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Part XXXXIV: Lumped-element model of an electrostatic loudspeaker

15.10. Electro-mechano-acoustical circuit

In Section 14.10 we developed an analytical (distributed-element) model of a circular electrostatic loudspeaker. Here we will develop a simpler lumped-element model which is valid when there is sufficient resistance (usually in the form of a dust screen) to suppress the membrane modes. Using an analogous circuit, we will then develop useful design formulas.

The analogous circuit of the electrostatic loudspeaker shown in Fig. 15.1 is given by Fig. 15.16. Although this is a general circuit, we shall assume for this analysis that the loudspeaker is circular with radius a and has no enclosure whatsoever.

The symbols have the following meanings:

${\tilde{e}}_{in}$ ${\tilde{e}}_{in}$ is the voltage of the generator (audio amplifier) in volts (V).
${\tilde{i}}_{in}$ ${\tilde{i}}_{in}$ is the total input current in amperes (A).
Figure 15.16 Electro-mechano-acoustical analogous circuit of the electrostatic loudspeaker shown in Fig. 15.1.
${\tilde{i}}_{s}$ ${\tilde{i}}_{s}$ is the static part of the input current in amperes (A).
${\tilde{i}}_{m}$ ${\tilde{i}}_{m}$ is the motional part of the input current in amperes (A).
C _E is the static capacitance between the electrodes in Farads (F).
− C _E represents the negative capacitance due to electrostatic attraction in Farads (F).
E _P is the polarization supply voltage in volts (V).
d is the separation distance between the membrane and each electrode in meters (m).
${\tilde{f}}_{D}$ ${\tilde{f}}_{D}$ is the mechanical force driving the membrane in Newtons (N).
${\tilde{u}}_{D}$ ${\tilde{u}}_{D}$ is the average velocity of the membrane in m/s.
$Z_{M} = {\tilde{f}}_{D} / {\tilde{u}}_{D}$ $Z_{M} = {\tilde{f}}_{D} / {\tilde{u}}_{D}$ is the total mechanical impedance in N·s/m.
C _MD is the mechanical compliance of the membrane in m/N due to its tension.
M _MT is the total moving mass of the membrane and stator perforations in kg.
R _MS is the mechanical resistance due to viscous flow losses through the stator electrode perforations and dust screen.
S _D = πa ² is the surface area of the membrane in m², where a is the radius in m.
${\tilde{U}}_{D} = S_{D} {\tilde{u}}_{D}$ ${\tilde{U}}_{D} = S_{D} {\tilde{u}}_{D}$ is the total volume velocity in m³/s.
${\tilde{p}}_{0}$ ${\tilde{p}}_{0}$ is the pressure in N/m² driving the radiation load 2Z _AR on both sides of the membrane.
M _AR is the acoustic radiation mass on one side of the membrane in kg/m⁴.
C _AR is the acoustic radiation compliance on one side of the membrane in m⁵/N.
R _AR is the acoustic radiation resistance on one side of the membrane in N⋅s/m⁵.

For simplicity, we assume that the output impedance of the amplifier and resistance of the cables are negligible, and we also ignore any stray capacitance in the cables. There are two transformers: the first acts as an interface between the electrical domain and the mechanical one converting voltage to force

{\tilde{f}}_{D}

${\tilde{f}}_{D}$

and current

{\tilde{i}}_{m}

${\tilde{i}}_{m}$

to velocity

{\tilde{u}}_{D}

${\tilde{u}}_{D}$

, while the second acts as an interface between the mechanical and acoustical domains, converting force to pressure

{\tilde{p}}_{0}

${\tilde{p}}_{0}$

and velocity

{\tilde{u}}_{D}

${\tilde{u}}_{D}$

to volume velocity

{\tilde{U}}_{D}

${\tilde{U}}_{D}$

Notice how the input current

{\tilde{i}}_{in}

${\tilde{i}}_{in}$

divides into two: one is the static current

{\tilde{i}}_{s}

${\tilde{i}}_{s}$

while the other is the motional current

{\tilde{i}}_{m}

${\tilde{i}}_{m}$

. The static current still flows when the membrane is blocked (or the polarization voltage E _P is turned off), but the motional current is dependent on the membrane velocity

{\tilde{u}}_{D}

${\tilde{u}}_{D}$

. Unfortunately, in most practical electrostatic loudspeakers

{\tilde{i}}_{s} ≫ {\tilde{i}}_{m}

${\tilde{i}}_{s} ≫ {\tilde{i}}_{m}$

, so that the electrical input impedance is defined almost entirely by C _E, although it is possible to measure the motional current by “balancing out” the static current with a capacitor [7]. The stator resistance R _MS is technically an acoustic flow resistance, but it is included on the mechanical side of Fig. 15.16 for convenience.

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Part XXXXIV: Lumped-element model of an electrostatic loudspeaker

15.10. Electro-mechano-acoustical circuit

Table of Contents for
Acoustics