Download Laboratory Exercise 5

Laboratory Exercise 5
Resonance
1. First, a reminder of some theory . . .
1 (a) Complex impedance
The impedance of a series RLC circuit is a complex quantity can be expressed
1
in the form: Z  R  jL  jC
It can also be expressed in polar form:
where
Z  Z e
j
Z is the magnitude given by: Z  R 2  L  1C 
2
(ohms)
The angle  is the phase angle between the voltage applied and the current
drawn by the circuit. It is given by:
1 

L 
C   arccos R 
  arctan 
Z 
 R



Imaginary
1
If the three components (R, jL and jC ) are plotted
on the complex plane they form a phasor diagram.
The diagram opposite is a phasor diagram of the
voltages in such a circuit. Note how the phase of the
total voltage, v(t), can be determined by adding
vectorially the other three phasors.
vC
v(t)
v
L

v
vx
Real
R
1 (b) When ZL and ZC cancel . . .
Author: Jon Pearce, 1998
Lab 5: Resonance
1 vary in exactly opposite ways as frequency is changed, there will be a
jC
frequency at which they “cancel” and the resultant impedance is simply R—a minimum. This
frequency is called the resonant frequency and must be given by:
Since jL and
jL +
1 = 0 or
jC
0 
1
LC
At this frequency, clearly the phase angle will be zero (think of the phasor diagram when the 2
imaginary components cancel out).
A typical plot of the way current varies with frequency for various resonant circuits is shown.
These plots are “normalised” to a maximum current of 1.0. Your plots not be normalised and
hence will look different. Can you predict how?
A plot of the changing phase angle is also shown.
i/i
max

2
1.00
Qo = 100
Q = 1
o
Qo = 10
Qo = 5
0.707
Q = 5
o
Qo = 1
0
0.9
1.0
1.1

o
Q = 10
o
0.00
0.8
0.9
1.0
1.1
Q = 100
o
 o
1.2
-

2
 o
f or Qo = 5
1 (c) A measure of quality
The quality factor, Q, is a measure of how “sharp” this resonance is. That is, how quickly the
current rises to its peak and falls again as you scan through frequency values. Its value, can be
1
obtained from the above graph by measuring the width of the resonance peak at 2 of its
maximum (“half power”). It is defined as:
Q
o

Its value can also be obtained from component values and can be shown to be:
Q
X L  XC   o L 1 L

or
 
R 
R 
R
R C
As you can see, the higher the Q, the sharper the peak; the higher the R the broader the peak.
Lab 5: Resonance
1 (d) Don’t forget the coil resistance
An important consideration when calculating the quality factor of a resonance is the intrinsic
resistance of the inductor. As was seen in the lab 3, real inductors always have a DC resistance,
which is the resistance of their windings, which is in series with the inductor. This resistance, RL,
this must be added to any other resistance in series with the inductor when calculating the
expected value for Q.
Author: Jon Pearce, 1998
Lab 5: Resonance
2. Investigating a series RLC resonant circuit
(1 1/2 hr)
In this section you will explore the resonance of your circuit and then measure the quality factor, Q.
2 (a) Find and measure the resonance
Preliminary data
(i)
Calculate the resonant frequency for the circuit shown, so that you know where about to
look for it when you construct the circuit. Measure the DC resistance of the inductor for use
later. Now you will take a quick look at the resonant current and voltage.
L
Sig
Gen
0.01 F
1 k
Check the resonant current
An easy way to view the current is to monitor the voltage across the resistor using the CRO (are
there any phase problems doing this? Explain.).
(ii)
Set up the RLC circuit, with the signal generator set to a low output voltage, and watch the
voltage (i.e. current) across the resistor as you sweep through a range of frequencies
around the resonant value. (Make sure your resistor is on the earth side of your circuit.
Why?) Record a value for the resonant frequency, o. No need to plot a curve.
Check voltage at resonance
(iii)
Rearrange the circuit so that you can monitor the voltage across the LC combination as you
sweep through the resonance region. How does this differ from the current? Check your
resonant frequency is consistent with the one above.
2 (b) Measuring the quality factor, Q
To measure the Q of your circuit, you need to obtain a plot of current vs frequency. To compare
the effect of different values of R on the value of Q, it is convenient to “normalise” all current
measurements so that the maximum current is always the same—say, 1 mA. So, for each circuit
below, begin by going to the resonant frequency, and setting the signal generator to give a current
in the circuit of 1 mA.
Lab 5: Resonance
measure current-frequency data
(i)
Construct your VI so that you can accumulate plots of current vs frequency on the one set
of axes. Set your maximum current to 1 mA and then obtain plots for about one decade
either side of the resonant frequency, using values or R of 500 , 1 k and 2.2 k. Keep an
eye on the voltage from the signal generator to ensure that it doesn’t drop in value as the
current gets larger.
Plot graphs of I vs  for 3 different values of R. You will vary the frequency of the signal
generator by hand (slowly).
Calculate Q
(ii)
For each plot, use the value of at 0.707 mA to calculate Q (0.707 because this is imax/√2).
Don’t forget to include all resistances in the circuit (coil, current meter resistance, etc.).
Compare these values with the expected values.
Questions
1. Is the resonance symmetric ? If not why not ? (Look at the equation for impedance; can you see
what must happen at very low frequencies compared with what must happen at very high
frequencies?)
2. From your resonant current value, determine the maximum voltage across the capacitor and
inductor, respectively, at resonance (use Ohm’s law: v = iZ). Express the result in terms of Q
(that is, by what factor, compared with Q, is the voltage magnified each time?). What are
consequences of this relationship for the maximum voltages tolerated by capacitors in series
resonant circuits?
Author: Jon Pearce, 1998
Lab 5: Resonance
3. Using an RLC circuit to tune-in radio
signals (1 hr)
3 (a) About AM radio
An AM (amplitude modulation) radio signal is a superposition of a high frequency wave (100’s
to 1000’s of kHz) called a carrier wave (this is the “RF” signal) and a signal in the audio
frequency range (only about 100 Hz to 7.5 kHz for the AM band in Australia). The carrier
frequency range in the standard commercial AM radio band is between about 550 kHz and
1600 kHz.
In order to “tune” to a station we need to filter out all unwanted carrier frequencies either side of
the desired station. This is easily done with a parallel resonant circuit—the resonant peak being
narow enough to tune only one station, yet wide enough to encompass the range of signal
frequencies (carrier ± signal).
For high fidelity sound it is necessary to have a bandwidth of 20 kHz. However, to pack more
radio stations into the limited AM radio frequency range, the maximum modulating frequency is
limited to 7.5 kHz. Incidentally, FM radio, being transmitted at around 100 MHz can have a
higher modulating frequency range, up to 20 kHz, which is why FM radio is higher fidelity than
AM.
A very simple AM radio receiver circuit is shown in the diagram. It contains a
parallel LC circuit with a variable capacitance to tune the resonant frequency to
that of the desired carrier wave frequency (most radios use variable
capacitance tuning but the inductor could be used for tuning instead). To a first
order approximation, the resonant frequency and quality factor of a parallel
RLC circuit are given by
0 
1
and Q 
Antenna
oL
, where the R is the resistance in series with the
R
LC
inductor. These formulae are the same formulae as used for series RLC circuits.
In order to extract the modulating signal from
the carrier signal, the signal is first passed
through a germanium diode, which passes only
the positive half-wave. This is called
rectification and will be investigated in Lab 9.
The signal is then passed through a low-pass
filter to smooth out the ripple from the carrier
signal. That is, just the amplitude of the signal that
you wish to listen to is remaining. This
demodulating of the signal cuts out the carrier
wave leaving only the audio signal to be
amplified.
Tuning
Why a germanium diode?
As will be seen in Lab 9, diodes pass current
only in the forward direction (Anode to
Cathode), provide the voltage is greater than
the turn on voltage. The RF signal will be very
weak so a germanium diode (Vturn on = 0.2 V)
will be used rather than a silicon diode (Vturn on
= 0.6 V). To improve the quality of the radio
an RF amplifier is usually used between the
filter and the diode.
A strong signal is needed in order to overcome the germanium diode‘s turn-on voltage. For this
reason an antenna of a few metres length is used.
Lab 5: Resonance
3 (b) Constructing a tuner
In this section you will construct an unusual radio tuner—one that is tuned using a variable
capacitor (If you possess an old great-grandfather, ask him how he used to make a radio using
only a crystal, a cat’s whisker and a headset!). Note that some lab locations give a poor
reception—you might need to move around to get a strong signal!
Calculating components
Choose an AM station that you want to design your radio to tune.
Local Radio Frequencies
(i)
Use a meter to measure the range of capacitance of your
rubbery ruler. Choose a middle value of capacitance and
calculate the value of inductance that you require to tune to
your station. Stray capacitances in your circuit might
amount to about 100pF.
(ii)
Your demonstrator will give you an inductor close to that
value (check its value using the RLC meter—there are two of
these meters in the lab). Measure its DC resistance and
calculate the bandwidth of your LC circuit.
(Bandwidth:   
o
Q

o
oL R

R
)
L
Will this give the desired tuning range of carrier ± signal?
3RN
Magic
3LO
3CR
3UZ
News Radio
3BM
3SBS
3AW
3MP
3XY
3AK
3RG
621 kHz
693 kHz
774 kHz
855 kHz
927 kHz
1026 kHz
1116 kHz
1224 kHz
1278 kHz
1377 kHz
1422 kHz
1503 kHz
1593 kHz
Constructing the radio
(iii)
Construct a tuned circuit using the inductor (calculated above), your
rubbery ruler as a tuning capacitor and an antenna several metres long
(draped below the ceiling of the lab). Use the most sensitive setting on
the CRO and observe the feint RF signal as you “tune” your receiver—
you are looking for small modulations as you tune near to the radio
station’s frequency. A time-base of about 1 sec per cm might be a good
scale to start with.
(iv)
With your circuit “tuned” to a station, add a diode and resistor to rectify
Tuning
the signal (why do you need the resistor as well as the diode?) and a
capacitor to filter out the RF frequency. The signal will be weaker due to
the current now being drawn through the resistor. You can expect only 1 or 2 mV with
features about 0.2 msec apart.
Antenna
Antenna
Ge diode
use co-axial
cable here to
connect the ruler
to the circuit
47 M 
Tuning
Author: Jon Pearce, 1998
Rectification
10 pF
Filtering
Lab 5: Resonance
(v)
Finally, connect your demodulated signal to an earphone and listen to the beautiful music!
You can’t do this directly as your weak signal cannot provide the power, so you will need
to use your amplifier module. Connect it first to the unity gain amplifier (gain = 1; very high
input impedance; low output impedance so that it won’t load down your signal) and then
to the gain 1 -> 100 amplifier (to increase the voltage) and finally to the audio amplifier (to
boost the current to power the speaker).
Radio
Receiver
co-axial
x1
x 50
+
Unity Gain Amplifier
Audio Amplifier
Which station is that?
A neat way to measure the frequency of your station is to generate beats using your signal
generator as a second RF source. When the frequency of the signal generator is adjusted to stop
the beating, you have found the RF frequency.
(vi)
Attach another wire to the antenna connection of your radio and lie it against your signal
generator (it will pick up the signal generator signal this way). Set the signal generator to a
few volts sine output.
(vii)
With your radio station tuned, slowly sweep the signal generator frequency across the AM
frequency range (600 kHz to 1600 kHz) and listen for a sharp whistle in the headset—then
you have found the station’s frequency!
(viii) Now measure directly the capacitance that your rubbery ruler is stretched to and do a
calculation to compare your calculated frequency with the one you are actually tuned to.
Assume any discrepancy is due to additional capacitance, calculate the magnitude of this
capacitance and think hard as to its physical origin.
Lab 5: Resonance
Electronics & Instrumentation
A-5-1
The Loudspeaker: An Electromechanical
Transducer
Author: Jon Pearce, 1998
Appendix
A-5-2
Electronics & Instrumentation
The Loudspeaker: An Electromechanical
Transducer
1. Introduction and Aims
The loudspeaker is an example of a device which is both electrical and mechanical, and the electric and
magnetic systems are strongly coupled. Other examples are piezoelectric, electrostatic and
magnetostrictive transducers. The general ideas of this laboratory study also apply to them.
Because the electric and magnetic systems of the loudspeaker are so strongly coupled, the electrical
behaviour of the loudspeaker depends strongly on the mechanical / acoustical environment of the
loudspeaker, and the mechanical behaviour depends strongly on the electrical loading of the speaker.
The aim of this investigation is to study this interconnection, which is characterised by the elements of a
2x2 matrix. In abstract, the device may be regarded as a 2-port in which one port is electrical and the
other mechanical . (See the diagram below.)
2. Theory (Short Version)
The extended theory of the loudspeaker is given in Appendix 2. Appendix 1 gives the AC conventions
used below.
The voltage across the speaker is given by
V = ZeI + GU
(1a)
As usual Ze = R + IXe is the electrical impedance of
S
the speaker, R is the electrical resistance, and Xe is
N
the reactance which is essentially inductive ( 2fLe).
U
Ze is the electrical impedance you measure when
the velocity of the speaker U = 0.
The term proportional to U arises from the fact that
when the speaker assembly moves, the magnetic
field in the gap induces an EMF in the speaker coil,
proportional to the velocity, U. The proportionality
factor, G, is the electromechanical coupling constant.
Appendix
F
S
I
I
V
0
V = voltage applied to speaker
I = current in the speaker coil
F = force applied to the speaker cone assembly
U = velocity of the speaker cone assembly
(assumed rigid at low frequencies)
A-5-3
Electronics & Instrumentation
S
The external mechanical force, F, is given by
F = Zm U - G I
(1b)
I
S
The term GI is the (internally supplied) force due to the current I
flowing in the magnetic field B. (A model shows that this is the
same G as before, a fact required by energy conservation.)
These two equations in the matrix form of a two-port read:
V   Z e G I 
   
 
F  G Zm U 
B
N
S
S
Radial magnetic field
in the gap
(2)
where the two-port is represented by the diagram below:
I
V
F
0
U
2 - Port
The overall electrical impedance of the system, Z, when it is loaded with an extra mechanical impedance
Zm is given by equation B7 as
3a
Z  V / I  Z e  G 2 /(Z m  zm )
 (R  jL) 
 R  j2 fL 
2
G
jMo  D  K / j
G2
2fr Mo j( f / fr  fr / f )  1/ Q
3b
3c
where the resonant frequency r = √(K/Mo) = 2fr
4a
and 1/Q = D/Mor.
(Q is mechanical oscillator quality factor.)
4b
The resonance width is (fr/Q) in frequency
4c
Author: Jon Pearce, 1998
Appendix
A-5-4
Electronics & Instrumentation
3. Qualitative Investigations
The speaker as a microphone
First convince yourself that the speaker will behave as a microphone by attaching it to the CRO
and speaking into it (the speaker, not the CRO, that is!).
Finding resonances
There are three ways to mount the speaker:
(a)
dangling in the open air,
(b)
mounted in the box, with speaker front uncovered, and
(c)
mounted in the box, with speaker front covered with a rigid plate.
For each of these configurations the stiffness constant is very different.
(a)
Open air: K = Ko
(b)
Mounted, open front: K = Ko + Ka
The calculated value of Ka for your speaker and box is given on the data sheet.
(c)
Mounted, closed front: K >> Ko
The air between the speaker and plate is so hard to compress that the speaker can
hardly move at all at low frequency. Thus U ≈ 0, and equation (1a) yields V = ZeI.
Equations 3 show that a mechanical resonance will appear as an electrical resonance in Z.
By scanning slowly through the frequency range, say 0 - 1.7 kHz, find and record the
approximate resonant frequencies for cases a, b and c above.
4. Quantitative investigations
Investigation of the mass of the speaker
We can determine the mass of the speaker by measuring the various resonance frequencies.
Resonance frequency of open speaker, fo
Resonance frequency of speaker in box (open front), fb
The rough relationship is:
(2fo)2M = Ko
(2fb)2M = Ko + Ka
where Ka is the extra stiffness due to the air entrapped in the box (see data sheet).
Appendix
A-5-5
Electronics & Instrumentation
Hence you can find M and Ko from your measured frequencies.
This is only rough due to the air loading on the speaker. Compare M with the given value in the
data sheet.
Data Sheet
Diameter of speaker:
d = 5.50 cm
Area of speaker:
A = (d/2)2 = 23.8 cm2
Volume of box (inside):
V = 10.85 x 8.2 x 5.3 = 471.5 cm3
Adiabatic bulk modulus of air:
(P is the air pressure)
P = 1.421 x 105 Pa
Stiffness constant of air in sealed box:
Ka = P A2/V = 1.71 x 103 Nm-1
Mass of the moving assembly:
M = 1.030 gm
Length of wire:
S = 2.56 m
Diameter of coil:
14.0 mm
References
Fraden J, AIP Handbook of Modern Sensors: Physics, Designs, Applications, American Institute of
Physics, 1993.
Lion K. S Elements of Electrical and Electronic Instrumentation. 1st edit, McGraw-Hill, 1975.
Horowitz P and Hill, The Art of Electronics, 2nd edit, Cambridge UP, 1989.
Neubert H. K. P Instrument Transducers, Oxford UP, 1963.
Diefenderfer A. J. and Holton B. E, Principles of Electronic Instrumentation, 3rd edit, Saunders, 1994.
Storey N., Electronics: A Systems Approach, 1st edit, Addison-Wesley, 1992.
Author: Jon Pearce, 1998
Appendix