PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Modeling of a Compression Driver using Lumped Elements
Joerg PANZER1
R&D Team, Germany

ABSTRACT
Modeling linear response functions of a typical electro-dynamic compression driver with the help of
lumped elements should be an easy task. However, it turns out that acoustic channels and slits inside such a
device can go into strong coupling with the fundamental mechanical resonance. This paper demonstrates anextended electro-mechano-acoustical network and compares its response functions to measurement.
Keywords: Electro-acoustic, Lumped Element, Compression Driver

INTRODUCTION
Modeling is a calculation technique, which in our context yields transfer functions in thefrequency domain such as the sound pressure response or the electrical driving point impedance.
Modeling with lumped parameters involves using building blocks or symbols, which represent
dominant properties of the device, such as the mass and stiffness of a fundamental resonator.
Lumped elements result from forming mean-values of parameters such as potentials and flows bymeans of spatial integration. For example, the lumped pressure inside a waveguide can be regarded
as a mean-value of the cross-section. In the frequency range, where the wavelength is much larger
than the device, the lumped parameter features an approximation of this mean-value. In themechanic domain lumped parameters refer to the rigid body motion in a certain direction.

Electrical domain

Mechanical domain

Acoustical domain

Figure 1 - Lumped network parameters

A lumped parameter is either a potential or a flow. There are the voltage V, the current I, thevelocity v, the force F, the sound-pressure p and the volume-velocity q. The lumped parameters are
selected in such a way that the product yields power: Pe = 1/ 2⋅V ⋅ I ,Pm = 1/2⋅ v ⋅ F ,Pa = 1/2 ⋅ p ⋅q or in more general terms: P = 1 /2 ⋅ potential ⋅ flow . Pe, Pm and P a are the electric,mechanic and acoustic power. Figure 1 graphs the Kirchhoff rule, which says that the sum of flows
should be zero. In between the electrical, mechanical and acoustical domains there are transducers.
A compression driver is an electro-acoustic transducer. It converts electric energy into an
acoustic one. The electric terminals are typically connected to the output stage of an amplifier or afilter-network of low impedance. The acoustic outlet drives a waveguide, which can be a large hornor a beam-forming device. A symbolic network is shown in figure 2. This paper reports on theprocess of modeling with the help of mainly lumped elements. We want to write down the string ofthoughts, which leads to particular design steps of the model.

1 j.panzer@randteam.de7280

Figure 2 - Electro-acoustic transducer

Figure 3 - Compression driver cut-view

Modeling Compression Driver
The driver we want to model has an outlet of 1.4 in, which is approx 36 mm. The diameter of themembrane is 86 mm. It has the shape of a dome. Figure 3 displays the circular symmetric cut-view.
The motion of the membrane is in axial direction, which produces sound waves. The inner area ofthe dome S 0 is opposite a range of radial slits which form the entrance to a system of concentric
ducts, which is called phase-plug. Between the membrane and the phase-plug there is a spacing,which should be large enough to allow for membrane excursion. The spacing together with the areaof the membrane S 0 forms the volume V 0 of the so-called compression chamber. The design of thephase-plug and the size of V 0 should be designed carefully in order to provide a large bandwidth of
the sound pressure response at high frequency. Because the effective area into the phase-plug S 3 istypically smaller than the area of the membrane S 0 there is a transformation of flux and pressure
according to the compression ratio = S 0/S 3. The outer area of the dome is not used and typicallyshielded by an enclosure (not shown in figure 1). S1 stands for a membrane area, which encircles the
main-membrane and functions as suspension of the vibrating assembly. Between the suspension andthe main-membrane there is attached the voice coil former. The voice coil consists of a number ofwire-turns arranged in the flux of a magnetic field B in such a way that a force F drives the voicecoil former in axial direction. Between the voice coil former and the iron of the magnetic device a
small path of air is formed. This so-called voice coil slit should be as narrow as possible for reasonsof motor-efficiency. Underneath the suspension there is a small chamber of volume V 1. It is
important to note that volume V 1 and the compression chamber V 0 are acoustically coupled via thevoice coil slit (see also [1,2]). Later, we will see that the response exhibits a high sensitivity to thisacoustic arrangement.
One can argue that the driver should be analyzed when loaded with a plane-wave tube, so that thetest is close to operation of horn-load. The plane-wave-tube would then provide an acoustic load of−1
Z a = ρ ⋅c ⋅ S 4 with radiation impedance Z a = p/q, density of air ρ, speed of sound c and radiationarea S 4 . Of advantage for analysis is that this value would be real and frequency independent.
Hence, the plane-wave-tube-load would be an interesting alternative, and it will be investigated in afurther work.
This paper investigates the behavior of the driver under free-radiation condition, i. e. there is nowaveguide attached at the outlet. The advantage of this particular radiation condition is that there islittle damping due to radiation. This in turn will yield to strong resonance among reactive
components within the driver, which are easy to observe.
In Vacuum
Under no-air condition we can neglect all acoustic components and concentrate on the electricaland mechanical domains. In a mechanical mobility network the potential is the velocity in aparticular direction. The flow is the force. We can apply standard network analysis [3] withimpedance and admittance components:
Y mob =

Mass

Ym = j ω⋅ M

Stiffness

Yk = K / j ω

1
flow
F
=
=
Z mob
potentialv

Dashpot(1)
Yr = R(2)

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The mobility type network facilitates the identification of the symbols. Typically, one looks firstfor the velocities, i. e. one may ask which components share a particular motion. In our simple
example there is only one velocity, which is shared by the mass, the suspension and the dashpot. M
symbolizes the total mass of the vibrating assembly assuming infinite stiffness. It includes the massof the membrane, the suspension and the voice coil former without acoustic load. K is the stiffnessof the suspension in vacuum assuming a linear spring without a mass. R stands for a velocity related
mechanic loss-factor typically caused within the suspension. The schematic is displayed in figure 4.
The mechanic mass is always grounded. The suspension is attached to a very large mass, so largethat we can connect the spring and the dashpot also to ground. The mass and the spring form aresonance at ω s = √ K / M . In loudspeaker jargon we call this the fundamental resonance.

Figure 4 - Mechanical mobility network

Figure 5 - Electro dynamic motor

Our compression driver features an electro-dynamic motor with Lorentz Force F = BL ⋅ I ,induction V e = BL ⋅ v and voice coil impedance Z e. The schematic is displayed in figure 5. At very
low frequency Z e is equivalent to a resistance R e. At higher frequencies the value of R e increases
because of losses caused by eddy currents. The reactance of Z e behaves similar to an inductance[4,5]. For network analysis it is useful to model the motor as an electro-mechanical four-pole withadmittance transfer matrix, which couples the electrical impedance network to the mechanicalmobility network.

[

1[Y ] = 1 ⋅
Ze

]

− BL
BL − BL 2[ ] [][ ]I1V= Y ⋅ 1
F2
v2

(3)

The force of the motor drives the mechanic structure. Its response is a velocity, which in turninduces a voltage at the electric terminals. This coupling can be symbolized by combining the modelof the motor-four-pole with the mechanic circuit shown in figure 6.

Figure 6 - Electro-mechanical network

This network is ready for inspection of transfer-functions. An easy to measure and to calculate
parameter is for example, the electrical driving point function Z 0 = V/I. In order to measure thisfunction the example-driver was placed into a vacuum chamber. Measurements can be used to verify
the model and to identify the electro-mechanical parameters of the components. Figure 7 displays anoverlay of the simulation and measurement curves. The fundamental resonance is at f s = 625 Hz.

Figure 7 - Electrical driving point function of driver in vacuum

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Acoustic Coupling - Simple Model
The mechanical domain is coupled to the acoustical domain with the help of membranes of area
S, which are assumed to vibrate in axial rigid body motion. In the acoustical impedance network thepotential is the sound-pressure p. The flow is the volume-velocity q in a certain direction. As before
we can apply standard network analysis [3] with impedance and admittance components, where
Y imp =

1
flow
q
=
=
Z imp
potentialp

The three basic acoustic elements are:Mass
Y ma = 1/( j ω ⋅ M a )(4)Resistance

Y ra = 1/ Ra(5)Compliance Y ca = j ω⋅Ca
In the simplified model of the compression driver we take into account only the two sides of themain membrane. The particular shape of the membrane is ignored. Each side of the membrane canbe symbolized as a gyrator, which transforms the mechanical mobility network into the acoustical
impedance network as shown in figure 8.

Figure 8 - Mechano-acoustical coupler

Figure 9 - Acoustic four-pole

Because the sound pressure is p = F/S and the volume-velocity is q = v∙S with effective area S[6], the admittance matrix yields:[Y ] =[ ] [][ ][−S0 S0 ]F1v= Y ⋅ 1q2
p2

(6)

Between the membrane and the input to the phase-plug there is the compression chamber. The
actual shape of this chamber is notoriously complicated, because typically there is curvature and a
distribution of slits as sketched in Figure 3. However, at frequencies where the wavelength is muchlarger than these details we can allow for some approximation. For example, the compression
chamber can be modeled with the help of a simple acoustic compliance2

C a = V /( ρ⋅c )(7)with volume V, medium density ρ and speed of sound c. The acoustic compliance is always
grounded.
Because of low frequency approximation we ignore the complex structure of concentric ducts ofannular cross-section of the phase-plug and simply calculate with the cross-section areas. The
symbolic representation of this waveguide is a four-pole as shown in figure 9.
The admittance matrix for a duct of linear rising cross-section area is (following ideas of [7]):[[ Y ] = y 0⋅ y11 y12
y 21

y22

]

2

[] [][ ]q1p= Y ⋅ 1q2
p2

y11 = y0 L⋅(cos (k L) + y 20⋅sinc( k L))

y12 = − y0 L

y21 = − y0 L

y22 = cos ( k L) − y 2 L⋅sinc ( k L)

y0 =

S mo
j⋅ρ c⋅sin ( k L)

sinc ( x) = sin( x )/ x(8)y20 = √( S mo / Sth ) − 1
y0 L = 1/( y 20 + 1)
y 2 L = y 20⋅y 0 L

With wave-number k = ω / c , length of waveguide L, cross-section areas at the throat S th andat the mouth S mo.

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The output of the phase-plug would normally be connected to a horn or similar device. However,
here, in a test-case, we arrange for free radiation, this means there is no further waveguide attachedto the outlet S 4 of the driver. Also, the rear of the membrane radiates free because we do not want
any influence from the rear-enclosure. This condition was found to be optimum for measurement aswell as for the simulation, and it provides low damping of the resonances of the driver.
The assembly of the acoustic network follows the same recipe as in the mechanical or electrical
domains. As a first step we try to identify zones of equal potential which is the sound-pressure. We
assume a large wavelength for approximation. For example, we assume identical sound pressure at
the membrane, the compression chamber and at the inlet of the phase-plug. Hence, all associatedterminals of the acoustic lumped elements are connected.
The acoustic of radiation is modeled with the help of the boundary element method (BEM),
which is coupled to the lumped element part. As a result we should get excellent values for theradiation impedance at the outlet and at the rear membrane. The radiation impedance can becalculated from the mean-value of pressure distribution over the radiator-area. Details are explainedin a dedicated paper [8]. Further the BEM also helps in predicting radiation to the exterior spacebecause it takes into account most of the diffraction and reflection conditions of the measurement
setup. A sketch of the boundary conditions and an example field is shown in figure 10.

Figure 10 - Free radiation. Example of pressure field at 3900 Hz (range 50 dB).

At this stage we have all ingredients available in order to model the whole electro-mechanoacoustical network. Going from left to right we have a voltage source S 1 , which feeds the electrodynamic-motor via a generator-resistance R g . The motor makes the membrane, voice-coil former andthe suspension system vibrate axially with velocity v.

Figure 11 - Electro-mechano-acoustical network - simple modelingIts mass and stiffness go into resonance. However, because of acoustic loading there can be
multiple resonances. One side of the membrane radiates into the phase-plug and from there into theexterior domain. The rear of the membrane radiates directly into the exterior domain. There is alsomutual coupling between the radiators in the exterior domain. We can place a microphone andmeasure the sound pressure response. Results are shown in figure 12.

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Figure 12 - Simple model: Electric driving point impedance andsound pressure level at 180 mm on-axis in front of aperture

Comparing simulation to measurement shows good agreement between our model and the device.
We can say this with confidence because we obtain a good match at two distinct observation
parameters.
Clearly visible are peaks and troughs, which indicate coupled resonances. The peak at 380 Hz
most likely belongs to the resonance, which was at f s = 625 Hz in vacuum (figure 7). The strong
alteration to low frequency is an indication of mass-load. The additional mass is caused by acoustic
elements such as the phase-plug and the radiation impedance. The deviation at very high frequencyof the sound-pressure curves seems to be caused by bending-vibration of the membrane, which is
not taken into account (see below).
Acoustic Coupling - Refined Model
However, there is still a difference between the simulated and the measured curves. It seems that
we need even more mass to take into account for our model. Candidates for acoustic mass are
structures were air is moved, preferably around bends and corners. An additional mass can be foundat the voice-coil-path which connects the acoustic field generated by the suspension and the one
generated by the main membrane. The refined model is shown in figure 13.

Figure 13 - Electro-mechano-acoustical network - refined model

In the refined model of the compression driver we model also the radiating area of thesuspension. It is excited by the same velocity as the main membrane because the suspension is
attached to the voice-coil-former, too. On the acoustic side of the suspension-membrane we canidentify a spatial point were components share the same pressure-potential. This point is underneaththe suspension. To this potential there is another acoustic compliance attached, which is caused by7285

the suspension chamber with volume V 1 as shown in figure 3. This compliance can be calculatedwith the help of equation 7.
The voice coil-path was initially modeled with the help of another waveguide with transferfunctions given by equation 8. However, experiments yield the best curve-fit by using a seriesconnection of a simple acoustic mass and resistance component.
The output of the voice-coil slit is at the side of the compression chamber. Noting that the
wavelength is larger than the radius of the membrane we assume the same sound pressure as inside
the chamber. This is, of course, an approximation and may need further investigation. Simulation
results are displayed in figure 14.

Figure 14 - Refined model: Electric driving point impedance andsound pressure level at 180 mm on-axis in front of aperture

The resonance peak is now properly aligned as the impedance curves of figure 14-leftdemonstrate. The curves of the acoustic observation are close. Why there are still deviations
between curves of simulation and measurement is unclear at this stage.
Figure 12 and figure 14 both display a measured sound pressure curve of high level at the upper
frequency band between 10 kHz and 20 kHz. This would be impossible to achieve with only therigid body motion of the membrane. We suspect bending vibration at high frequency. In order tocheck this the velocity distribution was measured with the help of a Doppler laser. Figure 15 shows
measurements taken at various points along the membrane.

Figure 15 - Velocity distribution of main membrane
Center at r = 0. Voice-coil-former at r = 45 mm

Figure 16 - Refined model: Mechanic velocity close to
the voice-coil former

Bending-vibration starts at 7-8 kHz as indicated by fine ripples. The amplitude of the velocity of
the rigid body mode should decrease by 6 dB/oct in this frequency range. Superimposed is thebending vibration. The highest amplitude was measured at the center (red curve).
The curve which we measured at the voice-coil former (r = 45 mm) should be close to the rigid
body velocity. Hence, we can compare this measurement to the mechanical velocity v of our model
as shown in figure 16.
The simulation shows resonances at peaks of the measurement curve. Deviations are likely to becaused by missing damping in the model. At high frequencies, residuals of bending vibration are
clearly visible in the measurement curve.
We can normalize all velocity measurements to the curve taken at the voice coil former. This
would give us a regulating filter which allows to weight the driving-source. Figure 17 shows theregulated sound pressure response and the applied amplitude of the filter (green).

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Figure 17 - Refine model. Weighted by normalized velocity distribution
Sound pressure level at 180 mm on-axis in front of aperture

CONCLUSIONSThis paper reports on the process of setting up a model with the help of lumped elements for theinterior acoustics and boundary element method for external radiation. The device to be modeled is atypical compression driver. The modeling demonstrates the complexity of the device, especially at
the acoustical domain. The advantage of lumped element modeling is the speed of calculation and àpriory knowledge applied. The disadvantage is the reduced accuracy due to the limitation to rigid
body motion and to simple wave mechanics.
The paper provides all ingredients in order to set up the lumped element analysis, using forexample open SciLab or Octave calculation software. Alternatively, one may use dedicated software
packages such as Akabak.
Further investigations, which study different radiation loads and specialized lumped elements,
are under way.

ACKNOWLEDGMENTS
The author sincerely acknowledges the company NEXO SA, and special thanks go to R&D
department of the company B&C Speakers SpA for support of this paper.

REFERENCES
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4. Dodd M, Klippel W, Oclee-Brown J. Voice coil impedance as a function of frequency anddisplacement. Preprint 6178; AES Convention 117; 2004.
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7. Morse P. Vibration and sound. USA; ASA; 1981.
8. Panzer J. Coupled lumped and boundary element simulation for electro-acoustics. Acoustics2012,
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