Fringe Counting

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SUCCESSIVE-APPROXIMATION METHOD FOR MEASUREING LINEAR ACCELERATION ��

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Abstract

The CSIR-NML developed its first primary calibration system for the calibration of linear acceleration devices during 1997. The system, based on the fringe counting technique, limits the NML’s current capabilities for primary calibration to the frequency range 50 Hz to 500 Hz. The NML is currently in the process of expanding its primary calibration capabilities to cover the frequency range 1 Hz to 10 kHz using the Successive Approximation Method.

Introduction As custodian of the National measurement standards the NML maintains the National measurement standard for vibration through its Acoustics, Ultrasound & Vibration project. Up till 1998, the standard was realized by reciprocity calibration at a single frequency point, E F GIH�JIK LME G G GIN O P Q R S TOther frequency points were covered by importing traceability from other NMI’s. During 1997 the NML developed the first primary calibration facility for linear acceleration in South Africa. The main focus of the project was to free the NML from importing traceability as well as building its knowledge base and expertise in this area. After investigating different configurations and techniques, a Michaelson interferometer system using the fringe counting or frequency ratio technique was implemented. This laid a solid foundation for future developments. The fringe counting method that was implemented provided in all of the initial requirements:

1. It provided primary calibration over the frequency range 50 Hz to 500 Hz, where the reciprocity system was at 160 Hz only.

2. It is a fairly easy system to implement; yet it provided the knowledge base to expand to other systems with ease.

3. It can be used as a reference for new developments.

4. It improved the uncertainties of measurement for this parameter.

The question was: What is the most suitable system to implement next?

An Overview of 3 Interferometer Systems

Following is a brief description of the two most common systems currently used by NMI’s and a third one that is gaining interest. All these systems are described in more detail in reference [1]. The Interferometer All the system described in the following discussions essentially uses a Michealson interferometer. The basic interferometer consists of:

1. A stable light source, normally a HeNe laser with a wavelength U3V W X<Y Z3[

2. A non-polarizing beam splitter. 3. A reference mirror. 4. A photodector. 5. A polarizer (if required). 6. A beam expander.

The basic construction for a Michaelson interferometer is shown in figure 1. Any deviation required for a specific technique will be discussed in the appropriate section. Figure 1 shows only the interferometer and conditioning equipment. None of the equipment required for performing the actual measurements or controlling the system is shown. Fringe counting method This method can be applied for sensitivity magnitude calibration in the frequency range 1 Hz to 800 Hz. The sensitivity of the transducer at the applied frequency and applied acceleration, is determined either by:

1. Counting the fringes at the output of the photo detector using a counter.

2. Measuring the ratio between the vibration frequency and the fringe frequency (output of the photo detector) using a ratio counter.

The acceleration amplitude, â, is calculated from the fringe frequency using the formula:

fff ••×= -6103,123â (m·s-2) (1)

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Figure 1 Interferometer setup for fringe counting & minimum point methods

The sensitivity magnitude is calculated from the formula:

fffS

••×= û

103202,0 6 (V· m-1·s-2) (2)

where: û is the accelerometer output in volts; f is the vibration frequency in hertz; f f is the fringe frequency in hertz;

If a ratio counter is used then the acceleration amplitude, â, is calculated from the fringe frequency using the formula: fRf 2-6103,123â •×= (m·s-2) (3)


fRfS

26 û

103202,0 •×= (V· m-1·s-2) (4)

where: û is the accelerometer output in volts; f is the vibration frequency in hertz; Rf is the ratio of the fringe frequency (f f) to the vibration frequency (f);

Minimum-point method This method can be applied for sensitivity magnitude calibration in the frequency range 800 Hz to 10 kHz. The signal from the detector is bandpass filtered with the center frequency equal to the frequency of the vibrator. This filtered signal has a number of minimum points at accelerometer displacements corresponding to zero crossings of Bessel functions of the first kind. Set the vibration frequency. Increase the vibration amplitude from zero until the filtered detector signal

reaches a minimum. This value corresponds to the first minimum point. The acceleration amplitude, â, is calculated from the following formula:

2-61039,478â f•×= (m·s-2) (5)


25

sf

û1025331,0 •×=S (V· m-1·s-2) (6)

where: û is the accelerometer output in volts; f is the vibration frequency in hertz; is the displacement amplitude in

micrometers for the different minimum points;

Successive-approximation method This method can be applied for sensitivity magnitude and/or phase calibration in the frequency range from 1 Hz to 10 kHz. The laser interferometer is adjusted to give output signals U1 and U2 in phase quadrature within ± 5º.

Figure 2 Interferometer setup for the sine-approximation method

The quadrature signals are equidistantly sampled during a measurement period \ 0 < \ <( \ 0 + TMeas) The sampling rate should be constant. The sampled series of quadrature output values are { ] 1( \ i)} and { ] 2( \ i)} while the series of accelerometer output values are { ] ( \ i)} . Data processing The magnitude and phase lag of the accelerometer sensitivity are obtained as follows:

a) ^�_ ` a b ` _ c d�_�e d f g d e�h i�j k _ e d�l _ ` b d e m Mod(ti), from the sampled interferometer output

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values { u1(ti)} and { u2(ti)} using the relationship

πϕ n+=)(tu

)(tuarctan)(t

i2

i1iMod

(7)

where: i = 0,1,2,…I note: n is calculated as described in reference [2].

b) Approximate the obtained series of n�o p q r s t u o v<w x s y z�{ s r q z y |

Mod(ti), by solving the following system of N+1 equations for the three unknown parameters A, B and C using the least squares sum method:

CtBtA ii +−= ωωϕ sincos)(t iMod (8)

i = 0,1,2,…,N

where: sMA ϕϕ cosˆ=

sMB ϕϕ sinˆ=

C = a constant }3~ �

s is the initial phase angle of the displacement;

The modulation phase amplitude Mϕ and the

displacement initial phase angle sϕ is calculated

from the following formulae:

22 BAM +=ϕ (9)

A

Bs arctan=ϕ (10)

c) Calculate the acceleration amplitude, â, and

acceleration initial phase angle, aϕ , from

Mϕ and sϕ using the formulae:

Mfa ϕπλ ˆˆ 2= (11)

πϕϕ += sa (12)

d) Approximate the series of sampled

accelerometer output values, { u(ti)} , by solving the following system of N+1 equations for the three unknown parameters A, B and C using the least squares sum method:

CtBtAu ii +−= ωω sincos)(t i

(13)

i = 0,1,2,…,N

where: uu uA ϕcosˆ=

uu uB ϕsinˆ=

Cu = a constant

�3� � û is the accelerometer output amplitude;

u is the accelerometer output initial phase angle;

The accelerometer output amplitude, u , and

the initial phase angle, uϕ , is calculated from

the following formulae:

22ˆ uu BAu += (14)

u

uu A

Barctan=ϕ (15)

e) Calculate the magnitude a and the phase lag � � � ��

a

uSa ˆ

ˆˆ = (16)

)( ua ϕϕϕ −=∆ (17)

Selected method After discussions with Dr Von Martens of the PTB the NML decided to implement the third method described below – successive approximation. The decision was based on the following:

1) The minimum point method can only be used at certain displacements, which is undesirable.

2) The NML needed to improve the uncertainty of measurement further if it was to participate in international key comparisons.

3) It was preferred to have a single system that can cover the complete range of 1 Hz to 10 kHz.

4) Satisfying future requirements for phase calibration had to be considered.

5) Development cost should be kept low.

Successive-Approximation Method at the NML The SAM system implemented at the NML is based on a homodyne interferometer [3]. Implementing a heterodyne system [3] holds further advantages, but such a system was not considered because of its complexity as well as high cost implications. SAM systems are new in the metrology arena, but they have been implemented in various laboratories with success. It is a more expensive system to implement (compared to the fringe counting and minimum point methods), requiring more optical components and a special data acquisition system. The system implemented by the PTB is based on a VAX computer and can store 50xe 6 samples [3].

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Other institutes use Wave memory recorders [4] and digital oscilloscopes [5] with a capacity of more or less 4000 samples. With reference to figure 2, the SAM system at the NML consists of the following: Vibration Exciter The drive signal to the Brüel & Kjær 4808 shaker is obtained from the signal source of a HP 35665A dual channel analyzer. The power amplifier is a MB Dynamics model SS530. Accelerometer Output During the development of the system a Brüel & Kjær 8305s, laboratory standard accelerometer was used as the UUT. The top of the accelerometer was polished by Opticon to obtain a mirror quality finish. The conditioning amplifier used is a Brüel & Kjær 2650 charge amplifier. Interferometer A frequency stabilized, red HeNe laser, is used as the light source. The laser is manufactured by Meles Griot. The laser beam is folded through 90º as the laser is mounted horizontally and a vertical beam is required. The beam passes through a 10 mm3 non polarizing beam splitter providing a 50/50 split of the beam. The two beams form the reference beam and the measuring beam. The reference beam travels through a quarter-wave plate, which is mounted at 45º to the incoming reference beam. The quarter-wave plate converts the linearly polarized beam into a circularly polarized beam as per figure 3. The reference mirror reflects the beam and the circularly polarized beam is converted into a linearly polarized beam. The reference beam is now 90º out of phase with the measurement beam. The combined beams exiting the non polarized beam splitter travels through a beam expander before it enters the polarized beam splitter. The polarized beam splitter splits the two polarizations, providing the two interference beams in quadrature.

Figure 3

Photo Detectors The photo detectors were developed at the NML. Each detector consists of a C30608E photodiode and a single gain stage amplifier. The gain of the amplifier is adjustable, enabling one to balance the output voltage of the two photo detectors. Data Acquisition All three the signals, u1, u2 and u is fed into separate channels of the A to D card. The NML acquired a special PC based A to D card for the system. The system consists of two SMT 340 dual channel A to D converters, capable of sampling rates up to 40 MHz. Each A to D module is linked to a SMT 332 DSP control unit with 8 M byte of RAM on board. The DSP modules have a data throughput of 20 MHz. In short, the system can store up to 4 million 12 bit samples per channel, of the 4 simultaneously sampled channels, at 20 MHz. Digital signal processing All the signal processing is done on a Pentium 100 PC. The PC is fitted with 256 M of RAM to speed up the amount of data to be processed. The software is developed using Delphi. Originally processing speed was thought to be the biggest problem during the signal processing stage. In actual fact disk storage could be a bigger problem. A measurement at a single frequency point, consisting of 3 channels of 4 M samples each, generates a text file of 32 M. Considering that a full calibration is made over more than 10 frequency points, hard disk space can soon become a problem. Considering the cost of hard disks and the compression ratio of text files, this is not a serious problem. Modulation phase values ( Mod) Delphi has a mathematical unit that can determine the arctan within the correct quadrant. Performing the phase unwrapping as described in [2] proved to be complicated. A more convenient method, based on the technique used in MatLab, was implemented. Figure 6 shows the modulation phase values obtained from the sampled interferometer outputs U1 and U2, shown in Figure 4 & Figure 5 respectively. Approximation using least squares fit Implementing the sine fit for solving the 3 unknown variables as per equations 8 and 13, where there is more equations (N+1 in this case) than unknowns, is best done using matrix mathematics. Solving equation 8 using matrixes is as follows:

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-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Figure 4: Interferometer output U1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Figure 5: Interferometer output U2

-300

-200

-100

0

100

200

300

400

Figure 6: Modulation Phase Values

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Figure 7: Accelerometer Output Values

1) Determine the elements of the 3x3 matrix X.

•••••••••

=Χ333231

322221

312111

xxxxxx

xxxxxx

xxxxxx

where: itx ωcos1 =

itx ωsin2 =

itx =3

i = 0,1,2,…N

2) Determine the elements of the vector Y.

[ ]Tyxyxyx •••=Υ 321

where: )( iMod ty ϕ=

i = 0,1,2,…N

3) Solve the three unknowns (A,B,C) from

YX

C

B

A

•=

−1 (16)

All this processing is achieved using software algorithms. After applying the least squares fit the magnitude and phase is calculated as described earlier.

Conclusion The paper gave a brief overview of the requirements and execution steps for the implementation of various laser based acceleration measurement systems. The NML is in the process of implementing a SAM system for acceleration calibration over the frequency range 1 Hz to 10 kHz. Though no actual results have been obtained up till now, the NML is confident that the system would be fully operational soon. With the successful implementation of the SAM system the NML would be able to perform measurements comparable with other NMI’s internationally. The NML hopes to expand this technology for shock calibration in the not to distant future.

References [1] Methods for the calibration of vibration and

shock transducers, ISO 5347-1, 1998.

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[2] José M. Tribolt, “A New Phase Unwrapping Algorithm” , in IEEE TRANSACTIONS ON ACOUSTICS, SPEECH AND SIGNAL PROCESSING, Vol. Assp-25, No 2, April 1977.

[3] Wabinski W. and Von Martens H.J., “Time interval Analyses of interferometer signals for measuring amplitude and phase of vibrations” , Proc SPIE 2868, 1996.

[4] Dobosz M, Usuda T and Kurosawa T , “The methods for calibration of vibration pick-ups by laser interferometer: part 3 Phase-lag evaluation” , Meas.Sci. Technol. 9 1672-1677, 1998.

[5] Wu C.M., Su G.S. and Huang Y.J., “Polarimetric, nonlinearty-free, homodyne interferometer for vibration measurement” , Metrologia, Vol 33, P 533-537, 1996.

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