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Nonlinear Behavior of Piezoelectric Accelerometers
J. Putner
1
, P.B. Grams
1
and H. Fastl
1
1
AG Technische Akustik, MMK, TU München, 80333 München, Germany, E-mail: putner@tum.de
Introduction
Piezoelectric accelerometers are widely used for vibration
measurements, because of their robustness and reliability,
which lead to stable transfer characteristics. Typical piezoe-
lectric accelerometers show good linearity over wide fre-
quency and dynamic ranges. While the linear behavior is
understood and documented, specifications of the nonlinear
behavior are scarce and, if present, state generally low dis-
tortions.
Since the limitations of the transducer have to be considered
in the analysis of the measured vibrations, the nonlinear
behavior of a typical piezoelectric accelerometer was further
investigated. For the measurements a setup for vibration
calibration by comparison to a reference transducer, accord-
ing to ISO 16063, was enhanced by laser vibrometry, to
resolve limitations of the vibration exciter. In order to calcu-
late the vibration amplitudes, methods for vibration calibra-
tion by comparison to a reference transducer, and primary
vibration calibration by laser interferometry, from ISO
16063, were combined. Since current standards for vibration
transducers do not regard nonlinear behavior, further analy-
sis has been performed according to IEC 60268 for sound
system equipment. In the frequency range, limited by the
setup, distortions, added to the measurements by the piezoe-
lectric accelerometer, were typically below 0.5 %, which
leads to the conclusion that these can be neglected for typical
applications.
Measurement Setup
An existing and accredited system for secondary calibration
of accelerometers by comparison to a reference transducer
was extended by a laser vibrometer and additional data re-
cording equipment in order to be able to analyse the total
harmonic distortions.
Calibration
In order to calibrate the measurement setup a method ac-
cording to ISO 16063-21 for vibration calibration by com-
parison to a reference transducer has been implemented with
sine approximation of the magnitude and phase values ac-
cording to ISO 16063-11. The magnitude of the complex
sensitivity was calculated using equation 1 and the phase of
the complex sensitivity using equation 2. Input values were
calculated using the accurate sine approximation method.
S
Sensor
, S
Reference
Magnitude of the complex sensitivity of
the transducer
X
Sensor
, X
Reference
Magnitude of the transducer output
φ
Sensor-Reference
Phase shift between the sensor to be cali-
brated and the Reference transducer
φ
Sensor ,
φ
Reference
Phase of the complex sensitivity of the
transducer
The results of the sensor calibration illustrated in Figure 2
were validated by comparison to the values of the accredited
calibration system which has been extended for the total
harmonic distortion measurements. Of course also the laser
vibrometer was calibrated accordingly, in order to have
reliable measurement values for further analysis.
Figure 1: Measurement setup for the true harmonic distor-
tion measurements.
Figure 2: Complex sensitivity of the piezoelectric accel-
erometer, the magnitude S
Sensor
shown in the top graph and
the phase φ
Sensor
in the bottom graph.
Amplifier
Measurement
frontend
Conditioning
amplifier
Computer
Shaker
Reference
Sensor
Laser
vibrometer
5
10
15
S
S
e
n
s
o
r
/
p
C
m
/
s
2
0.02 0.05 0.1 0.2 0.5 1
2 5 10
-3
-2
-1
0
Frequency / kHz
ϕ
S
e
n
s
o
r
/
d
e
g
=
(1)
=
+
(2)
Excitation
Since it is expected that the piezoelectric accelerometer
produces only little harmonic distortions, it cannot be as-
sumed that the distortions of the vibration exciter can be
neglected, as it is done in IEC 60268 for the measurement of
sound system equipment To correct the influence of the
electrodynamic shaker used for the vibration excitation dur-
ing the measurements, the behavior of the shaker has been
measured using a laser vibrometer. The calibrated signal of
the laser vibrometer represents the vibration excitation and is
used to correct imperfections in the excitation of the piezoe-
lectric accelerometer for which the distortions are calculated.
An example of both signals for an excitation frequency of
160 Hz is given in Figure 3.
Total Harmonic Distortion
Since the standards for measurements of accelerometers do
not include procedures to analyze the total harmonic distor-
tion, the calculation followed IEC 60268 with an extension
to correct the influence of the vibration excitation according
to equation 3.
A
i
Magnitude of i-th order harmonic of the calibrated
sensor output
L
i
Magnitude of i-th order harmonic of the calibrated
LDV output
The calculated total harmonic distortions (THD) of the pie-
zoelectric accelerometer are shown in Figure 4 for a constant
excitation acceleration of 6 m/s
2
. High THD values at fun-
damental frequencies less than 50 Hz can be explained by
rocking motions of the shaker. At low frequencies large
displacements of the sensor are necessary to keep the excita-
tion acceleration constant. The electrodynamic shaker of this
setup was therfore used slightly out of its specifications at
low frequencies. For frequencies greater than 500 Hz the
harmonics reach the resonance frequency of the accelerome-
ter, thereby increasing the magnitude of the harmonics. In
typical measurement situations this is not a problem, since
the resonance frequency is above the specified upper fre-
quency limit of the sensor and is cut off by a low-pass filter.
At frequencies greater than 1 kHz, less harmonics are used
for the calculation since the Nyquist frequency is reached. In
the frequency range between 50 Hz and 500 Hz were the
validity of the THD values is not influenced by the meas-
urement setup the THD is below 0.2 %. Higher excitation
amplitudes up to 20 m/s
2
were also investigated and a small
increase of the THD was found, resulting in THD values
typically less than 1 %. Since the basic principle of the pie-
zoelectric accelerometer does not change, it can be assumed
that the THD is generally low if the accelerometer is used
within its specifications.
An additional analysis of total, second and third order differ-
ence frequency distortions according to IEC 60268 with an
extension to correct the influence of the vibration excitation
similar to equation 3 showed values that were typically less
than 0.5 %.
Acknowledgments
This work has been supported by the Bavarian Research
Foundation as part of the FORLärm research cooperation for
noise reduction in technical equipment. The authors would
also like to thank Müller-BBM for access to the test facili-
ties.
References
IEC 60268-4: Sound system equipment - Part 4: Micro-
phones. (International Electrotechnical Commission,
Geneva, June 2010)
ISO 16063-11: Methods for the Calibration of vibration and
shock transducers - Part 11: Primary vibration calibration
by laser interferometry. (International Organization for
Standardization, Geneva, December 1999)
ISO 16063-21: Methods for the calibration of vibration and
shock transducers - Part 21: Vibration calibration by
comparison to a reference transducer. (International Organ-
ization for Standardization, Geneva, August 2003)
Figure 3: Magnitude of the calibrated laser vibrometer
(LDV) output spectrum shown in the top graph and of the
calibrated sensor spectrum shown in the bottom graph for
an excitation frequency of 160 Hz and an excitation accel-
eration of 6 m/s
2
.
Figure 4: Total harmonic distortions (THD) of the piezoe-
lectric accelerometer for a constant excitation acceleration
of 6 m/s
2
.
-50
0
50
LDV
Magnitude / dB
0.02 0.05 0.1
0.2
0.5
1 2 5 10
-50
0
50
Sensor
Magnitude / dB
Frequency / kHz
0.02
0.05 0.1
0.2
0.5
1 2 5 10
0
0.2
0.4
0.6
0.8
1
Frequency / kHz
Total harmonic distortion / %
=
∑
(
)
(
)
100%
(3)
