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5.5. Electrostatic microphone (capacitor microphone)

General features

The electrostatic type of microphone is used extensively as a standard microphone for the measurement of sound pressure and as a studio microphone for the high-fidelity pickup of music. It can be made small in size so it does not disturb the sound field appreciably in the frequency region below 1000 Hz.

Sound-pressure levels as low as 10 dB and as high as 185 dB re 0.0002 μbar can be measured with standard instruments. The mechanical impedance of the diaphragm is that of a stiffness and is high enough so that measurement of sound pressures in cavities is possible. The electrical impedance is that of a pure capacitance.

The temperature coefficient for a well-designed capacitor microphone is less than 0.025 dB for each degree Celsius rise in temperature.

Continued operation at high relative humidities may give rise to noisy operation because of leakage across the insulators inside. Quiet operation can be restored by desiccation.

Construction

In principle, the electrostatic microphone consists of a thin diaphragm, a very small distance behind which there is a back plate (see Fig. 5.18). The diaphragm and back plate are electrically insulated from each other and form an electric capacitor. In precision measuring-microphones, commonly used diaphragm materials are nickel, stainless alloy, and titanium. Sometimes there can be problems with pin holes in nickel and wrinkles in stainless alloy. Titanium suffers from neither of these problems. The thickness of the diaphragm is typically a few micrometers, and the tension is usually greater than 2000 N/m.

A commercial form of this type of microphone is shown in Fig. 5.19. The holes in the back plate form an acoustic resistance that serves to damp the diaphragm at resonance. One manner in which the microphones are operated is shown in Fig. 5.18b. The resistances R ₁ and R ₂ are made very large to keep the membrane charge constant at low frequencies and thus preserve the bass response. The direct voltage E is several hundred volts and acts to polarize the microphone. A JFET buffer amplifier is usually located close to the microphone capsule; otherwise the capacitance of the microphone leads would exceed that of the microphone itself and therefore attenuate its output.

Figure 5.18 (a) Cross-sectional sketch of an electrostatic microphone. (b) Simple FET circuit for use with capacitor microphone.

Figure 5.19 Cutaway view of the B&K type 4190 capacitor microphone. The perforated back plate serves both as the second terminal of the condenser and as a means for damping the principal resonant mode of the diaphragm. The cap with holes in it serves both for protection and as an acoustic network at high frequencies. This microphone has a polarizing voltage of E = 200 V.
Courtesy of Brüel & Kjær Sound & Vibration Measurement A/S.

Electromechanical relations

Electrically, the electrostatic microphone is a capacitor with a capacitance that varies with time so that the total charge Q(t) is

Q (t) = q_{0} + q (t) = C_{E} (t) (E + e (t)),

$Q (t) = q_{0} + q (t) = C_{E} (t) (E + e (t)),$

(5.30)

where q ₀ is the quiescent charge in coulombs, q(t) is the incremental charge in coulombs, C _E(t) is the capacitance in farads, E is the quiescent polarizing voltage in volts, and e(t) is the incremental voltage in volts.

The capacitance C _E (t) in farads is equal to (see Fig. 5.18a)

\begin{matrix} C_{E} (t) = C_{E 0} + C_{E 1} (t) = \frac{ε_{0} S}{d - η (t)} \approx \frac{ε_{0} S}{d} (1 + \frac{η (t)}{d}), \\ \approx C_{E 0} (1 + \frac{η (t)}{d}) \end{matrix}

$\begin{matrix} C_{E} (t) = C_{E 0} + C_{E 1} (t) = \frac{ε_{0} S}{d - η (t)} \approx \frac{ε_{0} S}{d} (1 + \frac{η (t)}{d}), \\ \approx C_{E 0} (1 + \frac{η (t)}{d}) \end{matrix}$

(5.31)

where C _E0 is the capacitance in farads for η(t) = 0 and C _E1(t) is the incremental capacitance in farads, ε ₀ is a factor of proportionality that for air equals 8.85 × 10− ¹², S is the effective area of one of the plates in square meters, d is the quiescent separation in meters, and η(t) is the average incremental separation in meters. It is assumed in writing the right-hand term of Eq. (5.31) that the square of the maximum value of η(t) is small compared with d ².

If we similarly assume that

{[e (t)]}_{\max}^{2} < < E^{2}

${[e (t)]}_{\max}^{2} < < E^{2}$

, then Eqs. (5.30) and (5.31) yield

q_{0} + q (t) = C_{E 0} E + C_{E 0} E (\frac{e (t)}{E} + \frac{η (t)}{d})

$q_{0} + q (t) = C_{E 0} E + C_{E 0} E (\frac{e (t)}{E} + \frac{η (t)}{d})$

(5.32)

so that

q (t) = C_{E 0} (e (t) + \frac{E}{d} η (t)) .

$q (t) = C_{E 0} (e (t) + \frac{E}{d} η (t)) .$

(5.33)

The total stored potential energy W(t) at any instant is equal to the sum of the stored electrical and mechanical energies,

\frac{1}{2} Q {(t)}^{2} / C_{E} (t) plus \frac{1}{2} η {(t)}^{2} / C_{M S},

$\frac{1}{2} Q {(t)}^{2} / C_{E} (t) plus \frac{1}{2} η {(t)}^{2} / C_{M S},$

where C _MS is the mechanical compliance of the moving plate in m/N. That is,

W (t) = \frac{1}{2} \frac{{(q_{0} + q (t))}^{2}}{C_{E 0} + C_{E 1} (t)} + \frac{1}{2} \frac{η {(t)}^{2}}{C_{M S}} \approx \frac{1}{2} \frac{q_{0}^{2} + 2 q_{0} q (t)}{C_{E 0} (1 + \frac{η (t)}{d})} + \frac{1}{2} \frac{η {(t)}^{2}}{C_{M S}} = \frac{1}{2} \frac{q_{0}}{C_{E 0}} (q_{0} + 2 q (t)) (1 - \frac{η (t)}{d}) + \frac{1}{2} \frac{η {(t)}^{2}}{C_{M S}} .

$W (t) = \frac{1}{2} \frac{{(q_{0} + q (t))}^{2}}{C_{E 0} + C_{E 1} (t)} + \frac{1}{2} \frac{η {(t)}^{2}}{C_{M S}} \approx \frac{1}{2} \frac{q_{0}^{2} + 2 q_{0} q (t)}{C_{E 0} (1 + \frac{η (t)}{d})} + \frac{1}{2} \frac{η {(t)}^{2}}{C_{M S}} = \frac{1}{2} \frac{q_{0}}{C_{E 0}} (q_{0} + 2 q (t)) (1 - \frac{η (t)}{d}) + \frac{1}{2} \frac{η {(t)}^{2}}{C_{M S}} .$

(5.34)

The force in N at any instant acting to move the plate is, from the equation for work, dW = f dη,

f_{0} + f (t) = \frac{d W (t)}{d η}

$f_{0} + f (t) = \frac{d W (t)}{d η}$

(5.35)

so that, by differentiation of Eq. (5.34),

\begin{matrix} f_{0} + f (t) \approx - \frac{q_{0}}{2 d C_{E 0}} (q_{0} + 2 q (t)) + \frac{η (t)}{C_{M S}} \\ = - \frac{q_{0}^{2}}{2 d C_{E 0}} + (\frac{η (t)}{C_{M S}} - \frac{q (t) q_{0}}{d C_{E 0}}) . \end{matrix}

$\begin{matrix} f_{0} + f (t) \approx - \frac{q_{0}}{2 d C_{E 0}} (q_{0} + 2 q (t)) + \frac{η (t)}{C_{M S}} \\ = - \frac{q_{0}^{2}}{2 d C_{E 0}} + (\frac{η (t)}{C_{M S}} - \frac{q (t) q_{0}}{d C_{E 0}}) . \end{matrix}$

(5.36)

Hence, because E = q ₀/C _E0,

f (t) = \frac{η (t)}{C_{M S}} - \frac{E q (t)}{d} .

$f (t) = \frac{η (t)}{C_{M S}} - \frac{E q (t)}{d} .$

(5.37)

Rearranging Eq. (5.33) gives

e (t) \approx - \frac{E η (t)}{d} + \frac{q (t)}{C_{E 0}} .

$e (t) \approx - \frac{E η (t)}{d} + \frac{q (t)}{C_{E 0}} .$

(5.38)

In the steady state,

\begin{matrix} j ω \tilde{q} = \tilde{i} \\ j ω \tilde{n} = \tilde{u} \end{matrix}

$\begin{matrix} j ω \tilde{q} = \tilde{i} \\ j ω \tilde{n} = \tilde{u} \end{matrix}$

(5.39)

where

\tilde{q}

$\tilde{q}$

\tilde{i}

$\tilde{i}$

\tilde{n}

$\tilde{n}$

, and

\tilde{u}

$\tilde{u}$

are now taken to be complex harmonically varying quantities; so Eqs. (5.37) and (5.38) become, in z-parameter matrix form,

[\begin{matrix} \tilde{f} \\ - \tilde{e} \end{matrix}] = [\begin{matrix} \frac{1}{j ω C_{M S}} & \frac{E}{j ω d} \\ \frac{E}{j ω d} & \frac{1}{j ω C_{E 0}} \end{matrix}] \cdot [\begin{matrix} \tilde{u} \\ - \tilde{i} \end{matrix}]

$[\begin{matrix} \tilde{f} \\ - \tilde{e} \end{matrix}] = [\begin{matrix} \frac{1}{j ω C_{M S}} & \frac{E}{j ω d} \\ \frac{E}{j ω d} & \frac{1}{j ω C_{E 0}} \end{matrix}] \cdot [\begin{matrix} \tilde{u} \\ - \tilde{i} \end{matrix}]$

(5.40)

with

\tilde{e}

$\tilde{e}$

and

\tilde{f}

$\tilde{f}$

also being complex harmonically varying quantities.

Analogous circuits

Eq. (5.40) may be represented by either of the networks shown in Fig. 5.20, or the simplified versions shown in Fig. 5.21 where

\begin{matrix} {C^{'}}_{E 0} = \frac{- C_{E 0} C_{M S} (d^{2} / E^{2} C_{M S}^{2})}{C_{E 0} - C_{M S} (d^{2} / E^{2} C_{M S}^{2})} = \frac{C_{E 0} d^{2}}{- E^{2} C_{M S} C_{E 0} + d^{2}}, \\ = \frac{C_{E 0}}{1 - (E^{2} / d^{2}) C_{M S} C_{E 0}} \end{matrix}

$\begin{matrix} {C^{'}}_{E 0} = \frac{- C_{E 0} C_{M S} (d^{2} / E^{2} C_{M S}^{2})}{C_{E 0} - C_{M S} (d^{2} / E^{2} C_{M S}^{2})} = \frac{C_{E 0} d^{2}}{- E^{2} C_{M S} C_{E 0} + d^{2}}, \\ = \frac{C_{E 0}}{1 - (E^{2} / d^{2}) C_{M S} C_{E 0}} \end{matrix}$

(5.41)

Figure 5.20 Alternate electromechanical analogous circuits for electrostatic microphones (impedance analogy).

Figure 5.21 Simplified alternate electromechanical analogous circuits for electrostatic microphones (impedance analogy).

{C^{'}}_{M S} = \frac{C_{M S}}{1 - (E^{2} / d^{2}) C_{M S} C_{E 0}} .

${C^{'}}_{M S} = \frac{C_{M S}}{1 - (E^{2} / d^{2}) C_{M S} C_{E 0}} .$

(5.42)

The negative polarity of

\tilde{e}

$\tilde{e}$

makes sense when we consider that in Fig. 5.18a: a positive incident pressure causes a reduction in separation d, which in turn increases the capacitance C _E(t) in Eq. (5.30). If the total charge remains constant, then the incremental voltage e(t) must be negative. If we were to reverse the polarity of the bias voltage E, however,

\tilde{e}

$\tilde{e}$

would become positive.

Note in particular that:

C_E0 is electrical capacitance measured with the mechanical “terminals” blocked so that no motion occurs ( $\tilde{u}$ $\tilde{u}$ = 0).
C^′ _E0 is electrical capacitance measured with the mechanical “terminals” operating into zero mechanical impedance so that no force is built up ( $\tilde{f}$ $\tilde{f}$ = 0).
C_MS is mechanical compliance measured with the electrical terminals open circuited ( $\tilde{i}$ $\tilde{i}$ = 0).
C′_MS is mechanical compliance measured with the electrical terminals short circuited ( $\tilde{e}$ $\tilde{e}$ = 0).

The negative elements − C _E0 and − C _MS in Fig. 5.20a,b respectively are due to the force of electrostatic attraction toward the back plate. These circuits were first shown as Fig. 3.37 and Fig. 3.38, and the element sizes were given in Eqs. (3.36) and (3.37). In practice, the circuit of Fig. 5.21b is ordinarily used for electrostatic microphones.

When the microphone shown in Fig. 5.19 is radiating sound into air, the force built up at the face of the microphone when a voltage is applied to electrical terminals (3–4 of Fig. 5.21b) is very small. Hence, when an electric-impedance bridge is used to measure the capacitance of the microphone, the capacitance obtained is approximately equal to C ^′ _E0.

By Thévenin's theorem, the capacitor microphone in a free field can be represented by Fig. 5.22. The quantity

{\tilde{e}}_{o}

${\tilde{e}}_{o}$

is the open-circuit voltage produced at the terminals of the microphone by the sound wave and equals (from Eq. (5.40) and Fig. 5.21)

{\tilde{e}}_{0} = - \frac{\tilde{u} E}{j ω d} = - \frac{C_{M S} S {\tilde{p}}_{B} E}{d},

${\tilde{e}}_{0} = - \frac{\tilde{u} E}{j ω d} = - \frac{C_{M S} S {\tilde{p}}_{B} E}{d},$

(5.43)

where the force

{\tilde{f}}_{B}

${\tilde{f}}_{B}$

, acting on the microphone with the diaphragm blocked so that

\tilde{u}

$\tilde{u}$

= 0, is equal to the blocked pressure

{\tilde{p}}_{B}

${\tilde{p}}_{B}$

times the area of the diaphragm S.

Figure 5.22 Thévenin's circuit of a capacitor microphone of the type shown in Fig. 5.18 situated in free space.

Acoustical relations

The microphone of Fig. 5.19 has a diaphragm with the property of mass M _MD in addition to the mechanical compliance C _MS assumed so far. However, unlike with the dynamic microphone, the two parameters are not separable and it is difficult to evaluate them with any accuracy over the whole frequency range because of the localized nature of the loading on the membrane, which is strongly coupled to the motion of the air in the gap and back-plate openings. However, we will use lumped-parameter elements with approximations to gain insight. For the 4190 microphone, the complete acoustical and mechanical circuit in the impedance-type analogy is seen in Fig. 5.23. The internal acoustical circuit consists of an air gap directly behind the diaphragm with an acoustic compliance C _AG, a back plate, including holes and a slot, with acoustic resistance and mass R _AS and M _AS, and a back cavity behind the plate with an acoustic stiffness C _AB. The radiation impedance looking outward from the front side of the diaphragm is R _AA + jωM _AA, where R _AA and M _AA are found from Table 4.6. In this circuit

{\tilde{p}}_{B}

${\tilde{p}}_{B}$

is the incident pressure at the diaphragm when it is restrained from moving, M _AD = M _MD/S ² is the acoustic mass of the diaphragm, S is the effective area of the diaphragm, and

{\tilde{U}}_{D} = S {\tilde{u}}_{D}

${\tilde{U}}_{D} = S {\tilde{u}}_{D}$

is the volume velocity of the diaphragm.

When Fig. 5.21b is combined with Fig. 5.23, the complete circuit for the electrostatic microphone shown in Fig. 5.24 is obtained.

Figure 5.23 Acoustical circuit of a capacitor microphone including the radiation mass and the acoustical elements behind the diaphragm (impedance analogy).

Figure 5.24 Complete electroacoustical circuit of a capacitor microphone (impedance analogy).

Performance

The performance of the capacitor microphone shown in Fig. 5.19, viz., the B&K type 4190, can best be understood by reference to Figs. 5.25 and 5.26, which are derived from Fig. 5.24. At low frequencies the circuit is essentially that of Fig. 5.25a. From this circuit, the open-circuit voltage

{\tilde{e}}_{0}

${\tilde{e}}_{0}$

is equal to

{\tilde{e}}_{0} = \frac{E}{d} \frac{C_{A B} C_{A S}}{C_{A B} + C_{A S}} \frac{{\tilde{p}}_{B}}{S} .

${\tilde{e}}_{0} = \frac{E}{d} \frac{C_{A B} C_{A S}}{C_{A B} + C_{A S}} \frac{{\tilde{p}}_{B}}{S} .$

(5.44)

At low frequencies, therefore,

{\tilde{e}}_{0}

${\tilde{e}}_{0}$

is independent of frequency. This is the frequency region shown as (a) in Fig. 5.26. Note that C _AS is inversely proportional to the diaphragm tension T, and C _AB is proportional to the back cavity volume V but inversely proportional to the atmospheric pressure P ₀ (see Eq. (4.13)). In a measuring microphone the tension is set high enough so that C _AB >> C _AS, which makes the microphone relatively insensitive to changes in atmospheric pressure.

In the vicinity of the first major resonance, the circuit becomes that of Fig. 5.25b. At resonance, the volume velocity through the compliance C _AB is limited only by the magnitude of the acoustic resistance R _AS. In general, this resistance is chosen to be large enough so that the resonance peak is less than 2 db (26%) higher than the response at lower frequencies. The response near resonance is shown at (b) in Fig. 5.26.

Above the resonance frequency, the circuit becomes that of Fig. 5.25c. The volume velocity is controlled entirely by the mass reactance. Hence,

{\tilde{e}}_{0} = \frac{E}{ω^{2} (M_{A A} + M_{A D} + M_{A S}) d} \frac{{\tilde{p}}_{B}}{S} .

${\tilde{e}}_{0} = \frac{E}{ω^{2} (M_{A A} + M_{A D} + M_{A S}) d} \frac{{\tilde{p}}_{B}}{S} .$

(5.45)

Figure 5.25 Capacitor microphone—simplified circuits (impedance analogy) for low frequencies (a), near resonance (b), and above resonance (c). The excess pressure produced by the sound wave at the diaphragm of the microphone with the microphone held motionless is ${\tilde{p}}_{B}$ ${\tilde{p}}_{B}$ , and the open-circuit voltage is ${\tilde{e}}_{0}$ ${\tilde{e}}_{0}$ .

Figure 5.26 Free-field response of the B&K type 4190 capacitor microphone shown in Fig. 5.19.
Courtesy of Brüel & Kjær Sound & Vibration Measurement A/S.

In this frequency region the response decreases at the rate of 12 dB per octave (see region (c) of Fig. 5.26).

At higher frequencies, further resonances could occur, but if they are not completely damped by the radiation resistance R _AA (which is no longer negligible compared with jωM _AA), the resonance peaks are likely to be limited by the viscous air flow resistance R _AS in the holes or gap.

In a detailed analysis, which we shall not reproduce here, Zuckerwar [1] finds that the membrane mass and compliance elements can be approximated by Ref. [1].

C_{A S} = \frac{S^{2}}{8 π T},

$C_{A S} = \frac{S^{2}}{8 π T},$

(5.46)

M_{A S} = \frac{4 ρ_{M} h}{3 S},

$M_{A S} = \frac{4 ρ_{M} h}{3 S},$

(5.47)

which gives a resonant frequency of

ω_{0} = \frac{\sqrt{6}}{a} \sqrt{\frac{T}{ρ_{M} h}} .

$ω_{0} = \frac{\sqrt{6}}{a} \sqrt{\frac{T}{ρ_{M} h}} .$

(5.48)

The fundamental resonant frequency of a membrane is

ω_{1} = \frac{α_{1}}{a} \sqrt{\frac{T}{ρ_{M} h}},

$ω_{1} = \frac{α_{1}}{a} \sqrt{\frac{T}{ρ_{M} h}},$

(5.49)

where α ₁ = 2.4048 is the first zero of the Bessel function J ₀(x). The two frequencies ω ₀ and ω ₁ differ by only 2%, which implies that the lumped-element model is an accurate representation of the membrane up to ω ₀. If we assume the back plate is virtually as large as the membrane, the membrane deflection

\tilde{η} (w)

$\tilde{η} (w)$

at low frequencies is given by

\tilde{η} (w) = (1 - \frac{w^{2}}{a^{2}}) {\tilde{η}}_{0},

$\tilde{η} (w) = (1 - \frac{w^{2}}{a^{2}}) {\tilde{η}}_{0},$

(5.50)

where

{\tilde{η}}_{0}

${\tilde{η}}_{0}$

is the maximum deflection at the center. Hence the average deflection is given by

< \tilde{η} > = \frac{2 π}{S} \int_{0}^{a} \tilde{η} (w) w d w = \frac{{\tilde{η}}_{0}}{2} .

$< \tilde{η} > = \frac{2 π}{S} \int_{0}^{a} \tilde{η} (w) w d w = \frac{{\tilde{η}}_{0}}{2} .$

(5.51)

Thus the average deflection is half of that at the center, which means that the effective membrane area is half of the total area S. The optimum back-plate radius b is given by Ref. [1].

b = \sqrt{\frac{2}{3}} a .

$b = \sqrt{\frac{2}{3}} a .$

(5.52)

When designing a capacitor microphone, it is desirable to minimize the air gap width d to maximize the sensitivity and hence also the signal to noise ratio. Obviously this limits the diaphragm excursion and thus also the maximum sound pressure that can be detected, but this is not generally a problem unless the microphone is designed to record very high sound pressure levels such as those produced by jet engines. Using a small gap, Paschen's law [2] works in our favor because larger electric field strengths (E/d) can be obtained before break down than in larger gaps. Once the gap width and maximum electric field strength have been established, we have to apply enough tension to the membrane for it to resist the electrostatic force of attraction toward the back plate. In other words, the positive mechanical compliance C _MS in Fig. 5.20a must be less than the negative electrical compliance − C _E0 when referred to the mechanical side

C_{M S} < \frac{d^{2}}{E^{2} C_{E 0}} .

$C_{M S} < \frac{d^{2}}{E^{2} C_{E 0}} .$

(5.53)

However, because C _MS = 1/(8πT) and C _E0 = ε ₀ S/d from Eqs. (5.46) and (5.31) respectively, we can write

T > \frac{ε_{0} a^{2} E^{2}}{8 d^{3}},

$T > \frac{ε_{0} a^{2} E^{2}}{8 d^{3}},$

(5.54)

which is similar to a more rigorous solution [3] based on the static membrane wave equation

T > \frac{ε_{0} a^{2} E^{2}}{α_{1}^{2} d^{3}} .

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

(5.55)

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Table of Contents for Acoustics

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5.5. Electrostatic microphone (capacitor microphone)

General features

Construction

Electromechanical relations

Analogous circuits

Acoustical relations

Performance

Table of Contents for
Acoustics