The ThieleSmall Parameters for Measuring, Specifying
and Designing Loudspeakers
 Neville Thiele
Introduction.
The loudspeaker
parameters were first described by Thiele in 1961[1], but were not accepted widely until after they were republished
in the Journal of the Audio Engineering Society in 1971, and Small, over the
next three years, published a series of papers [28] that expanded them and made them much more understandable and
useful. Over the last thirty years,
they have been used widely, almost universally, to characterise loudspeaker
drivers and thus facilitate the design of loudspeaker systems. However, some aspects of their measurement
procedure and their use have attracted a degree of misunderstanding. The paper will try to correct these misunderstandings
and put some aspects of the measurements into a better perspective.
Limitations .
First of all, it
is often emphasised that the parameters provide only an approximate
understanding to the performance of a loudspeaker. It must be understood that the approximation, with these
parameters, is sufficiently close as to predict performance within a dB or
so. Nevertheless it is an approximation. Any engineering
theory is, after all, a simplification of what happens in the “real world”, and
to that extent an approximation. For
this reason “theoretical predictions” sometimes fail, to the great glee of
those who do not fully understand them, not from failure of the basic theory
but more usually through its application to circumstances where its ideal, simplifying,
assumptions no longer hold.
We consider now some
of the assumptions inherent in applying the parameters. First, it is assumed that the driver’s
efficiency is negligibly small. Some
drivers with light cones and large magnets that produce a high flux can provide
efficiencies as high as 4%, i.e. they produce acoustic output power 14 dB lower
than the electrical input power. Since
one acoustic watt radiated isotropically into half space, i.e. on one side of a
flat baffle, produces a sound pressure level relative to reference level (SPL)
of +112 dB at 1 metre, a loss of 14 dB corresponds to an SPL of +98 dB. But small as that may be, the majority of drivers
are even less efficient. Drivers have
become less efficient generally since the 1961 paper as transistor amplifiers
with higher electrical output power (e.g. 150 watts, compared with the 15 watts
that was considered large from a valve amplifier) have become available at a
reasonable expense and made it practical for drivers to use heavier cones and
overhung voice coils in the quest for lower distortion. Nowadays a driver that produces an SPL of
+85dB at 1 metre from a 1 watt input. i.e 1s 0.2% efficient, is not at all unusual.
Second, because a
loudspeaker driver presents an impedance that varies widely with frequency, it
has become conventional to express its sensitivity in terms of its Power
Available Efficiency (PAE), i.e. the ratio of its acoustic power output to the
“available” electrical power input that it would
have absorbed if it had presented to the input a pure resistance equal to
its d.c. resistance R_{E} .
Third, the
parameters apply only over the “piston range” of frequencies where driver acts
as a true piston, i.e. at the lower frequencies where the wave number ka is less
than 1.
_{}_{}_{}
where w = 2πf
f
is the operating frequency
r
is the radius of the piston, and
c
is the velocity of sound
Taking c, the
velocity of sound, as 344 m/s or 1130 ft per sec then ka is 1 at 4300 Hz for a
piston of 1 inch diameter, i.e. ½ inch radius, or at 360 Hz for a piston of 12 inch
diameter, and the parameters apply at frequencies lower than these limits.
This shows that the
frequency range over which these “low frequency” parameters apply varies with
the radius, or diameter, of the cone, so they are equally useful in assessing
the performance of tweeters as of woofers.
Beranek [9] incidentally
places the limit of the piston range an octave lower, where ka = ½, where the
frequency response has deviated from the ideal by 0.2 dB. At ka = 1, however, the
error is still only 0.7 dB. At
frequencies above this the power radiated by a piston falls towards an ultimate
slope of 6dB per octave. However the
pressure response of a piston also tends to focus into an increasingly narrow
beam as the frequency rises so that with skilful design, a driver can radiate a
comparatively constant, ”flat”, pressure
response at much high frequencies to a listener on its axis even though the
total power that it radiates into the
whole space, is falling.
Fourth, the
parameters characterise only the “small signal” performance of a
loudspeaker. As the signal level
increases, a driver becomes more and more nonlinear, and much of the skill in
designing a driver goes into making it perform linearly to a high level. A number of researchers, particularly
Klippel [10], have shown how the
different sources of nonlinearity, in the effective flux through the voice
coil, in the compliance of the restraint by the edge surround and the spider,
in the effective inductance of the voice coil, and even to a smaller extent in
frequency modulation of higher frequencies by lower frequencies due to the
Doppler effect, can be separated and hence dealt with separately in the design.
However, because much
of the programme signals that a loudspeaker handles are at the lower levels,
and because a good loudspeaker can handle these signals linearly to
sufficiently high levels, these parameters remain among the most important
tools in specifying loudspeaker performance and, for forty years, have stood
the test of time.
The
Parameters
Four main
parameters characterise the performance of a loudspeaker driver and are used to
calculate its performance when it is mounted in an enclosure or box.
f_{S} the
resonance frequency of the driver in Hz
Q_{E} the ‘electrical’
quality factor, the ratio of the d.c. resistance of the voice coil to reactance
at resonance of the drivers motional impedance. It is a pure, dimensionless, number
Q_{M} the
‘mechanical’ quality factor, the ratio of the shunt resistance of the driver’s
motional impedance to its reactance at resonance, another pure number. In some early publications, Thiele has called
this parameter Q_{A} for ‘acoustical’.
The Q values,
quality factors, affect the damping of the driver around its resonance. The higher the Q’s the more the frequency
response at resonance will peak compared with the inband response at higher
frequencies. If the Q’s are too low the
frequency response will sag around resonance.
In applications where the input impedance affects the response, as in
passive crossover networks, Q_{E} and Q_{M} affect the input
impedance of the driver in different ways and are therefore significant
separately.
However when, as
is more usual, the driver is fed from an amplifier whose output that presents a
low impedance, the two Q’s act together to control damping near resonance, and it is convenient to
combine them into Q_{T} , the “total” quality factor
_{}
in the manner of
resistances in parallel, which is of course, how they act.
When these
parameters f_{S} Q_{M} Q_{E } and Q_{T }are
measured under different conditions and their values change somewhat, i8t is
convenient to characterise each of them with a second subscript. Thus
f_{SA} is
the value of f_{S} that is measured when the driver is unbaffled,
hanging “in air”.
f_{SB},
slightly lower, is the value of f_{S} when the driver is mounted on an
effective baffle, i.e. in any box,
f_{SC} is
the resonance frequency of the system
of a driver mounted in a closed box.
Similarly Q_{MA}
Q_{MB} Q_{MC}, Q_{EA} Q_{EB} Q_{EC} and
Q_{TA} Q_{TB} Q_{TC}.
V_{AS},
the volume of air
equivalent to the acoustical compliance of the driver. It may be specified in litres (i.e. cubic
decimetres or millicubicmetres) or in Imperial units of cubic feet or cubic
inches. This parameter V_{AS } affects the response through its ratio with V_{B }, the
volume of the box that the driver is mounted in. That ratio may be written as V_{AS}/V_{B} , but
more often as the ratio of the two compliances C_{AS}/C_{AB }. The user must be warned that because the
volume (compliance) of the box forms the denominator of the expression, the bigger the box the smaller is the figure for C_{AS}/C_{AB }. In his publications, Small has called this
quantity α.
These four
parameters of the driver (or three once Q_{E} and Q_{M} are
combined into Q_{T}) , along with V_{B} , the volume of the box,
control the smoothness of response that is obtained from a loudspeaker
driver/box combination and to them must be added, whenever the box is vented
(ported), the box resonance f_{B} , the frequency where the mass of the
vent resonates with the compliance of the air in the box. It is most easily visualised (!), for
example, as the frequency of the note that results from blowing across the top
of a bottle, which is of course can be considered as another variant of the
Helmholz resonator.
Two other
parameters are of lesser, but still significant, importance in some
designs. One Q_{L }, the “leakage”
Q, has a small effect in vented box designs.
The other, L_{E} , the voice coil inductance in series with R_{E
},must be treated with care. It
is not invariant, as would usually be expected from an inductance, but varies
inversely with the square root of frequency.
These two parameters will be discussed in detail later.
The driver is
specified in terms of the electrical impedance diagram of Fig. 1. In this, R_{E}
is the d.c resistance of the driver. The
rest of the network, consisting of the three elements C_{MES} L_{CES}
and R_{ES} in parallel, is the
“motional impedance” of the driver. It
is produced by the interaction of the motion of the cone, through the voice
coil, with the flux of the magnet. If the
motion is blocked, this motional impedance falls to zero.
We specify the
driver in terms of the network of Fig. 1 with the four parameters, R_{E}
f_{S} Q_{M} and Q_{E} .
_{} _{} and _{}
where _{} _{}
f_{S} being the resonance frequency of the
driver. We also need to determine V_{AS}
, the volume of air equivalent to the
acoustical compliance of the driver.
The Transfer Functions of Loudspeakers
In the next
section, we show how the performance of a loudspeaker system may be found using
these parameters. In this paper we do
not derive the expressions using the electrical equivalents of the acoustical
properties of the system. That is found
in other papers. We present here the
results of those derivations, the transfer functions that, applied to the
reader’s own calculations, or better and more easily still a computer program,
can be taken as a recipe for finding the responses. A set, or constellation, of parameters needed to produce a given
result is often called an ‘alignment’.
The transfer
function relating the acoustical output pressure to the electrical input
voltage is written most compactly in the “operational” form of the Laplace
transform, from which the amplitude, phase and transient responses can be
calculated.
Closed Box.
The transfer
function of the loudspeaker in a totally enclosed box is, in the operational
form,
_{}
where _{}
and f_{S } is the resonance frequency of the
driver. In early papers, written before
1965 approximately, “s” is written as “p”.
The meaning is the same, only the convention has changed. Also C_{AS}/C_{AB}, the
compliance ratio of the driver to the box is written more appropriately for
calculation as V_{AS}/V_{B}, the ratio of the effective
acoustic volume of the driver to the volume of the box. Q_{T} , as described earlier, is the
“total” Q (quality or loss) factor of the box, obtained by combining the
“mechanical” Q_{M} with the “electrical” Q_{E }.
We can write the
expression in the “jw” form from which the amplitude and phase responses
can be calculated more easily. We simply replace “s” with “jw” and remember
that ( j )^{2} = 1, ( j )^{3}
= j, ( j )^{4} = 1 etc. and that w = 2pf, where f is the frequency being considered
_{}
To calculate the
squared magnitude of that expression we square the sum of the “real” components
(those not multiplied by j) and
square the sum of the “imaginary” components (those that are multiplied by j) and add the squares together. Then 10 times
the logarithm of that squared magnitude is the response in dB. Thus
_{}
The expression
shows that, for a driver mounted in an enclosed box, the lowest cutoff
frequency is
f_{S}Ö(1 + C_{AS}/C_{AB}). Thus it can never be lower than f_{S}
, and then only when C_{AS}/C_{AB} is zero, i.e. when the box
is infinitely large. When C_{AS}/C_{AB}
is 1, the cutoff frequency is 1.414 f_{S}; when C_{AS}/C_{AB}
is 3, it is 2 f_{S}.
Vented Box.
The transfer
function of the loudspeaker in the vented box is, in the operational form,
_{}
where Q_{L}
is the “leakage Q”, considered as a parameter of the driver. In the 1961 paper,
Thiele calculated a similar parameter Q_{B}, the “box Q”, as a
parameter of the box. When Small showed later that such leakage
losses were much greater in the driver than those in the box, so much so that
they could be virtually ignored, he still called the changed parameter Q_{B}
initially, but later called it Q_{L}.
Thus in the expression above, Q_{L} is the loss parameter for
the driver, as calculated below. If Q_{L} is not known,_{ }it
may be taken as 7 without too much error.
If on the other hand Q_{L} is ignored, as in some of the earlier
papers, it is effectively taken as infinite and all the terms containing it
disappear in the expression above.
Also _{} _{}_{}
where f_{B}
is the resonance frequency of the box.
The expression
becomes, in the jw form,
_{}
from which the
response is calculated, first as the squared magnitude and then as decibels,
_{}
When the vent is
closed up, then f_{B} diminishes towards zero, and only the terms that
contain f_{B}^{2} contribute significantly to the denominator
of the expression above. If then the
insignificant terms are removed and the remaining terms multiplied by (f_{B}/f)^{2}
and, remembering that the denominator terms are later squared, the numerator is
multiplied complementarily by (f_{B}/f)^{4}, the dB response
becomes, not surprisingly, the same as for the closed box. Thus the same calculation as for the vented
box can be used for the closed box, so long as a box resonance figure much
lower than f_{S}, e.g. 1 Hz, is entered for f_{B}.
Fig 4 shows that, with a vented box system, unlike a closed box, the
response cam be extended half an octave below the driver resonance f_{S},
if the box is larger (C_{AS}/C_{AB} smaller), its tuning
frequency f_{B} is lower than f_{S} and Q_{T} is
comparatively high. The response could
be extended even lower but the response ripple, that can be seen already in
curve no 8, would increase excessively. When the box is small (C_{AS}/C_{AB} greater than
2) f_{B} tuned higher than f_{S} and Q_{T}
comparatively low, a response free of peaks can be realised, but its cutoff
frequency f_{3} rises rather higher.
The compliance of a driver may increase
with time, particularly after it has been “exercised” by use. When this happens, both f_{S} and the
Q’s will decrease in inverse
proportion to the square root of the compliance. In a vented box system, this has surprisingly little effect on
the response, so long as it is the only parameter that has changed.
Equalization. *
Once the transfer
function of a loudspeaker has been found, as above, it can be used to equalize
an undesirable alignment to a desirable one [11,
12]. The second order transfer
function of a closed box loudspeaker, or the two second order factors of the
fourth order transfer function of a vented box loudspeaker are taken as the
numerator(s) of biquadratic active
filter(s) whose denominator(s) realise the desirable transfer function(s)
*This section is extracted from ref.12
In this way, any
loudspeaker transfer function can, in principle, be equalized to any desired
function. We say “in principle” as a
reminder of the practical limitations.
The parameters of the loudspeaker or the equalizer may change with production
tolerances or age. It is therefore
important not to try to equalize too
high or narrow a response peak with a complementary notch, otherwise a small
drift in the responses of loudspeaker or equalizer may make the equalized
result worse than it was in the first place.
Similarly, an equalizer that restores a falling loudspeaker response too
enthusiastically may well demand excessive power from the amplifier at some
frequencies and make it distort at unexpectedly low levels of output signal.
Again, it is
important to remember that, in a vented system, the acoustic output of the vent
assists the acoustic output from the cone at frequencies above the box
resonance, but works against it at frequencies below. At these lower frequencies the cone must execute larger and
larger excursions to produce smaller and smaller acoustic outputs. The box resonance should therefore always be
placed at the bottom useful limit of the desired equalized response.
In loudspeakers
with both vented and sealed enclosures, the excursion of the voice coil rises
to a maximum at lower frequencies where they no longer radiate useful acoustic
power, as shown in Fig 5. All its
excursions result from Butterworth responses, which are 3 dB down when f/_{ }f_{B}
= 1 and fall rapidly at frequencies below.
It is therefore beneficial to both the driver and its associated
amplifier to provide additional highpass filtering in that region.
Fig 5 shows how the
addition of further second order filtering, that takes the transfer function of
a loudspeaker with a closed box from 2^{nd} order to 4^{th}
order Butterworth (dashed curves 2¢, 3¢ & 4¢) or with a vented box from 4^{th}
order to 6^{th} order Butterworth (solid curves 4, 5 & 6), provides
protection against excessive excursion [11]. Two extra orders are the maximum
needed. A brickwall filter, even were
it feasible, would provide little extra effective protection.
Fig 5 shows in
fact that even first order auxiliary filtering, which takes the transfer
function of a closed box to 3^{rd} order or a vented box to 5^{th}
order, affords very worthwhile protection.
It can be provided with the utmost simplicity, by suitable proportioning
of a CR coupling network and adjustment of the associated transfer function(s),
to ensure the desired, usually maximally flat, overall response.
Increasing the
order of filtering in any system increases the group delay error at the band
edge, and is often deprecated for that reason. However the increase in group
delay error occurs very much in the last halfoctave before cutoff. Further when the cutoff frequency of a
highpass system is lowered, e.g. by equalization, its group delay error at
higher frequencies is actually diminished [13].
Measurement of the Parameters.
The parameters of
a driver will often be published by the manufacturer, but if they are not or if
there is any doubt, e.g. due to spread of values during manufacture or a
suspicion of optimism in the manufacturer’s specification, the designer may
wish to measure them himself. They may
be measured in a number of different ways, for which various advantages are
claimed. However the writer believes
that the original method given in Fig. 3 from ref. 1, is still as satisfactory as
or better than any, so long as the readings are taken with proper care and
using the modifications to the procedures described below.
Measurement “In
Air”
First of all the
driver is measured in air, preferably at least 50 cm from reflecting
surfaces. The equipment required is
simple. It comprises 
(i) an
oscillator, a source of sine wave signals of known frequency. If a frequency counter is not provided
integral with the oscillator a separate frequency counter is required. The oscillator should be capable of producing
signals up to 10 times below and 20 times above the resonance frequency of the
driver. Thus for a driver with a
resonance of 50 Hz, the oscillator should cover a range from 5 Hz to 1000 Hz
(ii)
a voltmeter
and
(iii) an
ammeter, both accurate across the range of frequencies used in the tests
(iv)
a testing box
of known volume in which the driver may be mounted. Preferably, it should be vented and resonant at a frequency near
that of the driver, though neither is absolutely necessary. It should not contain acoustically absorbent material.
The first time three
instruments provide electrical measurements.
The fourth, the testing box, is the only “acoustic” parameter needed in
the measurements.
The oscillator
must be capable of being read to a precision rather better than 1 Hz,
preferably
0.1 Hz.
For this reason a frequency counter is a most desirable accessory. If the counter can read frequency only, a
measurement with a precision of 0.1 Hz requires a gate period of 10 seconds for
each measurement. But if the counter reads
“time”, the period of one cycle can be measured and its reciprocal taken as the
frequency. When, as is usual, the
period of one cycle is measured to a resolution of 0.1 ms, the frequency of a 50 Hz signal can be
read to within 1 part in 200,000. Some
of the better oscillators measure the period at low frequencies and then
convert it internally into a display of frequency
It will be seen that,
in the calculations below, two steps depend on the differences between two frequencies and that if these frequencies
are, for example, 30 Hz apart, even a precision of 0.1 Hz in their difference
only ensures an accuracy in that calculation of 1 in 300, i.e. 0.3%. For
this reason care in observation and precision of instrumentation is needed at
all times in the measurement of frequency.
The absolute calibration of the ammeter is
not important, but its reading must be linear
across its whole scale and it must maintain that linearity over the whole range
of frequencies being measured so as to ensure accuracy of the ratios of its measurements.
In an alternative
method, the ammeter is replaced by a variable resistance that is adjusted for
each reading so as to keep the voltmeter reading constant and is then measured
to a good precision after each reading, e.g. by a digital ohmmeter.
The voltmeter is
needed simply to maintain a constant voltage during the measurement and to draw
a current that is very much smaller than that drawn by the device under
test. Its indication must also remain
constant with frequency. An
oscilloscope can perform this function excellently.
The box needs to
be free of leaks  a colleague once suggested that it should be specified as
not so much airtight as watertight  and to have an opening that seals well to
the driver during the reading. It must
be rigid and free of internal damping material.
Measurements of
frequencies at peaks of impedance can be made more easily and accurately by
searching for zero phase angle between the voltage from the oscillator and that
across the driver, using the xy facility of an oscilloscope. The sensitivity of such a reading that
incorporates phase measurement can be seen in Fig 6, where the frequencies of
the maxima and minima of impedance are measured much more sensitively as the
zero crossings of the plot of phase angle.
Fig 7 shows that, even when the phase difference between the X and Y
deflection voltages is as small as 1°, a (highly squeezed) ellipse can still be
seen.
The reading of R_{E}
is taken preferably at d.c. with the oscillator replaced by a battery, but
alternatively at a frequency very much, say 10 times, lower than the driver
resonance. The next reading is taken at
the driver resonance frequency f_{S} and its resistance R_{ES},
which the writer now prefers to call R_{0} , noted. When an ammeter is used, the reciprocal of
the current reading (1 / I) is noted, as if it was the resistance with a
notional 1 volt signal.
Then two
frequencies are found either side of resonance where the resistance or current
has the same intermediate value, R_{1}
or I_{1} , and their two values f_{1} and f_{2} noted.
Unfortunately, in Fig 2 which is reproduced from Fig 16 of ref. 1, R_{1}
was drawn rather high up the bellshaped curve, which has led a number of readers
to imagine than it is intended to be at 0.707 times the peak impedance, the
socalled
“3dB point” where
readings of Q such as this are taken conventionally in radio frequency
measurements. Even Fig 8 could give a
misleading impression unless we assume that the scale for its Y impedance axis
is logarithmic.
However the “3
dB” method is suitable only for measurement of Q‘s greater than 5. The method proposed below measures Q’s of any value. It will be seen that an accurate calculation of Q_{M} ,
and hence of Q_{E} and Q_{T} , depends critically on an
accurate measurement of the frequency difference f_{2}  f_{1} , which are taken for the same
value of r_{1}. This requires
both accurate readings of f_{1} and f_{2} and a testing procedure
in which the two frequencies are separated as far as possible.
The optimum value
of R_{1} or I_{1} for taking these frequencies is at the
geometric mean between their readings at d.c and at resonance, in other words
when either
_{} or _{}
Such a reading not
only produces good separation between f_{1} and f_{2}. It is also the point on the curve of Z_{E} vs. frequency in Fig. 2 where
the slope is a maximum and thus the frequencies can be read most sensitively.
These are all the “In Air” readings that are required. but it is
worthwhile to check also for consistency that
_{}
Now the ratios are calculated
_{} or _{}
_{} or _{}
and then the
parameters
_{} : _{} : _{} _{}
It should be noted
that
_{}
i.e. Q_{E} and Q_{M} combine to form Q_{T} following the same law as resistances in
parallel.
If additionally,
we choose r_{1} at the optimum value described above where
_{ }
then Q_{M} is simplified to
_{}
We have discussed
already how Q_{M} and Q_{A} are different names for the same quantity.
Measurement “In Box”
If the driver is
mounted already in a sealed box and is to be used that way, then the “In Air”
measurement is all that is needed. The
same applies generally to closedback tweeters, whose “low frequency”
parameters will be measured around their resonances in the region of 1 kHz to 2
kHz.
However in the
more usual case, and in any case if we also want to know the driver’s
sensitivity, we will need to know the additional parameter V_{AS} which
expresses the compliance of a driver in terms of an equivalent volume of
air. Other more elaborate measurements
have been proposed, but the original method is simple and delivers the
parameter directly in the form in which it is most useful.
Two methods are
proposed in ref.1 for measuring V_{AS}.
The first, in its eqns
(100) to (104) makes the measurement in a totally enclosed box, without a vent,
and was preferred at the time of writing the paper. However, after experience using that method, it became clear
that if the box was not completely sealed, through leakage at the joints or
around the mounting hole for the driver, some quite anomalous results could be
obtained. For those reasons the writer
now prefers the second method using a vented box and outlined in eqns (105) to
(108) of ref. 1. Although leaks are
still undesirable in this method, they contribute only subsidiary, if somewhat
resistive, vents that impair the measurements less. The tuning of the box is
not critical, but should not be too far away from the resonance frequency f_{S} of the driver.
Testing in a Vented Box
With the driver
mounted in the box, the impedance exhibits two peaks, as in Fig 9, at f_{H}
and f:_{L} . Then f_{B} the resonance frequency of the box, is
measured. In the original paper, this
was taken as the frequency of minimum impedance, called f_{B} in Fig. 9. However two difficulties arise when f_{B
}is measured in this straightforward manner. The first is that the minimum of such a shallow trough is
difficult to read.
The second is that
the actual minimum read in this way is not
the true f_{B} but rather the frequency where the voice coil inductance
is in series resonance with the capacitive component of the motional impedance
that is rising with decreasing frequency towards f_{L}.
This had produced
little error in the initial work. At
that time, voice coils were underhung, so as to ensure maximum efficiency when
they were driven by valve amplifiers, and their inductance was comparatively
small. However, with the advent of
overhung voice coils the inductance increased and the error became
significant. The problem was overcome
by Benson [14], who took a further
reading f_{C} at the frequency of the peak impedance when the vent is
closed, as shown in the dashed curve which has been added in Fig 9, with other
changes, to Fig. 5 of ref. 1. Then the
true f_{B} is found from
_{}
_{ }
It is worthwhile
checking this figure against f_{B} , the frequency of the impedance
minimum near f_{B} where the driver impedance Z_{E} goes
through zero phase angle, which was taken as f_{B} in ref. 1. The true value for f_{B} should be
the higher by a small amount.
Then the
compliance V_{AS} , expressed as a volume of air, is found from
_{}
where V_{B}
is the box volume. Also f_{SB}
, the effective resonance frequency of the driver where it is mounted in the
box, is found from
_{}
When a driver is
mounted in a box the air mass loading on its cone doubles compared with its
value unbaffled, so the resonance frequency f_{SB} falls, usually around
5% lower than f_{SA} , the resonance frequency previously measured in
air with the driver unbaffled. This
again is a useful check for consistency in the results, see eqns (105) and
(106) in ref. 1. The effective Q’s with
the driver mounted in the box, which we call Q_{MB} Q_{EB} and
Q_{TB,} are also increased slightly from the earlier values, which we
now call Q_{MA} Q_{EA} and Q_{TA,} to
_{} _{} _{}
At the same time,
the leakage parameter Q_{L} is
derived from the resistance R_{B} measured at f_{B} in Fig. 5,
which is a little greater than R_{E}.
This is calculated
as
_{}
This expression again is different from eqn (107) for Q_{B} in ref.
1, omitting the term (w_{B}/w_{S} )( C_{AB}/C_{AS} ). Initially Thiele had considered Q_{B } to be a parameter of the box, but Small found it to be primarily a
property of the driver, a “leakage parameter”
Q_{L} , which is independent of the box parameters, or the ratios
of f_{B} to f_{S} or of V_{B} to V_{AS}. Note that when the expression for Q_{L}
is derived directly from measurements of currents, it includes Q_{TB}
. When the expression is derived from
resistance measurements, or from the “r” ratios, it includes Q_{M}. Note that the ratio of compliances C_{AS}/C_{AB}
is identical, in acoustic terms, to the ratio of air volumes V_{AS}/V_{B}
.
Testing in an Enclosed Box.
In spite of the
disadvantages described earlier of testing with a totally enclosed box,
occasions arise when it may be unavoidable.
A particular case is the measurement of V_{AS} in a closedback
tweeter.
The first set of
readings is taken “in air” and f_{SA} Q_{MA }Q_{EA} and
Q_{TA} calculated as before.
Then the driver is placed in a box of volume V_{B} (in the case
of the closedback tweeter, the box, quite a small one, is placed in front of it) with suitable
precautions that they seal tightly.
Then a second set of similar readings are taken using the same
procedures described earlier as “in air”, and parameters calculated which we
will call f_{SC} Q_{MC} and Q_{EC}. From these we find
_{} _{ }
Then we find the
ratio M_{ASA}/M_{ASB} of the masses of cone plus air load “in
air” to “in box”
_{}
and use its square
root as a multiplier to find f_{SB}
Q_{MB} Q_{EB}
and Q_{TB}
_{} : _{} : _{} : _{}
For the
closedback tweeter, which effectively has a baffle in both measurements, M_{ASA}/M_{ASB}
should be unity, a useful check on the accuracy of the measurements.
Voice Coil Inductance L_{E}
Fig 10 (a) shows
the voice coil inductance as it is usually represented, a pure inductance L in
series with R_{E} the d.c. resistance and Z_{M} the motional
impedance. However ref. 1 suggested a better
model of driver impedance whose increase at the higher frequencies was limited
by a resistance R_{1} shunting the inductance L_{1} in Fig 10
(b). This simplistic model may have
been based on faulty observation, but it seemed to describe the drivers of the
time, which had generally shorter, underhung, voice coils that had smaller
inductances and in which the mechanisms described below were less significant.
Fifteen years
later, Thiele used this model in developing a Zobel network to compensate the
driver’s input impedance to produce a resistive termination for passive
crossover networks [15], a shunt
network containing R_{E} in series with a dual of the remainder of the
driver impedance. However, Small found
that model inadequate for the more recent drivers with longer voice coils, and
produced a better fit to measured values using the model of Fig 10 (d),
including its R_{1} L_{1} R_{2} and L_{2} but
omitting R_{3} and L_{3} .
Finally, Dash, a student of Small’s, developed an even better model [16] as in Fig 10 (c), a continuous
structure comprising an infinite number of very large resistances and very small
inductances. Adopted and published some
years later by Vanderkooy [17], it is
recognised today as the best available.
In this model the
“pure” voice coil inductance is modified by eddy current losses in the pole
piece to become a “semiinductance”.
This semiinductance has the special quality that its reactance
increases with Öw rather than w and that, at all frequencies, its impedance
has series components of resistance and reactance that are equal, i.e. they
present an impedance of the form R + jR that varies as the square root of frequency. Thus it can be written as KÖw + jKÖw and has a
constant phase angle of 45°. When
the resistance is written as KÖw, the
inductance is K /Öw.
It will be seen
from this model that a measurement of L_{E} at any frequency is valid at that frequency alone, and gives only
a very rough approximation to the true, smaller, reactance at higher
frequencies, and that there is an equal series resistance, both of them varying
with Öf. Nevertheless,
this figure for L_{E} , and even better the time constant T_{E}
, provide a useful guide to the performance of a driver, from which its
properties at other frequencies may be inferred.
We calculate L_{E}
at f_{ZMIN} as _{} and _{}
Secondary Parameters.
Once the primary parameters
have been found, as above, they may be combined into secondary parameters that
convey further useful information about a driver.
Force Factors [18] provide
figures of merit that split the efficiency (sensitivity) of a driver into two
parts
F_{ME}, the magnetoelectrical force factor, which depends
purely on the motor, and
F_{AM} , the acoustomechanical force factor, which depends
purely on the area of the cone and its mass, including that of the voice coil.
The two factors
are not used directly for the design of loudspeaker systems, but allow
variations between different models of driver or inconsistencies in production
to be evaluated precisely in the two separate aspects.
The
magnetoelectrical force factor is
__
F_{ME} = Bl /Ö R_{E}
where B is the
flux density in the airgap, l is the length of wire in the voice coil lying effectively within the airgap and R_{E}
is the resistance of the voice coil.
Rather than
measuring B and establishing l, which is difficult particularly with an overhung
voice coil, part of which lies in the fringing magnetic field, it can be
estimated from the primary parameters
__________
F_{ME} = 149.4 S_{D} /Ö f_{SB} V_{AS} Q_{EB}
when all
quantities are specified in SI units, i.e. the cone area S_{D} is in m^{2},
the driver resonance frequency f_{SB} in Hz, V_{AS} in m^{3} and F_{ME} in Newtons per Watts^{1/2}. It should be noted that the square of this
parameter, namely
F^{2}_{ME} = ( B l )^{2} / R_{E}
is described as
“thrust” by loudspeaker engineers in
the U.K. The second,
acoustomechanical, force factor is ________
F_{AM} = Ö(r_{o} / 4 p c)( S_{D}/M_{MS})
where M_{MS} is the mass of the cone in kilograms and F_{AM} is in units of Watts^{1/2} per
Newton. This factor also is most easily
derived from the primary parameters as
F_{AM} = 4.671
f_{SB}^{2}V_{AS} /10^{6} S_{D}
In more familiar units, when V_{AS}
is in litres (dm^{3}) and S_{D}_{
}(mm^{2}) is expressed indirectly in terms of d_{C} the effective cone diameter in mm, then
__________
F_{ME} =
0.00371 d_{C}^{2} / Ö f_{SB }V_{AS} Q_{EB} and F_{AM} =
0.00595 f_{SB}^{2}_{
}V_{AS} / d_{C}^{2}
Small has shown
that S_{D} can usually be taken with little error as the area measured
to the middle of the surround.
Power Available Efficiency (PAE).
By multiplying F_{ME}
and F_{AM } together and
squaring that product we find the Power Available Efficiency, PAE or
h
h = ( F_{ME} F_{AM} )^{2} = 8.0 f_{SB}^{3}V_{AS} /10^{12} Q_{EB} when V_{AS} is in in^{3}
=
487 f_{SB}^{3}V_{AS} /10^{12} Q_{EB}
when V_{AS} is in
litres (dm^{3})
Efficiency is
usually used in electrical engineering to describe the ratio of the energy or
power going into a device to the energy or power that comes out of it,
calculated as
Efficiency h = W_{OUT }/ W_{IN}
Power is defined
conventionally in terms of the voltage E applied to, or the current I flowing
into, a resistance R, thus
W
= E^{2}/R =
I^{2} R
However, a driver
presents an impedance to the external circuit whose resistive component varies
widely with frequency, even while its output remains substantially
constant. For this reason it is
impractical to use the actual, varying, value of the driver impedance but to
rate its efficiency in terms of PAE, where W_{IN }is not the power
going into the driver but rather the power that would have gone into a pure resistance equal to the d.c. resistance
R_{E} of the voice coil.
Sound Pressure Level.
Rather than by
efficiency, the sensitivity of a driver is more often rated by the sound
pressure level (SPL) that it
produces at a distance of 1 metre in halfspace from an electrical input power
(available power as defined above) of 1 Watt.
This is derived from the PAE value as follows.
An acoustic power of
1 acoustic Watt radiated nondirectionally into 2p steradians halfspace, i.e. full space
divided into two by an infinite plane (e.g. a vertical wall) produces an
acoustic pressure of
Ö r_{o}c /2p
Pascal (N/m^{2})_{ }at
1 metre distance. When the ambient temperature
T is 22° C and the barometric pressure is 10^{5}
Pascal (751 mm Hg), then r_{o} the density of air is 1.188 kg/m^{3}, c the
velocity of sound is 344.5 m/s and the acoustic pressure thus generated is
8.060 Pascal. Expressed in dB relative
to a reference sound pressure of 20 mPascal, which
is taken as a standard pressure level at the threshold of human hearing, this
is 112.1 dB SPL.
Once the Power
Available Efficiency of a driver is found in decibels
dB PAE = 10 log_{10}PAE
which is a negative
quantity since PAE is less than unity, then the driver can be rated as
dB SPL, 1W, 1m
= 112.1 dB + dB
PAE
Thus a driver with
a PAE of 1% is rated as having either an efficiency h of 20.0 dB, or alternatively, an output
SPL at 1 metre distance, with 1 Watt input, of +92.1 dB.
Further, the
convention has arisen of specifying SPL in terms of an input voltage of 2.83
Volts rather than an input power of 1 Watt.
A voltage of 2.83 Volts produces 1 Watt power in a resistance of 8 ohms,
and such a specification obviates the need to establish the value of the voice
coil resistance R_{E }. It
also makes a driver with a low voice coil resistance rate as more sensitive
because it draws more power from a transistor amplifier, which performs as a
source of voltage. By this criterion
then, which accords with practice when it is fed from an amplifier with a very
low output impedance, a driver with a d.c. resistance of 4 ohms is 3.0 dB more sensitive than if its
resistance had been 8 ohms with nothing else changed, and
dB
SPL, 2.83 V, 1m = 112.1 dB + dB PAE + 10 log_{10} (8/R_{E})
Note that this
expression includes R_{E} , which should not be confused with the rated
impedance of the driver. That, conventionally,
is higher than R_{E }, being usually taken at the frequency f_{ZMIN}
, where the impedance is resistive but also includes, in series with R_{E}
, resistive components from the motional impedance and the semiinductance of
the voice coil.
_{ }
Conclusion.
The paper has sought to present,
in as compact a form as possible, the loudspeaker parameters that allow the
amplitude, phase and transient responses of a system to be presented using the
same techniques as for an electrical highpass filter. Then a loudspeaker’s performance can be
estimated by wellestablished techniques of computation.
The parameters are estimated from
measurements of the input impedance of the driver, again using wellestablished
electrical techniques, with the only ‘acoustical’ device employed a testing box
of known internal volume.
It has not been shown how the
background theory was derived from the electrical equivalents of acoustical
quantities, or how devices can be modified to achieve more favourable
designs. The prime object has been to
collect into the one place the techniques of measurement and calculation from
the divers places where they were described and present them in a form best
suited to their effective use.
Acknowledgement.
The author
gratefully acknowledges the assistance of Graeme Huon in helpful discussions
and in editing.
References.
1. A. N. Thiele  Loudspeakers in Vented Boxes 
Proc. IRE Australia, Vol. 22,
No. 8, 1961 August, pp. 487–508
reprinted J Audio Eng Soc, Vol. 19, Nos. 5
& 6, 1971 May & June, pp. 382–392
& 471–483
2. Richard H. Small – Direct Radiator Loudspeaker System Analysis –
J Audio Eng Soc, Vol 20 No 5, 1972 June, pp.
383395
3. Richard H. Small – Closed Box Loudspeaker Systems, Part I Small Signal Analysis – J Audio Eng Soc, Vol 20 No 5, 1972 December
4. Richard H. Small – Closed Box Loudspeaker Systems, Part II Large Signal Analysis – J
Audio Eng Soc, Vol 21 No 1, 1973 January/February
5. Richard H. Small – Vented Box Loudspeaker Systems, Part I Small Signal Analysis – J
Audio Eng Soc, Vol 21 No 2, 1973 March
6. Richard H. Small – Vented Box Loudspeaker Systems, Part II Large Signal Analysis – J
Audio Eng Soc, Vol 21 No 6, 1973 July/August
7. Richard H. Small – Vented Box Loudspeaker Systems, Part III
Synthesis – J
Audio Eng Soc, Vol 21 No 6, 1973 July/August
8. Richard H. Small – Vented Box Loudspeaker Systems, Part IV Appendices – J
Audio Eng Soc, Vol 21 No 7, 1973 September
9. L. L. Beranek – Acoustics – McGrawHill, London, 1954 reprinted American Institute of
Physics, New York, 1986, Figs 5.3 & 5.9, pp.119 & 127
10. Wolfgang Klippel – Dynamic Measurements and Interpretation of the
Nonlinear Parameters of Electrodynamic Loudspeakers – J Audio Eng
Soc, Vol 38 No 10, 1990 December, pp.944955
11. A. N. Thiele  Loudspeakers, Enclosures and Equalisers 
Proc. IREE Australia,
Vol. 34, No. 11, November 1973, pp. 425–448
12. Neville Thiele  An
Active Biquadratic Filter for Equalizing Overdamped Loudspeakers –
AES 116^{th} Convention,
Berlin, Germany, 2004 May 811, Preprint No. 6153, 14 pp.
13. Neville Thiele  Phase
Considerations in Loudspeaker Systems

AES 110^{th} Convention, Amsterdam, The Netherlands, 1215
May 2001, Preprint No. 5307, 13pp.
14. J. E. Benson – Theory and Design of Loudspeaker Enclosures Part 3 – Introduction to Synthesis of
Vented Systems – AWA Technical Review, Vol 14 No 4, 1974,
Appendix 4, eqn A46, p.471
reprinted as Theory and Design
of Loudspeaker Enclosures –
Synergetic Audio Concepts, Don & Carolyn Davis,
1993, 244 pp.
15. A. N. Thiele  Optimum Passive Loudspeaker Dividing Networks 
Proc. IREE Australia, Vol. 36, No. 7, July 1975, pp. 220–224
16. I. M. Dash – An Equivalent Circuit Model for the Moving Coil
Loudspeaker –
honours thesis 1982, School of Electrical
Engineering, University of Sydney, Australia
17. John Vanderkooy 
A Model of Loudspeaker Driver Impedance including Eddy Currents in the Pole
Structure – J Audio Eng Soc, Vol. 37, No. 3,
March 1989, pp. 119128
18. A. N. Thiele  Force Conversion Factors of a Loudspeaker Driver
– JEEEA,Vol. 13, No.2, June 1993, pp.129131 &
reprinted J Audio Eng Soc, Vol.
41, No. 9, September 1993, pp. 701703
Note: References nos. 18
reprinted in “Loudspeakers, An Anthology, Vol.1 – Vol.25 (19531977)”,
ed. R.E.Cooke, Audio Engineering Society,
New York, 1978
References nos. 10 & 16 reprinted in “Loudspeakers, An
Anthology, Volume 4, Transducers,
Measurement and Evaluation, Vol.32 – Vol.39
(19841991)”,
ed. Mark R Gander,
Audio Engineering Society, New York, 1996
References 1, 58 & 11 reprinted IREE, Australia in “Vented
Loudspeakers – An Anthology”.
oOo
Figures.
Figure 1  Simplified electrical equivalent circuit
of the loudspeaker
Figure 2  Typical impedance curve of loudspeaker (Modulus of Ze in
figure 1)
Figure 3  Test circuit schematic for measurement of
loudspeaker parameters
Curve
No. 
f_{3}/f_{S} 
f_{B}/f_{S} 
C_{AS}/C_{AB} 
Q_{T} 
3 
1.77 
1.42 
4.46 
0.26 
5 
1.00 
1.00 
1.42 
0.38 
8 
0.64 
0.76 
0.56 
0.52 
Figure 4  Typical curves for a vented loudspeaker.
Driver with the same resonance frequency fs, but different alignments
comprising box resonance f_{B}, box size C_{AS}/C_{AB}
and total Q_{T}
Figure 5  Curves of cone
excursion
_{}
Figure 6a  Driver impedance in
vented and enclosed box
_{}
Re = 8.0 ohms: Qe = 0.50: Qm = 2.00: Ql = 7
.00: Fs = 50.0 Hz: Fb = 40.0 Hz: Vas/Vb = 1.20: Le = 800 uH
Figure 6b  Driver phase angle
of driving impedance for vented and enclosed box
_{}
Dashed
curve  1 deg: Solid curve  15
deg: Dotted curve  30 deg: Dashdot curve  90 deg
Figure 7  Oscilloscope XY
phase plot
Figure 8  Driver measurements “In air”
Figure 9  Vented box
measurements “in box”
(a)
(b) (c) (d)
Figure 10  Simplified
electrical equivalent circuits of the loudspeaker voice coil.
Z_{M} is the
motional impedance modelled by L_{ces} C_{mes } and R_{es} in Fig 1.