Use of reciprocity calibrated accelerometer standards for performing routine laboratory comparison calibrations

Technical paper 226

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IntroductionThe early reciprocity calibrations were performed using separate electrodynamic velocity transducers not mounted on shakers. These transducers were connected mechanically as required by the reciprocity procedure. Later, the reciprocity method was used at low frequencies to calibrate an electrodynamic velocity standard built into an electrodynamic shaker. Further research made it possible to perform reciprocity calibrations at high frequencies near and above shaker resonances as well as at low frequencies. This method has been used at the National Institute of Standards and Technology (NIST) to calibrate primary vibration standards.

This paper deals with the reciprocity calibration of piezoelectric accelerometer standards mounted temporarily on an electrodynamic shaker. The reciprocity calibration procedure is simplified for use at low frequencies below resonances. Instead of performing the reciprocity calibration at high frequencies, the constancy of the piezoelectric standard sensitivity as a function of frequency is verified by a comparison calibration performed with another piezoelectric standard previously calibrated at NIST.

Both reciprocity calibrated electrodynamic and piezoelectric vibration standards are used routinely to calibrate accelerometers by the comparison method. Simply measuring the ratio of the test accelerometer and standard accelerometer outputs completes the comparison calibration. The use of the piezoelectric standard has the advantage that the comparison calibration of other accelerometers may be performed at frequencies up to 10,000 Hz and higher. With the electrodynamic standard, good results are obtained only up to 2,000 Hz. Another advantage is that the piezoelectric standard is portable and can be used on various shakers.

Reciprocity theoryThe reciprocity theory for calibrating vibration standards requires that the shaker and standard operate in the linear amplitude range; i.e., standard voltage is proportional to motion, etc. The theory shows that the reciprocity relation exists for the driver coil or, more specifically, that the ratio of the driver coil current to force produced at the mounting table (when the shaker is constrained from movement) equals the ratio of the velocity at the mounting table to the driver coil voltage produced (where the coil is open circuit). The driver coil in the electrodynamic shaker provides the reciprocal transducer necessary for the calibration.

The theory also indicates that the sensitivity of the electrodynamic vibration standard in the shaker is determined from the measurements described by the following equations:

Sv = a+bYp

Where: Sv = sensitivity of electrodynamic standard, volts/in./sec

a= 0.01711(√i2πfJR)

b = 6.601 Q (√R/j2πfJ

Yp = mechanical impedance of test accelerometer, lb/in./sec.

i= √(-1),the90°vector

f = frequency, Hz

J = Intercept of plot of weight to transfer admittance ratio lb/mho

Q = slope of plot of weight transfer admittance ratio, ohms

R = voltage ratio, volts/volt

The terms Q and b become zero at frequencies below shaker resonance where there is no relative motion between the vibration standard and the mounting surface where the test accelerometers are attached. In

The reciprocity calibration of piezoelectric accelerometer standards is described in detail. The reciprocity calibration method was studied and developed in the 1960’s, and it remains a highly accurate and repeatable method. These standards are now routinely used to calibrate other accelerometers by the comparison method at frequencies up to 10,000 Hz. This method is seldom found in the typical laboratory, however many accelerometer manufactures and calibration laboratories use this highly accurate method. This paper is intended to provide a better understanding of this method.

Use of reciprocity calibrated accelerometer standards for performing routine laboratory comparison calibrations

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addition, it becomes unnecessary to measure Yp, the mechanical impedance of the test accelerometer. The test accelerometer is not attached during the reciprocity calibration.

To make the above equations applicable to piezoelectric accelerometerstandards,thetermaisdividedbyi2πf.Also, the term a is multiplied by 1,000 and by 386 to convert the sensitivity to mV and g units of acceleration. These operations convert the above equation for Sv. to:

Sa=2635(√JR/if) (1)

Sa = piezoelectric accelerometer standard sensitivity, mV/g

The other terms are the same as defined above.

The term J is computed from transfer admittance measurements made with various weights attached to the shaker. Best accuracy is achieved by making this measurement with each of ten different weights attached. In addition, the transfer admittance is measured with no weight attached. The differences between the transfer admittances with each weight and no weight attached are computed. The errors in these differences decrease as the weight is increased. This decrease in error is taken into account by the following equations:

Jr∑n2+Qr∑n2Wn=∑n2Ynr Jr∑n2 Wn + Qr∑n

2 Wn2=∑n2 Wn Ynr

Where: n = integer values; 1,2,3,. . . . . . . . . . Wn = weights of equal increments and the other terms described below

The weights selected are determined by the shaker characteristics. It is preferable to have the largest weight equal to about two times the weight of the moving parts of the shaker as specified by the manufacturer. For the reciprocity calibration described in this paper, ten weights are used, the largest weighing one pound. Therefore, Wn=0.1, 0.2, 0.3,…1 pound. The values of n and Wn are substituted in the above equations. The equations are solved simultaneously eliminating Qr:

J = 0.04204 Σn2 Ynr – 0.0502 Σn

2 Wn Ynr (2)

Where: Ynr = Wn/(Gn-Go) (3)

And where Gn = transfer admittance, ratio of driver coil current to vibration standard voltage output, obtained with each weight attached to the shaker

Go = transfer admittance with no weight attached to the shaker

The subscript r of the term J is omitted in equation 2 since only the real part of J is required for low frequency reciprocity calibrations. Also, equation 3 includes more terms if the calibration is performed at high frequencies near or above axial resonances in the shaker moving parts. In other words, equations 2 and 3 assume there is no phase shift between the driver coil current and the output of the piezoelectric accelerometer standard being calibrated. This assumption is easily satisfied at frequencies up to at least 500 Hz for most commercially available shakers.

The voltage ratio term R in equation 1 is the voltage output of the piezoelectric accelerometer standard divided by the voltage output of the driver coil when the shaker is driven by a second shaker and the driver coil in the first shaker is open circuit. An electronic counter is used to measure the frequency at which this voltage ratio is measured. Equation 1 assumes that thephaseangleshiftofthisvoltageratiois90°.Againthis assumption is satisfied because the calibration is performed at frequencies that are small compared to the axial resonant frequency of the shaker.

Reciprocity procedure and resultsThe calibration procedure is summarized as follows:

1. Calibrate the ten weights, 0.1 to 1 pound on a scale balance. 2. Measure the transfer admittance, ratio of driver coil current to piezoelectric accelerometer voltage output, with each of the weights and with no weight attached to the shaker. 3. Make the summations of the transfer admittance measurements as indicated by equations 2 and 3. 4. Measure the ratio of the piezoelectric standard output to open circuit driver coil output when the shaker is driven by a second shaker. 5. Measure the frequency at which Step 4 is performed. 6. Compute the sensitivity of the piezoelectric accelerometer standard from equation 1 using the values of J, R, and f from steps 3, 4 and 5 respectively.

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The reciprocity calibration procedure can be illustrated by the following sample calibration.

Figure1:Blockdiagramoftheset-upusedfortransferadmittancemeasurements.

Transfer Admittance MeasurementsThe setup and block diagram for the transfer admittance measurements are illustrated in Figure 1. The voltage dividerisadjusteduntiltheoutputofthevoltagedivideris exactly equal to the voltage across the precision 1 ohm resistor. The oscilloscope is required to verify that the phase angle is zero degrees. The shaker selected is satisfactory if the phase angle is zero degrees, if it is free of excessive transverse motion, and if the distortion in the sinusoidal motion is negligible, preferably less than one per cent. The shaker should have these characteristics throughout the frequency range up to approximately 500 Hz. The transfer admittance,

Gn = 1/Dn (4)

where Dn is the setting of the voltage divider. This transfer admittance is measured and computed with each of the ten weights attached to the shaker. The values of Gn from a recent reciprocity calibration and the corresponding values of Ynr are given in Table. 1. The summations of the products of n2Ynr and n2WnYnr are made for all the values of n. These summations are substituted into equation 2 and J is computed. For the measurementslistedinTable1,J=0.01503lb-volt/amp.

Table 1: Transfer admittance measurements on piezoelectric standard number 10.

Voltage Ratio MeasurementThe block diagram for the voltage ratio measurement is illustrated in Figure 2.

Figure 2: Block diagram for voltage ratio measurements

Thevoltagedividerisadjusteduntilitsoutputisequaltothe output of the piezoelectric accelerometer standard. Theoscilloscopeindicatesthephaseangleis90°.Thecounter is used to measure the frequency. The voltage ratio is,

R = Dr (5)

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whereDristhesettingonthevoltagedivider.ThevalueofRobtainedonPX-SP46piezoelectricstandardS/N10was0.0479volt/voltat50Hz.

Computation These values of R, f, and J determined above are substituted into equation 1. This computation indicates the sensitivity of piezoelectric accelerometer standard No. 10 is 10.00 mV/g.

In addition to the above reciprocity calibration, this accelerometer standard was calibrated at NIST. The average of the 38 calibration points reported in the frequency range from 10 Hz to 4,000 Hz was 10.02 mV/g with a standard deviation of 0.5 percent,NBLab.No.213.23/180699.

This agreement between the two independent calibrations is better than expected.

Reciprocity calibration accuracyThe calibration results described above are listed in column 8 of Table 2. Also included in this table are calibration results obtained on other vibration standards calibrated by the reciprocity method. All the results are reciprocity calibrations on piezoelectric accelerometer standards except those in the second column that are reciprocity calibrations on the electrodynamic velocity standard. The accelerometers are specially designed to minimize calibration error due to environmental effects of temperature, strain, etc. The sensitivity of each of the standards is a multiple of 10 to simplify the use of the standard. These sensitivities are obtained by inserting the required capacity between the standard accelerometer and standard amplifier. For use with voltage amplifiers, the capacity is placed in parallel with the accelerometer. For those standards used with charge amplifiers, the required capacity is inserted in series with the accelerometer. In the case of the electrodynamic standard, the proper value of resistance is put in series with the coil to obtain the desired sensitivity.

Table 2 also lists the standard deviation expressed as a percent of the average sensitivity for the various reciprocity calibrations performed on each standard.

Table 2: Summary of reciprocity calibration results on vibration standards.

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Table2wasgeneratedinthe1960’sduringqualificationof the reciprocity method. These results are applicable today.

Table 3 gives an error analysis of the reciprocity calibration method. The individual errors are estimated for the various error sources present. The final estimated error is determined from the square root of the sum of the individual errors squared.

1)Assumes90°phaseshift2)Assumes0°phaseshift3) Temperature, strain, acoustic, etc.4) Determined from the root mean square of individual errors listed in the table

Table 3: Analysis of sensitivity errors for primary vibration standards calibrated by the reciprocity method.

The error of 0.4% in Table 3 for the sensitivity of the piezoelectric accelerometer standard agrees closely with the standard deviations listed in Table 2. Also, the difference between the estimated error in Table 3 and standard deviation in Table 2 for the electrodynamic velocity standard is considered insignificant. In addition to the low frequency reciprocity calibration, it is necessary to verify the frequency response of the piezoelectric accelerometer standards. This is done by a comparison calibration referenced to an accelerometer standard previously calibrated at NIST. These calibrations are performed at frequencies up to 10,000 Hz.

Sample comparison calibrationThereciprocity-calibratedstandardsdescribedaboveareusedroutinelyinseveralcalibrators.The2958Afixtureused with these standards is satisfactory for calibrating all single axes piezoelectric accelerometers up to 4,000 Hz. However, a special fixture must be used for the calibration of triaxial accelerometers. In order to use

these piezoelectric standards up to 10,000Hz, a small shaker is used. First, the piezoelectric standard is used to calibrate the accelerometer in the small shaker and second, the test accelerometer is calibrated against the accelerometer in the small shaker.

To accomplish this, a standard accelerometer built into a case suitable for attaching test accelerometers was developed. This standard accelerometer is reciprocity calibrated and used directly to calibrate single axis piezoelectric accelerometers at frequencies up to 10,000 Hz.

To demonstrate its use, the new piezoelectric standard was reciprocity calibrated and used to perform comparison calibrations on typical accelerometers. The setup and block diagram for performing these comparisoncalibrationsisshowninFigure3,set-up3A,3B.Thevoltagedividerisadjusteduntilthetestaccelerometer output is equal to the output of the voltage divider. The sensitivity of the test accelerometer is equal to the sensitivity of the standard accelerometer multiplied by the setting on the voltage divider. If the output of the test accelerometer is larger than the standard accelerometer, the voltage divider is connected to the test accelerometer and the procedure is changed accordingly.

Figure 3A

Figure3B:Set-upandblockdiagramforroutinecomparisoncalibrations using the reciprocity method. 3A is the actual shaker used in the Meggitt calibration laboratory.

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Typical calibration results are shown in Figure 4. The accelerometers selected are representative of the various piezoelectric accelerometers presently in wide use. The weight of these three different accelerometers ranges from 2 gm to 78 gm, and the sensitivity ranges from approximately 4 to 250 mV/g. The calibration results indicate the suitability of the piezoelectric standard accelerometer illustrated in Figure 3 for performing routine comparison calibrations at frequencies up to 10,000 Hz.

Figure 4: Typical comparison calibration results obtained on various accelerometers using a reciprocity calibrated piezoelectric accelerometer standard. The increase in sensitivity at the high frequencies agrees with theoretical response corresponding to the accelerometer’sresonance.

Comparison calibration accuracyAn analysis of the errors of the comparison calibration resultsisgiveninTable4.Theestimatederrorof0.9%in the sensitivity of the test accelerometer sensitivity is previously determined by the reciprocity method, which is an absolute calibration. This error is applicable at 50 Hz, the frequency of the reciprocity calibration. The estimated error in the test accelerometer sensitivity at other frequencies is larger because the calibration errors of the NIST are included.

Measurement % Error Standard sensitivity (50Hz) 0.4 Standardsensitivity(10-900Hz) 1.0 Standardsensitivity(900–10,000Hz) 2.1 Voltage ratio (10 – 10,000 Hz) 0.2 Relativemotion(900–10,000Hz) 0.5 External capacity 0.5 Amplifier gain 0.2 Amplifier frequency response 0.5 Frequency -- Environmental effects 0.5 Estimatederror(50Hz) 0.9* Estimatederror(10–900Hz) 1.5* Estimatederror900–10,000Hz) 2.3**Determinedbyrootmeansquareofapplicableindividualerrors.

Table 4: Estimated error of sensitivity for accelerometers calibrated by the comparison method with piezoelectric accelerometer standards

These errors are typical of those achieved in modern calibration laboratories. Calibration differences between laboratories of less than 2% at low frequencies are common. The slightly larger errors near 10,000 Hz can be consistently maintained by careful selection of accelerometer standards and shakers.

SummaryThe reciprocity calibration procedure described provides an accurate means for establishing primary vibration standards. The procedure is simplified by performing the reciprocity calibration at low frequencies below resonances in the shaker. The sensitivity of the reciprocity calibrated piezoelectric standard is verified at higher frequencies by a comparison calibration to a standard previously calibrated at NIST. The sensitivity of the piezoelectric standard is determined with an estimated error of 0.4%.

© Meggitt (Orange County), Inc. 2014. All rights reserved. TP 226 111715