Technical Review
Technical
Review
No. 1 – 1996
Calibration Uncertainties & Distortion of Microphones.
Wide Band Intensity Probe. Accelerometer Mounted Resonance Test
ISSN 007 – 2621
BV 0048 – 11
bv004811.book : InsideCovers Black 44
Previously issued numbers of
Brüel & Kjær Technical Review
2 – 1995 Order Tracking Analysis
1 – 1995 Use of Spatial Transformation of Sound Fields (STSF) Techniques in the
Automotive Industry
2 – 1994 The use of Impulse Response Function for Modal Parameter Estimation
Complex Modulus and Damping Measurements using Resonant and
Non-resonant Methods (Damping Part II)
1 – 1994 Digital Filter Techniques vs. FFT Techniques for Damping
Measurements (Damping Part I)
2 – 1990 Optical Filters and their Use with the Type 1302 & Type 1306
Photoacoustic Gas Monitors
1 – 1990 The Brüel & Kjær Photoacoustic Transducer System and its Physical
Properties
2 – 1989 STSF — Practical instrumentation and application
Digital Filter Analysis: Real-time and Non Real-time Performance
1 – 1989 STSF — A Unique Technique for scan based Near-Field Acoustic
Holography without restrictions on coherence
2 – 1988 Quantifying Draught Risk
1 – 1988 Using Experimental Modal Analysis to Simulate Structural Dynamic
Modifications
Use of Operational Deflection Shapes for Noise Control of Discrete
Tones
4 – 1987 Windows to FFT Analysis (Part II)
Acoustic Calibrator for Intensity Measurement Systems
3 – 1987 Windows to FFT Analysis (Part I)
2 – 1987 Recent Developments in Accelerometer Design
Trends in Accelerometer Calibration
1 – 1987 Vibration Monitoring of Machines
4 – 1986 Field Measurements of Sound Insulation with a Battery-Operated
Intensity Analyzer
Pressure Microphones for Intensity Measurements with Significantly
Improved Phase Properties
Measurement of Acoustical Distance between Intensity Probe
Microphones
Wind and Turbulence Noise of Turbulence Screen, Nose Cone and
Sound Intensity Probe with Wind Screen
3 – 1986 A Method of Determining the Modal Frequencies of Structures with
Coupled Modes
Improvement to Monoreference Modal Data by Adding an Oblique
Degree of Freedom for the Reference
2 – 1986 Quality in Spectral Match of Photometric Transducers
Guide to Lighting of Urban Areas
(Continued on cover page 3)
bv004811.book : bv004811_TOC.doc Black 1
Technical
Review
No.1 – 1996
bv004811.book : bv004811_TOC.doc Black 2
Contents
A Sound Intensity Probe for Measuring from 50 Hz To 10 kHz .............. 1
by F Jacobsen, V Cutanda and P M Juhl
Measurement of Microphone Free-field Corrections and Determination of
their Uncertainties ...................................................................................... 9
by Erling Frederiksen and Johan Gramtorp
Reduction of Non-linear Distortion in Condenser Microphones by Using
Negative Load Capacitance ...................................................................... 19
by Erling Frederiksen
In Situ Verification of Accelerometer Function And Mounting .............. 32
by Torben R. Licht
Copyright © 1994, Brüel & Kjær A/S
All rights reserved. No part of this publication may be reproduced or distributed in any form, or
by any means, without prior written permission of the publishers. For details, contact:
Brüel & Kjær A/S, DK-2850 Nærum, Denmark.
Editor: Harry K. Zaveri
Layout: Judith Sarup
Photographer: Peder Dalmo
Printed by Nærum Offset
bv004811.book : SoundIntensityProbe Black 1
A Sound Intensity Probe for Measuring
from 50 Hz To 10 kHz
by F. Jacobsen, V. Cutanda*) and P. M. Juhl, Department of
Acoustic Technology, Technical University of Denmark,
Building 352, DK-2800 Lyngby, Denmark
Abstract
The upper frequency limit of a p-p sound intensity probe with a certain microphone separation distance is generally considered to be the frequency at
which an ideal probe would exhibit an acceptably small finite difference error
in a plane wave of axial incidence. This article shows that the resonances of
the cavities in front of the microphones in the usual ‘face-to-face’ configuration give rise to a pressure increase that to some extent compensates for the
finite difference error. Thus the operational frequency range can be extended
to an octave above the limit determined by the finite difference error, if the
length of the spacer between the microphones equals the diameter.
Résumé
Pour une sonde d’intensité acoustique à deux microphones séparés par une distance donnée, la limite supérieure de fréquence est généralement considérée
comme la fréquence à laquelle une sonde idéale présenterait, pour une onde
plane et une incidence de 0°, une erreur de différence finie acceptable. Cet article montre que le phénomène de résonance dû aux cavités frontales des microphones configurés “face à face” entraîne un accroissement de pression tendant
à compenser cette erreur. La gamme de fréquence opérationnelle peut donc
être élargie d’un octave au-dessus de la limite ainsi imposée, si le bloc d’espacement présente une longueur égale au diamètre.
*) Present address: U.P.M., E.U.I.T. de Telecomunicación, Department of Audiovisual
Engineering and Communications, Ctra de Valencia km 7, E-28031 Madrid, Spain
1
bv004811.book : SoundIntensityProbe Black 2
Zusammenfassung
Die obere Grenzfrequenz einer Zwei-Mikrofon-Schallintensitätssonde mit einem bestimmten Mikrofonabstand wird allgemein als diejenige Frequenz betrachtet, bei der die ideale Sonde einen noch akzeptablen Fehler, bedingt
durch den endlichen Abstand der beiden Mikrofone, für eine axial einfallende
ebene Welle zeigt. Dieser Artikel zeigt, daß die Resonanzen der Hohlräume
vor den Mikrofonen bei der üblichen Anordnung (Mikrofone einander gegenüber) einen Druckanstieg verursachen, der den Abstandsfehler teilweise kompensiert. Der Arbeitsfrequenzbereich kann daher auf eine Oktave über der
durch den Abstandsfehler definierten Grenze erweitert werden, wenn die
Länge des Mikrofonabstandstücks gleich dem Mikrofondurchmesser ist.
Introduction
Sound power determination is a central point in noise control engineering,
and the method of sound power determination based on measurement of
sound intensity has the significant advantage over other methods that it
makes it possible to determine the sound power of a source of noise in situ,
even in the presence of other sources.
Existing sound intensity probes in commercial production are based on the
“two-microphone” (p-p) measurement principle in which the intensity is determined from the signals from two closely spaced pressure microphones.
One of the obvious limitations of this measurement principle is the frequency range; the fact that the method relies on the finite difference approximation clearly implies an upper frequency limit that is inversely proportional
to the distance between the microphones. Unfortunately, the influence of
phase mismatch and several other measurement errors is also inversely proportional to the distance between the microphones; therefore, one cannot
extend the frequency range simply by placing the microphones very close
together.
One can extend the frequency range by combining measurements with two
sets of microphones. The purpose of this paper is to examine whether it is possible to cover a significant part of the audible frequency range, from 50Hz to
10 kHz, with one single probe configuration.
2
bv004811.book : SoundIntensityProbe Black 3
Numerical Results
Error in intensity (dB)
In what follows it is assumed that the intensity probe is a p-p probe with the
two microphones in the usual ‘face-to-face’ arrangement with a solid ‘spacer’
between them.
0
–5
0.25
0.5
1
2
4
8
Frequency (kHz)
960348e
Fig. 1. Finite difference error of an ideal intensity probe which does not disturb the sound
field in a plane wave of axial incidence for different values of the separation distance —,
5 mm; ---. 8.5 mm; ···, 12 mm; – – , 20 mm; – - – - , 50 mm
Error in intensity (dB)
5
0
–5
0.25
0.5
1
2
4
8
Frequency (kHz)
960349e
Fig. 2. Error of an intensity probe with 12 mm long half-inch microphones in a plane wave
of axial incidence for different spacer lengths, — , 5 mm; ---, 8,5 mm; ···, 12 mm; – –, 20 mm;
– - – - , 50 mm
3
bv004811.book : SoundIntensityProbe Black 4
The operational frequency range of an intensity probe depends on the particulars of the sound field conditions [1]. Nevertheless, the highest frequency at
which an ideal p-p probe with a certain microphone separation distance would
exhibit an acceptably small finite difference error in a plane wave of axial incidence has usually been regarded as the upper frequency limit [1]. This finite
difference error is shown in Fig.1. According to this reasoning a probe with
half-inch microphones separated by a 12mm spacer (which is a very common
configuration) should not be used above, say, 5 kHz. However, more than ten
years ago Watkinson and Fahy pointed out that the resonance of the cavities in
front of the microphones in this configuration gives rise to a pressure increase
that to some extent might compensate for the finite difference error [2] A
recent investigation based on a boundary element model of an axisymmetric pp probe has confirmed Watkinson and Fahy's observation [3]. Fig.2, which corresponds to Fig. 1, shows the error calculated for a probe with two 12mm long
half-inch microphones. The error is essentially the result of the combined
effect of the finite difference approximation and the pressure increase. It is
3
(a)
Error in intensity (dB)
0
–4
3
(b)
0
–3
0.25
0.5
1
2
4
8
Frequency (kHz)
960350e
Fig. 3. Error of an intensity probe with 12 mm long half-inch microphones in a plane
wave. (a) 8.5 mm spacer; (b) 12 mm spacer. Angle of incidence: — , 0°; ---, 20°; ···, 40°; – –,
60°; – - – - , 80°
4
bv004811.book : SoundIntensityProbe Black 5
apparent that the optimum length of the spacer is about 12 mm, and that a
probe with this geometry performs very well in the case of a plane wave of
axial incidence up to 10 kHz. It can also be deduced from the figure that such a
probe is superior to a probe with quarter-inch microphones separated by a
12 mm spacer, owing to the fact that the compensating pressure increase is
shifted an octave upwards for the latter configuration. In Fig.3 is shown the
corresponding error for non-axial incidence, calculated for two different spacer
lengths.
Experimental Results
Actuator
response(dB)
The numerical results briefly summarised in the foregoing imply that the frequency range of a probe with the conventional combination of half-inch microphones and a 12 mm spacer is wider than hitherto believed. To test this
conclusion a series of experiments have been carried out: the sound power of a
loudspeaker driven with pink noise, Brüel& Kjær Type 4205, was determined
in a large (240 m3) reverberant room with a reverberation time of about 4s.
The source was placed on the floor about 1.5m from the nearest wall, and the
ratiated sound power was estimated by scanning manually with an intensity
probe over the five faces of a cubic surface of 1 × 1 × 1 m.
A frequency analyser of Brüel& Kjær Type 3550 was used in combination
with an intensity probe of Brüel& Kjær Type 3548, either with half-inch microphones of Brüel & Kjær Type 4181 or with quarter-inch microphones of
Brüel & Kjær Type 4178. Since these microphones are so-called free-field
microphones it is necessary to compensate for the drop of the pressure sensitivity at high frequencies. Fig.4 shows the pressure response of the two sets of
microphones, determined with an electrostatic actuator. All the results presented in the following have been corrected with the corresponding actuator
response.
0
–5
0.063
0.125
0.25
0.5
1
2
4
8
Centre Frequency (kHz)
960351e
Fig. 4. Electrostatic actuator response of microphones. — , Brüel & Kjær Type 4178; --- ,
Brüel & Kjær Type 4181
5
bv004811.book : SoundIntensityProbe Black 6
80
(a)
70
Sound Power (dB re 1 pW)
60
80
(b)
70
60
80
(c)
70
60
0.063
0.125
0.25
0.5
1
2
4
8
Centre Frequency (kHz)
960352e
Fig. 5. Sound power of source, estimated with different combinations of microphones and
spacers. (a) No extraneous noise, (b) strong diffuse background noise from an extraneous
source, (c) strong diffuse and non-diffuse noise from an extraneous source. –- , Half-inch
microphones, 8.5 mm spacer; --- , half-inch microphones, 12 mm spacer; ··· , half-inch
microphones, 50 mm spacer; – – , quarter-inch microphones, 6 mm spacer; – - – -, quarterinch microphones, 12 mm spacer
The measurements were carried out under three conditions: i) without
extraneous noise, ii) with strong diffuse background noise from a distant
source (Airap A14 from Électricité de France), and iii) with strong non-diffuse
and diffuse background noise from the same source placed about 2.5m from
the surface. In the last mentioned case the partial sound power of the nearest
1 m2 segment was negative in the entire frequency range.
6
bv004811.book : SoundIntensityProbe Black 7
The measurements with quarter-inch microphones were carried out with a
6 mm spacer and with a 12 mm spacer. The former measurement, which can be
expected to be reliable at high frequencies, served as the reference in the frequency range from 4 to 10 kHz. The measurements with half-inch microphones
were carried out with an 8.5mm spacer, a 12 mm spacer and a 50 mm spacer.
In order to reduce the effect of transducer phase mismatch as far as possible,
all measurements were repeated with the two microphones interchanged [4].
∆pl (dB)
10
0
0.063
0.125
0.25
0.5
1
2
4
8
Centre Frequency (kHz)
960353e
Fig. 6. Pressure-intensity index. — , Quarter-inch microphones, 6 mm spacer, no extraneous noise; ---, half-inch microphones, 12 mm spacer, no extraneous noise‚ ···, quarter-inch
microphones, 6 mm spacer, diffuse noise; – – , half-inch microphones, 12 spacer, diffuse
noise; – · –. quarter-inch microphones 6 mm spacer, non-diffuse and diffuse noise: – - – -,
half-inch microphones, 12 mm spacer, non diffuse and diffuse noise
The results of the sound power measurements are presented in Fig.5; and
Fig. 6, which shows the pressure-intensity index, gives an impression of the
acoustic conditions. It can be seen from Fig.5 that practically all measurements are in agreement from 50Hz to 1.25 kHz. An exception is the measurements with quarter-inch microphones at 50 Hz under the most difficult sound
field condition. (This is probably the result of random errors due to electrical
noise [5]; however, without compensation for phase mismatch significant
errors occurred with the quarter-inch microphones in most of the frequency
range.) From 1.6 kHz and upwards the combination of half-inch microphones
and the 50 mm spacer underestimates, but it is worth noting that the error is
less than predicted by the idealised expression for an axial plane wave (Fig.1),
and that the size of the error depends on the sound field conditions, which
leads to the conclusion that one cannot compensate for the finite difference
error. The combination of quarter-inch microphones and a 12mm spacer leads
to underestimation from 5 kHz and upwards, more or less as expected. The
7
bv004811.book : SoundIntensityProbe Black 8
measurements with half-inch microphones and the 12mm spacer are in fair
agreement with the reference measurements, confirming the predicted advantage of this combination. In fact, only the combination of half-inch microphones and the 8.5 mm spacer behaves unexpectedly.
As can be seen, it overestimates slightly under mild measurement conditions, but underestimates under more difficult conditions. It seems as if the
ability of suppressing extraneous noise at high frequencies deteriorates if the
spacer is significantly shorter than the diameter of the microphones. A possible
explanation is that the error depends more on the angle of incidence for this
configuration, cf. Fig. 3.
Conclusions
One cannot compensate for the finite difference error of p-p intensity probes
by using the theoretical plane wave expression, and one cannot extend the frequency range by using a spacer appreciately shorter than the diameter of the
microphones. Moreover, existing quarter-inch microphones are not suitable
for measurement of sound intensity at low frequencies. However, a numerical
and experimental study of diffraction effects has demonstrated that the operational frequency range can be extended to an octave above the limit determined by the finite difference error if the length of the spacer between the
microphones equals the diameter. This means that a probe with half-inch
microphones can cover the frequency range from 50Hz to 10 kHz.
References
[1]
FAHY, F.J.: “Sound Intensity” (E & FN Spon (second ed.), London 1995)
[2]
WATKINSON, P.S.: and FAHY, F.J.:J. Sound Vib. 94, 299-306 (1984)
[3]
CUTANDA, V.: JUHL P.M. and JACOBSEN, F. : “A numerical investigation of the performance of sound intensity probes at high frequencies”
Proc. Fourth Int. Congress on Sound and Vib., 1996, pp. 1897 – 1904
[4]
REN, M.: and JACOBSEN, F.: Noise Control Eng. J. 38, 17-25 (1992)
[5]
JACOBSEN, F.: J. Sound Vib. 166, 195-207 (1993)
8
bv004811.book : MicFree-field Black 9
Measurement of Microphone Free-field
Corrections and Determination of their
Uncertainties
by Erling Frederiksen and Johan Gramtorp
Abstract
Modern measurement techniques and international cooperation on calibration research have made it possible to obtain more accurate free-field and diffuse-field corrections. In this article the methods and measurements are
described and calculations of the uncertainties for the free-field corrections for
microphone Type 4191 (12.5 mV/Pa) are shown.
Résumé
Les techniques de mesure modernes et la coopération internationale dans la
recherche sur le calibrage ont contribué à améliorer la précision des corrections
de champ libre et de champ diffus. Cet article décrit les différentes méthodes de
mesure utilisées et présente un calcul d’incertitude pour les corrections champ
libre du Microphone Type 4191 (12.5mV/Pa).
Zusammenfassung
Moderne Meßtechnik und internationale Zusammenarbeit auf dem Gebiet
der Kalibrierung ermöglichen eine höhere Präzision bei Freifeld- und Diffusfeldkorrekturen. Dieser Artikel beschreibt die verschiedenen Meßmethoden
sowie die Berechnung der Unsicherheit der Freifeldkorrektur für das Mikrofon Typ 4191 (12,5 mV/Pa).
Introduction
The most commonly used method for frequency response calibration of measurement microphones is still to measure the individual electrostatic actuator
response and to add corrections to this. The corrections are the same for all
microphones of the same type. For well-documented microphone types, free-field
9
bv004811.book : MicFree-field Black 10
and diffuse-field corrections are available from the microphone manufacturers.
Brüel&Kjær has recently developed a new line of 1/2″ microphone types (Falcon
Range) and has determined their corrections which are to be used for calibration
at the factory and at calibration service laboratories abroad.
The principles applied for the determination of these free-field and diffusefield corrections are the same as those used in the past but modern measurement techniques and international co-operation on calibration research have
significantly improved the possibilities for obtaining more accurate corrections. Today, many countries and companies are improving their calibration
systems. Therefore, there is an increasing demand for accurate corrections
with documented uncertainty. The methods, the measurements and the uncertainties related to the resulting corrections for Type 4191 (12.5 mV/Pa) are discussed in this article.
Description of Free-field and Diffuse-field
Corrections
The free-field correction is the ratio between the free-field response and the
response of the microphone diaphragm system, Fig.1. The correction is dominantly determined by sound reflection and refraction caused by the microphone body. There are two different types of free-field corrections. They refer
2
dB
0
Free-field Response
–2
Free-field Correction
–4
–6
Actuator Response
–8
– 10
– 12
100
1000
10000
Frequency (kHz)
100000
960368e
Fig. 1. Free-field Frequency Response of a Type 4191 microphone obtained by adding the
Free-field Correction to an individually measured Actuator Response
10
bv004811.book : MicFree-field Black 11
to the slightly different pressure and actuator responses. Both responses
account for the individual properties of the microphone diaphragm systems.
They are relatively easy to measure in comparison with the free-field
response. Actuator response calibrations are especially simple and require no
special acoustic facilities. The Diffuse-field Response can be determined in the
same way by applying other corrections to the actuator response.
Free-field Response Measurement
Three microphones were calibrated together. Pair-wise they were mounted in
an anechoic room where one was transmitting sound to another. Microphone
(a) transmitted to receiver (b), (b) to (c) and (c) to (a).
In order to minimise the influence of room reflections a rather large room of
approx. 4 m × 4.5 m × 5 m open space and a relatively short measurement distance (0.23 m) were chosen.
The international standard IEC1094-3 defines the free-field sensitivity
product, Mf , a. Mf,b by the formula (A). A modification of this eliminates the
transmitter current and leads to the applied formula (B).
M f, a ⋅ M f, b = − j ⋅
M f, a ⋅ M f, b = −
U R, b 2d ab αdab
⋅
⋅e
I T, a ρ ⋅ f
U R, b
⋅
d ab
U T, a ρ ⋅ π ⋅ f 2 ⋅ C
a
(A)
⋅e
αd ab
(B)
UR,b : Receiver output voltage
UT,a : Transmitter driving voltage
dab: Distance between acoustic centres of the microphones
ρ:
Air density
f:
Frequency
Ca : Transmitter Capacitance
∝:
Sound attenuation of air
For all three pairs of microphones the output voltage of the receiver microphone, the voltage driving the transmitter and the transmitter capacitance
11
bv004811.book : MicFree-field Black 12
dB re. 1 µV
80
70
60
50
40
30
20
10
0
1
10
Frequency (kHz) 100
960369e
Fig. 2. Receiver Microphone Output Voltage as a function of frequency
were measured. During the measurement of the receiver voltage, the voltage
across the transmitter was kept constant as a function of frequency. For flat
microphone frequency responses this leads to a sound pressure and a receiver
output which increase by 40 dB/decade and are very low at 1000Hz where the
measurements should preferably start; see Fig.2.
IEC-bus
Controller
Measurement
Monitor
Measurement
Amplifier
Type 2636
Modified
Narrow
Band
Analyzer
Type 2010
Microphones
Receiver
Transmitter
Terminal
Digital
Interface
for Type 2010
Amplifier
20 dB
A/D
Converter
VAX
Computer
Free-field
Measurement System
960370e
Fig. 3. Free-field Calibration System operating to 200 kHz. The possible accuracy is better
than 0.025 dB at 0 dB SPL
12
bv004811.book : MicFree-field Black 13
To overcome problems with the signal to noise ratio Brüel& Kjær built a
dedicated free-field calibration system some years ago. The system can work
up to 200 kHz and thus cover the frequency range of most types of measurement microphones (sizes from 1″ to 1/8″). A block diagram of this system is
shown in Fig. 3. The measurements were performed with this system at frequency steps of 100 Hz with a spread and a resolution better than 0.025dB.
Synchronous sampling and measurement times of up to half an hour at each of
the lowest frequencies were used for improving the S/N-ratio.
After determination of the above parameters for the three pairs of microphones, the sensitivity products and the individual free-field sensitivity module were calculated using the formula below.
M f, a =
U T, b
d ab ⋅ d ca
 U R, a ⋅ U R, c
⋅
⋅
U
d bc
U T, a ⋅ U T, c

R, b
1
⋅
Cb
1
⋅
⋅e
Ca ⋅ Cc ρ ⋅ π ⋅ f 2
α ( d ab − d bc + d ca )
2

Electrostatic Actuator Response Calibration
The actuator responses were measured with 0.01dB resolution and with the
same measurement system as that used for the free-field measurements. This
type of measurement is relatively easy to perform. The uncertainty of actuator
calibration is high with respect to the absolute sensitivity. This is typically of
the order 1 dB while it is very low for the frequency response calibration.
The response measured with an electrostatic actuator is generally influenced by the radiation impedance which loads the microphone diaphragm. For
Type 4191, which has a relatively high diaphragm impedance, the influence
ranges from essentially zero at low frequencies to about 0.3dB at the highest
frequencies (40 kHz). As the influence may be modified by the mechanical configuration of the actuator, the actuator type used for calibration service should
be equal to that used for determination of the corrections.
13
bv004811.book : MicFree-field Black 14
Measured and Calculated Free-field Corrections
As mentioned above, the absolute sensitivity cannot be measured accurately
with an electrostatic actuator. Therefore, there is no reason for also spending
great efforts on obtaining an absolute measurement of the free-field response.
The division of the free-field response by the actuator response will, anyway,
give a result which contains a significant error. However, as this error makes
a constant factor over the entire frequency range, some methods are available
for its elimination.
From the laws of physics it can be concluded that the true corrections are
small and gradually approaching zero dB at low frequencies. The resulting
free-field correction may thus be determined visually from the shape of the
measured and calculated curve which covers the frequency range from 1kHz
to 50 kHz. The accuracy obtained by this method is assumed to be significantly
better than 0.1 dB. The curve shown in Fig.4 gives the resulting free-field corrections determined by the above described method.
To verify the measured free-field correction results the microphone and the
sound field were simulated by a mathematical model. The simulation
accounted for the microphone dimensions, the diaphragm impedance and the
properties of the ambient air. The simulation was made by Peter Juhl who also
developed the applied boundary element model at the Acoustics Laboratory of
10
dB
9
8
7
6
5
4
3
2
1
0
1
10
Frequency (kHz)
100
960371e
Fig. 4. Measured (curve) and calculated (points) free-field corrections for Type 4191 (0 °
incidence). The results are valid without protection grid. The calculated results support
the measurement results
14
bv004811.book : MicFree-field Black 15
the Technical University in Lyngby, Denmark. The 1/3 octave calculation
results are shown by the points in Fig.4.
These results support the measurement results, especially at low frequencies while they are probably less reliable at the high frequencies due to lack of
sufficient spatial resolution of the model.
Measurement of Directional Responses
Turntable
Controller
Type 5949
Audio
Analyzer
Type 2012
Power
Amplifier
Type 2706
Microphone
Turntable
Type 5960
TDS Measurement System
960372e
Fig. 5. The time selective measurement system for directional responses eliminates
reflections
The free-field correction discussed above is valid for 0° incidence (reference
direction). The sensitivities and corrections for other angles of incidence were
determined relative to 0° for angle steps of 5° by the set-up shown in Fig.5. The
sound source was mounted in a fixed position while the receiving microphone
was rotated to its different angles by an automatic turntable. The necessary,
rather long, measurement distance and the turntable arrangement will generally lead to disturbing reflections. Therefore, a TDS measurement system
based on the Brüel &Kjær Type 2012 was applied. This system makes time
selective measurements and can eliminate reflections. The directional characteristics obtained are shown in Fig.6.
15
bv004811.book : MicFree-field Black 16
Fig. 6. Free-field corrections for Type 4191 without protection grid. The angles are stepped
by 5 degrees
Random Incidence Corrections
The random incidence correction is calculated in accordance with the International Standard IEC 1183 (1994/95) which specifies how the calculation should
be made. The free-field corrections shown in Fig.6 were applied with weighting factors which account for that fraction of total power coming to the microphone from the specific directions. The resulting random incidence correction
is also shown in Fig. 6.
Estimated Uncertainty of Free-field Corrections
The resulting uncertainties of the 0° free-field correction were estimated from
the uncertainties of the parameters applied for its calculation. Their uncertainties were separated into groups of random and systematic errors as their
weight in the reciprocity calculations are different. The systematic errors do
partly eliminate each other while the non correlated or random errors add up
statistically. The relative uncertainty of Free-field Response was determined
by using the formula below:
16
bv004811.book : MicFree-field Black 17
∆M f
Mf
2
∆U R  2  ∆U T  2  ∆C  2  ∆f 2 
 ∆d  +
= A ⋅ 
+
+
+
+
 f2 
  UR 
 UT 
C 
d 
2
 ∆ρ 
ρ 
2
1
∆α  2  ∆d  2  2
+  αd
+ αd
 d  
 α 
where the weighting factor ‘A’ equals 1/2 for the systematic and 3 ⁄ 2 for the
random uncertainties respectively. After determination of other uncertainties
related to the actuator response, the low frequency normalisation of the freefield corrections to 0 dB and to the influences of ambient pressure and temperature, the resulting uncertainty was calculated.
The uncertainty (tentative) of the free-field corrections for Type 4191 without grid was estimated to:
Frequency
1kHz
2kHz
4kHz
8kHz
16kHz
32kHz
40kHz
Uncertainty (2×σ)
0.03dB
0.06dB
0.08dB
0.10dB
0.12dB
0.15dB
0.20dB
Notes:
The uncertainty (95% confidence level) is valid for the 0°-correction at
101.3 kPa, 23°C and 50% RH.
To calculate the uncertainty of a calibration for which the correction is
applied, the uncertainty of the electrostatic actuator calibration must be taken
into account. It should also be noted that the uncertainty of calibrations valid
for microphones with protection grid are generally higher.
Conclusion
Free-field corrections were worked out for a new line of 1/2″ measurement
microphones. The uncertainty of the Type 4191 free-field correction (0°-incidence) was analysed and calculated. The result agrees well with the uncertainty estimated from practical experience. The new corrections represent
improvements in comparison with the corrections and the uncertainty estimated for earlier microphone types. The analysis revealed possibilities for further improvements in the future.
17
bv004811.book : MicFree-field Black 18
References
[1]
IEC 486, “Precision method for free-field calibration of one-inch standard condenser microphones by the reciprocity technique and the draft for
the succeeding IEC document which includes 1/2″ microphones
[2]
RASMUSSEN, K.: and SANDEMANN OLSEN, E.: “Intercomparison
on free-field calibration of microphones”, Final report (PL-07), the
Acoustics Laboratory, Technical University of Denmark
[3]
JUHL, P.: “Numerical Investigation of Standard Condenser Microphones” Journal of Sound and Vibration 1994 Vol. 177 (4) p. 433-446
18
bv004811.book : Non-linear Black 19
Reduction of Non-linear Distortion in
Condenser Microphones by Using
Negative Load Capacitance
by Erling Frederiksen
Abstract
An analysis was made of the distortion produced by condenser measurement microphones which operate with stiffness controlled diaphragms. A
calculation formula which was theoretically derived was verified by experiments. Analysis of this formula indicated that the distortion can be
reduced by loading the microphone with a proper capacitance which has to
be negative and is a function of the ratio between the backplate and diaphragm diameters. Experimental results confirmed this.
Résumé
Nous avons analys la distorsion produite par les microphones de mesure condensateur quips de diaphragmes dont la tension tait contrle. Une formule,
rsultat dune approche thorique, indiquait que lon peut rduire le phnomne de
distorsion en chargeant le microphone avec une capacit approprie, qui doit tre
ngative et tenir compte du rapport entre les diamtres du diaphragme et de la
plaque arrire. Cest ce quont confirm les expriences ralises.
Zusammenfassung
Es wurde die Verzerrung analysiert, die Kondensatormeßmikrofone aufweisen, wenn die Auslenkung der Membran durch ihre Steifigkeit
bestimmt wird. Eine theoretisch abgeleitete Berechnungsformel wurde
experimentell bestätigt. Die Analyse der Formeln wies darauf hin, daß sich
die Verzerrung reduzieren läßt, wenn das Mikrofon mit einer geeigneten
Kapazität belastet wird. Diese Kapazität ist negativ und eine Funktion des
Quotienten aus den Durchmessern von Gegenelektrode und Membran. Das
wurde durch Versuchsergebnisse bestätigt.
19
bv004811.book : Non-linear Black 20
Introduction
Condenser measurement microphones have very wide dynamic ranges, typically 140 dB. At low sound levels the range is limited by inherent noise of the
microphone and/or preamplifier. At high levels it is generally limited by nonlinear distortion which is proportional to the sound pressure and is produced
by the microphone itself. This distortion has been analysed in theory and in
practice for some frequently used types of microphones. The good agreement
which was found between measured and calculated results verifies the
derived distortion formulae and points clearly at the load capacitance and
mode of diaphragm displacement as being the dominating reasons. The formulae were used for calculation of the capacitive loading which would lead to
the lowest possible distortion. This appeared to be negative and to be a function of the ratio between the backplate and diaphragm diameters. Tests made
with an experimental preamplifier with negative input capacitance gave
promising results.
Theoretical Distortion Analysis of Transduction
General. The operation of most condenser measurement microphones is
based on the application of a constant electrical charge stored on the active
microphone capacitance and on its parallel (stray) capacitance. The constant
charge may either be supplied from an external voltage source via a resistor
(typically 109 Ω) or by a built-in electret. The formulae below describe the
transduction of capacitance variation to voltage:
C0 + Cp
E ⋅ ( C a + C p ) = Q 0 = E 0 ⋅ ( C 0 + C p ) or E = E 0 ⋅
Ca + Cp
(1)
E : Voltage across capacitances with diaphragm displaced by sound pressure
E0 : Voltage across capacitances with diaphragm at rest position
Ca : Active diaphragm-backplate capacitance (varies with sound pressure)
C0 : Active diaphragm-backplate capacitance with diaphragm at rest position
Cp :Parallel capacitance (passive)
Q0 : Constant charge stored on the active and passive capacitances
20
bv004811.book : Non-linear Black 21
As the charge (Q0) is kept constant the voltage (E) will vary with the variation of the active capacitance (Ca) which is caused by the sound pressure. The
voltage produced depends on the microphone configuration and on the diaphragm deflection mode.
Flat Diaphragm Displacement Mode
If the microphone diaphragm is considered to be parallel to a circular backplate and displaced like a flat piston, then the active capacitance (Ca) and its
rest capacitance (C0) can be expressed by the equations:
ε ⋅ π ⋅ R 2b
Ca = Rs ⋅
D+d
C0 = Rs ⋅
ε :
Rb :
D :
d :
Rs :
Ab :
Ah :
y :
ε ⋅ π ⋅ R 2b
D
= C0 ⋅ ( 1 + y ) −1
where R s =
Ab −Ah
Ab
and y =
d
D
Dielectric constant of air
Radius of backplate
Rest distance, backplate to diaphragm
Displacement of diaphragm
Ratio of effective and total backplate area
Total backplate area (includes area of holes)
Area of holes in backplate (a uniform hole distribution is considered)
Relative diaphragm displacement
D
d
Ca
Cp
960363e
Fig. 1. Microphone with flat diaphragm displacement mode and parallel capacitance
21
bv004811.book : Non-linear Black 22
Insertion of the above expressions into Equation (1) leads to Equation (2)
which defines the output voltage of a microphone with a flat diaphragm displacement mode:
E fm = E 0 ⋅
C0 + Cp
C0 ⋅ ( 1 + y ) −1 + Cp
(2)
Series expansion of equation (2) leads to:
1
Cp  2
  Cp 

2 
E fm = E 0 + E 0 ⋅
⋅ y−
⋅y +
⋅ y 3 − ..... (2a)

 C0 + Cp 
C0 + Cp   C0 + Cp 
C0
For a sinusoidally varying diaphragm displacement with time y, y2 and y3
become:
y = y
m
1
1
⋅ sin ωt; y 2 = ⋅ y 2
⋅ ( 1 − cos 2ωt ) ; y 3 = ⋅ y 3
⋅ ( 3 sin ωt − sin 3ωt )
2 m
4 m
where ym : Maximum value of relative diaphragm displacement.
This leads to the following second (D2) and third (D3) harmonic distortion
components of the output voltage (relative to the fundamental frequency component):
D2 =
Cp  1
Cp  2
1
1
⋅ ym ⋅
⋅ ym ⋅
⋅ 100% and D 3 =
⋅ 100%
2
2
C0 + Cp 
C0 + Cp 
The dominating second harmonic component (D2) is proportional to the diaphragm displacement and thus to the sound pressure while the third harmonic
(D3) is proportional to the square of the pressure. Notice, that the distortion
decreases with the parallel capacitance and that it becomes zero if this capacitance becomes zero (Cp = 0).
22
bv004811.book : Non-linear Black 23
Parabolic Diaphragm Displacement Mode
Generally condenser microphones use foil diaphragms with a high internal
mechanical tension which gives the diaphragm its required stiffness and
determines the displacement mode at lower frequencies. At higher frequencies the air damping and the foil mass do also influence the mode. This discussion covers only the low frequency mode which can be considered to occur up
to a frequency which is 0.2 to 0.5 times the diaphragm resonance frequency.
The displacement mode of an ideal circular diaphragm, which is purely stiffness controlled, is defined by the formula below and illustrated in Fig.2:
r2 
d ( r) = d0 ⋅ 1 −

R 2d 
where
d0 : centre displacement
r : distance to centre
Rd : Radius of diaphragm
Fig. 2. Calculated foil diaphragm displacement mode
To verify this the displacement mode of a one inch microphone Type 4144
was measured. This was done by scanning its diaphragm along a diameter.
The scanning was made with a small microphone (Type 4138, 1.8 mm backplate diameter) with its own diaphragm dismantled. This microphone was
moved at a fixed distance in front of the large diaphragm in such a way that it
detected the local displacement of the larger diaphragm while this was
exposed to sound pressure. The measured and the calculated displacement
magnitude were found to be in very good agreement; see Fig.3.
Considering the above displacement mode the active capacitance (Ca ) and
its rest capacitance (C0 ) may be expressed by the equations below:
23
Displacement re. Centre
bv004811.book : Non-linear Black 24
100
%
80
60
40
20
0
– 10
–8
–6
–4
–2
0
2
Distance from Centre (mm)
4
6
8
10
960365e
Fig. 3. Diaphragm displacement as a function of distance to centre for a one inch microphone with a diaphragm of 18 mm diameter. The points are measured at 40 Hz and at
130 dB and 150 dB SPL. The curve is calculated
Rb
Ca = Rs ⋅
where
Rs =
ε⋅2⋅π⋅r
∫
r2 
0 D + d ⋅ 1−
0

R 2d 
C0 = Rs ⋅
dr
ε ⋅ π ⋅ R 2b
D
Ab − Ah
Ab
Simplification and integration of the above equation leads to:
Ca = Rs ⋅
ε ⋅ π ⋅ R 2b
D
Rb
⋅ R −b2 ⋅
∫
0
2⋅r
r2 
1 + y0 ⋅ 1 −
 R2 
d
24
dr
bv004811.book : Non-linear Black 25
Rb
C a = C 0 ⋅ R −b2 ⋅
∫
2⋅r
r2 
0 1 + y ⋅ 1−
0
 R2 
dr
d
 R 2b 
Ca = Co ⋅ 
2
 Rd 
−1
⋅ y −01 ⋅ ln
1 + y0

R 2b 
1 + 1 −
 ⋅ y0
R 2d 

1 + y0
C a = C 0 ⋅ ( 1 − k ) − 1 ⋅ y −01 ⋅ ln
1+k⋅y
where k = 1 −
0
R 2b
d0
and y 0 =
D
R 2d
Insertion of Ca and C0 in Equation (1) gives the equation valid for parabolic
mode:
E pm = E 0 ⋅
C0 + Cp
1 + y0
+ Cp
C 0 ⋅ ( 1 − k ) − 1 ⋅ y −01 ⋅ ln
1 + k ⋅ y0
(3)
Series expansion of Equation (3) leads to:
E pm = E 0 + E 0 ⋅
C0
k+1
⋅
⋅ ( y 0 + F 2 ⋅ y 20 + F 3 ⋅ y 30 + .... )
2
C0 + Cp
(3a)
where
25
bv004811.book : Non-linear Black 26
F2 = −
F3 =
C0 ⋅ ( k2 − 2 ⋅ k + 1) + 4 ⋅ Cp ⋅ ( k2 + k + 1)
6 ⋅ ( C0 + Cp) ⋅ ( k + 1)
and
( C 20 + 4 ⋅ C 0 ⋅ C p ) ⋅ ( k 2 − 2 ⋅ k + 1 ) + 6 ⋅ C 2p ⋅ ( k 2 + 1 )
12 ⋅ ( C 0 + C p ) 2
The factors F2 and F3 are functions of the parameter k and thus of the ratio
between the backplate and diaphragm radii. The larger the backplate
becomes, the higher becomes the distortion. Equation (2a) indicates that this
effect was to be expected as area added along the outer circumference represents a less active capacitance which loads the more active capacitance located
at the diaphragm and backplate centres.
Notice, that the smaller the backplate becomes in comparison with the diaphragm, the more flat does the active part of the diaphragm becomes. Therefore, for k equal to ‘1’, Equation (3a) becomes equal to Equation (2a) which is
valid for flat diaphragm mode. In practice, distortion should be defined as a
function of Sound Pressure (p) rather than relative diaphragm displacement
at the centre (y 0 ). The relation between displacement and sound pressure can
be obtained from the following equations:
e1 = E0 ⋅
C0 + Cp − Ci  −1
C0
k+1

⋅
⋅ y 0 and p = e 1 ⋅ S 0 ⋅

C0 + Cp 
2
C0 + Cp
e1 : Microphone output voltage according to Equation (3a)
(1st term with y0)
p : Sound pressure
S0 : Measured open circuit sensitivity
For a sinusoidal diaphragm displacement with time, the ratios (D2 and D3)
between the second and third harmonic components and the fundamental
component become:
26
bv004811.book : Non-linear Black 27
1
D2 =
 ym 
⋅ F 2 ⋅ 100%
 2 
(4)
and
2
D3 =
 ym 
⋅ F 3 ⋅ 100%
 2 
where y m =
2 ⋅ p RMS ⋅
(5)
S0 C0 + Cp − C i
2
⋅
⋅
k+1
E0
C0
Comparison of Calculated and Measured Distortion
Calculated and measured distortion data were evaluated by comparison. See
input data and distortion results for some commonly used microphones in the
tables below.
B&K Type No.
4144
4133
4165
4135
4190
4192/93
Dimension
1/1″
1/2″
1/2″
1/4″
1/2″
1/2″
S0
mV/Pa
50
12
50
4.0
50
12.5
E0
V
200
200
200
200
200
200
Rd
mm
9.1
4.6
4.6
2.1
4.6
4.6
Rb
mm
6.65
3.60
3.65
1.75
3.45
3.45
D0
µm
23.5
21.0
22.2
18.0
24.5
19.0
Ah
mm2
20.3
1.70
4.24
0
3.96
5.50
Ce
pF
2.1
0.5
0.9
0.2
0.8
0.8
Ch
pF
2.1
1.1
1.8
1.6
1.6
1.5
27
bv004811.book : Non-linear Black 28
B&K Type No.
4144
4133
4165
4135
4190
4192
4193
Load
Capacitance
pF
0.2
0.2
0.2
0.2
0.2
0.2
100
Sound Pressure
Level
dB
140
150
140
160
140
150
150
2. harm.
calculated
%
0.51
0.49
0.97
1.52
0.97
0.63
3.0
2. harm.
measured *
%
0.56
0.50
1.08
1.55
0.90
0.54
3.4
0.91
0.97
0.98
0.98
1.08
1.17
0.88
0.01
0.01
0.02
0.04
0.02
0.01
0.09
Ratio - Calc.
re. HB-data
3. harm.
calculated
%
* Sources: Brüel & Kjær’s blue Microphone Handbook and the Falcon Handbook. The data applied
for Types 4133 and 4165 originates from measurements performed in January 1994
There is very good agreement between the calculated and the measured distortion data. The calculated ratio (see lower table) is very close to one. As the
calculations only account for the transduction itself this seems to be the only
source of low frequency distortion which is of importance for the analysed
types of microphones.
Distortion Reduction By Negative Capacitance
Loading
Theory
Equation (4) shows that the 2nd harmonic component is proportional to the
factor F2. It is a function of the ratio between the active and the passive parallel capacitances as well as of the ratio between the backplate and diaphragm
radii. For a certain radii ratio (i.e. a certain value of ‘k’) F2 and the second harmonic distortion become zero if the microphone is loaded with a proper negative capacitance. Calculated distortion is shown for two extreme microphone
configurations in Fig.4.
28
bv004811.book : Non-linear Black 29
10
%
2. Harmonic
1
Distortion
Large Backplate (k = 0)
Small Backplate (k = 1)
0.1
3. Harmonic
0.01
0.001
– 0.5 – 0.4 – 0.3 – 0.2 – 0.1
0
0.1 0.2 0.3 0.4 0.5
Ratio between Cp and Co
0.6
0.7 0.8
0.9
1
960366e
Fig. 4. Harmonic distortion calculated as a function of the ratio between the passive and
active capacitances
Typical microphone/preamplifier combinations have a Cp /C0-ratio of + 0.1
to + 0.4. Fig. 4 shows that the optimum with respect to distortion is between
‘− 0.25’ and ‘0’. The ideal Cp /C0-ratio is defined by the following equation:
 Cp 
 C0 
= −
ideal
1 k2 −2 ⋅ k + 1
⋅
4 k2 + k + 1
Negative Capacitance
In principle, negative load capacitance may be created by a circuit like that
shown in Fig. 5. A capacitor (C) is connected between the input and output of
the microphone preamplifier whose gain (>+ 1) can be adjusted to give the
proper input capacitance (Ci); see the formula:
Ci =
where
e1 − e2
e1
⋅ C = ( 1 − A) ⋅ C
e1: input voltage,
e2: output voltage
29
bv004811.book : Non-linear Black 30
C
e1
e2
A
960367e
Fig. 5. Principle of negative input capacitance circuit
Experimental Results
An experimental preamplifier with adjustable negative input capacitance was
designed and tested with three different microphones at 100Hz. For each
microphone the capacitance was adjusted to give the lowest possible 2nd harmonic distortion. The results are shown in the table below:
Initial Experimental
Results obtained with
Negative Input
Capacitance
Unit
Type 4165
Type 4133
Type 4135
Sound Pressure Level
dB
134
146
156
Nominal 2. harmonic
distortion
%
0.61
0.32
0.98
Minimised 2. harmonic
distortion (negative cap.)
%
0.10
0.16
0.18
Reduction factor
–
6.1
2.0
5.4
Measured increase of
sensitivity (approximate)
dB
2.0
1.5
4.3
Calculated increase of
sensitivity
dB
2.0
1.4
3.6
Optimal input
capacitance (calculated)
pF
− 3.7
− 2.6
− 2.2
30
bv004811.book : Non-linear Black 31
The idea (patented) of using negative input capacitance for distortion reduction seems to work well in practice but further experiments have to be made to
clarify all aspects of its use. The Brüel& Kjær High Pressure Calibrator Type
4221 was used for the distortion measurements.
Conclusion
Harmonic distortion of condenser microphones using constant electrical
charge has been analysed for the frequency range where the diaphragm displacement is stiffness controlled. Distortion formulae have been derived for
the transduction from diaphragm displacement to output voltage. The formulae were applied for calculating distortion of some commonly applied types of
measurement microphones. The results were compared with data supplied by
the manufacturer and very good agreement was found. This verified the formulae and pointed at parallel capacitance and displacement mode as being
the dominating reason for the distortion. The distortion analysis indicated the
possibility of reducing harmonic distortion by loading the microphone with
negative capacitance. This was confirmed by experiments.
Further work has to be done to analyse the practical possibilities which in
addition to distortion reduction might be improvement of high level peak
measurements and extension of the applicable dynamic range of condenser
microphones.
31
bv004811.book : In-situ Black 32
In Situ Verification of Accelerometer
Function And Mounting
by Torben R. Licht
Abstract
Piezoelectric accelerometers are used for vibration measurements in a huge
variety of measurement situations. Many different ways of checking the
integrity of a measurement channel have been devised, from tapping on the
structure to very sophisticated measurement schemes such as complex
impedance measurements. A new practical charge amplifier input system
which permits in situ check of accelerometer functionality and mounting performance has been developed. (Patent pending). A description of the system
and performance examples will be given.
Résumé
Les accéléromètres piézoélectriques interviennent dans de nombreux types de
mesures vibratoires. De nombreuses méthodes de vérification de l’intégrité de
la chaîne de mesure ont été mises en œuvre, des simples impacts sur la structure jusqu’à des dispositifs de mesure d’impédance complexe. Un nouveau système d’entrée d’amplification de charge très pratique vient d’être mis au point
pour permettre la vérification in situ des fonctionnalités de l’accéléromètre et
des performances de l’installation (en instance de brevet). Il est décrit dans
ces pages et ses performances sont illustrées par des exemples.
Zusammenfassung
Piezoelektrische Beschleunigungsmesser werden für eine Vielzahl von
Schwingungsmessungen verwendet. Um die Integrität des Meßkanals zu
prüfen, wurden zahlreiche Methoden entwickelt, vom Beklopfen der Struktur
bis zu hochkomplizierten Meßverfahren wie die Messung der komplexen
Impedanz. Es wurde ein neues praktisches Ladungsverstärker-Eingangssystem entwickelt, mit dem sich Funktionstüchtigkeit und Befestigung von
Beschleunigungsaufnehmern in situ prüfen lassen. (Patent angemeldet). Es
folgen eine Beschreibung des Systems und Beispiele für seine Leistungsfähigkeit.
32
bv004811.book : In-situ Black 33
Introduction
Piezoelectric accelerometers are used extensively to measure vibration on
many different structures and with many different vibration sources.
A number of different techniques are used to ensure the proper functioning
of each measurement channel, but only a few very complicated methods have
been used to verify the mechanical integrity of the transducer and its mounting.
This is even more serious because the handling and mounting are often carried out by people without knowledge of vibration measurement techniques.
Both statistics and an educated guess indicate that, on a list of common
problems with piezoelectric accelerometers, the first is cabling and the second
is mounting.
Therefore it is believed that the simple method for testing mounted transducers, described in the following, can make a significant contribution to the
quality of future vibration measurements.
Accelerometer Theory
Normally an accelerometer is described as a seismic transducer with the simple model shown below. k is the stiffness of the spring, c the damping and m
the seismic mass.
k
m
c
Moving Part
960354e
Fig. 1
The ratio between the motion amplitude of the moving part and the relative
motion of the mass with respect to the housing can be described in magnitude
and phase by the formulae
33
bv004811.book : In-situ Black 34
Rd =
ω 
 ωr 
ω 2

1− 
ω
 r 
2
2
2d
θ = tan− 1
and
ω
+ 2d
ω
2
ω
ω
r
ω 2

1− 
 ωr 
r
where d = c/cc is the fraction of critical damping ( c c = 2 km = 2mω ) and
r
k
is the resonant angular frequency.
ω =
r
m
For accelerometers the input is taken as acceleration of the moving part yielding Ra = − Rd/ω2. The resulting general curves are shown here.
Seismic Transducer Phase Response
Seismic Transducer Response
100
150
120
d = 0.01
d = 0.1
d = 0.2
d = 0.707
90
60
30
0
0.1
1
10
Frequency/Resonant Frequency
Rel. Displacement/Input Acceration
Rel. Displacement/Input Acceration (degrees)
180
d = 0.01
10
d = 0.1
d = 0.2
1
d = 0.707
0.1
0.01
0.1
1
10
Frequency/Resonant Frequency
960355e
Fig. 2
34
960356e
Fig. 3
bv004811.book : In-situ Black 35
Practically all piezoelectric accelerometers have a negligible damping in the
order of 0.01, which means that the resonance peak has a high Quality Factor
Q, making it possible to excite the resonance and observe the decaying signal.
To include the mounting, the model has to be extended to the following where
the base and mounting stiffness is included as separate items and the damping
has been omitted.
m
k
Base M
K
Structure
960357e
Fig. 4
The normal modes of such a system is described by the roots of the equation
Mω 2 − ( k + K )
k
k
mω 2 − k
= 0
or
Mmω 4 − ( m ( k + K ) + Mk ) ω 2 + kK = 0
ω2 = ω2
m 1+
R
R 
± 1+
2 
2
giving the roots
2
where R = K/k is the ratio between the two spring constants and ω =
m
is the mounted resonance frequency.
k
m
35
bv004811.book : In-situ Black 36
The two modes described by the formula are shown in the graph below.
Mode Frequency/Mounted Resonance Frequency
System Modes
10
Mode 2
1
Mode 1
0.01
0.1
1
10
Spring Constant Ratio
960358e
Fig. 5
Test Method
The basic idea of the test method is based on the reciprocal nature of the piezoelectric material, i.e. if a voltage is applied to the piezoelectric discs they will
change shape, following the general equation
∆x = d xy ⋅ V
where dxy is the appropriate piezoelectric constant, mostly d33 (compression
constructions) or d15 (shear constructions).
This implies that on transducer designs with low damping and relatively
high coupling (high d) an electrical pulse can be used to excite the structure
and make it “ring” or vibrate at its resonance frequency for an extended period
of time.
The decrease in amplitude is given by the logarithmic decrement
∆ ≈ 2πd
36
bv004811.book : In-situ Black 37
which for a damping d = 0.01 gives ∆ = 0.06 i.e. the amplitude decreases 6% per
oscillation, leaving a number of oscillations in the order of 50 to be observed.
The frequency observed will be the dominant mode of the two shown above.
A well mounted accelerometer, i.e. where a high spring constant ratio is
obtained, will then resonate at the mounted resonance frequency. However,
the more loose the mounting becomes, the lower will that mode frequency be
and the more the free hanging resonance will become dominant. This gives 1.4
times the mounted resonance in the typical case shown in the figure.
The amplitude of the signal can be used as a check on the accelerometer sensitivity squared.
Implementation
In the Brüel & Kjær Measuring Amplifier Type 2525 a function has been
implemented to exploit the idea described previously.
The instrument contains a generator which can generate a single square
pulse, as well as the logical circuitry to switch the generator in and out in front
of the charge amplifier. The time signal is shown schematically below.
A counter with gating is also included in the amplifier. This permits the
direct display of the measured resonance frequency of the system tested.
Input and Output time records
Input and Output Signal Amplitudes
20
15
10
5
0
–5
– 10
– 15
– 20
0
200
400
600
800 1000
Time, typically microseconds
960359e
Fig. 6
37
bv004811.book : In-situ Black 38
Examples of Application
An accelerometer Type 4382 with a specified mounted resonance frequency of
28 kHz ± 10% and a mass of 17 grams was mounted on a large steel block. A
good, smooth surface was present at the mounting position.
The measured resonance frequency as a function of the mounting torque
was measured. The results are shown in Table 1.
Mounting torque
[Nm]
Resonance frequeny
[kHz]
0.4
25.76
0.8
26.87
1.2
26.87
1.6
26.87
2.0
26.87
Table 1. Dependence of resonance frequency on mounting torque
The results show that when a good mounting surface is made the torque is
not critical for low level measurements (at higher vibration levels the transducer might lose contact to the surface with dramatic errors as a result).
To simulate a bad surface a single 0.18mm (0.007″) strand of copper wire
was introduced between the accelerometer and the mounting surface. A frequency of 21.28 kHz was measured, which shows the dramatic change from
26,87 to 21.28 kHz on the resonance.
Another experiment was made to show the kind of information which can be
obtained.
A plate has a certain local stiffness, i.e. in principle, a certain mechanical
impedance. When we mount an accelerometer of a certain mass M, i.e. a certain mechanical impedance equal to jωM, we load the plate and change its
parameters, e.g. the resonance frequencies according to the ratio between the
two impedances.
If the plate impedance is large compared to the accelerometer impedance
this will also mean that the accelerometer mounted resonance will be close to
the specified typical value. Whereas if the plate is thin and has an impedance
which is low compared to that of the accelerometer, the mounted resonance
found will tend to approach the free hanging frequency. This will indicate that
we have introduced important changes to the structure.
38
bv004811.book : In-situ Black 39
A number of measurements were made on steel plates with thicknesses
from 1 to 10 mm (0.04 to 0.4″). The results are shown in Table 2.
It is seen that the accelerometer starts to affect the vibration even at relatively large plate thicknesses.
Plate Thickness
[mm]
Measured Resonance
Frequency [kHz]
1.0
36.96
1.5
34.72
2.0
33.60
3.0
31.36
10.0
29.11
Table 2. Resonance frequency as function of plate thickness
Other Ways To Implement This Method
When using frequency analyzers more information can be obtained. If the
2525 is used, the output signal can be analysed even when resonances are not
pronounced and several different frequencies may be observed.
When using advanced FFT (Fast Fourier Transform) analyzers containing
special generators like the Brüel&Kjær Type 3550, other possibilities exist. A
transformer can be inserted between the accelerometer and the charge amplifier
input. This allows a pulse to be injected into the system using the generator.
The system is shown schematically below.
Generator
Analyzer
Transformer
Sensor
960360e
Fig. 7
39
bv004811.book : In-situ Black 40
Fig. 8 The pulse used for testing and its spectrum on a 40 dB display range
Fig. 9 Knock transducer measured mounted (upper) and unmounted (lower)
40
bv004811.book : In-situ Black 41
The generator signal, its frequency content and the response from a
mounted and unmounted knock sensor is shown in Figs.8 and 9.
The pulse applied has the important properties of no frequency content at
DC and a broad continuous spectrum. This is similar to the pulse produced by
built-in generator of the Measuring Amplifier Type 2525.
The unmounted response shows a first resonance at 63kHz and some more
resonances at higher frequencies.
The mounted response shows a rather complicated response (in contrast to
most accelerometers) with the most pronounced resonance at 58kHz.
It is seen that a lot of information can be obtained from FFT analysis. In particular, it is believed to be a viable method for quality control of transducers
made in large numbers, like the knock sensors shown here.
Conclusion
A new, simple method for checking the performance of piezoelectric transducers has been described.
A simple instrument implementing this method has been tested and the
results showed good agreement with the theory.
An advanced FFT analyzer has been demonstrated to be able to exploit the
method even further.
It is believed that this method can help the vibration measuring community
to improve the total quality of its work in the future.
References
[1]
HΑRRIS & CREDE: “Shock and Vibration Handbook”, McGraw-Hill,
Inc., 1976
[2]
SERRIDGE, M. & LICHT T.: “Piezoelectric Accelerometers and Preamplifiers”, Brüel & Kjær, 1987
[3]
Measuring Amplifier Type 2525, Instruction Manual, Brüel& Kjær,
1994
[4]
WISMER, N.J. and KONSTANTIN-HANSEN, H.: “Mounted Resonance
Measurements using Type 2525”. Application Note, BO0413,
Brüel & Kjær, 1994
41
bv004811.book : InsideCovers Black 45
Previously issued numbers of
Brüel & Kjær Technical Review
(Continued from cover page 2)
1 – 1986 Environmental Noise Measurements
4 – 1985 Validity of Intensity Measurements in Partially Diffuse Sound Field
Influence of Tripods and Microphone Clips on the Frequency Response
of Microphones
3 – 1985 The Modulation Transfer Function in Room Acoustics
RASTI: A Tool for Evaluating Auditoria
2 – 1985 Heat Stress
A New Thermal Anemometer Probe for Indoor Air Velocity
Measurements
1 – 1985 Local Thermal Discomfort
4 – 1984 Methods for the Calculation of Contrast
Proper Use of Weighting Functions for Impact Testing
Computer Data Acquisition from Brüel & Kjær Digital Frequency
Analyzers 2131/2134 Using their Memory as a Buffer
3 – 1984 The Hilbert Transform
Microphone System for Extremely Low Sound Levels
Averaging Times of Level Recorder 2317
2 – 1984 Dual Channel FFT Analysis (Part II)
1 – 1984 Dual Channel FFT Analysis (Part I)
4 – 1983 Sound Level Meters — The Atlantic Divide
Design Principles for Integrating Sound Level Meters
3 – 1983 Fourier Analysis of Surface Roughness
2 – 1983 System Analysis and Time Delay Spectrometry (Part II)
Special technical literature
Brüel & Kjær publishes a variety of technical literature which can be obtained
from your local Brüel & Kjær representative.
The following literature is presently available:
❍
❍
❍
❍
❍
❍
Modal Analysis of Large Structures – Multiple Exciter Systems (English)
Acoustic Noise Measurements (English), 5th. Edition
Noise Control (English, French)
Frequency Analysis (English), 3rd. Edition
Catalogues (several languages)
Product Data Sheets (English, German, French, Russian)
Furthermore, back copies of the Technical Review can be supplied as shown in the
list above. Older issues may be obtained provided they are still in stock.
Technical
Review
No. 1 – 1996
Calibration Uncertainties & Distortion of Microphones.
Wide Band Intensity Probe. Accelerometer Mounted Resonance Test
ISSN 007 – 2621
BV 0048 – 11
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement