No. K20.260e Ed. 10.01 P. 1 ... 3

Written by. Hans W. Hugli

FREQUENCY RESPONSE OF PIEZOELECTRIC TRANSDUCERS A PRACTICAL APPROACH

1.0 IntroductionThe purpose of this paper is to give the instrumentation engineer asimple means of estimating the frequency response of piezoelectrictransducers. This can easily be done from two data points or the res-onant frequency which is usually specified in data sheets of instru-ment manufactures. Accuracy, as a function of frequency, will also bediscussed.

Frequency response is defined as the ratio of the transducer outputsignal applied sinusoidal stimulus as a function of frequency.

The useful frequency response of the piezoelectric transducersrange typically from near DC to nearly one MHz. The resonant fre-quencies of pressure sensors range from approximately 100 kHz tomore than 700 kHz; accelerometer range from 20 kHz to 125 kHzand load cells may resonate at as little as 100 Hz.

These high natural frequencies are the result of the extraordinarystiffness of some piezoelectric materials, particularly of quartz andtourmaline. The high modulus of elasticity, the high compressivestrength and the high insulation resistance of quartz allows theseinstruments to be statically calibrated and dynamically used from 1gto 100000g, one psi to 150000 psi or from one pound to 100000pounds full scale.

Some typical applications are:

· Machine vibration monitoring, high g-level impact studies, recoilacceleration and forces, vibration calibration standards, aircraft, rail-road and surface effect vehicle component vibration studies.

· Shock wave propagation studies in shock tubes, above ground,underground and in water; ballistics and closed bomb pressure mea-surements; rocket engines instability detection, denotations dieselengine fuel injection pressure measurements.

· Safety helmet impact force, rocket engine thrust force and keyimpact force studies among many other applications.

Frequency response of dynamic transducers can be characterizedby specifying two points of the curve, the -5% point at low frequen-cy and the +5% point at high frequency, shown in figures 1 and 3.Arbitrary points such as these are specified in most data sheets.Other points, such as the 1%, 10% or 3 dB points may be shown inother data sheets to express the frequency range of interest of a par-ticular transducer. The usable frequency range of a piezoelectricquartz transducer is not limited by the two specified points.Depending on the acceptable amplitude error more or less of the fre-quency range near DC to resonance can be utilized.

2.0 Theoretical Frequency ResponseThe theoretical frequency curve; shown in Figure 1, is governed bythe formula of forced vibration of a system with a single degree offreedom as a function of system damping and exiting frequency.

Where:

Quartz transducers are typically under-damped, with a factor ofamplitude increase at resonance, Q, of approximately 10 to 40.

F1, f2 = frequencies on either side of a resonance peak for which theamplitude is 0.707 times the amplitude of the resonance.These datapoints are termed the half power points.

Figure 1: Typical frequency response curve, no scale.

The factor P/K is the zero frequency or static deflection by a force ona spring with the stiffness K. This factor of the equation is calledmagnification factor, it determines the amplitude increase versus fre-quency.

Table 1 shows the calculated frequency response of a transducerhaving resonance of 100 kHz and a Q of 20. With this table, one canestimate the response of any piezoelectric pressure transducer,force transducer or accelerometer. Simply correct the frequency val-ues by the ratio resonant frequency of the subject transducer to 100kHz.

Example; Calculate the frequency for which the amplitude increases∆Y is 5% of a pressure transducer, with a resonant frequency of 250kHz.

Find amplitude increase ∆Y = 5% or nearest value in Table 1, shift tofrequency column and read value for f which is 22 kHz.

f+5% = f x 250/100 = 22 x 2.5 = 55 kHz

This is theoretical value, an amplitude tolerance must be allowed.Refer to paragraph 3.0.

Kistler Instrument Corporation , 75 John Glenn Drive, Amherst NY 14228Phone 716-691-5100, Fax 716-691-5226, e-mail: kicsales@kistler.com, www.kistler.com

Kistler Instrument Corporation reserves the right to discontinue or change specifications, designs or materialswithout notice consistent with sound engineering principles and quality practices.

Y =1

1 - + 4ζωωn

( (2 ωωn

( (222[ [PK

ω = undamped natural frequencyωn = frequency at any given point of the curveζ = damping ratio

12ζ

fnf2 - f1

Q = =

y

1 2

∆Y <5%

fn

5fn f0

No. K20.260e Ed. 10.01 P. 2

Kistler Instrument Corporation , 75 John Glenn Drive, Amherst NY 14228Phone 716-691-5100, Fax 716-691-5226, e-mail: kicsales@kistler.com, www.kistler.com

Kistler Instrument AG Winterthur, Postfach 304, CH-8408 WinterthurTel + 41 - 052 - 224 11 11, Fax + 41 - 052 - 224 14 14, sales@kistler.ch, www.kistler.com

3.0 Amplitude Tolerance vs. FrequencyThe actual frequency response of a quartz traducer deviates fromthe theoretical curve due to variations in transducer rigidity, measur-ing errors and due to the presence of sub-resonance. In general, thetransducer response falls within the dashed boundaries shown inFigure 2. Toward higher frequencies, the amplitude deviation isincreasingly influenced by the damping ratio as well as by sub-reso-nance of a particular transducer or transducer installation site.

Figure 2. Curve depicting amplitude increase and tolerance versusfrequency, no scale.

Sub-resonance usually occur at frequencies below fn/5. They sel-dom can be completely eliminated. However, if a resonance is evi-dent below fn/5, it indicates either faulty transducer installation oractual vibration signature of the measuring site. Sub-resonance areparallel resonance of transducer or installation components. Thenatural frequency of those components, when excited, produce amechanical feedback to the transducer's signal amplitude.

As a general rule, it can be said that the deviation from the theoret-ical frequency response is within ±2% at fm/5. At higher frequencies,the tolerance increases approximately as shown in Figure 2.

For high frequency applications, it is recommended that a transduc-er's frequency response be optimized by means of filters. Either byusing compensating low pass filters to extend the flat frequencyrange or by using low pass notch filters to eliminate resonance ring-ing, which allows highest possible frequency content with little over-shoot.

4.0 Low Frequency Response of Charge TransducersQuartz charge transducers are quasi-static transducers. Their fre-quency response at room temperature is flat to DC and is increas-ing at higher frequencies per Figure 1. They can be statically cali-brated and the calibration factor or sensitivity thus obtained, is validto fn/10.

In actual usage environments, it may be desirable to limit the low fre-quency response to avoid zero drift due to temperature fluctuations.This can be done by switching the charge amplifier time constantswitch to MEDIUM or SHORT for piezoelectric ceramic transducersit may be necessary to add an even shorter time constant positionto avoid drift.

4.1 Calculation of Time ConstantAll new charge transducer transducers have an insulation resistanceof 1013 W or more. Thus, the low frequency determining time con-stant is dependent on charge amplifier settings only.

Time Constant t (s) = R (W) x C (F),

where: R is the value of the time constant resistor

C is the value of the range capacitor

The time constant tolerance is normally within ±20%.

Contamination can be degrade the insulation resistance and low fre-quency response of charge transducer/charge amplifier systems.Using a HI-Meg. Insulation Tester, it can be easily and quickly deter-mined which part of the systems needs to be cleaned or replaced.

The time constant of Piezotron® transducer, patent number Re28.596 is specified in data sheets. Their Time Constant tolerance is-50% to + 100% of the listed values. To obtained specified low fre-quency roll-off characteristics, it is recommended to use high passor band filters in the signal conditioning electronics.

4.2 Calculation of Low Frequency ResponseFigure 3 depicts the low frequency roll-off for a transducer/chargeamplifier system which has a time constant of 5 seconds. The cor-ner frequency fc … -3 dB is calculated as follows:

Other points of the low frequency curve can easily be obtained fromFigure 4, on the last page, as, multiples of the corner frequency fc.

Example: Calculate the frequency at which the output drops 1% or asystem with a time constant of 5 seconds. First calculate the cornerfrequency or -3 dB point.

Second, look for the 1% point on the ordinate of Figure 4, follow thearrows to abscissa and read the multiplication factor which is 7.

Third calculated f 1% = fc x 7 = 0.22 Hz.

The -1% and other points have been plotted in figure 3.

Figure 3. Low Frequency Roll-off for a transducer or system with atime constant of 5 seconds, no scale.

1% point 5% point

Sub-resonances

±1%±2%

fn

10fn

5fn f

12πτ(s)fc (Hz) =

11 π 5

fc = = 0.032

No. K20.260e Ed. 6.01 P. 2

Kistler Instrument Corporation , 75 John Glenn Drive, Amherst NY 14228Phone 716-691-5100, Fax 716-691-5226, e-mail: kicsales@kistler.com, www.kistler.com

Kistler Instrument AG Winterthur, Postfach 304, CH-8408 WinterthurTel + 41 - 052 - 224 11 11, Fax + 41 - 052 - 224 14 14, sales@kistler.ch, www.kistler.com

5.0 Calculation of Phase AngleThe phase angle given by the equation:

Determines the angle by which the transducer mass displacementlags behind the driving force.

1 - ωωn

( (22ζ

ωωnφ = tan-1

Figure 4. Signal roll-off in % vs. multiples of corner frequency fc.