltage always remains the same, zero. Very useful thing the ground!
12 - 6
Electromagnetic Resonance: Building A Radio
Physics 226 Lab
Radio waves are just time varying electromagnetic fields. When they hit the antenna,
they will cause the electrons in the wire to oscillate and this causes a time dependent
current to flow I(t). The changing magnetic field created by this current produces an
EMF within the secondary coil which is part of the resonant circuit. That’s it! You
have captured energy from the radio wave. Now to hear the signal… we have to get
the energy out?
The diode:
In the simplest terms a diode behaves in a circuit in a manner like a resistor, with the
exception that they conduct current easily only in one direction. The symbol used for
a diode is →|─ which is intended , via the arrow, to show which way the current will
flow from left to right in this case. We need to add on an earphone and a diode to the
circuit and then we’ll be able to get sound out. See diagram below;
This is roughly how to set it all up. The knob is attached to a variable capacitor. This
is how you will tune the circuit.
A typical amplitude modulated (AM) signal ( current) which might be excited within
your resonant circuit and the corresponding current that would flow through the
earphone/ diode would look like the figure on the next page.
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Electromagnetic Resonance: Building A Radio
Physics 226 Lab
The reason why this wave form is called amplitude modulated (AM) is probably clear
from the figure… the amplitude is constantly changing! The frequency of the wave,
called the carrier frequency, is a constant and falls in the range from 535 kHz to
1605 kHz for the commercial AM broadcasts.
The carrier frequency is well above the highest audible sound 20 kHz. Note that while
the frequency is constant the amplitude varies continuously at a frequency much
lower than that of the carrier. The amplitude changes at a frequency in the audible
range! That’s what we want, the modulation frequency not the carrier frequency.
Enough talk… assemble the circuit as shown.
A) Your radio LCR circuit is supplied to you mounted on an acrylic base. The
inductor is hand wound copper wire.
B) Connect the antennas coming from the ceiling to the rear terminal on the board.
Please clip the antenna to the terminal not to the wire. The copper wire is fragile and
not insulated.
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Electromagnetic Resonance: Building A Radio
Physics 226 Lab
C) Use the short wires from the other terminals to connect your board to the spring
board, connect the diode and earphone and make sure you run a ground wire to the
green terminal at your lab station.
Hidden close by is a secret radio transmitter which is broadcasting a secret message.
Your mission is to tap into the radio broadcast and take note of what it says.
You have been given an oscilloscope to assist you.
You can adjust the capacitance.. to tune the circuit, watch for a strong response on the
scope, then listen in on the earphone.
Under no circumstances are you to divulge to anyone, beside your lab instructor, what
you hear in these secret messages!!
In the picture above you can see the antenna from the ceiling and the ground, green
socket in the wall. You will have one at your station.
The Post Lab Activity is on the next page.
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Electromagnetic Resonance: Building A Radio
Physics 226 Lab
Post-Lab Activity
Plot the curve from the data
on this page
A)
Frequency (Hz)
10,000
15,000
20,000
24,000
26,000
28,000
29,000
30,000
31,000
32,000
32,400
32,800
33,130
33,400
33,800
34,200
35,000
36,000
37,000
38,000
40,000
45,000
50,000
55,000
Voltage
Voltage
Circuit #1 (V)
Circuit #2 (V)
0.038
0.038
0.065
0.065
0.106
0.109
0.165
0.174
0.212
0.234
0.280
0.338
0.325
0.427
0.377
0.572
0.432
0.848
0.478
1.560
0.491
2.280
0.498
3.760
0.500
5.000
0.499
4.080
0.492
2.480
0.482
1.690
0.451
1.020
0.405
0.682
0.359
0.514
0.320
0.415
0.258
0.301
0.172
0.184
0.130
0.135
0.106
0.108
For each of the 2 circuits shown, using the values of C and L given, predict the
following from the equations:
1) The resonant freq’s, f0 .
2) The time constants for decay, τ.
3) The band widths for each circuit, Δf.
4) The quality or Q-factor for each circuit.
Now plot the curves of Volts vs Freq and get all the same 4 results from the graph.
An answer sheet is provided for your results.
This may be kept as a post lab exercise, up to the instructor.
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Name:_____________________________
Physics 226L
Experiment#12
Electromagnetic Resonance: Building a Radio
Post-Lab Answer Sheet
Theory Results:
Circuit
#1 (R=20Ω)
#2 (R=2Ω)
fo (fresonance) (Hz)
τ (s)
∆f (Hz)
Q
fo (fresonance) (Hz)
τ (s)
∆f (Hz)
Q
Graph Results:
Circuit
#1 (R=20Ω)
#2 (R=2Ω)