Instrumentation for Noise Measurement and Analysis
3.1 MICROPHONES
3.1.1 Condenser Microphone
A condenser microphone consists of a diaphragm that serves as one electrode of a condenser, and a polarised backing plate, parallel to the diaphragm and separated from it by a very narrow air gap, which serves as the other electrode. The condenser is polarised by means of a bound charge, so that small variations in the air gap due to pressure-induced displacement of the diaphragm result in corresponding variations in the voltage on the condenser.
The bound charge on the backing plate may be provided either by means of an externally supplied bias voltage of the order of 200 V, or by use of an electret, which forms either part of the diaphragm or the backing plate. Details of the electret construction and its use are discussed in the literature (Frederiksen et al., 1979). For the purpose of the present discussion, however, the details of the latter construction are unimportant. In either case, the essential features of a condenser microphone and a sufficient (but simplified) representation of its electrical circuit for the present purpose are provided in Figure 3.1. A more detailed treatment is provided by Brüel and Kjær (1996)
Referring to Figure 3.1, the bound charge Q may be supplied by a d.c. power supply of voltage E0 through a very large resistor Rp. Alternatively, the branch containing the d.c. supply and resistor Rp may be thought of as a schematic representation of the electret. The microphone response voltage is detected across the
Q
C%Cs ' E0 (3.1)
E ' & Q
C%δC%Cs % E0 (3.2)
C%δC
C ' h
h&x (3.3)
δC ' C 1
1&x/h&1 (3.4)
E ' &Q C
1&x/h
1%(Cs/C)(1&x/h) & 1
1%Cs/C (3.5)
E ' &Q C
1&x/h
1%Cs/C 1% (x/h)(Cs/C)
1%Cs/C % ... & 1
1%Cs/C (3.6) load resistor R. A good signal can be obtained at the input to a high internal impedance detector, even though the motion of the diaphragm is only a small fraction of the wavelength of light.
An equation relating the output voltage of a condenser microphone to the diaphragm displacement will be derived. It is first observed that the capacitance of a condenser is defined as the stored charge on it, Q, divided by the resulting voltage across the capacitance. Using this observation, it can be seen by reference to Figure 3.1, where C is the capacitance of the microphone and Cs is the stray capacitance of the associated circuitry, that for the diaphragm at rest with a d.c. bias voltage of E0:
The microphone capacitance is inversely proportional to the spacing at rest, h, between the diaphragm and the backing electrode. If the microphone diaphragm moves a distance x inward (negative displacement, positive pressure) so that the spacing becomes h - x, the microphone capacitance will increase from C to C + δC and the voltage across the capacitors will decrease in response to the change in capacitance by an amount E to E0 - E. Thus:
The microphone capacitance is inversely proportional to the spacing between the diaphragm and the backing electrode, thus:
Equation (3.3) may be rewritten as:
Substitution of Equation (3.4) into Equation (3.2) and use of Equation (3.1) gives the following relation:
Equation (3.5) may be rewritten in the following form:
K1 ' 1
Ch (3.7)
E . K1Q x
1%Cs/C & K1Q x2
h(1%Cs/C)2 . K1Q x
1%Cs/C (3.8a,b)
fco ' 1
2πRvCdc (3.9)
Rv ' 8µR
πa4 (3.10)
µ % T0 7 (3.11)
The empirical constant K1 is now introduced and defined as follows:
By design, Cs/C << 1 and x/h << 1; thus in a well-designed microphone the higher order terms in Equation (3.6) may be omitted and by introducing Equation (3.7), Equation (3.6) takes the following approximate form:
Equation (3.8b) reflects the fact that a positive pressure, which causes a negative displacement x (inward motion), results in a negative value of the induced voltage E.
Reference to Equation (3.7) shows that the constant K1 depends upon the spacing at rest between the microphone diaphragm and the backing electrode, and the capacitance of the device, both of which are generally very difficult to determine by design, and consequently the constant must be determined by calibration. For good linear response, the capacitance ratio Cs/C must be kept as small as possible, and similarly, the microphone displacement relative to the condenser diaphragm backing electrode spacing, x/h, must be very small. Thus, in a well-designed microphone, the second term in Equation (3.8a) is negligible.
The vent shown in Figure 3.1 is an essential element of a microphone in that it allows the mean pressure to be equalised on both sides of the microphone diaphragm.
Without the presence of this vent, any changes in atmospheric pressure would act to push the diaphragm one way or the other and drastically change the frequency response of the microphone. Unfortunately, the presence of the vent limits the low frequency response of the microphone so that below the cut off frequency caused by the vent, the microphone will be insensitive to acoustic pressure. The cut-off frequency is proportional to the size of the vent; the larger the vent the higher will be the cut-off frequency, which is defined as (Olsen, 2005):
where Rv is the resistance of the vent and for a vent consisting of a tube of radius, a, and length, R, it is given by (Beranek, 1954):
where µ is the dynamic viscosity for air and is equal to 1.84 × 10-5 N-s/m5 at 20EC.
The coefficient of viscosity varies with absolute temperature, T, in degrees Kelvin such that:
(a) (b) grid
vent
diaphragm
conductor insulator
piezo- electric material
E E
C
Cs Rs
0
R
Figure 3.2 A schematic representation of a piezoelectric microphone and equivalent electrical circuit.
1 Cdc ' 1
Cc % 1
Cd ' Cd % Cc
CdCc (3.12)
Cc ' Vc
γP (3.13)
A typical value for Rv is 36 × 109 N-s/m5. The quantity, Cdc in Equation (3.9) is the compliance of the diaphragm cavity system and is the reciprocal of the sum of the reciprocals of the compliance, Cc, of the cavity behind the microphone diaphragm and the compliance, Cd, of the diaphragm as these latter two compliances act together in parallel. Thus:
A typical value for the diaphragm compliance, Cd, is 0.3 × 10-12 m5/N (Brüel and Kjær, 1996). The compliance of the cavity of volume Vc (m3 ) is given by:
where γ = 1.4 is the ratio of specific heats for air and P is the mean atmospheric pressure (Pa). A typical value for Cc is 2.8 × 10-12 (Brüel and Kjær, 1996), which thus defines a typical value for Vc.