Basta!
Technical documentation
Ó Tolvan
Data 20052009
20090111
Basta! is a computer
program for simulation and measurement of loudspeaker systems. It can simulate
closed, bassreflex, 1port bandpass and 2port bandpass systems, or use the measured
frequency response and impedance data and connect these to simulated electrical
components. It can derive amplitude and phase response as a function of
frequency for these systems. It manages multiple elements, both in parallel and
isobaric operation.
The model used in
Basta! assumes a signal (voltage) source followed by an optional set of active
crossover filters, a power amplifier, a passive electric circuit which feeds a
voltage to the loudspeaker. The loudspeaker feeds an acoustic flow into the
box, which in turn feeds an acoustic flow into the radiation resistance. Some
of this flow may originate directly from the loudspeaker. The power produced in
the radiation resistance is the acoustic power generated by the system.
Block diagram for the Basta! simulation.
Basta! can also use
measured data for the loudspeaker driver. In this setting, the response of the driver
is measured by some other software (for example Sirp from Tolvan Data). Also,
the electrical impedance of the driver is measured. These two sets of data are
used to simulate the combined behaviour of the electrical components in the
loudspeaker system and the measured driver.
Block diagram for the Basta! simulation, using measured data for driver response and impedance.
In the following,
the parts of the simulation are described in detail.
The signal source is a
simple voltage source. By changing the voltage, the output level of the loudspeaker
will change. The voltage can also be set to a negative value in order to
simulate reversed polarity of the speaker.
Active filters are
typically connected before the power amplifier. Basta! allows high and lowpass
filters of order 1 to 4. A lookup table for butterworth (odd order) or linkwitz
(even order) filters, is included.
The filters are
realised as follows:
First order One first order link
Second order One second order link
Third order One second order link and one first order
link
Fourth order Two second order links
First and second order
links can typically be realised by means of operational amplifiers and a few
passive components. In the following, examples are given on how to realise the
circuitry.
The first order filter
is realised by means of an RC link and a voltage follower.
Example realisation of a first order low pass filter
The transfer function
is
_{}
The cutoff frequency
is given by
_{}
Select R_{1}=10
kW, and calculate C_{1} = 1/(2pf_{0}R_{1})
The first order filter
is realised by means of an CR link and a voltage follower.
Example realisation of a first order high pass filter
The transfer function
is
_{}
The cutoff frequency
is given by
_{}
Select R_{1}=10
kW, and calculate C_{1} = 1/(2pf_{0}R_{1})
The second order
filter is realised by means of an RC network and a voltage follower.
Example realisation of a second order low pass filter
The transfer function
is
_{}
The cutoff frequency
and Q value are given by
_{} _{}
Select R_{1} =
R_{2} = R =10 kW, calculate C_{1} = Q/(pf_{0}R), C_{2} = 1/(4pf_{0}RQ)
The second order
filter is realised by means of an RC network and a voltage follower.
Example realisation of a second order high pass filter
The transfer function
is
_{}
The cutoff frequency and
Q value are given by
_{} _{}
Select R_{2}
=10 kW, calculate R_{1} = R_{2}/(4Q^{2}) and
_{}
The third order filter is realised by cascading a first order filter with a second order filter.
The fourth order
filter are realised by cascading two second order filters.
For the normal
configuration, the power amplifier has little purpose in Basta!. It is assumed
to be an ideal voltage follower, ie gain =1 and output impedance = 0. However,
when the ACbass* configuration is simulated, an ACbass network is connected
to the output of the voltage follower.
It can be seen that if
R_{acneg} equals R_{E} of the voice coil, they will cancel,
and L_{ac} and C_{ac} will have the same effect on the response
as a spring and a mass on the mechanical side. R_{ac} will become the
new effective voice coil resistance. Thus, the ACbass principle can be used to
control the apparent mechanical mass, compliance and lossiness, or in other
words, f_{s}, V_{as} and Q_{ts} can be selected freely.
The ACbass network is
typically not built like the network above; it is normally included in the
power amplifier using current feedback to design an output impedance like in
the figure.
When the ACbass network
is used, all other filters are typically disabled. The only exception is the
conjugate link, which can be utilised to reduce the effects of the voice coil
inductance.
*ACbass was first described in a master thesis by
KarlErik Ståhl at the department of Speech, Music and Hearing, Royal Institute of Technology, Sweden. The
principle was patented in the late 1970's and is user by Audio Pro AB under the
name ACEbassä.
There are also products from Yamaha that use the principle. The patents have
now expired, and as far as I understand anyone is free to use the concept.
Between the driving
amplifier and the loudspeaker, some passive electrical components can be added.
This network consists of a set of optional parts; a freely configurable passive
"advanced RLC network", a lowpass filter of order 0 to 4, a
highpass filter of order 0 to 4, an attenuation network and a conjugate link.
Circuit diagram of the passive network used in Basta! Some or all parts may be excluded.
This network can be
any combination of resistors, capacitors and/or inductors. The network has
three predefined nodes, "in", "out", and "gnd",
corresponding to its input, output and ground connections. Apart from these
nodes, the network may contain additional internal nodes, which are identified
by userselected names.
In some cases, like
the conjugate link, no separate in and outputs are desired. If so, the input
and outputs can be shorted. In this case "in" and "out" are
treated as synonyms for the common node that is connected to the
"hot" side of the network.
In principle, the
remaining parts of the passive electric circuit can also be built as part of
the advanced RLC network, but in many cases it is easier to use the predefined
circuits, as follows.
Passive low pass and
high pass sections up to order 4 can be connected. The low pass section is
formed by L_{L1}, C_{L1}, L_{L2} and C_{L2}.
Series resistances in the two coils are modelled through R_{L1} and R_{L2}.
The low pass filter is followed by a high pass filter consists of C_{H1},
L_{H1}, C_{H2} and L_{H2}, and the series resistances
of the coils are modelled through R_{H1} and R_{H2}. For lower
filter orders, some of the components are removed
Lowpass 
0^{th} order 
1^{st} order 
2^{nd} order 
3^{rd} order 
4^{th} order 
L_{L1} 
Short 
USED 
USED 
USED 
USED 
R_{L1} 
Short 
USED 
USED 
USED 
USED 
C_{L1} 
Open 
Open 
USED 
USED 
USED 
L_{L2} 
Short 
Short 
Short 
USED 
USED 
R_{L2} 
Short 
Short 
Short 
USED 
USED 
C_{L2} 
Open 
Open 
Open 
Open 
USED 
Highpass 
0^{th} order 
1^{st} order 
2^{nd} order 
3^{rd} order 
4^{th} order 
C_{H1} 
Short 
USED 
USED 
USED 
USED 
L_{H1} 
Open 
Open 
USED 
USED 
USED 
R_{H1} 
Open 
Open 
USED 
USED 
USED 
C_{H2} 
Short 
Short 
Short 
USED 
USED 
L_{H2} 
Open 
Open 
Open 
Open 
USED 
R_{H2} 
Open 
Open 
Open 
Open 
USED 
Design of the passive filters. For lower order filters, some of the components are shorted or left open.
A lookup table for Butterworth
(odd order) or Linkwitz (even order) filters, is included The tables assume a
resistive load, which means that the values of the components probably will
need some manual tweaking to achieve the intended response of the loudspeaker
system.
The signal to the
loudspeaker can be attenuated by the resistors Rs and Rp. The resistors are not
used if theirs values are set to 0. These resistances together with R_{L1}
and R_{L2} will affect the effective Q_{ts} value of the
loudspeaker and can thus be utilised to fine tune the Q_{ts} value. It
will, however also deteriorate the efficiency of the system since some power is
lost in the resistances.
To compensate for the
voice coil inductance a conjugate link formed by R_{EC} and C_{EC}
can be used. Their values can be calculated automatically from voice coil
resistance and inductance, but given the lossy nature of the voice coil
inductance, these values mostly need some manual tweaking to achieve an
approximately flat and resistive impedance curve.
The purpose of the
conjugate link is to provide the crossover filters with an approximately
resistive load towards higher frequencies.
The loudspeaker is
modelled using the electrical impedance of the voice coil via the resistance R_{E}
and inductance L_{E}. This inductance can be modelled as lossy, see
below. The electrodynamic transducer is modelled by means of a gyrator with
the gyration constant T=Bl. The mechanical system is modelled by the moving
mass M_{MS}, the suspension compliance C_{MS} and mechanical
damping R_{MS}. The mechanical velocity is then converted to an
acoustic flow Q_{s} via the equivalent piston area S_{s}.
Equivalent circuit diagram of the loudspeaker element. To the left, variables and impedances are electrical, in the middle, they are mechanical, and to the left they are acoustic.
The following
equations are used to determine the component values in the diagram:
_{} _{}
where
f_{s} is the resonance of the loudspeaker element
in free air
V_{as} is the equivalent volume of the cone
suspension
T is the force factor AKA Bl
Q_{ts} is the total Q value of the loudspeaker
element
R_{E} is the DC resistance of the voice coil
S_{s} is the equivalent piston area of the
loudspeaker element
r_{r} is the piston radius corresponding to S_{s}
Note that the
mechanical mass M_{MS} is reduced by the amount of the cooscillating
air for a piston in free air. Cooscillating air is later added in terms of M_{AL}.
The voice coil
inductance can be modelled as lossy. Measurement of real voice coils show that
the impedance behaves far from a simple resistor in series with an inductance.
A more appropriate model also takes into account "eddy currents"
induced in the magnetic pole pieces in the loudspeaker. A much better model is
to use this equation for the voice coil impedance
_{}
Where n is the loss
factor. If n=1 the voice coil is lossless and the impedance is R_{E}+jwL_{E}, however most loudspeakers have a
n value of 0.6 to 0.7. The main drawback with using this equation is that
manufacturers rarely specify the voice coil inductance in this way. Thus in
order to take advantage of the improved precision provided by the refined
model, the simulation has to be matched against measured data, e.g. in terms of
an impedance curve.
The voice coil
resistance R_{e} varies with temperature. Basta! can simulate this by
assuming that the voice coil resistance is proportional to the absolute
temperature. The Actual R_{e} is modelled as
where T is the
temperature in °C. Modifying the temperature setting is
equivalent to adding a series resistance, e.g. by the Lpad.
The box can be either
of a closed box, bassreflex box, 1ported band pass box or a 2ported band
pass box.
The closed box is
simulated by an acoustic compliance C_{AV} simulating the cavity and an
acoustic resistance R_{AV} corresponding to resistive losses within the
box, e.g. from damping material. Two masses M_{AL} corresponding to the
cooscillating air on the in and outside of the box are included, as well as
one radiation resistance R_{AL}. M_{AL} and R_{AL} are
derived from the radiation impedance of a pulsating sphere. The power
dissipated in the radiation resistance corresponds to the radiated acoustic
power of the system. The component values are calculated as follows:
_{} _{}
where
V_{b} is
the box volume
Q_{b} is
the Qvalue that would occur if f_{s} was determined by C_{AV} and
R_{AV} was the only loss
r_{s} is
the equivalent radius of the pulsating halfsphere.
The closed box and the equivalent circuit diagram of its acoustic load.
The simplified
transfer function of the closed box design behaves like a second order high
pass filter with a slope of 12 dB/octave at low frequencies.
The bassreflex box is
simulated using the same connection of the acoustic compliance C_{AV}
and damping R_{AV} as for the closed box. Part of the flow into the box,
passes out through the tube, and thus the vent is connected in parallel with C_{AV}
and R_{AV}. The radiation resistance is connected in the box branch
since the net flow into the surroundings is Q_{s}Q_{B}, ie the
difference between the flow out of the loudspeaker element and the flow out of
the vent. This net flow forms the useful flow of the system, and thus the
radiation resistance is connected there. In case of the more advanced vent
model, the third connection of the vent ensures that Q_{s}Q_{B}
rather than Q_{s}Q_{A} flows through the radiation resistance,
see diagram. The radiation resistance is now calculated using the radius of the
loudspeaker, since the loudspeaker delivers the major part of the flow at high
frequencies. The part of the radiation resistance that contains the radiator
radius is only important towards higher frequencies.
The bassreflex box and the equivalent circuit diagram of its acoustic load.
The simplified transfer
function of the bassreflex design behaves like a fourthorder high pass filter
with a slope of 24 dB/octave at low frequencies.
The simulation of the
1ported band pass box is similar to the bassreflex box, but the radiation
resistance is moved to the flow coming out of the vent (Q_{B}) and an
extra cavity and damping represented by C_{AV2} and R_{AV2} is
added. There are two main differences from the bassreflex box; since Q_{B}
now determines the flow to the surrounding air, the radiation resistance is
moved to this branch. Also, C_{AV2} and R_{AV2} add extra
compliance and resistance to the loudspeaker element, and thus provide an extra
possibility for the designer to affect the response of the system.
The radiation
resistance is now calculated using the radius of the port, rather than the
radius of the loudspeaker.
_{} _{}
The 1ported band pass box and the equivalent circuit diagram of its acoustic load.
The simplified
transfer function of the 1ported design behaves like a fourthorder band pass
filter with slopes of 12 dB/octave at low frequencies and 12 dB/octave at high
frequencies.
The simulation of the
2ported band pass box is similar to that of the 1ported band pass box, but
also has a vent in the second cavity. The second vent is connected in parallel
with C_{AV2} and R_{AV2} and forms a symmetrical diagram,
corresponding to the symmetrical design of the box. However, the two vents must
be tuned to different frequencies. Just as the vent in the bassreflex design
provides the advantage of an extended lowfrequency response compared to the
closed box design, the second vent in the 2ported design provides an extended
lowfrequency response as compared to the 1ported box.
The radiation
resistance is now calculated using the radius of the Vent1, so the highest
helmholtz frequency should be assigned to Vent1, in this way this vent will
dominate the radiation at higher frequencies.
The 2ported band pass box and the equivalent circuit diagram of its acoustic load.
The simplified
transfer function of the 2ported design behaves like a sixthorder band pass
filter with slopes of 24 dB/octave at low frequencies and 12 dB/octave at high
frequencies.
The tube vent can be modelled as a lumped mass or as a tube. The lumped mass model is accurate enough for low frequencies, and provides quick calculation of the response curves. The tube model provides extra information regarding resonances that occur in the tube, typically when tube the length corresponds to multiples of l/2. The vent can also be realised as a passive radiator.
Note that the cooscillating air M_{ALP} is included in the model of the vent, but that the radiation resistance is treated separately in the other parts of the analogue circuit diagrams.
_{}
Symbol for the vent model used in Basta! It symbolises the vent mass and losses, and a compliance distributed along these. The vent can be modelled as a lumped mass, a tube or as a passive radiator.
The lumped model is
activated by setting the number of sections of a vent to zero. In this case the
vent is modelled by a mass M_{AP} and a resistance R_{AP}, the flow
out of the tube is the same as the flow into the tube and thus the air in the
tube is considered as being incompressible.
_{}
The equations for the acoustic mass and resistance of the port. Note that two end corrections first are removed from the mass, to model the air contained inside the tube. Thereafter the better model is added, this yields a better precision at higher frequencies.
The lumped mass model and the equivalent circuit diagram of its acoustic properties.
In this case the air
inside the vent is assumed to be incompressible and thus if air that flows into
the vent at one end the same amount immediately flows out of the other end.
This model is fast and accurate for low frequencies.
If the air inside
the tube is allowed be compressed, the model can simulate standing waves
("pipe resonances") within the tube. In its simplest form, the mass
and resistance are split in two, and an acoustic compliance C_{AP}
corresponding to the volume of air inside the tube is connected in between the
two halves. This model can in principle simulate the first pipe resonance, but
the resonance frequency will come out slightly too low.
_{}
Setting the number
of sections of the vent to one activates this simple tube model. The difference
between incoming flow Q_{A} and outgoing flow Q_{B} represents
the compression of the air inside the tube and flows out of the third branch.
The simple tube model of the vent and the equivalent circuit diagram of its acoustic properties.
The tube model can
be expanded to simulate higher order resonances as well. In this case the tube
is split in several consecutive sections according to the figure. A higher
number of tube sections will make the calculation process slower, but will
increase the accuracy of the resonance frequencies.
The generalised tube model of the vent and the equivalent circuit diagram of its acoustic properties.
The vent can also be
realised as a passive radiator. In this case, no tube resonances will appear.
An extra compliance C_{ASP} is added, originating from the suspension
of the cone. Just as for the lumped model the flow into the inside the radiator
is the same as the flow out of the outside. Contrary to the lumped model of a
tube, this model is accurate also for higher frequencies, as there are no tube
resonances.
The passive radiator model of the vent and the equivalent circuit diagram of its acoustic properties.
Basta! can simulate acoustic
coupling of multiple loudspeaker elements, both in parallel and isobaric
operation. By connecting n elements in parallel the maximum sound pressure of
the system is increased a factor n^{2}, or 6 dB for two loudspeakers,
12 dB for four loudspeakers, etc. The efficiency is increased a factor n in the
same configuration, but the box volume must be increased a factor n in order to
maintain approximately the same frequency response as for the single
loudspeaker system. As an alternative the loudspeakers may be mounted in the
isobaric configuration. Using this configuration for two loudspeakers, the box
volume can be halved, at the cost of a halved efficiency. However, since the
electrical power handling capacity is doubled, and the maximum cone excursion remains
the same, the maximum output sound pressure also remains the same.
In Basta! the
electrical connection of the loudspeakers is expressed as the number of
loudspeakers that are connected in series. They are always connected in such a
way that each driver receives the same voltage. For example, if six drivers are
used and three are connected in series, two such branches of three drivers are
connected in parallel.
For multiple elements,
the equivalent V_{as}, T, S_{s}, R_{E} and L_{E}
values of the combined driver are derived from the single element. Given that
a is the number of elements that are
connected in series, electrically,
b is 1 for parallel configuration 2 for
isobaric configuration and
n is the total number of elements,
the new values are
calculated as:
_{} _{}
where subindex
"1" corresponds to the parameter of a single loudspeaker element.
b=1, n=2 
b=2, n=2 
b=2, n=4 



Acoustic connection of loudspeakers for some configurations. To the left, the loudspeakers are connected in parallel configuration, middle; isobaric configuration and to the right a combination of parallel and isobaric configuration.
a=2, b=1, n=2 
a=2, b=2, n=2 
a=1, b=1, n=2 



a=1, b=2, n=2 
a=3, b=1, n=6 



Electrical connection of some configurations. Note the reversed polarity for half of the elements in isobaric configuration.
When a driver is mounted
on a baffle, the driver will roughly radiate in half space at high frequencies,
but in full space at low frequencies. The result of this is an increase of 6 dB
of the high frequencies. The response curve, starting at 0 dB at low
frequencies and ending at + 6 dB at high frequencies, is commonly called the baffle
step.
Basta! can model the
baffle step, and uses a simplified version of the Geometric Theory of
Diffraction (GTD). In short, a number of secondary sources are placed around the
edge of the baffle, each having an amplitude and phase shift depending on the
baffle shape. The resulting baffle step is thereafter added to the other
response curves from Basta!.
The room gain is
represented by two poles and two zeroes and is added to the different response
curves. The red explanatory curve below illustrates a pole pair at 20 Hz, Q=5
and a zero pair at 100 Hz, Q=5. Normally lower Q values are used; the default
(black) curve has a smooth lift of the response towards lower frequencies.
Basta! implements
three commonly found design equations for vented boxes. These suggest the box
volume and the vent tuning. The equations are:
Öhman:
Keele:
Margolis/Small:
For the closed box,
the box volume that results in Butterworth response under free field conditions
(ie Q=0.7071) can be suggested from
Basta! allows for
calculation of the maximum output level of the system. It is calculated based
on these limits:
Maximum peak cone excursion
Maximum electric RMS power in R_{E}
Maximum RMS voltage from the power amplifier
Maximum RMS velocity in the vent(s)
Maximum RMS excursion of the vent(s)
The maximum output
level is the highest level at which none of these limits are exceeded.
Measured data can be
imported from Tolvan Data Sirp files or from text files. Text files should have
the following syntax:
[<any text>…]
<freq header><colsep>…<amp
header><colsep>…<phase header>[<colsep>[<any text>]]
<freq><colsep>…<amp><colsep>…<phase>[<colsep>[<any
text>]]
…
where <any text>,
<freq header>, <amp header> and <phase header> are any text strings.
<colsep> is the
column separator character, which can be tab, semicolon, space or comma.
<freq>,
<amp>, <phase> are numerical strings, possibly containing a decimal
separator. The decimal separator can be a period (.), a comma (,) or the
character set in windows as the decimal separator. The numerical value can be
on a linear scale (ohm or pascal) or on a logarithmic scale (dB relative to 20
µPa).
An example file containing
only five points may look like this:
This is the response
of LS12345
Measured 20090103
By Albert Einstein
Freq;Amp;Phase
20.0;70.2;178.1
180;82.7;44.2
1393;84.2;12.3
5304;79.4;40
20000;76.2;160.2
Make sure to adjust
the settings appropriately in the import dialog.
Curve 
Unit 
Note 
Explanation 
System response 
dB 
@ 1 m re
20 mPa 
Sound pressure level
as it would be measured straight in front of the loudspeaker 
Max output level
(MOL) 
dB 
@ 1 m re
20 mPa 
Maximum possible SPL
straight in front of the loudspeaker 
Speaker voltage 
V 

The voltage across
the speaker terminals 
Speaker voltage at
MOL 
V 

The voltage across the
speaker terminals required to reach MOL 
Amplifier voltage 
V 

The voltage across
the amplifier terminals 
Amplifier voltage at
MOL 
V 

The voltage across
the amplifier terminals required to reach MOL 
Box (2) pressure 
dB 
re 20 mPa 
The sound pressure
inside the box. To measure this pressure with a microphone and compare it
with the Basta! simulation can be a way to verify the response of a system,
without having access to an anechoic chamber. 
Box (2) pressure at
MOL 
dB 
re 20 mPa 
The sound pressure
inside the box at max output level. This SPL is commonly very high, typically
140160 dB. 
Speaker response 
dB 
@ 1 m re
20 mPa 
The part of the
sound pressure level originating from the loudspeaker element. 
Cone excursion 
mm 

The RMS movement of
the loudspeaker cone 
Cone velocity 
m/s 

The RMS velocity of
the loudspeaker cone 
Cone excursion at
MOL 
mm 

The RMS movement of
the loudspeaker cone at max output level 
Cone velocity at MOL 
m/s 

The RMS velocity of
the loudspeaker cone at max output level 
Vent (2) response 
dB 
@ 1 m re
20 mPa 
The part of the sound pressure level
originating from the vent as it would be measured straight in front of the
speaker. 
Vent (2) excursion 
mm 

The RMS movement of
the vent 
Vent (2) velocity 
m/s 

The RMS velocity of
the vent 
Vent (2) excursion
at MOL 
mm 

The RMS movement of
the vent at max output level 
Vent (2) velocity at
MOL 
m/s 

The RMS velocity of
the vent at max output level 
Speaker baffle step 
dB 
94 dB added. 
The baffle step that
results from the dimensions of the baffle and the driver placement. 
Vent (2) baffle step 
dB 
94 dB added. 
The baffle step that
results from the dimensions of the baffle and the vent placement. Usually
this curve is of little practical concern, since the vent mostly radiates
well below the frequencies where the baffle step is active. 
Electrical impedance 
W 

This impedance is
calculated as u_{a}/i_{a}, thus excluding the ACbass
circuit, but including the passive
electric circuit. 
Electrical
inductance 
mH 

The reactive part of
the electrical impedance, seen as an inductance ie divided by w. This curve can be useful if the
manufacturer has specified the voice coil inductance as the inductance at two
different frequencies. 
Electrical
resistance 
W 

The resistive part
of the electrical impedance 
Electrical reactance 
W 

The reactive part of
the electrical impedance 
R_{E} power
margin 
dB 

The level increase
allowed to reach the power limit in R_{E}. 
Cone excursion
margin 
dB 

The level increase
allowed to reach the cone excursion limit. 
Vent (2) excursion
margin 
dB 

The level increase
allowed to reach the vent excursion limit. 
Vent (2) velocity
limit 
dB 

The level increase
allowed to reach the vent velocity limit. 
Overall margin 
dB 

The level increase allowed
without exceeding any of the limits. This curve is the difference between the
MOL and system response curves. 
Room gain 
dB 

The room gain (bass
lift) approximation as selected on the room gain tab. 
Linkwitz transform 
dB 

The response of the
Linkwitz transform circuit as selected on the Linkwitz transform tab. 
<xxx> 
dB 

The response of a
measured system, using the response curve <xxx> on the measured response
tab. 
Note: If the baffle
step is not enabled, all sound pressure levels outside the box are based on
that the loudspeaker is mounted in a wall or floor and thus radiates in half
space (2p). If the baffle step is enabled, the loudspeaker box is assumed to radiate
in free field.
Symbol 
Explanation 
R_{ac} 
Resistance determining the resistive losses
in the ACbass system 
R_{acneg} 
Negative output
resistance of the ACbass circuit 
L_{ac} 
Inductance of the ACbass
circuit. Determines the effective compliance of the ACbass system 
C_{ac} 
Capacitance of the
ACbass circuit. Determines the effective mass of the ACbass system 
L_{L1},L_{L2} 
Inductors in passive
lowpass filter 
C_{L1},C_{L2} 
Capacitors in passive
lowpass filter 
R_{L1},R_{L2} 
Resistances in
passive lowpass filter, eg in the inductors 
L_{H1},L_{H2} 
Inductors in passive
highpass filter 
C_{H1},C_{H2} 
Capacitors in
passive highpass filter 
R_{H1},R_{H2} 
Resistances in passive
highpass filter, eg in the inductors 
R_{s},R_{p} 
Series and parallel
resistances for passive attenuation 
R_{EC},C_{EC} 
Resistance and
capacitance of the conjugate link 
R_{E} 
DC resistance of the
voice coil 
L_{E} 
Inductance of the
voice coil 
n 
Loss factor of the
voice coil inductance 
T 
Force factor of the
loudspeaker. Also known as Bl 
R_{MS},C_{MS},M_{MS} 
Mechanical
resistance, compliance and mass of the loudspeaker. M_{MS} does not
include any cooscillating air. 
S_{s} 
Equivalent piston area
of the loudspeaker cone 
f_{s} 
Resonance frequency
of the loudspeaker in free air 
V_{as} 
Equivalent volume of
the loudspeaker. This volume would give a compliance equal to C_{MS},
given the equivalent piston area S_{s}. 
Q_{ts} 
Total Qvalue of the loudspeaker in free air,
and zero series resistance. 
r_{r} 
The radius of a
circular membrane that corresponds to S_{s}. 
Z_{E} 
The impedance of the voice coil, neglecting
the effects of the mechanical system. 
M_{ALS}, M_{ALP},
M_{ALP2} 
The acoustic mass of the cooscillating air
for the loudspeaker, first vent and second vent. This mass corresponds to the
reactive part of the radiation impedance. 
R_{AL} 
The radiation
resistance. 























