Junior Honours - Experimental Physics
Electromechanical Resonance
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Objectives
• Measure the complex impedance Z of a loudspeaker as a function of frequency;
observe resonance(s)
• Determine values for the loudspeaker parameters Re , Le , T, Rm , Lm and Cm
2
Introduction
If a loudspeaker coil is prevented from moving in the permanent magnetic field of the
loudspeaker structure, no electric current can be induced in it when it is powered by
an ac audio voltage signal – the current is produced only by the voltage signal. For
audio frequencies, the electrical capacitance of the coil can be ignored, and the electrical
impedance Z e of the loudspeaker coil has resistive and inductive parts only:
Z e = Re + jωLe = Re + jXe
where Re is the electrical resistance, Le the electrical inductance, ω the angular frequency (=2πf ) of the voltage signal (and of the resulting current) and Xe = ωLe is the
electrical reactance. The complex impedance
Z e = |Z e |ejα = |Z e | cos α + j|Z e | sin α
from which |Z e | cos α = Re and |Z e | sin α = ωLe Suppose that the voltage signal is
written in a compatible way:
V = |V |ejωt = V◦ ejωt
Since V = IZ e ,
I=
V
V◦ j(ωt−α)
V◦ ejωt
e
= I◦ ej(ωt−α)
=
=
jα
Ze
|Z e |e
|Z e |
Thus |Z e | can be measured from values of data pairs V◦ , I◦ for any chosen angular
frequency ω. A separate measurement of the phase difference α then allows a complete
specification of Z e .
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Loudspeaker Impedance
At its simplest, a loudspeaker consists of a conical diaphragm with a circular coil of wire
mounted at its (relatively free) narrow end concentrically. The rim of its outer end is
fixed to a rigid circular frame and the diaphragm between frame and coil is sufficiently
flexible to deform slightly, periodically, when the coil moves axially, periodically. The
periodic movement occurs because the coil is a current-carrying coil in a radial magnetic
field B produced by a permanent magnet of suitable shape. Because the coil is moving in
a magnetic field, “cutting” the magnetic field lines, an induced emf ε (sometimes called
a back emf) is set up. Once any transients have subsided, the net voltage V − ε and the
corresponding current I are related by V − ε = IZ e , or, alternatively, V = IZ e + ε = IZ.
The total impedance Z has an electrical component Z e and a component which must be
related to the mechanical properties of the loudspeaker. The same V as before produces a
different I from before because the total impedance is now different from Z e .
Just as for Z e , the frequency-dependent values of Z can be found from measurements
of V and the “new” I. In general, the frequency dependence of Z will be different from
the frequency dependence of Z e . In particular, because the motion of the loudspeaker coil
is the oscillatory motion of a driven, damped harmonic oscillator (simple harmonic for
small amplitudes) a resonant behaviour can be anticipated. This can be traced back to
the properties of ε.
It was assumed without comment, earlier, that each loop or turn of the coil experienced
the same constant radial magnetic field B, produced by the permanent magnet. Thus for
a current I in the coil the total force experienced is, in magnitude,
Z
F = n| Idl × B| = nIB × 2πr = IBl
where l is the total length of the wire coil, each loop having radius r and length 2πr, to a
reasonable approximation. Each loop moving axially through the radial B-field contributes
the same amount to the total induced emf ε. Let the coil move a small distance dx in the
short time interval dt. Each loop then cuts through 2πr × dx × B magnetic field lines in
dt. The magnitude of the induced emf is then ε = n × 2πr × dx × B/dt = Blv, where v is
the velocity magnitude and v has the same periodicity as that of the applied voltage V .
Introduce now the mechanical impedance Z m , defined (as is customary) by v = F /Z m .
Thus the larger Z m , the smaller is the velocity v produced by a given force F . Then
ε = Blv = Bl
F
BlI
(Bl)2
T2
= Bl
=
I=
I
Zm
Zm
Zm
Zm
where T is called the “transduction coefficient”. It follows from V = IZ e + ε that
T2
T2
V = IZ e +
I = Ze +
I = ZI
Zm
Zm
The justification of the definition of Z m is as follows. Suppose that the motion of the
loudspeaker coil plus cone can be modelled as the driven damped simple harmonic motion
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of an object of mass m subject to a restoring force proportional to its displacement x and
a dissipative force proportional to its velocity dx/dt. Introducing a driving force F , the
equation of motion is
d2 x
dx
=m 2
dt
dt
where k and b are constants. To keep the notation as simple as possible, show time
dependence explicitly by writing F ejωt in place of F and xejωt in place of x. Then v = jωx,
and x = v/(jω), and acceleration = jωv, leading to:
F − kx − b
F −k
v
− bv = mjωv
jω
or
F =
k
k
+ b + jωm v = b + j ωm −
v
jω
ω
Traditionally this is written
F = Rm + j ωLm −
1
ωCm
v = Z mv
mimicking the electrical notation. F /v is now the mechanical impedance Z m ; and Rm ,
Lm and Cm are the mechanical resistance, inductance and capacitance
It follows that
Z = Ze +
T2
Zm
T2
= Re + jωLe +
1
ωCm
− ωLm
Rm + j ωLm −
= Re + jωLe +
Rm − j
T2
1
ωCm
from which
T2
1
Rm + j
Z = Re + jωLe +
− ωLm
|Z m |2
ωCm
T2
T2
1
− ωLm
= Re +
Rm + j ωLe +
|Z m |2
|Z m |2 ωCm
= R + jX
Because |Z m |2 depends on ω, both the equivalent total resistance R and the equivalent
total reactance X are frequency dependent. As will be shown next, the shape of the locus
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of Z plotted in the complex plane can be used to determine the values of Re , Le , T, Rm , Lm
and Cm , and hence a complete specification of Z. Two sets of experimental data are
needed, one for the loudspeaker on its own and the second for the loudspeaker with a small
mass m′ attached to the coil, so that Lm is replaced by Lm + m′ .
Im
ωL e
ω
Ze
α
ω=0
Re
Re
Figure 1: Complex impedance Ze and its components in the complex plane
Figure 1 shows Z e and its components in the complex plane. The locus of Z e is a
straight line, starting on the real axis at the point (Re , j0). The mechanical impedance is
best represented by the term T 2 /Z m , which can also be plotted on the complex plane.
From the earlier analysis
T2
T2
=
Zm
Rm − j ωC1m − ωLm
T2
1
=
Rm + j
− ωLm
|Z m |2
ωCm


1
−
ωL
2
m
ωCm
T  Rm

+j
=
|Z m | |Z m |
|Z m |
=
T2
(cos φ + j sin φ)
|Zm |
Here, cos φ = Rm /|Z m |. From this, the frequency-dependent |Z m | in the first factor
can be replaced by Rm / cos φ, giving
T2
T2
=
cos2 φ + j cos φ sin φ
|Z m |
Rm
sin 2φ
T 2 1 + cos 2φ
+j
=
Rm
2
2
2
T
(1 + cos 2φ + j sin 2φ)
=
2Rm
4
Im
ω
T2
Zm
ω=0
φ
T2
2Rm
2φ
T 2sin2φ
2Rm
Re
T 2cos2φ
2Rm
T2
Rm
Figure 2: Locus of T 2 /Z m in the complex plane
The locus of T 2 /Z m (figure 2) is a circle of radius T 2 /2Rm whose centre is a constant
real distance from the origin. As ω increases from zero, the circle is described clockwise
starting from (0, j0). The initial value (at ω = 0) of 2φ is thus 180◦ = π, and with
increasing ω, 2φ decreases, becoming zero at Z m = (T 2 /Rm + j0), for which ω = ωr , the
resonant frequency. For ω > ωr , 2φ is negative.
The locus of Z = Z e + (T 2 /Z m ) is shown in figure 3. This is the curve which can be
obtained from plotted values of |Z| and θ, obtained from measurements of voltage V across
and current I through a loudspeaker coil.
Once a curve has been fitted to the experimental points, a start can be made on
evaluating the above set of parameters. Re is the limiting value of the real part of Z as
ω → 0 (and as ω → ∞). ωLe is the limiting value of the imaginary part of Z as ω → ∞,
2
ie ω ≫ ωr . At ω = ωr , the real part of Z has its maximum value Re + RTm ; hence the value
2
of RTm if Re is known. To evaluate Rm , further measurements are needed which yield the
values of Lm and Cm , see later. Assuming Lm and Cm are known, exploit the trigonometry
2
of the loci of Z and ZT . Let ω have the values ω1 and ω2 at 2φ = 90◦ and −90◦ (ie
m
φ = 45◦ and −45◦ ). Then Rm = ( ω11C1 − ω1 Lm ) and Rm = −( ω21Cm − ω2 Lm ). Without
further measurements, the product Lm Cm can be evaluated from the value of the resonant
frequency ωr for which (since 2φ = 0◦ ) ωr2 Lm Cm = 1. A convenient way to evaluate Lm and
Cm separately is to attach an object of small mass m′ to the loudspeaker coil, whereupon
Lm has to be replaced by Lm + lm where lm = m′ . The resonant frequency will decrease
to ωt , say, where ωt2 (Lm + lm )Cm = 1, which together with ωr2 Lm Cm = 1 allows Lm and
Cm to be evaluated. As described above, these may be used to evaluate Rm . Finally, from
2
the now known values of Rm , Re and Re + RTm , the value of T can be deduced. Once the
values of the six parameters are known, the real and imaginary components of Z can be
5
Im
ω
φ
Ze
α
T2
Zm
Z
ω=ω r
θ
Re
ω=0 T 2
Rm
Re
Figure 3: Locus of Z = Z e + (T 2 /Z m ) in the complex plane
predicted retrospectively — mathematical modelling! — as functions of ω. How well the
subsequently predicted locus of Z matches the real locus (determined by the observed
values of Z – essentially |Z| and θ, where, note well, θ 6= φ) should be of interest for several
reasons.
4
Experiment
I
R
DAQ channel #0
~
DAQ channel #1
Figure 4: Schematic of measurement of Z
The PC is equipped with a National Instruments NI PCI-6023E multifunction data acquisition (DAQ) card [2]. This can be used to measure the voltages across the loudspeaker
and a series resistor (figure 4). A National Instruments LabVIEW application [3] is provided which acquires data from two of the analogue input channels of the DAQ card and
measures the frequency, amplitude and phase difference of the signals. Measurement of the
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voltage across the resistor R determines the circuit current I. The circuit current I and
the voltage across the loudspeaker determine the magnitude of the complex impedance |Z|
of the loudspeaker, and the phase difference between the voltages across the loudspeaker
and a series resistor (voltage and current are in phase across the resistor) determines the
phase angle between the current and loudspeaker voltage.
Consider the following issues:
• What is an appropriate choice for the value of the resistor R?
• What is an appropriate choice for the DAQ card sampling frequency?
• How does the LabVIEW application determine the frequency, amplitude and the
phase difference of the signals?
• How can you estimate the frequency, amplitude and the phase difference measurement
errors ?
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References
1. Electromagnetism, I.S.Grant and W.R.Phillips, Second Ed., Wiley (JCMB Library
QC760Gra)
2. National Instruments NI PCI-6023E Multifunction DAQ Specifications
3. National Instruments LabVIEW Measurements Manual
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