# Engineering Acoustics/Transducers - Loudspeaker

# Acoustic transducer

The purpose of an acoustic transducer is to convert electrical energy
into acoustic energy. Many variations of acoustic transducers exist,
such as electrostatic, balanced armature and moving-coil loudspeakers.
This article focuses on moving-coil loudspeakers since they are the most
commonly used type of acoustic transducer. First, the physical
construction and principle of a typical moving coil transducer are
discussed briefly. Second, electro-mechano-acoustical modeling of each
element composing the loudspeaker is presented in a tutorial way to
reinforce and supplement the theory on electro-mechanical
analogies
and electro-acoustic
analogies
previously seen in other sections. Third, the equivalent circuit is
analyzed to introduce the theory behind Thiele-Small parameters, which
are very useful when designing loudspeaker enclosures. A method to
experimentally determine Thiele-Small parameters is also included.

## Moving-coil loudspeaker construction and principle

The classic moving-coil loudspeaker driver can be divided into three key
components:

1\) The magnet motor drive system, comprising the permanent magnet, the
center pole and the voice coil acting together to produce a mechanical
force on the diaphragm from an electrical current.

2\) The loudspeaker cone system, comprising the diaphragm and dust cap,
permitting mechanical force to be translated into acoustic pressure;

3\) The loudspeaker suspension, comprising the spider and surround,
preventing the diaphragm from breaking due to over excursion, allowing
only translational movement and tending to bring the diaphragm back to
its rest position.

The following illustration shows a cut-away view of a typical moving
coil-permanent magnet loudspeaker. A coil is mechanically coupled to a
diaphragm, also called cone, and rests in a fixed magnetic field
produced by a magnet. When an electrical current flows through the coil,
a corresponding magnetic field is emitted, interacting with the fixed
field of the magnet and thus applying a force to the coil, pushing it
away or towards the magnet. Since the cone is mechanically coupled to
the coil, it will push or pull the air it is facing, causing pressure
changes and emitting a sound wave.

```{=html}
<center>
```
![](MovingCoilLoudspeaker.png "MovingCoilLoudspeaker.png"){width="800"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 1: A cross-sectional view of a typical moving-coil loudspeaker

```{=html}
</center>
```
An equivalent circuit can be obtained to model the loudspeaker as a
lumped system. This circuit can be used to drive the design of a
complete loudspeaker system, including an enclosure and sometimes even
an amplifier that is matched to the properties of the driver. The
following section shows how such an equivalent circuit can be obtained.

## Electro-mechano-acoustical equivalent circuit

Electro-mechanico-acoustical systems such as loudspeakers can be modeled
as equivalent electrical circuits as long as each element moves as a
whole. This is usually the case at low frequencies or at frequencies
where the dimensions of the system are small compared to the wavelength
of interest. To obtain a complete model of the loudspeaker, the
interactions and properties of electrical, mechanical, and acoustical
subsystems composing the loudspeaker driver must each be modeled. The
following sections detail how the circuit may be obtained starting with
the amplifier and ending with the acoustical load presented by air. A
similar development can be found in \[1\] or \[2\].

### Electrical subsystem

The electrical part of the system is composed of a driving amplifier and
a voice coil. Most amplifiers can be approximated as a perfect voltage
source in series with the amplifier output impedance. The voice coil
exhibits an inductance and a resistance that may be directly modeled as
a circuit.

```{=html}
<center>
```
![](ElectricalSubsystemLoudspeaker.png "ElectricalSubsystemLoudspeaker.png"){width="400"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 2: The amplifier and loudspeaker electrical elements modeled as a
circuit

```{=html}
</center>
```
### Electrical to mechanical subsystem

When the loudspeaker is fed an electrical signal, the voice coil and
magnet convert current to force. Similarly, voltage is related to the
velocity. This relationship between the electrical side and the
mechanical side can be modeled by a transformer.

$\tilde{f_c} = Bl \tilde{i}$; $\tilde{u_c} = \dfrac{\tilde{e}}{Bl}$

```{=html}
<center>
```
![](_ElectricalToMechanicalLoudspeaker.png "_ElectricalToMechanicalLoudspeaker.png")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 3: A transformer modeling transduction from the electrical
impedance to mechanical mobility analogy

```{=html}
</center>
```
### Mechanical subsystem

In a first approximation, a moving coil loudspeaker may be thought of as
a mass-spring system where the diaphragm and the voice coil constitute
the mass and the spider and surround constitute the spring element.
Losses in the suspension can be modeled as a resistor.

```{=html}
<center>
```
![](MechanicalSubsystemModelingLoudspeaker.png "MechanicalSubsystemModelingLoudspeaker.png"){width="1000"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 4: Mass spring system and associated circuit analogies of the
impedance and mobility type.

```{=html}
</center>
```
The equation of motion gives us :

```{=html}
<center>
```
$\tilde{f_c} = R_m \tilde{u_c} + \dfrac{\tilde{u_c}}{ j \omega C_{MS}} + j \omega M_{MD} \tilde{u_c}$

```{=html}
</center>
```
```{=html}
<center>
```
$\dfrac{\tilde{f_c} }{\tilde{u_c}}= R_m+\dfrac{1}{ j \omega C_{MS}}+ j\omega M_{MD}$

```{=html}
</center>
```
Which yields the mechanical impedance type analogy in the form of a
series RLC circuit. A parallel RLC circuit may also be obtained to get
the mobility analog following mathematical manipulation:

```{=html}
<center>
```
$\dfrac{\tilde{u_c} }{\tilde{f_c}}= \dfrac{1}{R_m+\dfrac{1}{ j \omega C_{MS}}+ j\omega M_{MD} }$

```{=html}
</center>
```
```{=html}
<center>
```
$\dfrac{\tilde{u_c} }{\tilde{f_c}}= \dfrac{1}{\dfrac{1}{G_m}+\dfrac{1}{ j \omega C_{MS}} + \dfrac{1}{\dfrac{1}{j \omega M_{MD}}}}$

```{=html}
</center>
```
Which expresses the mechanical mobility type analogy in the form of a
parallel RLC circuit where the denominator elements are respectively a
parallel conductance, inductance, and compliance.

### Mechanical to acoustical subsystem

A loudspeaker's diaphragm may be thought of as a piston that pushes and
pulls on the air facing it, converting mechanical force and velocity
into acoustic pressure and volume velocity. The equations are as follow:

$\tilde{P_d} = \dfrac{\tilde{f_c}}{\tilde{S_D}}$;
$\tilde{U_c} = \tilde{u_c}{S_D}$

These equations can be modeled by a transformer.

```{=html}
<center>
```
![](MechanicalToAcousticalLoudspeaker.png "MechanicalToAcousticalLoudspeaker.png")

```{=html}
</center>
```
```{=html}
<center>
```
Figure 5: A transformer modeling the transduction from mechanical
mobility to acoustical mobility analogy performed by a loudspeaker\'s
diaphragm

```{=html}
</center>
```
### Acoustical subsystem

The impedance presented by the air load on the loudspeaker\'s diaphragm
is both resistive due to sound radiation and reactive due to the air
mass that is being pushed radially but does not contribute to sound
radiation to the far field. The air load on the diaphragm can be modeled
as an impedance or an admittance. Specific values and approximations can
be found in \[1\], \[2\] or \[3\]. Note that the air load depends on the
mounting conditions of the loudspeaker. If the loudspeaker is mounted in
a baffle, the air load will be the same on each side of the diaphragm.
Then, if the air load on one side is $Y_{AR}$ in the admittance analogy,
then the total air load is $Y_{AR}/2$ as both loads are in parallel.

### Complete electro-mechano-acoustical equivalent circuit

Using electrical impedance, mechanical mobility and acoustical
admittance yield the following equivalent circuit, modeling the entire
loudspeaker drive unit.

```{=html}
<center>
```
![](_LoudspeakerEquivalentCircuit.png "_LoudspeakerEquivalentCircuit.png"){width="1000"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 6: A complete electro-mechano-acoustical equivalent circuit of a
loudspeaker drive unit

```{=html}
</center>
```
This circuit can be reduced by substituting the transformers and
connected loads by an equivalent loading that would present the same
impedance as the loaded transformer. An example of this is shown on
figure 7, where acoustical and electrical loads and sources have been
\"brought over\" to the mechanical side.

```{=html}
<center>
```
![](LoudspeakerEquivalentCircuitMechanical.png "LoudspeakerEquivalentCircuitMechanical.png"){width="900"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 7: Mechanical equivalent circuit modeling of a loudspeaker drive
unit

```{=html}
</center>
```
The advantage of doing such manipulations is that we can then directly
relate electrical measurements with elements in the circuit. This will
later allow us to obtain values for the different components of the
model and match this model to real loudspeaker drivers. We can further
simplify this circuit by using Norton\'s theorem and converting the
series electrical components and voltage source into an equivalent
current source and parallel electrical components. Then, using a
technique called the Dot method, presented in section Solution Methods:
Electro-Mechanical
Analogies,
we can obtain a single loop series circuit which is the dual of the
parallel circuit previously obtained with Norton\'s theorem. If we are
mainly interested in the low frequency behavior of the loudspeaker, as
should be the case when using lumped element modeling, we can neglect
the effect of the voice coil inductance, which has an effect only at
high frequencies. Furthermore, the air load impedance at low frequencies
is mass-like and can be modeled by a simple inductance $M_{M1}$. This
results in a simplified low frequency model equivalent circuit, shown of
figure 8, which is easier to manipulate than the circuit of figure 7.
Note that the analogy used for this circuit is of the impedance type.

```{=html}
<center>
```
![](LowFrequencyLoudspeakerEquivalentCircuitMechanical.png "LowFrequencyLoudspeakerEquivalentCircuitMechanical.png"){width="800"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 8: Low frequency approximation mechanical equivalent circuit of a
loudspeaker drive unit

```{=html}
</center>
```
Where $M_{M1} = 2.67 a^3 \rho$ if $a$ is the radius of the loudspeaker
and $\rho$, the density of air. Mass elements, in this case the mass of
the diaphragm and voice coil $M_{MS}$ and the air mass loading the
diaphragm $2M_{M1}$ can be regrouped in a single element:

```{=html}
<center>
```
$M_{MS} = M_{MD}+2M_{M1}$

```{=html}
</center>
```
## Thiele-Small Parameters

### Theory

The complete low frequency behavior of a loudspeaker drive unit can be
modeled with just six parameters, called Thiele-Small parameters. Most
of these parameters result from algebraic manipulation of the equations
of the circuit of figure 8. Loudspeaker driver manufacturers seldom
provide electro-mechano-acoustical parameters directly and rather
provide Thiele-Small parameters in datasheets, but conversion from one
to the other is quite simple. The Thiele-Small parameters are as follow:

1\. $R_e$, the voice coil DC resistance;

2\. $Q_{ES}$, the electrical Q factor;

3\. $Q_{MS}$, the mechanical Q factor;

4\. $f_s$, the loudspeaker resonance frequency;

5\. $S_D$, the effective surface area of the diaphragm;

6\. $V_{AS}$, the equivalent suspension volume: the volume of air that
has the same acoustic compliance as the suspension of the loudspeaker
driver.

These parameters can be related directly from the low frequency
approximation circuit of figure 8, with $R_e$ and $S_D$ being explicit.

$Q_{MS} = \dfrac{1}{R_{MS}} \sqrt{\dfrac{M_{MS}}{C_{MS}}}$;
$Q_{ES} = \dfrac{R_g + R_e }{(Bl)^2} \sqrt{\dfrac{M_{MS}}{C_{MS}}}$ ;
$f_s= \dfrac{1}{2\pi\sqrt{M_{MS}C_{MS}}}$; $V_{AS}= C_{MS}S_D^2\rho c^2$

Where $\rho c^2$ is the Bulk modulus of air. It follows that, if given
Thiele-Small parameters, one can extract the values of each component of
the circuit of figure 8 using the following equations :

$C_{MS} = \dfrac{V_{AS}}{S_D^2 \rho c^2}$;
$M_{MS}= \dfrac{1}{(2 \pi f_s)^2 C_{MS}}$;
$R_{MS} = \dfrac{1}{Q_{MS}}\sqrt{\dfrac{M_{MS}}{C_{MS}}}$;
$Bl = \sqrt{\dfrac{R_e}{2 \pi f_s Q_{ES} C_{MS} } }$;
$M_{MD}= M_{MS}-2M_{M1}$;

### Measurement

Many methods can be used to measure Thiele-Small parameters of drivers.
Measurement of Thiele-Small parameters is sometimes necessary if a
manufacturer does not provide them. Also, the actual Thiele-Small
parameters of a given loudspeaker can differ from nominal values
significantly. The method described in this section comes from \[2\].
Note that for this method, the loudspeaker is considered to be mounted
in an infinite baffle. In practice, a baffle with a diameter of four
times that of the loudspeaker is sufficient. Measurements without a
baffle are also possible: the air mass loading will simply be halved and
can be easily accounted for. The setup for this method includes an FFT
analyzer or a mean to obtain an impedance curve. A signal generator of
variable frequency and an AC meter can also be used.

```{=html}
<center>
```
![](Setup_to_measure_impedance_of_loudspeakers.png "Setup_to_measure_impedance_of_loudspeakers.png"){width="300"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 9: Simple experimental setup to measure the impedance of a
loudspeaker drive unit

```{=html}
</center>
```
```{=html}
<center>
```
$Z_{spk}= R\dfrac{V_{spk}}{V_s \left( 1-\dfrac{V_{spk}}{V_s} \right)}$

```{=html}
</center>
```
```{=html}
<center>
```
![](ImpedanceCurveLoudspeaker.png "ImpedanceCurveLoudspeaker.png"){width="600"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 10: A typical loudspeaker drive unit impedance curve

```{=html}
</center>
```
Once the impedance curve of the loudspeaker is measured, $R_e$ and $f_s$
can be directly identified by looking at the low frequency asymptote of
the impedance value and the center frequency of the resonance peak. If
the frequencies where $Z_{spk}=\sqrt{R_e R_c}$ are identified as $f_l$
and $f_h$, Q factors can be calculated.

```{=html}
<center>
```
$Q_{MS}= \dfrac{f_s}{f_h-f_l} \sqrt{\dfrac{R_c}{R_e}}$

```{=html}
</center>
```
```{=html}
<center>
```
$Q_{ES} = \dfrac{Q_{MS}}{\dfrac{R_c}{R_e}-1}$

```{=html}
</center>
```
$S_D$ can simply be approximated by $\pi a^2$, where $a$ is the radius
of the loudspeaker driver. The last remaining Thiele-Small parameter,
$V_{AS}$ is slightly trickier to measure. The idea is to either increase
mass or reduce compliance of the loudspeaker drive unit and note the
shift in resonance frequency. If a known mass $M_x$ is added to the
loudspeaker diaphragm, the new resonance frequency will be:

```{=html}
<center>
```
$f^'_s= \dfrac{1}{2 \pi \sqrt{(M_{MS} + M_x) C_{MS} } }$

```{=html}
</center>
```
And the equivalent suspension volume may be obtained with:

```{=html}
<center>
```
$V_{AS} =  \left( 1- \dfrac{f^{'2}_s}{f_s^2} \right) \dfrac{S_D^2 \rho c^2}{(2 \pi f^'_s)^2 M_x}$

```{=html}
</center>
```
Hence, all Thiele-Small parameters modeling the low frequency behavior
of the loudspeaker drive unit can be obtained from a fairly simple
setup. These parameters are of tremendous help in loudspeaker enclosure
design.

### Numerical example

This section presents a numerical example of obtaining Thiele-Small
parameters from impedance curves. The impedance curves presented in this
section have been obtained from simulations using nominal Thiele-Small
parameters of a real woofer loudspeaker. Firsy, these Thiele-Small
parameters have been transformed into an electro-mechano-acoustical
circuit using the equation presented before. Second, the circuit was
treated as a black box and the method to extract Thiele-Small parameters
was used. The purpose of this simulation is to present the method, step
by step, using realistic values so that the reader can get more familiar
with the process, the magnitude of the values and with what to expect
when performing such measurements.

For this simulation, a loudspeaker of radius $a=6.55cm$ is mounted on a
baffle sufficiently large to act as an infinite baffle. Its impedance is
obtained and plotted in figure 11, where important cursors have already
been placed.

```{=html}
<center>
```
![](SimulatedImpedanceWoofer.png "SimulatedImpedanceWoofer.png"){width="1000"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 11: Simulated measurement of an impedance curve for a woofer
loudspeaker

```{=html}
</center>
```
The low frequency asymptote is immediately identified as $Re=6.6\Omega$.
The resonance is clear and centered at $fs=33Hz$. The value of the
impedance at this frequency is about $66\Omega$. This yields
$\sqrt{R_eR_c}=20.8\Omega$, which occurs at $f_l=19.5 Hz$ and
$f_h= 52.5 Hz$. With this information, we can compute some of the
Thiele-Small parameters.

```{=html}
<center>
```
$Q_{MS}= \dfrac{f_s}{f_h-f_l}\sqrt{\dfrac{R_c}{R_e}}=\dfrac{33}{52.5-19.5}*\sqrt{\dfrac{66}{6.6}}= 3.1$

```{=html}
</center>
```
```{=html}
<center>
```
$Q_{ES} = \dfrac{Q_{MS}}{\dfrac{R_c}{R_e}-1} = \dfrac{3.1}{\dfrac{66}{6.6}-1} = 0.35$

```{=html}
</center>
```
As a next step, a mass of $M_x=10g$ is fixed to the loudspeaker
diaphragm. This shifts the resonance frequency and yields a new
impedance curve, as shown on figure 12.

```{=html}
<center>
```
![](SimulatedImpedanceWooferAddedMass.png "SimulatedImpedanceWooferAddedMass.png"){width="1000"}

```{=html}
</center>
```
```{=html}
<center>
```
Figure 12: Simulated measurement of an impedance curve for a woofer
loudspeaker

```{=html}
</center>
```
```{=html}
<center>
```
$S_{D} =  \pi a^2 = 0.0135 m^2$

```{=html}
</center>
```
```{=html}
<center>
```
$V_{AS} =  \left( 1- \dfrac{27.5^2}{33^2} \right) \dfrac{0.0135^2*1.18*344^2}{(2 \pi 27.5)^2*0.01}= 0.0272 m^3$

```{=html}
</center>
```
Once all six Thiele-Small parameters have been obtained, it is possible
to calculate values for the electro-mechano-acoustical circuit modeling
elements of figure 6 or 7. From then, the design of an enclosure can
start. This is discussed in application sections Sealed box subwoofer
design
and Bass reflex enclosure
design.

# References

\[1\] Kleiner, Mendel. Electroacoustics. CRC Press, 2013.

\[2\] Beranek, Leo L., and Tim Mellow. Acoustics: sound fields and
transducers. Academic Press, 2012.

\[3\] Kinsler, Lawrence E., et al. Fundamentals of Acoustics, 4th
Edition. Wiley-VCH, 1999.

\[4\] Small, Richard H. \"Direct radiator loudspeaker system analysis.\"
Journal of the Audio Engineering Society 20.5 (1972): 383-395.
