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5.9. Dual-diaphragm combination of pressure and pressure-gradient microphones

A versatile microphone that is popular in recording studios and for recording ensembles on location is the dual-diaphragm variable-pattern capacitor microphone, the schematic of which is shown in Fig. 5.31. It has two diaphragms: one mounted in front (F) of and the other mounted at the back (B) of a common central plate (P). An array of holes in the central plate provides a mixture of resistance and reactance. When the slider of the potentiometer is at position “k,” the polarizing voltages on both diaphragms are equal so that they behave like a pair of back-to-back pressure microphones. Hence the resulting directivity pattern is omnidirectional. The low compliance of the air in the holes of the plate ensures that the resonance frequencies of the diaphragms are high to provide a suitably wide working bandwidth. As with all pressure microphones, it is stiffness-controlled and displacement-sensitive in this mode.

Figure 5.31 Schematic of a dual-diaphragm capacitor microphone with a variable directivity pattern. In switch positions “i,” “j,” and “k,” bi-, uni-, and omnidirectional directivity patterns are obtained respectively.

When the slider is at the other end of its range in position “i,” the polarizing voltages on the two diaphragms are of equal magnitude but opposite polarity. Therefore, the microphone is only sensitive to the difference in pressures at the two diaphragms so that they behave more or less as a single diaphragm. Hence the resulting directivity pattern is figure 8. The resonance frequency is now determined by the diaphragm tension instead of the stiffness of the trapped air, which now travels back and forth through the holes providing a high viscous damping resistance. It is this damping resistance that determines the bandwidth of the microphone, which is resistance-controlled and velocity-sensitive in this mode. Without this resistance, the frequency response would be just one sharp resonant peak.

When the slider is at position “j,” no polarizing voltage is supplied to the diaphragm at the back (B), which in turn no longer contributes to the output voltage. Because it has low mass and high compliance, the back diaphragm is also acoustically transparent so that we have essentially the same configuration as an acoustic combination of pressure and pressure-gradient microphones shown in Fig. 5.7.

The analogous circuit for this kind of microphone is shown in Fig. 5.32, where

{\tilde{p}}_{1}

${\tilde{p}}_{1}$

and

{\tilde{p}}_{2}

${\tilde{p}}_{2}$

are the pressures on the outer surfaces of the front and back diaphragms,

{\tilde{U}}_{1}

${\tilde{U}}_{1}$

and

{\tilde{U}}_{2}

${\tilde{U}}_{2}$

are their respective volume velocities, E ₁ and E ₂ are the front and back polarizing voltages, and

\tilde{e}

$\tilde{e}$

is the microphone output voltage. Also, C _E0 is the static capacitance of each diaphragm when blocked, S = πa ² is the surface area of each diaphragm, d is distance between each diaphragm and the central plate, C _AP is the compliance of the air in the holes of the plate, C _AG is the compliance of the air in the gap between each diaphragm and the plate, and Z _AD is the impedance of each diaphragm that includes the mass, compliance, radiation mass, and resistance. The holes in the plate are represented by the T-circuit impedances Z _AP1 and Z _AP2 using the tube model shown in Fig. 4.45 except that the holes are assumed to be so narrow that the pressure variations are effectively isothermal and hence R _T can be ignored. The elements C _E0 and C _AG are defined by

Figure 5.32 Analogous circuit for a dual-diaphragm capacitor microphone with a variable directivity pattern.

C_{E 0} = \frac{ε_{0} S}{d},

$C_{E 0} = \frac{ε_{0} S}{d},$

(5.69)

C_{A G} = \frac{S d}{ρ_{0} c^{2}},

$C_{A G} = \frac{S d}{ρ_{0} c^{2}},$

(5.70)

where ε ₀ = 8.85 × 10− ¹² is the permittivity of free space, ρ ₀ = 1.18 kg/m³ is the static density of air, and c = 345 m/s is the speed of sound in free space. From Fig. 5.32 we can write the following equations:

{\tilde{p}}_{1} = {\tilde{p}}_{a} + (Z_{A D} + \frac{1}{j ω C_{A G}}) {\tilde{U}}_{1} - \frac{1}{j ω C_{A G}} {\tilde{U}}_{3},

${\tilde{p}}_{1} = {\tilde{p}}_{a} + (Z_{A D} + \frac{1}{j ω C_{A G}}) {\tilde{U}}_{1} - \frac{1}{j ω C_{A G}} {\tilde{U}}_{3},$

(5.71)

{\tilde{p}}_{2} = {\tilde{p}}_{b} - \frac{1}{j ω C_{A G}} {\tilde{U}}_{4} + (Z_{A D} + \frac{1}{j ω C_{A G}}) {\tilde{U}}_{2},

${\tilde{p}}_{2} = {\tilde{p}}_{b} - \frac{1}{j ω C_{A G}} {\tilde{U}}_{4} + (Z_{A D} + \frac{1}{j ω C_{A G}}) {\tilde{U}}_{2},$

(5.72)

0 = - \frac{1}{j ω C_{A G}} {\tilde{U}}_{1} (Z_{A P 1} + Z_{A P 2} + \frac{1}{j ω C_{A G}}) {\tilde{U}}_{3} + Z_{A P 2} {\tilde{U}}_{4},

$0 = - \frac{1}{j ω C_{A G}} {\tilde{U}}_{1} (Z_{A P 1} + Z_{A P 2} + \frac{1}{j ω C_{A G}}) {\tilde{U}}_{3} + Z_{A P 2} {\tilde{U}}_{4},$

(5.73)

0 = - \frac{1}{j ω C_{A G}} {\tilde{U}}_{2} + Z_{A P 2} {\tilde{U}}_{3} + (Z_{A P 1} + Z_{A P 2} + \frac{1}{j ω C_{A G}}) {\tilde{U}}_{4},

$0 = - \frac{1}{j ω C_{A G}} {\tilde{U}}_{2} + Z_{A P 2} {\tilde{U}}_{3} + (Z_{A P 1} + Z_{A P 2} + \frac{1}{j ω C_{A G}}) {\tilde{U}}_{4},$

(5.74)

{\tilde{e}}_{1} = \frac{S d}{E_{1} C_{E 0}} {\tilde{p}}_{a} = (\frac{1}{- j ω C_{E 0}} + \frac{1}{2 j ω C_{E 0}}) {\tilde{i}}_{1} + \frac{1}{2 j ω C_{E 0}} {\tilde{i}}_{2},

${\tilde{e}}_{1} = \frac{S d}{E_{1} C_{E 0}} {\tilde{p}}_{a} = (\frac{1}{- j ω C_{E 0}} + \frac{1}{2 j ω C_{E 0}}) {\tilde{i}}_{1} + \frac{1}{2 j ω C_{E 0}} {\tilde{i}}_{2},$

(5.75)

{\tilde{e}}_{2} = \frac{S d}{E_{2} C_{E 0}} {\tilde{p}}_{b} = \frac{1}{2 j ω C_{E 0}} {\tilde{i}}_{1} + (\frac{1}{- j ω C_{E 0}} + \frac{1}{2 j ω C_{E 0}}) {\tilde{i}}_{2},

${\tilde{e}}_{2} = \frac{S d}{E_{2} C_{E 0}} {\tilde{p}}_{b} = \frac{1}{2 j ω C_{E 0}} {\tilde{i}}_{1} + (\frac{1}{- j ω C_{E 0}} + \frac{1}{2 j ω C_{E 0}}) {\tilde{i}}_{2},$

(5.76)

{\tilde{U}}_{1} = \frac{S d}{E_{1} C_{E 0}} {\tilde{i}}_{1},

${\tilde{U}}_{1} = \frac{S d}{E_{1} C_{E 0}} {\tilde{i}}_{1},$

(5.77)

{\tilde{U}}_{2} = \frac{S d}{E_{2} C_{E 0}} {\tilde{i}}_{2},

${\tilde{U}}_{2} = \frac{S d}{E_{2} C_{E 0}} {\tilde{i}}_{2},$

(5.78)

\tilde{e} = \frac{{\tilde{i}}_{1} + {\tilde{i}}_{2}}{2 j ω C_{E 0}} .

$\tilde{e} = \frac{{\tilde{i}}_{1} + {\tilde{i}}_{2}}{2 j ω C_{E 0}} .$

(5.79)

Firstly we solve Eqs. (5.73) and (5.74) for

{\tilde{U}}_{3}

${\tilde{U}}_{3}$

and

{\tilde{U}}_{4}

${\tilde{U}}_{4}$

and insert these into Eqs. (5.71) and (5.72). If we then insert

{\tilde{p}}_{a}, {\tilde{p}}_{b}, {\tilde{U}}_{1}

${\tilde{p}}_{a}, {\tilde{p}}_{b}, {\tilde{U}}_{1}$

, and

{\tilde{U}}_{2}

${\tilde{U}}_{2}$

from Eqs. (5.75–5.78) respectively into Eqs. (5.71) and (5.72), and solve for

{\tilde{i}}_{1}

${\tilde{i}}_{1}$

and

{\tilde{i}}_{2}

${\tilde{i}}_{2}$

before inserting the latter in Eq. (5.79), we obtain an expression for the output voltage

\tilde{e}

$\tilde{e}$

. Furthermore, we use the expression for

{\tilde{p}}_{2}

${\tilde{p}}_{2}$

given by Eq. (5.17).

Omnidirectional performance

If E ₂ = E ₁, which corresponds to the slider being at position “k” in Fig. 5.31, we have a pressure microphone and the sensitivity is given by

{\tilde{e}}_{k} = \frac{E_{1} (2 - j \frac{ω}{c} Δ l \cos θ)}{2 j ω S d (Z_{A D} + \frac{(Z_{A P 1} + 2 Z_{A P 2}) \frac{1}{j ω C_{A G}}}{Z_{A P 1} + 2 Z_{A P 2} + \frac{1}{j ω C_{A G}}})} {\tilde{p}}_{1},

${\tilde{e}}_{k} = \frac{E_{1} (2 - j \frac{ω}{c} Δ l \cos θ)}{2 j ω S d (Z_{A D} + \frac{(Z_{A P 1} + 2 Z_{A P 2}) \frac{1}{j ω C_{A G}}}{Z_{A P 1} + 2 Z_{A P 2} + \frac{1}{j ω C_{A G}}})} {\tilde{p}}_{1},$

(5.80)

where we let

Δ l = (l + π a / 4)

$Δ l = (l + π a / 4)$

(5.81)

in accordance with Eq. (5.15) for a resilient disk. Using this formula, the exact on-axis response with the switch in position “i” for an omnidirectional pattern is plotted in Fig. 5.33. We see from this formula that the directivity pattern is essentially omnidirectional provided that

f < < \frac{c}{3 a + π l}

$f < < \frac{c}{3 a + π l}$

(5.82)

so that the cos θ term becomes insignificant. The impedances can be expanded as follows:

Figure 5.33 Exact (solid) and approximate (dashed) curves of the on-axis responses of a dual-diaphragm condenser microphone in three different modes: omni-, uni-, and bidirectional. Exact results from Eqs. (5.80, 5.98, 5.103) are shown by black, dark gray, and light gray solid curves respectively. Approximate results from Eqs. (5.95, 5.99, 5.104) are shown by black, dark gray, and light gray dashed curves respectively. The parameters are given in Table 5.2.

Z_{A D} = j ω M_{A D} + \frac{1}{j ω C_{A D}} + \frac{1}{\frac{1}{j ω M_{A R}} + \frac{1}{R_{A R}}},

$Z_{A D} = j ω M_{A D} + \frac{1}{j ω C_{A D}} + \frac{1}{\frac{1}{j ω M_{A R}} + \frac{1}{R_{A R}}},$

(5.83)

Z_{A P 1} = \frac{R_{A P}}{2} + j ω \frac{M_{A P}}{2},

$Z_{A P 1} = \frac{R_{A P}}{2} + j ω \frac{M_{A P}}{2},$

(5.84)

Z_{A P 2} = \frac{1}{j ω C_{A P}} - \frac{R_{A P}}{6} - j ω \frac{M_{A P}}{6},

$Z_{A P 2} = \frac{1}{j ω C_{A P}} - \frac{R_{A P}}{6} - j ω \frac{M_{A P}}{6},$

(5.85)

where dynamic mass M _AD and compliance C _AD of the membrane are given by Ref. [1].

M_{A D} = \frac{4 ρ_{D} h}{3 S},

$M_{A D} = \frac{4 ρ_{D} h}{3 S},$

(5.86)

Table 5.2

Dual-diaphragm condenser microphone parameters
Membrane
Radius	a	12.6 mm
Thickness	h	2.5 μm
Density	ρ _D	1400 kg/m³
Tension	T	50 N/m
Air
Density	ρ ₀	1.18 kg/m³
Absolute viscosity	μ	17.9 μN s/m²
Mean free path	λ	60 nm
Accommodation coefficient	α	0.9
Adiabatic sound speed	c	345 m/s
Specific heat ratio	γ	1.403
Gap	d	50 μm
Permittivity	ε ₀	8.85 pF/m
Polarizing voltage	E ₁	100 V
Plate
Hole radius	a _p	6 μm
Hole pitch (center to center)	p	18 μm
Depth	l	1.15 mm

C_{A D} = \frac{S^{2}}{8 π T},

$C_{A D} = \frac{S^{2}}{8 π T},$

(5.87)

where ρ _D is the density of the membrane material, h is its thickness, S = πa ² is its surface area, a is its radius, and T is its tension. The acoustic radiation mass and resistance are given by

R_{A R} = \frac{ρ_{0} c}{S},

$R_{A R} = \frac{ρ_{0} c}{S},$

(5.88)

M_{A r} = \frac{ρ_{0}}{4 a} .

$M_{A r} = \frac{ρ_{0}}{4 a} .$

(5.89)

From Chapter 4 the plate mass, compliance, and resistance are given by

R_{A P} = \frac{8 μ l}{(1 + 4 B_{u}) a_{p}^{2} S f_{f}},

$R_{A P} = \frac{8 μ l}{(1 + 4 B_{u}) a_{p}^{2} S f_{f}},$

(5.90)

M_{A P} = \frac{1 + 3 B_{u}}{1 + 4 B_{u}} \frac{4 ρ_{0} l}{3 S f_{f}},

$M_{A P} = \frac{1 + 3 B_{u}}{1 + 4 B_{u}} \frac{4 ρ_{0} l}{3 S f_{f}},$

(5.91)

C_{A P} = \frac{l S f_{f}}{ρ_{0} c^{2}},

$C_{A P} = \frac{l S f_{f}}{ρ_{0} c^{2}},$

(5.92)

where l is the thickness of the plate, a _p is the hole diameter, μ = 18.6 × 10⁻⁶ N · s/m² is the viscosity of air, and f _f is the fill factor, which for a rectangular hole grid is given by

f_{f} = \frac{π a_{p}^{2}}{p^{2}},

$f_{f} = \frac{π a_{p}^{2}}{p^{2}},$

(5.93)

where p is the hole pitch. Also B _u is the boundary slip factor that is given by

B_{u} = (\frac{2}{α_{u}} - 1) \frac{λ_{m}}{a_{p}},

$B_{u} = (\frac{2}{α_{u}} - 1) \frac{λ_{m}}{a_{p}},$

(5.94)

where α _u = 0.9 is the accommodation coefficient and λ _m = 60 nm is the mean free path length of an air molecule between collisions. It is assumed that the holes are so narrow that the pressure variations within them are isothermal because of heat conduction through the walls. Hence the specific heat ratio γ is absent from the expression for C _AP. Because the membrane is flexible as opposed to rigid, the radiation mass M _AR is that of a resilient disk in free space as derived in Chapter 13. If we ignore M _AP, M _AR, R _AR, C _AD, and ω ³ in Eq. (5.80), we obtain the following approximate formula for the sensitivity

{\tilde{e}}_{k} \approx \frac{E_{1} (1 - j \frac{ω}{2 c} Δ l \cos θ) ((C_{A G} + \frac{C_{A P}}{2}) + j ω \frac{R_{A P} C_{A G} C_{A P}}{12})}{S d (1 - ω^{2} M_{A D} (C_{A G} + \frac{C_{A P}}{2}) + j ω \frac{R_{A P} C_{A P}}{12})} {\tilde{p}}_{1} .

${\tilde{e}}_{k} \approx \frac{E_{1} (1 - j \frac{ω}{2 c} Δ l \cos θ) ((C_{A G} + \frac{C_{A P}}{2}) + j ω \frac{R_{A P} C_{A G} C_{A P}}{12})}{S d (1 - ω^{2} M_{A D} (C_{A G} + \frac{C_{A P}}{2}) + j ω \frac{R_{A P} C_{A P}}{12})} {\tilde{p}}_{1} .$

(5.95)

Using this formula, the approximate on-axis response with the switch in position “k” for an omnidirectional pattern is plotted in Fig. 5.33 and also in Fig. 5.34 along with the 180 degrees off-axis response. We see that at low to mid frequencies, where ω → 0, the reference sensitivity is given by

{\tilde{e}}_{k (ref)} = \frac{E_{1} (C_{A G} + \frac{C_{A P}}{2})}{S d} {\tilde{p}}_{1} .

${\tilde{e}}_{k (ref)} = \frac{E_{1} (C_{A G} + \frac{C_{A P}}{2})}{S d} {\tilde{p}}_{1} .$

(5.96)

The upper limit of the working range is roughly determined by the resonance frequency

f_{U} = \frac{1}{2 π \sqrt{M_{A D} (C_{A G} + \frac{C_{A P}}{2})}} .

$f_{U} = \frac{1}{2 π \sqrt{M_{A D} (C_{A G} + \frac{C_{A P}}{2})}} .$

(5.97)

Figure 5.34 Plots of the on-axis responses at the front (black) and rear (gray) of a dual-diaphragm condenser microphone in unidirectional mode. The parameters are given in Table 5.2.

Bidirectional performance

If E ₂ = − E ₁, which corresponds to the slider being at position “i” in Fig. 5.31, we have a pressure-gradient microphone and the exact sensitivity is given by

{\tilde{e}}_{i} = \frac{E_{1} Δ l \cos θ}{2 c S d (Z_{A D} + \frac{Z_{A P 1} \frac{1}{j ω C_{A G}}}{Z_{A P 1} + \frac{1}{j ω C_{A G}}})} {\tilde{p}}_{1},

${\tilde{e}}_{i} = \frac{E_{1} Δ l \cos θ}{2 c S d (Z_{A D} + \frac{Z_{A P 1} \frac{1}{j ω C_{A G}}}{Z_{A P 1} + \frac{1}{j ω C_{A G}}})} {\tilde{p}}_{1},$

(5.98)

which gives a bidirectional directivity pattern at all frequencies. Using this formula, the on-axis response is plotted in Fig. 5.33 and also in Fig. 5.34 along with the 180 degrees off-axis response. If we ignore M _AP, M _AR, and R _AR, we obtain the following approximate formula for the sensitivity

{\tilde{e}}_{i} \approx \frac{E_{1} Δ l \cos θ}{2 c S d (\frac{R_{A P}}{2 + j ω R_{A P} C_{A G}} + j ω M_{A D} + \frac{1}{j ω C_{A D}})} {\tilde{p}}_{1},

${\tilde{e}}_{i} \approx \frac{E_{1} Δ l \cos θ}{2 c S d (\frac{R_{A P}}{2 + j ω R_{A P} C_{A G}} + j ω M_{A D} + \frac{1}{j ω C_{A D}})} {\tilde{p}}_{1},$

(5.99)

which is also plotted in Fig. 5.33. We also see that at mid frequencies, where jωC _AD > 2/R _AP but jωM _AD < R _AP/2 and jωR _AP C _AG < 2, the reference sensitivity is given by

{\tilde{e}}_{i (ref)} = \frac{E_{1} Δ l}{c S d R_{A P}} {\tilde{p}}_{1} .

${\tilde{e}}_{i (ref)} = \frac{E_{1} Δ l}{c S d R_{A P}} {\tilde{p}}_{1} .$

(5.100)

The lower cut-off frequency is

f_{L} = \frac{1}{π R_{A P} C_{A D}}

$f_{L} = \frac{1}{π R_{A P} C_{A D}}$

(5.101)

and the upper limit of the working range is roughly determined by the resonance frequency:

f_{U} = \frac{1}{2 π \sqrt{M_{A D} C_{A G}}} .

$f_{U} = \frac{1}{2 π \sqrt{M_{A D} C_{A G}}} .$

(5.102)

Unidirectional performance

If E ₂ = 0, which corresponds to the slider being at position “j” in Fig. 5.31, we have a combination of pressure and pressure-gradient microphones and the exact sensitivity is given by

{\tilde{e}}_{j} = \frac{\frac{1}{2} ({\tilde{e}}_{i} + {\tilde{e}}_{k})}{1 - \frac{C_{E 0} E_{1}^{2}}{4 j ω S^{2} d^{2}} {{(Z_{A D} + \frac{(Z_{A P 1} + 2 Z_{A P 2}) \frac{1}{j ω C_{A G}}}{Z_{A P 1} + 2 Z_{A P 2} + \frac{1}{j ω C_{A G}}})}^{- 1} + (Z_{A D} + \frac{Z_{A P 1} \frac{1}{j ω C_{A G}}}{Z_{A P 1} + \frac{1}{j ω C_{A G}}})}^{- 1}},

${\tilde{e}}_{j} = \frac{\frac{1}{2} ({\tilde{e}}_{i} + {\tilde{e}}_{k})}{1 - \frac{C_{E 0} E_{1}^{2}}{4 j ω S^{2} d^{2}} {{(Z_{A D} + \frac{(Z_{A P 1} + 2 Z_{A P 2}) \frac{1}{j ω C_{A G}}}{Z_{A P 1} + 2 Z_{A P 2} + \frac{1}{j ω C_{A G}}})}^{- 1} + (Z_{A D} + \frac{Z_{A P 1} \frac{1}{j ω C_{A G}}}{Z_{A P 1} + \frac{1}{j ω C_{A G}}})}^{- 1}},$

(5.103)

which is plotted in Fig. 5.33. The expression in the numerator is a pure summation of the pressure and pressure-gradient responses obtained with the switch in positions “k” and “i” respectively. In those positions, all terms containing C _E0 are balanced out, but in position “j” there is no such balance, which explains the presence of the complicated denominator term in Eq. (5.103). However, the denominator only contributes at low frequencies so that after removing all the high-frequency terms we can make the following approximation:

{\tilde{e}}_{j} = \frac{\frac{2}{R_{A P} C_{A D}} + j ω}{\frac{2}{R_{A P}} (\frac{1}{C_{A D}} - \frac{C_{E 0} E_{1}^{2}}{4 S^{2} d^{2}}) + j ω} \cdot \frac{{\tilde{e}}_{i} + {\tilde{e}}_{k}}{2},

${\tilde{e}}_{j} = \frac{\frac{2}{R_{A P} C_{A D}} + j ω}{\frac{2}{R_{A P}} (\frac{1}{C_{A D}} - \frac{C_{E 0} E_{1}^{2}}{4 S^{2} d^{2}}) + j ω} \cdot \frac{{\tilde{e}}_{i} + {\tilde{e}}_{k}}{2},$

(5.104)

which produces a shelf at low frequencies that is determined by the amount of negative stiffness produced by the force of electrostatic attraction. This in fact helps to equalize the amount of low-frequency attenuation, which would otherwise be 6 dB because of taking the half sum of the pressure and pressure-gradient responses, where the former is flat at low frequencies and the latter rolls-off at a rate of 6 dB per octave. The low-frequency shelf starts to rise at the upper frequency of

f_{S U} = \frac{1}{π R_{A P} C_{A D}}

$f_{S U} = \frac{1}{π R_{A P} C_{A D}}$

(5.105)

and levels off at the lower frequency of

f_{S L} = \frac{1}{π R_{A P}} (\frac{1}{C_{A D}} - \frac{C_{E 0} E_{1}^{2}}{4 S^{2} d^{s}}) .

$f_{S L} = \frac{1}{π R_{A P}} (\frac{1}{C_{A D}} - \frac{C_{E 0} E_{1}^{2}}{4 S^{2} d^{s}}) .$

(5.106)

The approximate on-axis response with the switch in position “j” for a unidirectional pattern is plotted in Fig. 5.33 where the approximate expressions for

{\tilde{e}}_{i}

${\tilde{e}}_{i}$

and

{\tilde{e}}_{k}

${\tilde{e}}_{k}$

obtained from Eqs. (5.95) and (5.99) respectively are inserted into Eq. (5.104).

Condition for equal sensitivity in all three switch positions

Ideally we would like the sensitivity of the microphone to be the same at all three switch positions “i”, “j”, and “k”. It turns out that this is also the condition for obtaining the optimum cardioid directivity pattern in position “j,” which is met by setting

{\tilde{e}}_{i} = {\tilde{e}}_{k}

${\tilde{e}}_{i} = {\tilde{e}}_{k}$

in Eqs. (5.95) and (5.99) to yield

R_{A P} = \frac{Δ l}{(C_{A G} + \frac{C_{A P}}{2}) c},

$R_{A P} = \frac{Δ l}{(C_{A G} + \frac{C_{A P}}{2}) c},$

(5.107)

which after inserting the path length difference from Eq. (5.15) gives

R_{A P} = \frac{l + \frac{π a}{4}}{(C_{A G} + \frac{C_{A P}}{2}) c} .

$R_{A P} = \frac{l + \frac{π a}{4}}{(C_{A G} + \frac{C_{A P}}{2}) c} .$

(5.108)

Quite a large resistance is needed to meet this condition so that the holes through the plate have to be very narrow.

Condition for stability

Another condition that must be met is for the restoring force of the membrane tension to be greater than the force of electrostatic attraction towards the plate. We see from the schematic that this is met if

C_{A 0} = {(\frac{S d}{C_{E 0} E_{1}})}^{2} C_{E 0} > C_{A D}

$C_{A 0} = {(\frac{S d}{C_{E 0} E_{1}})}^{2} C_{E 0} > C_{A D}$

(5.109)

or using the expressions from Eqs. (5.69) and (5.87), we obtain the minimum tension value:

T > \frac{ε_{0} a^{2} E_{1}^{2}}{8 d^{3}} .

$T > \frac{ε_{0} a^{2} E_{1}^{2}}{8 d^{3}} .$

(5.110)

Typically the tension should be about three times the minimum value to allow for slackening through age and environmental conditions.

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