Flute acoustics: measurement, modelling and design - School of ...

CHAPTER 4. FINGER HOLE IMPEDANCE SPECTRA AND LENGTH CORRECTIONS 55
4.2.3 Calibration
4.2.3.1 Calibration of impedance heads
Each impedance head is calibrated separately against known impedance loads, using the
methods described in §3.4.2 with some modifications. Most importantly, since the reference
plane for each head is 50 mm further from the microphones than the coupling point, the ‘infinite
impedance’ and the ‘infinite flange’ calibration loads are in reality pipes of length 50 mm
with the appropriate termination. Further, in §3.4.2, we were only interested in measuring the
input impedance rather than pressure and flow. Consequently, only three calibration loads
were required and one element (A 11 ) in the matrix A was left undetermined, with no effect on
impedance measurements. In determining the equivalent circuit for a finger hole, pressure
and flow differences enter into the equations, so the impedance heads should be calibrated
absolutely for pressure and flow.
This can be achieved by flush-mounting a reference microphone in the infinite impedance
calibration load. Assuming that the microphone has negligible compliance, the elements A j 1
are derived from the pressure p measured at the face of this impedance load as
A j 1 = b j /p (4.3)
where, as in (3.7), b j is the signal from microphone j (j = 1,2,...,n where n is the number of
microphones). The remaining elements in A are determined using the methods of §3.4.2 from
measurements of the semi-infinite pipe and the infinite flange calibration loads.
4.2.3.2 Calibration of entire system
The two heads are attached to either end of a 100 mm tube with no finger hole (thereby
sharing the same reference plane). For this configuration, p 1 = p 2 and U 1 = −U 2 (the sign difference
is a consequence of the definition of positive flow being into the measurement load).
However, allowing for small differences in the microphones comprising each array, the measured
pressures and flows may be somewhat different. This is accounted for by making small
changes to the matrix A for each head.
A measurement of each of Z s and Z a is performed on the ‘no-hole’ system. In each case, the
pressure at the reference plane p is taken as the mean of the pressure seen by each measurement
head at the reference plane. Likewise the flow U is taken as the mean of the flows (with a
sign change due to the way flow is defined). Then for each head we have the following matrix
equation:
A
[ ]
ps p [
a
=
Z 0 U s Z 0 U a
c s c a
], (4.4)
where c s and c a are the vectors of microphone signals obtained during the measurement of Z s
and Z a respectively. This equation may be solved for the matrices A for each head.
The splitting ratio r may also be improved by calibration. As mentioned above, the voltage
signal applied to each loudspeaker is V r or V r
. The pressure and flow at the reference plane, p