AN ALTERNATIVE CALIBRATION METHOD FOR THE ULYSSES FLUXGATE MAGNETOMETER

Claire Pinches Space and Atmospheric Physics Group

Blackett Laboratory Imperial College of Science and Technology

London SW7 2BZ UK

Abstract

The Fluxgate magnetometer on the Ulysses mission will be used to collect data on the heliospheric magnetic field originating from the sun. A new method of inflight calibration involving Fourier transformation of a finite length square wave will be implemented along with the standard single step method. This paper examines both calibration methods, and demonstrates the potential for increased accuracy using the square wave method.

Introduction

The heliospheric magnetic field originating from the sun is very complex, and although much has been learned about the interplanetary field in the ecliptic plane, very little is known about the heliolatitude dependence.

The Ulysses mission (formerly the Inter- national Solar Polar Mission) will pass out of the ecliptic using a Jupiter swingby, and will proceed to fly over both poles of the sun. On board are nine scientific instruments which will collect data on a wide range of aspects of the solar environ- ment. The results obtained will greatly increase our understanding of the mechanisms at work in the sun and in interplanetary space. One of the instruments is a dual magnetometer consisting of two triaxial sensors, a Vector Helium magnetometer provided by the Jet Propulsion Laboratory and a Triaxial Fluxgate magnetometer provided by Imperial College of Science and Technology, London (reference I). This instrument will be used to measure spatial and temporal variations in the magnetic field. It is hoped that simultaneous in- ecliptic observations will be available to complement the data and help to distinguish between variations due to changes in solar latitude and those due to solar disturbances.

Calibration of the Fluxgate Magnetometer

This project was concerned with an innovative new method for calibrating the Fluxgate magneto- meter.

Associated with each of the three magnetometer sensors (x, y and z orientations) is a primary, a secondary and a calibration winding. When an external magnetic field is present, a signal is generated in the secondary coil that is propor- tional to both the magnitude and the direction of the component of the field along the sensor's magnetic axis. The field is sampled twice per second at twice the bandwidth of the magnetometer and translated into despun coordinates at ground level processing. The calibration coil is used for one of the inflight calibration methods.

The Standard Calibration Method

The standard method of calibration is to use a fixed current step in the calibration winding to generate a known field step in the sensor. The calibration step is usually applied over a long duration. Referring to figure I, the average background field is found before and after the calibration step is applied (sections A and C respectively). The value of the calibration step plus the background field is then determined by suitable averaging over section B. The difference between the field strength in section B and that in sections A and C gives two values of the step height, which can be averaged to provide the final estimate. By comparing this value to the pre- flight calibrated value of the step height, the calibration constant of the system can be deter- mined. The accuracy associated with the value of the calibration constant obtained with this method is quite low since it is difficult to separate the noise from the step function. (During calibration, noise is taken to mean the noise associated with the system plus the background field).

The New Calibration Method

For this method a square wave current pulse sampled four times per period is applied over a finite number of samples. The pulsed wave is bandwidth limited by a low pass filter at I Hertz, thus satisfying the Nyquist criterion (i.e. the bandwidth must be less than or equal to half the reciprocal of the sample spacing). The frequency of the square wave is half the Nyquist frequency. By performing a discrete Fourier transform (DFT) on the sampled data the frequencies present in the signal can be separated. The dominant frequency will be that of the square wave, and its amplitude in the frequency domain is proportional to the height of the calibration square wave. Thus the square wave signal is extracted from the noise and a more accurate value of the calibration constant can be determined.

The Fourier Transform Technique

A BASIC program was written to perform the DFT on an Olivetti microcomputer using a Fast Fourier Transform (FFT) routine. (FFT's involve fewer calculations and so reduce computational time).

As a first step the calibration signal was taken as a square wave on a constant baseline of zero. The square wave can be considered as a rect function convolved with a comb (shah) and multi- plied by a much wider rect of unit height to account for the finite length of the pulsed calibration signal (figure 2).

Released to AIAA to publish in all forms.

AA 18~Y 3 7 9

380

(a) THE SINGLE STEP CALIBRATION OATA

50 ~00 O-

= -i Lu LL

D<3 SECTION A

TIME (SECI

150 200 250

SECTION 8 SECTION C -2,

(b) THE SQUARE WAVE CALIBRATION OATA TIME (SEC)

50 100 i50 200 250

w

R -I w

Fig. |(a) The single step calibration data. The average background field is found in sections A and C. The average background field plus calibration step is found in section B. The differences between the values in section B and sections A and C gives two values for the step height. These are averaged to give the final estimate.

Fig. l(b) The pulsed square wave calibration data. A pulsed square wave of frequency 0.5 Hertz sampled four times per period is applied to the data.

b • c c

Fig. 2 The pulsed square wave of finite length. The pulsed square wave can be considered as a rect function convolved with a comb, to give an infinite square wave, and multiplied by a very wide rect function to account for the finite length.

The frequency spectrum is the Fourier trans- form of the first rect multiplied by that of the comb all convolved with the transform of the wider rect (see appendix). This gives the amplitude of the maxima in the frequency domain as

DMAX = b x ]/a x c x h

where h is the height of the square wave (see appendix). Thus, to obtain the height of the square wave, the maximum amplitude in frequency

space, calculated by the program, must be divided by a factor hc/a which is a characteristic of the square wave (b, c and a are all known accurately). The calculated step height should be approximately

]nT.

The sampled data points are half a second apart. In the program they are contained in a one dimensional array spaced at unit intervals. Increasing the separation in this way introduces more spatial frequencies but does not affect the maximum amplitude, and so it is acceptable to work with this array. However, care is needed when labelling the frequency axis to take account of this. The transformed sampled data is stored in the same one dimensional array of unit spacing but, since the real spacing in time is half a unit, the actual spacing in frequency is two units.

The next consideration is that the calibration

step applied is a low pass filtered square wave with the 3dB point at | Hertz. Figure 3 shows the form of the signal applied for unit step height (this is similar to the real data but is over sampled relative to the actual form of the data).

o ~1

-2

TIME (SEC) l 2 3 4 5 6

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0

- 1

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TIME

Fig. 3 The filtered square wave. This shows an over sampled representation of the low pass filtered square wave. The actual square wave is of unit height and displaced from zero.

Fig. 4 The offset from zero baseline. This figure shows the square wave is not applied to a zero baseline.

Ground based test data used in this project were negative and the square wave is thus offset from zero. Figure 4 shows a simplistic representation.

This is equivalent to the addition of a constant rect of height k running the whole length of the calibration. The Fourier transform is then the sum of the two individual transforms, therefore the maximum amplitude also includes a term due to the offset.

DMAX = (b x c x h x ]/a) + (c x k)

Now k = H - h where H is known so the step height h can be calculated.

For this work the calibration step input square wave was of height | nT, and was used with the +8 nT range of the magnetometer. There are four possible ranges +8 nT, +69 nT, +2048 nT and 44000 nT (the last range is used for ground testing when compensation for the Earth's field is not possible).

Using the unfiltered square wave the step height is recovered to within 0.2% and using the filtered square wave to within 2.0%. The errors are insensitive to the range of the background field variation (to two significant figures). With the single step method the error ranges from 13% to 4% depending on the variation in the background field.

The noise in the signal is simulated using a random number generator. Various levels are used ranging between 2.5% and 10% of the step height.

Finally it is necessary to account for the variation in background field which alters the height of the baseline during calibration (note that this variation is much less than the step height). Since we are unable to predict this variation, a simple approach to evaluating the offset was used. By averaging over every fourth data point, i.e. where the contribution of the calibration signal is approximately zero, the average baseline is determined and used as H in the equation for the step height:

step height h = cH - DMAX c ~

The frequency spectra in figure 5 show that the minimum signal to noise ratio (i.e. maximum noise) is of the order of 5 x 10 ^ 4. With a greater number of samples the noise contribution would be reduced (the real data contains 4096 samples compared to the 5]2 used here).

During testing it was noticed that a single data point at significantly greater than one standard deviation from the expected value had very little effect on the value of the step height obtained using the square wave method (a change of 0.4%). However there was a noticeable difference in the Fourier spectrum, and consequently plotting the Fourier transform could act as a simple check for errors in the data. The effect on the value of the step height obtained with the standard method was more significant (a change of 2%).

where c is the number of samples, b is the width of the rect function, a is the spacing of the comb function, DMAX is the maximum amplitude of the Fourier transform calculated by the program, and H is the average baseline height.

Results

The results are shown in Table ! for simul- ated data using the values

b = 2 a = 4 c = 512 h = I

Conclusion

The results show that the standard single step calibration method is reasonably accurate provided that the background field variation during the calibration run is small. By contrast, the new Fourier transform method is accurate even for large jumps in the background field. This new method is also less sensitive to bad data points. Thus from the work done for this paper with simul- ated data the Fourier transform calibration method appears to be more accurate than the standard calibration method. The next stage of testing is to use data from ground based testing in the

382

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The Fourier transform of the calibration data.

(a) With the noise level at 2.5% (b) With the noise level at 10%

where "noise" means the noise associated with the system plus the background field variation.

Acknowledgements

I would like to thank Mr A. Balogh for his supervision of this project.

References

]. Balogh, A. et al., The magnetic-field invest- igation on ISPM, ESA SP-]050, 27-45, ]983.

2. Otnes, R.K. and Enochson, L., Digital Time Series Analysis, John Wiley and Sons, New York, ]972.

3. The International Solar Polar Mission - Its Scientific Investigations, ESA SP-]050, ]983.

Background Noise Standard Square Wave Method field level method unfiltered filtered

]/40 0.8698 0.9996 0.9797

Medium ]130 0.8697 0.9940 0.9795 Variation

1/20 0.8693 0.9990 0.979]

]/]0 0.8684 0.9979 0.9780

1/40 0.9577 0.9994 0.9827 Very little

Variation ]/10 0.9607 0.9977 0.9811

]/40 0.9655 ].0026 0.9827 Large

Variation 1 /10 0.9640 1.0009 0.98]0

Table ] The recovered step height for data simulated using a step height of 1.0000. Three varying background fields are used. In all cases the variation in the background field is less than 0.25% of the step height.

Appendix

Definition of a Fourier transform +co

F(u) = f_ f(x) exp(-2~iux) dx

and

w

f(x) = I F(u) exp(+2niux) dx

Write as a transform pair F(u) ~ f(x). Note that

rect (x/a) ~ a.sinc (ua) and

uLk(xla) ~ laJ (ua)

whemm ~/.(xla) is a series of delta functions spaced at intervals of a. (Shah function).

Convolution:

/

h(x) = I f(x) g(x' - x) dx = f~ g

taking the Fourier transform

co co

f(x') { ~ g(x - X') exp(-2~iux) dx} dx'

ce+

f(x') exp(-2~iux') G(u) dx'

/-co

383

= F(u).G(u)

therefore a transform pair

f(x) * g(x) ~ F(u).G(u)

and in this paper the formula below was used

(f(x) * g(x)).l(x) ~ (F(u).G(u)) * L(u)