Introduction to Acoustics

iωctE0
Ct0
Cs
Fig. 24.3 Equivalent small-signal circuit
Rc
Ci
+
Ri }e0
are represented by their corresponding static and timevarying
components
ct = cstat + ct(t) , (24.1)
E = E0 + e(t) . (24.2)
The time-varying current it generated by the microphone
is
dct de
it = E0 + cstat . (24.3)
dt dt
For small electrical signals, Fig. 24.2 can be represented
by an equivalent circuit shown in Fig. 24.3 and for small
time-varying (ω) pressure at the diaphragm, the microphone
can be represented by a fixed capacitance C and
a current source it, and the output voltage eo is:
ct iωCRp
eo = E0
, (24.4)
C 1 + iωCRp
where Rp is the impedance of Rc in parallel with Ri,
and C is the sum of cstat, ci and cs. The microphone
open-circuit voltage eoc as a function of the varying
capacitance ct is
ct
eoc = E0
cstat
�
1 + ci + cs
cstat
�−1 . (24.5)
Based on an approximate equation derived by Hawley
([24.29], Chap. 2, Eq. 5.8) for the capacitance of
a microphone under the influence of the polarizing
voltage, and assuming that the diaphragm can be modeled
as a piston-like displacement, Zuckerwar ([24.29],
Chap. 3, Eq. 2.23) has shown that the electrical transfer
impedance Me is
�
Me = E0
h0
1 + r2 b
2r 2 a
� �
1 + ci + cs
cstat
� −1
, (24.6)
where ra and rb are the radii of the diaphragm and the
backplate, respectively.
Amplitude (dB)
5
0
–5
–10
–15
–20
Microphones and Their Calibration 24.2 Theory 1025
Amplitude
Phase lag
Phase lag (deg)
10 10
Frequency (Hz)
5
–30
0
10 2
10 3
10 4
180
150
120
Fig. 24.4 Sample amplitude and phase response of a oneinch
microphone (Brüel Kjær 4146) Solid lines: theoretical.
Symbols: experimental (after Zuckerwar [24.29], Chap. 3)
Both Hawley et al. ([24.29], Chap. 2) and Zuckerwar
([24.29], Chap. 3) arrived at the conclusion that the
optimum backplate size is
ra/rb = (2/3) 1/2 = 0.8165 . (24.7)
24.2.3 Mechanical Response
The membrane motion is coupled to the air between the
diaphragm and the backplate. The system exhibits damping
due to the damping holes in the backplate. Zuckerwar
([24.29], Chap. 3, Eq. 3.51) arrived at a solution for the
mechanical response as
Mm = [y(r)]
pi
90
60
30
= 1
TK2 J2(Ka)
, (24.8)
J0(Ka) + D
where K is the wave number of sound in the membrane,
given by
K = ω
� �1/2 σM
, (24.9)
Tm
ω is angular frequency, σM is the membrane surface
mass density, and Tm is the membrane tension. D is
a complex function ([24.29], Chap. 3, Eq. 3.50). With
(24.8) the amplitude and phase response can be realized;
some examples ([24.29],Chap.3,Fig.3.8)areshownin
Fig. 24.4.
Part H 24.2