Introduction to Acoustics

eB
e1
Reference
signal
(2)
(1)
SW
Attenuators
α ß
B A
(A–B)
Inphase quadrature
e2
eA
AC null detector
(lock-in amplifier)
Fig. 24.8 Interchange reference method (after Wong [24.29],
Chap. 4)
we have
e2β1 cos θ − e1α1 = 0 . (24.25)
Similarly, when e2 is selected as the reference by the
switch at position 2, the null condition with attenuator
readings β2 and α2 provides the following:
e1β2 cos θ − e2α2 = 0 . (24.26)
From (24.5)and(24.6), the general equations for the
amplitude ratio R = e2/e1 and cos θ are found
R =[(α1/α2)(β2/β1)] 1/2 , (24.27)
cos θ =[(α1α2)/(β1β2)] 1/2 . (24.28)
Although for symmetry, Fig. 24.8 shows two attenuators,
in practice only one attenuator is required. For
the condition in Fig. 24.9, wheree2 > e1, α1 = β2 = 1.
24.4 Corrections
24.4.1 Heat Conduction Correction
The alternating sound pressure induces compression and
expansion in the gas medium inside the closed cavity.
At sufficiently low frequencies, the induced temperature
changes occur isothermally, and the heat exchange
at the walls of the cavity has to be included when computing
the acoustic impedance of the cavity. As the
frequency increases, heat conduction between the gas
e1
Microphones and Their Calibration 24.4 Corrections 1029
a) With e1 as the reference b) With e2 as the reference
+j
+j
Quadrature
component
(eA– eB)
e1α1
–j
Θ
e2 ß1
e1
e2
e2
Quadrature
component
(eA– eB)
e2 α2
–j
Θ
e1 ß2
Fig. 24.9 Vector diagram for the in-phase null condition
(after Wong [24.29], Chap. 4)
The uncertainty of the attenuator that can be a sevendecade
ratio transformer (inductive voltage dividers) is
of the order of 0.1 ppm. The uncertainty of the measured
amplitude ratio is of the order of 1 ppm. Other circuits
employing inductive voltage dividers for voltage ratio
measurements are given in [24.17, 23].
Reference Impedance Selection
The reference impedance Zx showninFig.24.6 needs
to be carefully selected. The nature of the impedance,
either purely resistive or purely capacitive, or a combination
of electrical impedances, dictates the mode of
evaluation of the factor k, which is required for the computation
of microphone sensitivity in (24.24). If Zx is
a resistance, the angular frequency ω,shownin(24.19),
has to be measured. When the impedance is a capacitance
Cr the term Zx = 1/(iωCr), and the frequency
terms are canceled. The numerical value of Zx is chosen
such that the magnitudes of e1 and e2 are nearly equal in
order to enhance the measurement of voltage ratios using
null measurement techniques. For precise measurement
of Zx, the capacitance of the connecting cables to the
impedance should be taken into consideration.
and the walls decreases. Gradually, isothermal conditions
change to adiabatic conditions, at which virtually
no heat is exchanged with the walls.
Classical analytical approaches to calculate heat
conduction in cavities are given by Ballantine [24.2]and
Daniels [24.33]. Correction factors for heat conduction
were obtained by Biagi and Cook [24.34]. Theoretical
and experimental data, based on thermal diffusion, on
the acoustic impedance of the cavity, are given by Ger-
e1
e2
Part H 24.4