Acoustics

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5.3. Combination of pressure and pressure-gradient microphones

A combination of pressure and pressure-gradient microphones is one that responds to both the pressure and the pressure gradient in a wave. A common example of such a microphone is the one having a cavity at the back side of the diaphragm that has an opening to the outside air containing an acoustic resistance (see Fig. 5.7a).

Figure 5.7 (a) Sketch of a combination pressure and pressure-gradient microphone consisting of a right enclosure in one side of which is a movable diaphragm connected to a transducing element and in another side of which is an opening with an acoustic resistance R _A. (b) Acoustic-impedance circuit for (a).

The analogous circuit for this device is shown in Fig. 5.7b. If we let

{\tilde{p}}_{1} = {\tilde{p}}_{0} e^{- j k x}

${\tilde{p}}_{1} = {\tilde{p}}_{0} e^{- j k x}$

(5.16)

\begin{matrix} {\tilde{p}}_{2} = {\tilde{p}}_{1} + \frac{\partial ({\tilde{p}}_{0} e^{- j k x})}{\partial x} Δ l \cos θ \\ = {\tilde{p}}_{1} (1 - j \frac{ω}{c} Δ l \cos θ) . \end{matrix}

$\begin{matrix} {\tilde{p}}_{2} = {\tilde{p}}_{1} + \frac{\partial ({\tilde{p}}_{0} e^{- j k x})}{\partial x} Δ l \cos θ \\ = {\tilde{p}}_{1} (1 - j \frac{ω}{c} Δ l \cos θ) . \end{matrix}$

(5.17)

Let us say that

{\tilde{U}}_{D}

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

is the volume velocity of the diaphragm,

{\tilde{U}}_{0}

${\tilde{U}}_{0}$

is the volume velocity of the air passing through the resistance,

{\tilde{p}}_{D}

${\tilde{p}}_{D}$

is the net pressure acting to move the diaphragm, and Z _AD is the diaphragm impedance. In the case of an electrostatic or ribbon microphone, the radiation mass will have a significant effect, so for the sake of simplicity let us lump this in with Z _D. The acoustic resistance R _A will also have a mass component, but we assume that it is very small compared with the resistance. Then we can write the following equations from Fig. 5.7b:

\begin{matrix} {\tilde{U}}_{D} (Z_{A D} + \frac{1}{j ω C_{A}}) - \frac{{\tilde{U}}_{0}}{j ω C_{A}} = {\tilde{p}}_{1} \\ \frac{- {\tilde{U}}_{D}}{j ω C_{A}} + {\tilde{U}}_{0} (R_{A} + \frac{1}{j ω C_{A}}) = - {\tilde{p}}_{2}, \end{matrix}

$\begin{matrix} {\tilde{U}}_{D} (Z_{A D} + \frac{1}{j ω C_{A}}) - \frac{{\tilde{U}}_{0}}{j ω C_{A}} = {\tilde{p}}_{1} \\ \frac{- {\tilde{U}}_{D}}{j ω C_{A}} + {\tilde{U}}_{0} (R_{A} + \frac{1}{j ω C_{A}}) = - {\tilde{p}}_{2}, \end{matrix}$

(5.18)

which are solved for

{\tilde{U}}_{D}

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

. The pressure difference across the diaphragm is

{\tilde{p}}_{D} = {\tilde{U}}_{D} Z_{A D} = \frac{Z_{A D} ({\tilde{p}}_{1} R_{A} + \frac{{\tilde{p}}_{1} - {\tilde{p}}_{2}}{j ω C_{A}})}{Z_{A D} R_{A} - j ((R_{A} + Z_{A D}) / ω C_{A})} .

${\tilde{p}}_{D} = {\tilde{U}}_{D} Z_{A D} = \frac{Z_{A D} ({\tilde{p}}_{1} R_{A} + \frac{{\tilde{p}}_{1} - {\tilde{p}}_{2}}{j ω C_{A}})}{Z_{A D} R_{A} - j ((R_{A} + Z_{A D}) / ω C_{A})} .$

(5.19)

Substitution of (5.17) in (5.19) yields

{\tilde{p}}_{D} = {\tilde{p}}_{1} \frac{Z_{A D} (R_{A} + \frac{Δ l \cos θ}{c C_{A}})}{Z_{A D} R_{A} - j ((R_{A} + Z_{A D}) / ω C_{A})} .

${\tilde{p}}_{D} = {\tilde{p}}_{1} \frac{Z_{A D} (R_{A} + \frac{Δ l \cos θ}{c C_{A}})}{Z_{A D} R_{A} - j ((R_{A} + Z_{A D}) / ω C_{A})} .$

(5.20)

Let

\frac{Δ l}{c C_{A} R_{A}} = B,

$\frac{Δ l}{c C_{A} R_{A}} = B,$

(5.21)

where B is an arbitrarily chosen dimensionless constant. Because

{\tilde{f}}_{D} = {\tilde{p}}_{D} S

${\tilde{f}}_{D} = {\tilde{p}}_{D} S$

, where S is the effective area of the diaphragm, we have

| {\tilde{f}}_{D} | = {\tilde{p}}_{1} | A | S (1 + B \cos θ),

$| {\tilde{f}}_{D} | = {\tilde{p}}_{1} | A | S (1 + B \cos θ),$

(5.22)

where A is the ratio given by

A = \frac{Z_{A D} R_{A}}{Z_{A D} R_{A} - j ((R_{A} + Z_{A D}) / ω C_{A})} .

$A = \frac{Z_{A D} R_{A}}{Z_{A D} R_{A} - j ((R_{A} + Z_{A D}) / ω C_{A})} .$

(5.23)

A plot of the force |f _D| acting on the diaphragm as a function of θ for B = 1 is shown in Fig. 5.8a. The same pattern plotted in decibels is given in Fig. 5.8b. The directivity pattern for B = 1 is commonly called a cardioid pattern. Other directivity patterns are shown in Fig. 5.30 for B = 0,

\frac{1}{2}

$\frac{1}{2}$

, 1,

\sqrt{3}

$\sqrt{3}$

, 3, and ∞.

Figure 5.8 (a) Directivity characteristic of the combination pressure and pressure-gradient microphone of Fig. 5.7. (b) Same but with scale in dB.

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5.3. Combination of pressure and pressure-gradient microphones

Table of Contents for
Acoustics