Improving Dynamic Measurement Accuracy by Defining Limited by nikeborome


									              Improving dynamic measurement accuracy by defining a
                      limited usable frequency range (LUFR)
                                Michael D. Insalaco, R&D Engineer
                       PCB Piezotronics, 3425 Walden Ave., Depew, NY, 14043


Measurement accuracy expectations continue to spiral following the ongoing improvements in equipment
performance and enhancements in ISO guidelines. The protocol used to compute measurement uncertainty has
been well defined by the standards organizations and these methods can be applied to assess possible error
within specified test conditions. However, when the entire usable frequency range (UFR) of the dynamic
measurement system is considered, overall measurement error can be very large. In the dynamic measurement
community, industry standards have been adapted regarding acceptable limits of sensitivity deviation throughout
the UFR where ±5%, ±10%, and ±3dB are not uncommon. Significant accuracy improvement can be obtained
when the appropriate sensor is selected and a limited usable frequency range (LUFR) is defined. Factors
defining the UFR of dynamic systems are discussed in detail. The influence of sensor selection and methods to
minimize error using an appropriate LUFR are explored.


Dynamic measurement systems often bring interesting challenges to the mechanical test engineer. Output from
the system is a voltage proportionally scaled to a mechanical input. However, for dynamic sensors, the scale
factor is defined at a distinct frequency. This characteristic definition is different from the typical mechanical
measurement device where scale factor is usually a constant only altered by environmental influences. The
typical usable operating range is defined in terms of the measurand (i.e. lbf, ft, sec.) and environmental
restrictions are specified. A dynamic scale factor is also range limited with respect to the measurand, typically
qualified by a linearity specification, but also associated are additional ‘dynamic’ provisions. This scale factor or
sensitivity is specified at a prescribed frequency, valid within a stated percentage deviation and only applicable
throughout a defined UFR. For acceleration measurements, the dynamic test community has settled on two
different reference frequencies: 100 Hz or 159.2 Hz (1000 rad/s). A typical specification may be 101 (mV/g) +/-
1% at 100 (Hz) and +/- 5% from 0.5 to 10,000 (Hz). The characteristic ‘shape’ of the scale factor or sensitivity as
a function of frequency has not adapted to any industry standard and instead is a result of the construction of the
sensor and its interaction with necessary signal conditioning. It is quickly becoming obvious that this apparently
simple parameter can vary considerably in a routine measurement. If a very accurate measurement with low
uncertainty is required, more than the basic scale factor must be considered. An example is the common shaker
control loop scenario where ‘global’ sensor specifications within the UFR are the greatest contribution to the
uncertainty budget [1].

Environmental factors also influence the dynamic sensitivity. All of these influences must be kept in consideration
especially when measurements are approaching the extremes of the specified limits. The discussion of this paper
will address the system’s frequency response and will focus on the piezoelectric (P/E) based accelerometers
since they cover a very large dynamic range and have interesting characteristics at both high and low frequencies
(and sometimes between). The principles and correlation of the discussion adapt to any dynamic sensor
(temperature, pressure, force, flow, etc.) except each of these have their own particular anomalies that must be
understood in order to correct or knowingly accept the error.

Narrowing the discussion to P/E accelerometers will demonstrate the need to develop an understanding of the
various technologies available to achieve a specific measurement. “P/E accelerometers” appears to be a refined
scope of discussion but within this group are a variety of natural crystal families and a very large range of man
made piezoceramic alternatives. Each of these basic core technologies can be packaged in different shapes and
the matrix of options available to make a measurement becomes quite large. There are also many sensor
technologies to measure, for instance, force. Piezoresistive, strain gage, or P/E technologies all bring a solution
to measure dynamic force but each must be considered in detail for the best option. Within each technology
exists still another subset of options with their own specific attributes.


Materials capable of developing an electric surface charge when stressed are classified as piezoelectric. Some of
these are natural crystals and practical sensors have been constructed from many including tourmaline, quartz
and lithium niobate. During the mid 20 century, other P/E solutions were developed using blends of rare earth
elements molded into useful shapes and artificially polarized. Known as piezoceramics, a wide variety has
evolved and families of sensors are available using formulations of lead titanate, lead zirconate titanate, lead
metaniobate and bismuth titanate. Further technological advancements occurring late in the 20 century has
brought on an evolution of artificially grown crystals that have many superior attributes. Now common are ultra
high strength crystals with large piezoelectric charge coefficients and incredibly wide range thermal stability.
These artificially grown crystals are now readily available from the Gallium Orthophosphate and Langasite
classified crystal families.

Most of these materials can be shaped into several useful forms where either a compressive or shear load is
arranged to generate a proportional stress within the piezo element. The load may be transferred from an
imposed pressure applied to a diaphragm or a force applied directly to the element. A strain in a support may
transmit a resultant stress or, in the case of an accelerometer, an inertial mass loads the element proportional to
its motion. Sensor constructions exploit the unique features of the P/E sensing elements where each family offers
application specific advantages. Natural crystals tend to be extremely stable and piezoceramics often provide
versatile packages since their shape can be optimized. An overlap of basic features is always offered to the user
making the optimal selection a considerable task.

The UFR is an important parameter and it is a design optimization variable where the ideal solution would yield an
infinite bandwidth. A compromise must be established based on end use considerations since the widest
bandwidth will be available from the smallest, lightest sensor but also one with a relatively low sensitivity. An
increase in fundamental output requires larger seismic mass and therefore restricts its use to applications
involving larger specimens.

Describing a sensor’s frequency response in general terms can divide the characteristics into three realms of
classification: the high-end, mid-range and low-end response. It will be shown that the mechanical system, the
sensing element material characteristics’ and the electrical system parameters define these regions respectively.
There is also an interaction between mating regions that can be tailored to expand the UFR.


In its simplest form, the accelerometer is a single degree of freedom spring-mass system. The stiff P/E elements
represent the spring and the supported seismic mass follows the motion of the spring’s base as a classical
second order mechanical system. The damping characteristic of these extremely stiff systems is negligible
therefore equation 1 accurately describes the system’s amplification factor.

                                       A = 1/[(1-f/fn)2]                (1)

Considering mechanics only, the mass follows the spring’s base starting at DC and up to a frequency
approximately one tenth of the mechanical resonance (+1.01% rise @ 1/10 fn). Using equation 1 to derive the
frequency at which the mass moves five percent more than the base (f+5%, the +5% point), shows the common
rule of thumb: to use an accelerometer within a range up to 1/5 of resonance (f+5% ≈ 0.2 fn). Table 1 presents
typical error band percentages found in manufacturer specifications and some other useful narrow band
                                                   Table 1
                                  Amplitude Rise and Common Error Bands

        % Resonance             9.95           14.0            21.8           30.2              54.0
        Amplitude Rise          1%             2%              5%             10 %              3dB
From this brief summary, it is apparent that if accuracy in the range of magnitude of one percent is desired, the
high-end frequency is limited to ten percent of resonance. It should also be noted that there exists a very linear
range until the exponential effects of resonance start to significantly influence the response. It is this flat range
that is useful in the LUFR consideration.

The classical second order response describes the high-end characteristics of P/E sensors from a pure
mechanical standpoint. This must be used as a starting point since it should be noted that previous discussion
has been based on assumptions that the response from the piezo element is uniformly proportional to stress
throughout the frequency range explored. By design, the first resonance is very large and indeed dominates the
mechanical response. However, there is a subtle characteristic of piezoceramic materials that alters this
conventional approach and must also be considered. A frequency dependent reduced sensitivity is inherent to
the material itself. This will be discussed in the mid-range section following a discussion of the low-end response.


A piezoelectric sensor can have electronic circuitry internal to the construction or can be conditioned by an
external charge amplifier. There are benefits to each approach and choosing the optimal solution is the role of
the manufacturer’s application engineer. From a system perspective, the output voltage is an AC coupled signal
where any steady state component is removed by the characteristic electrical time constant. This realm of the
measurement is called the low-end frequency response and is governed by the interaction of all of the time
constants of the system components. Considering each component (sensor, conditioner, data acquisition system,
etc), taking the ratio of product to sum is the method to compute the system time constant but in ‘good’ practice
the shortest time constant should be established within the sensor. All other measurement components are
selected with relatively long time constants (factor of ten) such that they do not influence the low-end capability of
the sensor. A common mistake with low frequency measurements, when using an oscilloscope, is to use the AC
mode of the scope. When AC scope is selected, a relatively short time constant is imposed and the sensor’s time
constant is no longer the controlling factor in measurement capability.

For P/E sensors, the system time constant is the governing parameter that defines the low-end frequency limit
(f-5%). Any steady state condition applied to a sensor, such as an inversion in a gravitational field, will initially
present a proportional voltage but it will decay in a manner described as a first order ‘electrical’ system’s
response. A larger time constant will cause a slower decay and will increase the systems ability to accurately
respond to low frequency or slow moving events. In the frequency domain, this time dependent decay has a
larger influence on lower frequencies and is characterized as shown in Figure 1.
                           Amplitude Dev. (dB)

                                                 0.1                        1                           10

                                                                      Frequency (Hz)

                                                 Figure 1) Time Constant Effect on Frequency Response
                                                                                                9            12
Time constant is established by selection of very high magnitude resistors (10 to 10 Ω) acting parallel to the
sensing element on the front end of the electronic circuit. Their value is based on the sensing material where
piezoceramics have large capacitance characteristics compared to natural crystals thereby requiring less
resistance for the same system effect. Thermal transients tend to disturb the low frequency response of
piezoelectric sensors since the very stiff, high impedance, mechanical packages transmit any induced strain
immediately to the sensing element. The low-end frequency characteristic is designed to provide immunity to this
undesirable transient effect where sensor time constants in the range of a quarter to three seconds are typical.

Thermal transient response is a result of sensor construction, sensing material selection, and electronic circuit
design. A long time constant increases UFR but always with some influence on the thermal transient response.
When thermal activity is expected to be prevalent, and measurement capability in the very low frequency realm is
not required (less than 10 Hz), a fast time constant in the range of a tenth of a second is a better choice. For
voltage mode ICP type sensors, this requires special order configuration since the adjustment must be made on
the input side of the internal electronics. It should be understood that the catalog accelerometer is designed to
provide the widest UFR to capture most applications. It has not been optimized for your application.

High pass filtering could be applied to the output signal with a similar result regarding UFR. However, the
dynamic range within the voltage mode sensor is restricted to approximately five volts and the transient may
saturate the input stage of the internal electronic thereby corrupting any output signal. If thermal activity is
expected, and low frequency response is not critical, a short sensor time constant is warranted. Eliminating
thermally induced voltages allows better use of the full range capability of subsequent measurement components.
Matching the measurement range to the measurand’s magnitude yields the best resolution and best results. As
an example, the Environmental Stress Screening (ESS) process involves exposing specimens to known
temperature, humidity and vibration extremes while monitoring for failure. The vibration exposure requirement is
well away from DC and a very fast time constant would benefit this measurement.


The frequency realm between the high-end and low-end usually has very different characteristics dependent on
the technology employed. This range is configured by design to be useful for a specific application. For instance,
experimental modal analysis is used to characterize structural specimens often of significant size. The
accelerometer must have large voltage sensitivity since the measurand is typically very low level. This requires
relatively large seismic mass to generate appropriate stress within the piezo element. The design optimization
process yields the minimal acceptable sensitivity (minimal permissible mass) thereby maximizing the UFR (largest
fn, result of minimized mass). As can be seen, a compromise must be established and the ultimate goals (minimal
size, lightweight, largest UFR, highest sensitivity, low threshold, minimal cost) can never be achieved. The
optimization process involves a rigid definition of the bounding parameters and then a careful consideration of
available solutions. Since the optimal blend of these performance features is in itself not a single solution, a
variety of potential sensors evolve with slight differences. Sometimes the flexibility of range adjustment from an
external charge amplifier type system is appropriate. Another set of vast and complicated possibilities exist.

Narrowing the discussion to the details of the UFR, the generalized design goal is to maximize the frequency
span between the defined amplitude deviation limits. Once the basic feature requirements have been satisfied,
such as sensitivity and weight, an expansion of the resulting UFR is still possible in several ways. A simple,
inexpensive, method to extend the f+5% is to incorporate a series resistor on the input of the electronic circuit. This
introduces a moderate electrical low pass filter. This has the effect to increase the UFR by ‘pulling down’ the
resonant rise but also introduces a slight phase shift at the higher frequencies.          An example of a filtered
accelerometer is shown in Figure 2. The addition of more filter stages can cause significant and abrupt low pass
filtering. This type of response is common to sensors used in shock or ride quality measurements. The penalty is
an added phase shift and a non-linear ‘shape’ to the frequency spectrum. The advantage is improved resolution
in the frequency range of the measurement since out of band signals are suppressed allowing a lower ranged
sensor to be used.

Figure 2a) Unfiltered Frequency Response                   Figure 2b) Filtered to extend UFR & suppress resonance

With such a disparity within the options, it is essential to review an actual measured frequency response plot
before selecting a sensor for a critical application. Typically, sensors that incorporate a low pass filter have
favorable UFRs compared to unfiltered options. Only very special designs incorporate substantial filtering such
that the unfiltered UFR is wider than the filtered option. When known structural phenomena (such as a bearing
resonance) is desired to be removed from the measurement data, a strong low pass filter may be employed. This
will reduce the UFR but also suppress the out of band resonance as desired. The always-present design goal to
‘maximize’ the UFR usually results in some irregularity in the FRF similar to the exaggeration presented in Figure
2b. When optimized to counter the effect of the sensor resonance, a frequency response that reduces to ‘just’
less than –5% before rising to +5% can be realized yielding a wideband UFR. If the critical data to be analyzed is
in the range ‘of interaction’ between regions, careful attention should be applied to the possible sensitivity
variation. It should be noted that the importance of phase relationship in the application should be assessed.
Also, all other critical or important features must be common to the alternatives.

Incorporating the many advantages of piezoceramic designs also introduces further selection criteria. A
characteristic of these materials is a reduction in charge sensitivity as a function of frequency at a rate of
approximately two and a half percent per decade. From the reference frequency of 100 (Hz), the sensitivity at 10
(Hz) is 2.5% larger and at 1000 (Hz) its 2.5% percent less. Considering a sensor with a 50 (kHz) resonance and
applying the 1/5 fn rule of thumb for +5%, the interaction of the material droop and resonant rise results in minimal
deviation at 10 (kHz). This relationship drives the upper UFR (+5%) near 11 (kHz), although some wavering is
apparent. As shown in Figure 3b, adding some filtering further extends the UFR. If a five percent error band is
acceptable to the analysis, this is a very favorable sensor. If improved accuracy is a concern, there are several
correction factors that must be considered. The slope of the sensitivity shift is well behaved and applying a post
measurement, frequency domain, correction constant is relatively straightforward. Alternately, returning to the
example of ESS, the shaker control algorithm can be simply adjusted to ramp up the driving voltage at higher
frequencies thereby countering the sensitivity reduction effect. Methods to incorporate sensitivity coefficients into
uncertainty calculations have been established and lead to overall reduction in uncertainty of measurement [2].
Also a way of reducing uncertainty would be to develop a calibration data set to eliminate systematic errors
depending on test frequency and level [1].
                   Amplitude Deviation (%)

                                                  1 0 .0

                                                   5 .0

                                                   0 .0

                                                  -5 .0

                                                 -1 0 .0
                                                           10        100                1000             10000          100000

                                                                               F re q u e n c y (H z )

                                                 Figure 3a) Typical Piezoceramic Accelerometer Frequency Response
                       Amplitude Deviation (%)

                                                   1 0 .0

                                                     5 .0

                                                     0 .0

                                                    -5 .0

                                                  -1 0 .0
                                                            10       100                 1000            10000          100000

                                                                                F re que ncy (H z)

                                                   Figure 3b) Typical Piezoceramic Accelerometer with Moderate Filter

Now from the other end, the sensitivity reduction slope of the piezoceramic presents a nominal 5% rise at 1 (Hz)
but this is where the sensor’s time constant is causing an exponential roll off. The piezoceramic droop and high
pass filter characteristic of the electrical time constant combine to yield an expanded UFR.

The piezoceramic designs often provide a wide band UFR where the material characteristic droop interacts well
with the other response contributors. Also, miniature packages with significant output sensitivity are typically
available. The drawback is a post measurement correction is required if accuracy improvements are demanded
by a critical application. The natural crystal solutions have very flat mid range characteristics while the
piezoceramic solutions usually offer a wider UFR but with more irregularity. Sometimes, an internal impedance
converter or voltage amplifier is used with the piezoceramic designs instead of the common charge amplifier
approach. The capacitance of many piezoceramic materials change in a frequency dependent manner similar to
the charge sensitivity coefficient and the net result is a response similar to a natural crystal design as shown in
Figure 4.

                     Amplitude Deviation (%)




                                                     10     100                     1000            10000
                                                                  Frequency ( Hz)

                                     Figure 4) Typical Single Crystal Accelerometer Frequency Response

It should be emphasized that the preceding review of the variety of technologies has focused on the particular
attribute of frequency response. Other features, such as output to weight ratio, may be much more important in a
specific situation and may override any advantage of a LUFR. If a miniature low output sensor (with wide and flat
UFR) is used on an extremely large structure, the uncertainty due to poor signal to noise ratio will be excessive.
However, in many instances, a significant reduction in measurement error or improvement in uncertainty can be
realized by simply defining an appropriate LUFR or by incorporating an additional sensitivity coefficient (or
combination of both). The purpose of this paper is to highlight the fact that although dynamic sensitivity is not a
single value, it is often a linear constant throughout a large portion of its UFR. With proper sensor selection, and
possibly adding special filtering, a better match to your application is readily available.


Several of the approaches used to tailor the UFR of piezoelectric accelerometers have been reviewed. It has
been shown that the frequency response is defined by the electrical system time constant, sensing element
material characteristics and sensor construction for the low-end, mid-range and high-end respectively. The
interaction of these controlling factors can be configured to maximize the UFR within the scope of its defined
limits, often in the deviation range of ±5%, ±10% or ±3dB. When the UFR is optimized, significant sensitivity
variation within specific frequency zones may be realized. This variation must be recognized by close inspection
of the supplied calibration data when accurate measurements are essential. In some instances, post
measurement correction, using sensitivity coefficients, can be applied since the error is well defined. Often, in
practice, the error is unnecessarily tolerated and uncertainty budgets can be excessive. Proper consideration
towards defining a LUFR can lead to significant accuracy improvements particularly when the advantages of both
low and high pass filtering are optimized. Sensor selections with special features, adapted to a LUFR, can lead to
measurement accuracy of less than two percent throughout the entire measurement band.


    1. Lax, Richard, Analyzing the Vibration Controller Closed Loop Chain, SEE Convention Proceedings, 2003
    2. Expression of the Uncertainty of Measurement in Calibration, Publication Reference EA-4/02, European
       Co-operation for Accreditation, 1999

To top