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Tuned current-output fluxgate
Sensors and Actuators 82 Ž2000. 161–166 www.elsevier.nlrlocatersna
Tuned current-output fluxgate Pavel Ripka a
a,)
, Fritz Primdahl
b,1
Czech Technical UniÕersity, Faculty of Electrical Engineering, Department of Measurement, Technicka 2, 166 27 Prague 6, Czech Republic b Danish Technical UniÕersity, Department of Automation, DK-2800 Lyngby, Denmark Received 7 June 1999; received in revised form 27 October 1999; accepted 1 November 1999
Abstract The current-output fluxgate may be tuned by using a serial capacitor. Such tuning increases the sensor sensitivity in the situation when the pick-up coil has a low number of turns. We achieved a signalrfeedthrough ratio improvement by a factor of 5. The measured parameters fit the simplified theoretical model within 20% deviation. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Magnetic sensors; Magnetometer; Fluxgate
1. Introduction Fluxgate sensors serve for the measurement of DC and low-frequency AC magnetic fields. Their principle is based on modulation of the flux in the pick-up coil by changing the permeability of the ferromagnetic core by means of the AC excitation field w1x. In order to erase all the remanent magnetization of the sensor core and thus suppress memory effects, the core should be deeply saturated. High narrow peaks of the excitation current may be achieved by using a tuning capacitor either parallel to the excitation winding Žfor higher source resistance. or connected serially Žfor voltage-mode excitation.. Traditional fluxgates use the second harmonic component of the voltage induced in the pick-up coil. Large spurious components at odd harmonics coming from the transformer effect were suppressed by using symmetrical construction of the sensor Žtwo open cores excited in opposite directions, ring-core, or race track.. The voltagemode sensor sensitivity may be increased by parametric amplification using a parallel tuning capacitor w2,3x. The theoretical description and numerical results of a capacitively loaded fluxgate sensor are given in Ref. w4x and in previous works by Russell et al. More recently, an analytical theory of parallel tuned fluxgate sensors was given by Player w5x. If the quality factor of the non-linear tuning
) Corresponding author. Tel.: q42-2-243-53945; fax: q42-2-311-9929. E-mail: [email protected] 1 E-mail: [email protected].
circuit is high, then parametric amplification may even result in instability. There are indications that the tuning technique concentrates the information from higher-order even harmonic components of the induced voltage into the second harmonic, which may result in lowering of the sensor noise in case we use the classical synchronous detector w6x. According to the Serson–Hannaford solution of the tuning circuit, the only significant contributions come from the 2nd and 4th harmonics. In some cases, the tuning capacitor is formed by the parasitic capacitance of the pick-up coil. This may happen if the number of windings is high, the coil is wound without splitting and the excitation frequency is high. In such a case, the output voltage of the unloaded sensor is close to a sine wave. This mode is usually undesirable as the value of the parasitic capacitance may be unstable in time and with temperature. The mentioned tuning effect has to be taken into account in any sensitivity vs. frequency measurements of the fluxgate sensors. Stability of both tuning capacitors is important; surprisingly, the sensor offset is more sensitive to changes of the excitation capacitor than to changes of the output capacitor C2 w7x. Current-output Žor short-circuited. fluxgates were introduced in Ref. w8x. In contrast to the voltage output device, the sensitivity is inversely proportional to the number of turns of the pick-up coil with certain practical limitations w9x. Besides that, the sensitivity also depends on several geometrical and material factors. In the ideal case, the p y p value of the output current is directly proportional to the measured field. In real sensors, the output current waveform is distorted by spurious components coming
0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 9 9 . 0 0 3 3 2 - 5
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
162
from the sensor geometrical asymmetry and core non-homogeneity w10x. A practical way to detect the difference between the current value at the time when the excitation is zero and the core permeability is maximum, and the value at the instant of fully saturated core when the flux is minimum, is to use the gated integrator. This is in fact time-domain signal processing rather than frequency-domain, which is more typical for the voltage output device. In the present paper, we will show that the current-output fluxgate may be tuned by using a serial capacitor.
Apparent permeability is lower than the core material permeability because of the core demagnetisation w11x. It is dependent not only on the core size and properties and mode of excitation, but also on the geometry of the pick-up coil. In order to easily incorporate the measured DC field Bex into the circuit equations, we replace its effect by the equivalent coil current i ex , which flowing in the coil would create a field equal to the external field component along the coil axis. This ‘‘external field’’ current is related to the field by
2. Theory
Bex s m 0
The circuit diagram is shown in Fig. 1. The sensor core is excited by a periodical current i ex into saturation. The sensor measures the magnetic field component Bex in the direction of the pick-up coil axis. The pick-up coil is short-circuited by the current-to-voltage converter with the feedback resistor R. The capacitor C is used to prevent any DC input current of the op amp from flowing through the pick-up coil. In the traditional broadband case C is high; here, we propose to use C for tuning. The secondary pick-up coil has a time-changing induction given by LŽ t . s m0
N2 l
A ma Ž t .
Ž 1.
where l is the effective length of the coil, A is its cross sectional area, N is the number of turns, maŽ t . is the Žmodulated. apparent permeability of the sensor core. The effective length of the coil is here defined as l s m 0 NIrBex , where Bex is an external field which is cancelled by a DC compensation current I into the pick-up coil. We suppose that l is only dependent on the coil geometry, not on the sensor core properties nor on the mode of the excitation. The value of l is higher than the physical length of the coil and may also be determined from the inductance of the pick-up coil with removed Žor completely saturated. core w11x. The apparent permeability ma is defined as ma s m 0 BrBex , where B is the magnetic field inside the core.
N l
i ex .
Ž 2.
The total flux in the coil is then
F s i ex q i Ž t . L Ž t .
Ž 3.
and the equation for the input circuit in Fig. 1 becomes: dF dt
q i Ž t . rCu q
1 C
Hi Ž t . d t s 0
Ž 4.
and by inserting F and differentiating with respect to time, we have: i ex
d LŽ t .
d q
dt
dt
i Ž t . L Ž t . q i Ž t . rCu q
1 C
Hi Ž t . d t s 0. Ž 5.
Here, LŽ t . is a periodic function of period T s 2prŽ2 v ., half the fluxgate core excitation period, and LŽ t . may be approximated by a Fourier’s series: `
L Ž t . s Lo q
Ý Ln cos Ž n v t . Ž n even. .
Ž 6.
ns2
The zero point for the time is selected to give only cosine terms in the series. The current iŽ t . similarly contains the fundamental frequency 2 v and the higher even harmonics, and it is also represented by a Fourier’s series `
iŽ t . s
Ý Ž i a Ž n . cos Ž n v t . q i b Ž n . sin Ž n v t . . .
Ž 7.
ns2
Fig. 1. Short-circuited fluxgate input stage. The operational amplifier inverting input acts as a virtual ground, and the sensor secondary current flows through the feedback resistor R. The capacitor C is used for tuning the secondary pick-up coil, or for C ™`, it acts as a short-circuit in the relevant frequency band.
The task is now to find iŽ t . when LŽ t ., the circuit parameters and Bex are known. The product iŽ t . P LŽ t . complicates the solution for iŽ t ., because the pX th harmonic of iŽ t . contains contributions from, in principle, an infinite number of harmonics of LŽ t ., and consequently, an exact calculation of iŽ t . very quickly becomes intractable. Two approximations are then made: First, the untuned broadband sensor secondary case where C ™ `, and secondly, the Serson–Hannaford approximation for the tuned secondary case w2x, where the current contains only one frequency component.
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
2.1. The broadband case
has a limited number of terms contributing to the second harmonic, even if LŽ t . is an infinite Fourier’s series. The terms needed in this case are:
In the broadband case, we have: dFrdt q i Ž t . rCu s 0.
Ž 8.
If the coil losses are small, then the copper resistance rCu may be neglected, as discussed in Ref. w8x. The equation is then integrated to give:
F s Ž i ex q i Ž t . . L Ž t . s constant, or i Ž t . s yi ex q FrL Ž t . .
Ž 9.
The output current has no DC component, so by taking the time-average Žmarked by ² :. we get: 0 s yi ex q F ²1rL Ž t . : ´ F s i ex LG 0 ;
Ž 10 .
where LG0 is the geometric mean value of the pick-up coil induction 1rLG 0 s ²1rL Ž t . : .
Ž 11 .
Notice that LG0 differs from the arithmetic mean value L 0 s ² LŽ t .: from Eq. Ž6.. And from this we have the basic short-circuited fluxgate equation: i Ž t . s i ex
LG 0 LŽ t .
y1 .
Ž 12 .
The p y p value of the output current was derived in Ref. w9x: i py p s i ex
LG 0 Lmax
ž
Lmax Lmin
/
y1 .
Ž 13 .
The complete solution then follows from a Fourier’s analysis of Ž LG0 rLŽ t . y 1.. 2.2. The tuned secondary case The sensor secondary circuit may be parallel or series tuned to one of the even higher harmonics of the excitation frequency. This enhances the chosen harmonic signal and attenuates all other signals, in particular the omnipresent odd harmonics, which in some cases may be cumbersome. The parallel tuning was first described by Serson and Hannaford w2x and it is further discussed in Ref. w3x. Series tuning of the sensor secondary appears not to have been considered elsewhere, and it is briefly discussed theoretically in the following. In the second harmonic tuned case, the solution to iŽ t . is limited to the Serson–Hannaford approximation: i Ž t . ( i a cos Ž 2 v t . q i b sin Ž 2 v t .
163
Ž 14 .
because in the tuned circuit, the second harmonic is by far the dominant signal for any reasonable circuit Q-factor. In principle, the circuit could be tuned to any other even harmonic frequency to the same effect. The assumption of only second harmonic content in iŽ t . dramatically limits the number of harmonic components of LŽ t . entering the calculation, because the product iŽ t . P LŽ t .
L Ž t . ( Lo q L2 cos Ž 2 v t . q L 4 cos Ž 4v t . . Ž 15 . We substitute for iŽ t . and LŽ t . from Eqs. Ž14. and Ž15. into Eq. Ž5.: d Ž i ex q i a cos Ž 2 v t . q i b sin Ž 2 v t . .Ž Lo q L2 cos Ž 2 v t . dt qL4 cos Ž 4v t . . q rCu Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . 1
Ž 16 . Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . d t s 0. C Retaining only the terms leading to second harmonics, we get: d i L cos Ž 2 v t . q Lo Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . d t ex 2 q Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . L 4 cos Ž 4v t . q
H
qrCu Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . 1 q Ž 17 . Ž i sin Ž 2 v t . y i b cos Ž 2 v t . . s 0 2vC a applying the trigonometric product formulas: d i L cos Ž 2 v t . q i a Lo cos Ž 2 v t . q i b L o sin Ž 2 v t . d t ex 2 1 1 q i a L4 cos Ž 2 v t . y i b L 4 sin Ž 2 v t . 2 2 qi a rCu cos Ž 2 v t . q i b rCu sin Ž 2 v t . ia ib q sin Ž 2 v t . y cos Ž 2 v t . s 0 2vC 2vC and performing the differentiation:
Ž 18 .
y2 v i ex L 2 sin Ž 2 v t . y 2 v i a L o sin Ž 2 v t . q 2 v i b Lo cos Ž 2 v t . y v i a L 4 sin Ž 2 v t . y v i b L4 cos Ž 2 v t . q i a rCu cos Ž 2 v t . ia q i b rCu sin Ž 2 v t . q sin Ž 2 v t . 2vC ib y cos Ž 2 v t . s 0 Ž 19 . 2vC then, we divide through by 2 v L o and assume 2nd harmonic resonance: 1 Cs Ž 20 . 2 Ž 2 v . Lo . By considering the sine and cosine terms separately, we get the following matrix equation determining i a and i b : rCu
yL4
2 v Lo
2 Lo
yL4
rCu
2 Lo
2 v Lo
0 ia L 2 s i . ib Lo ex
Ž 21 .
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
164
The determinant is: 2
D s Ž rCu r2 v Lo . y Ž L4r2 Lo . s 0
for rCu s v L4.
For rCu - v L4 , i.e., for sufficiently small losses, the circuit becomes unstable as shown in Ref. w3x.
3. Measurements We performed our measurements on the standard ringcore fluxgate sensor similar to that described in Ref. w12x. The sensor core is made of 18r22 mm rings etched from 79Ni4Mn permalloy: 8 rings 50 mm thick form the core with cross-section of A Fe s 0.8 mm2 . The rings are stacked in the corundum ceramic holder. The sensor is excited by 4 kHz, 5 V rms sinewave from 50 V source. The excitation winding of N1 s 150 turns, f 0.3 mm is tightly wound on the ceramic core holder so it has the toroidal shape. Excitation winding is parallel tuned by C1 s 2.2 nF in order to increase the excitation field amplitude. The excitation winding resistance was R 1 s 0.7 V, its cross-section is A air s 16.2 mm2 . The pick-up coil has N2 s 100 turns of f 0.3 mm wire wound on glass-fiber bobbin. The coil has no split; its DC resistance is R 2 s 1.8 V, average cross-section is A pick s 196.3 mm2 and the coil length is l s 22 mm. The core holder with excitation winding is mounted in the middle of the pick-up coil bobbin. The pick-up coil was connected to the current-to-voltage converter with LT 1028 op amp, the feedback resistor value was R f s 5.1 k V. Our measurements were made using 1:10 voltage probe in order to fit the input range of the SR 830 lock-in amplifier. All the measurements were performed without feedback. The field sensitivity was measured at 1 and 5 mT for both untuned sensor and sensor tuned to 2nd harmonic by serial capacitor of C2 s 1.7 mF. The measured sensitivities for the first six even harmonics are shown in Table 1. Serial tuning increased the sensitivity at 2nd harmonic by the factor of 5, which is lower than the amplification factor achievable at voltage-output fluxgates. The signal at higher even harmonics is amplified approximately by the factor of 2, while in properly tuned voltage-mode sensors, the sensitivity at higher harmonics is decreased. Fig. 2 shows both the tuned and untuned current-output sensor for the measured field of 2 mT. Top trace is the Table 1 Field sensitivity of untuned and tuned sensor at first six even harmonics Žin mV, 1 mV corresponds to 2 mA. Harmonics
Untuned Tuned
1 mT 5 mT 1 mT 5 mT
2
4
6
8
10
12
9.1 43.4 47 219
6.7 38 11 69
2.6 30.5 7.3 50
4.2 22 7 36
0.6 13.5 3.5 25
1.1 6 2.3 15
Fig. 2. Ža,b. Excitation current Župper trace. and output current Žlower trace, 200 mArdiv.. of untuned Ža. and tuned Žb. fluxgate sensor. The measured field was 2 mT.
excitation current and bottom trace is the output current Žmeasured at the output of the IrU converter..
4. Calculated values With the ring-core completely magnetically saturated, the induction of the secondary pick-up coil is measured to be Lair ( Lsat s 0.1085 mH Ž0.1080 mH. at 1 kHz Ž10 kHz., and the maximum inductance for unsaturated core is L max s 0.2465 mH Ž0.2461 mH. at 1 kHz Ž10 kHz.. Let us approximate the LŽ t . waveform by a step function changing from Lmax to Lair w9x as shown in Fig. 3. The fraction of time spent in saturation by the core is estimated from the excitation currentroutput current traces shown in Fig. 1a,b to be d s 0.0686. The average induction L o is calculated from Eq. Ž8. as Lo s 0.22678 mH. 4.1. Broadband output current The current equivalent to the external field is i ex s Bex lr Ž mo N . s 9.151 mA for 50,000 nT external field along the sensor axis ŽFig. 4..
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
Fig. 3. The time variation of the fluxgate secondary pick-up coil inductance for two periods of the excitation frequency. During the times when the core is in magnetic saturation then the secondary inductance decreases to approximately the air-cored coil value. The average induction determining the resonance is L o .
An effective coil winding length l s 23 mm is adopted in order to match the calculated value to the measured secondary coil induction w3x. The waveform of the theoretical output current is also squarewave with a duty factor of d w9x. The Fourier’s analysis of a periodically repeated narrow square impulse of width d P T, period T and amplitude I yields the coefficients: i a Ž n . s 2 Ir Ž n P p . sin Ž n P p P d . i b Ž n. s 0
cosine terms
sine terms.
The six first even harmonic Fourier’s coefficients ŽFig. 5. of the computed broadband short-circuited output for 5000 nT are: 145.7, 142.3, 136.8, 129.3, 120.0 and 109.2 mA.For comparison with the measured numbers Žin mV. in Table 1, divide by 2.
165
Fig. 5. Broad-band even harmonics Fourier’s coefficients of the output currents. The external magnetic field is 50,000 nT.
4.2. The second harmonic tuned case This requires knowledge of the first few Fourier’s coefficients of the time-modulated secondary induction: L2 s 18.7874; L4 s 18.3528 mH. Lo s 0.22678 mH as used above. Using rCu s 1.8 V and 8 kHz as the second harmonic of the excitation frequency, we get: rCu s 0.15795 2 v Lo L4 2 L2
s 0.04048.
The circuit is stable because rCu ) v L4 . Solving the matrix Eq. Ž21. for i a,b gives: i a s 1.317 mA, i b s 5.139 mA and a total second harmonic current amplitude i 2 s 5.305 mA for Bex s 50,000 nT. Table 1 gives the measured second harmonic current in mA as twice the tabulated number in mV and for 5000 nT. The measured current i 2 s 0.438 mA and the calculated current i 2 s 0.531 mA are in a fair agreement. The tuned output current depends on the series resistance rCu . Any input impedance of the operational amplifier adds to rCu , and this may account for the slight difference between the measured and the calculated results.
5. Conclusion
Fig. 4. Calculated broad-band short-circuit output current from the sensor secondary pick-up coil. One excitation period is shown and the external magnetic field is 50,000 nT.
We have shown that the current-output fluxgate may be tuned by using a serial capacitor. Such tuning increases the sensor sensitivity in the situation when the pick-up coil has a low number of turns, and this may be advantageous in the process of the miniaturization of the fluxgate sensors.
166
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
We have increased the sensitivity of the ring-core shortcircuited fluxgate sensor five times while the level of the spurious feedthrough remained the same. Serial tuning is easily made only by decreasing the value of the input capacitor. The change of signal-to-noise ratio Žor effective noise in units of the measured field. with tuning is under investigation. The preliminary results are ambiguous: noise improvement with tuning was reported in Ref. w6x, while noise degradation by tuning was observed in Ref. w3x and recently in Ref. w13x.
The ring core fluxgate sensor null feed-through signal, Meas. Sci. Technol. 3 Ž1992. 1149–1154. w11x F. Primdahl, B. Hernando, O.V. Nielsen, J.R. Petersen, Demagnetising factor and noise in the fluxgate ring-core sensor, J. Phys. E: Sci. Instrum. 22 Ž1989. 1004–1008. w12x P. Ripka, P. Kaspar, Portable fluxgate magnetometer, Sensors and Actuators A 68 Ž1998. 286–289. w13x E.B. Pedersen, Digitalisation of Fluxgate Magnetometer, PhD Thesis, Academy of Technical Sciences, TERMA Elektronik and Department of Automation, Technical University of Denmark, DK-2800 Lyngby, Denmark, June 1999.
Biographies References w1x P. Ripka, Review of fluxgate sensors, Sensors and Actuators A 33 Ž1992. 129–141. w2x P.H. Serson, W.L.W. Hannaford, A portable electrical magnetometer, Can. J. Technol. 34 Ž1956. 232–243. w3x F. Primdahl, P. Anker Jensen, Noise in the tuned fluxgate, J. Phys. E: Sci. Instrum. 20 Ž1987. 637–642. w4x Z. Gao, R.D. Russell, Fluxgate sensor theory: sensitivity and phase plane analysis, IEEE Trans. Geosci. 25 Ž1987. 862–870. w5x M.A. Player, Parametric amplification in fluxgate sensors, J.Phys. D 21 Ž1988. 1473–1480. w6x P. Ripka, W. Billingsley, Fluxgate: tuned vs. untuned output, IEEE Trans. Magn. 34 Ž1998. 1303–1305. w7x A. Tipek, P. Ripka, P. Kaspar, Gradiometric sensor Žin Czech., Proc. ˇ of the Sensory-snimace-aplikace conf., Ostrava, 1998. w8x F. Primdahl, J.R. Petersen, C. Olin, K. Harbo Andersen, The shortcircuited fluxgate output current, J. Phys. E: Sci. Instrum. 22 Ž1989. 349–353. w9x F. Primdahl, P. Ripka, J.R. Petersen, O.V. Nielsen, The short-circuited fluxgate sensitivity parameters, Meas. Sci. Technol. 2 Ž1991. 1039–1045. w10x J.R. Petersen, F. Primdahl, B. Hernando, A. Fernandez, O.V. Nielsen,
PaÕel Ripka received an Ing. degree in 1984, a CSc Žequivalent to PhD. in 1989 and a doctoral degree in 1996 at the Czech Technical University, Prague, Czech Republic. He works at the Department of Measurement, Faculty of Electrical Engineering, Czech Technical University as a lecturer, teaching courses in Electrical Measurements and Instrumentation, Engineering Magnetism and Sensors. His main research interests are magnetic measurements and magnetic sensors, especially, fluxgate. He is a member of Elektra Society, Czech Metrological Society, Czech National IMEKO Committee and Eurosensors Steering Committee. Fritz Primdahl received an MSc in Electrical Engineering and Physics at the Technical University of Denmark in 1964. He worked as Research Scientist at the Department of Geophysics, Danish Meteorological Institute Ž1966–1968 and 1970–1980.. In 1968–1970, he was awarded a fellowship by the National Research Council of Canada. Since 1980, he has been with the Danish Space Research Institute, now on leave to lead the Ørsted satellite science instrumentation team at the Technical University of Denmark. He mainly works in space magnetometry and space plasma physics. He was a member or head of teams developing instrumentation for sounding rockets and satellites ŽCLUSTER, Oersted, ASTRID-2, CHAMP and SAC-C. and scientific teams planning the space experiments and analyzing the data. He is a member of Danish Geophysical Society.
Tuned current-output fluxgate Pavel Ripka a
a,)
, Fritz Primdahl
b,1
Czech Technical UniÕersity, Faculty of Electrical Engineering, Department of Measurement, Technicka 2, 166 27 Prague 6, Czech Republic b Danish Technical UniÕersity, Department of Automation, DK-2800 Lyngby, Denmark Received 7 June 1999; received in revised form 27 October 1999; accepted 1 November 1999
Abstract The current-output fluxgate may be tuned by using a serial capacitor. Such tuning increases the sensor sensitivity in the situation when the pick-up coil has a low number of turns. We achieved a signalrfeedthrough ratio improvement by a factor of 5. The measured parameters fit the simplified theoretical model within 20% deviation. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Magnetic sensors; Magnetometer; Fluxgate
1. Introduction Fluxgate sensors serve for the measurement of DC and low-frequency AC magnetic fields. Their principle is based on modulation of the flux in the pick-up coil by changing the permeability of the ferromagnetic core by means of the AC excitation field w1x. In order to erase all the remanent magnetization of the sensor core and thus suppress memory effects, the core should be deeply saturated. High narrow peaks of the excitation current may be achieved by using a tuning capacitor either parallel to the excitation winding Žfor higher source resistance. or connected serially Žfor voltage-mode excitation.. Traditional fluxgates use the second harmonic component of the voltage induced in the pick-up coil. Large spurious components at odd harmonics coming from the transformer effect were suppressed by using symmetrical construction of the sensor Žtwo open cores excited in opposite directions, ring-core, or race track.. The voltagemode sensor sensitivity may be increased by parametric amplification using a parallel tuning capacitor w2,3x. The theoretical description and numerical results of a capacitively loaded fluxgate sensor are given in Ref. w4x and in previous works by Russell et al. More recently, an analytical theory of parallel tuned fluxgate sensors was given by Player w5x. If the quality factor of the non-linear tuning
) Corresponding author. Tel.: q42-2-243-53945; fax: q42-2-311-9929. E-mail: [email protected] 1 E-mail: [email protected].
circuit is high, then parametric amplification may even result in instability. There are indications that the tuning technique concentrates the information from higher-order even harmonic components of the induced voltage into the second harmonic, which may result in lowering of the sensor noise in case we use the classical synchronous detector w6x. According to the Serson–Hannaford solution of the tuning circuit, the only significant contributions come from the 2nd and 4th harmonics. In some cases, the tuning capacitor is formed by the parasitic capacitance of the pick-up coil. This may happen if the number of windings is high, the coil is wound without splitting and the excitation frequency is high. In such a case, the output voltage of the unloaded sensor is close to a sine wave. This mode is usually undesirable as the value of the parasitic capacitance may be unstable in time and with temperature. The mentioned tuning effect has to be taken into account in any sensitivity vs. frequency measurements of the fluxgate sensors. Stability of both tuning capacitors is important; surprisingly, the sensor offset is more sensitive to changes of the excitation capacitor than to changes of the output capacitor C2 w7x. Current-output Žor short-circuited. fluxgates were introduced in Ref. w8x. In contrast to the voltage output device, the sensitivity is inversely proportional to the number of turns of the pick-up coil with certain practical limitations w9x. Besides that, the sensitivity also depends on several geometrical and material factors. In the ideal case, the p y p value of the output current is directly proportional to the measured field. In real sensors, the output current waveform is distorted by spurious components coming
0924-4247r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 Ž 9 9 . 0 0 3 3 2 - 5
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
162
from the sensor geometrical asymmetry and core non-homogeneity w10x. A practical way to detect the difference between the current value at the time when the excitation is zero and the core permeability is maximum, and the value at the instant of fully saturated core when the flux is minimum, is to use the gated integrator. This is in fact time-domain signal processing rather than frequency-domain, which is more typical for the voltage output device. In the present paper, we will show that the current-output fluxgate may be tuned by using a serial capacitor.
Apparent permeability is lower than the core material permeability because of the core demagnetisation w11x. It is dependent not only on the core size and properties and mode of excitation, but also on the geometry of the pick-up coil. In order to easily incorporate the measured DC field Bex into the circuit equations, we replace its effect by the equivalent coil current i ex , which flowing in the coil would create a field equal to the external field component along the coil axis. This ‘‘external field’’ current is related to the field by
2. Theory
Bex s m 0
The circuit diagram is shown in Fig. 1. The sensor core is excited by a periodical current i ex into saturation. The sensor measures the magnetic field component Bex in the direction of the pick-up coil axis. The pick-up coil is short-circuited by the current-to-voltage converter with the feedback resistor R. The capacitor C is used to prevent any DC input current of the op amp from flowing through the pick-up coil. In the traditional broadband case C is high; here, we propose to use C for tuning. The secondary pick-up coil has a time-changing induction given by LŽ t . s m0
N2 l
A ma Ž t .
Ž 1.
where l is the effective length of the coil, A is its cross sectional area, N is the number of turns, maŽ t . is the Žmodulated. apparent permeability of the sensor core. The effective length of the coil is here defined as l s m 0 NIrBex , where Bex is an external field which is cancelled by a DC compensation current I into the pick-up coil. We suppose that l is only dependent on the coil geometry, not on the sensor core properties nor on the mode of the excitation. The value of l is higher than the physical length of the coil and may also be determined from the inductance of the pick-up coil with removed Žor completely saturated. core w11x. The apparent permeability ma is defined as ma s m 0 BrBex , where B is the magnetic field inside the core.
N l
i ex .
Ž 2.
The total flux in the coil is then
F s i ex q i Ž t . L Ž t .
Ž 3.
and the equation for the input circuit in Fig. 1 becomes: dF dt
q i Ž t . rCu q
1 C
Hi Ž t . d t s 0
Ž 4.
and by inserting F and differentiating with respect to time, we have: i ex
d LŽ t .
d q
dt
dt
i Ž t . L Ž t . q i Ž t . rCu q
1 C
Hi Ž t . d t s 0. Ž 5.
Here, LŽ t . is a periodic function of period T s 2prŽ2 v ., half the fluxgate core excitation period, and LŽ t . may be approximated by a Fourier’s series: `
L Ž t . s Lo q
Ý Ln cos Ž n v t . Ž n even. .
Ž 6.
ns2
The zero point for the time is selected to give only cosine terms in the series. The current iŽ t . similarly contains the fundamental frequency 2 v and the higher even harmonics, and it is also represented by a Fourier’s series `
iŽ t . s
Ý Ž i a Ž n . cos Ž n v t . q i b Ž n . sin Ž n v t . . .
Ž 7.
ns2
Fig. 1. Short-circuited fluxgate input stage. The operational amplifier inverting input acts as a virtual ground, and the sensor secondary current flows through the feedback resistor R. The capacitor C is used for tuning the secondary pick-up coil, or for C ™`, it acts as a short-circuit in the relevant frequency band.
The task is now to find iŽ t . when LŽ t ., the circuit parameters and Bex are known. The product iŽ t . P LŽ t . complicates the solution for iŽ t ., because the pX th harmonic of iŽ t . contains contributions from, in principle, an infinite number of harmonics of LŽ t ., and consequently, an exact calculation of iŽ t . very quickly becomes intractable. Two approximations are then made: First, the untuned broadband sensor secondary case where C ™ `, and secondly, the Serson–Hannaford approximation for the tuned secondary case w2x, where the current contains only one frequency component.
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
2.1. The broadband case
has a limited number of terms contributing to the second harmonic, even if LŽ t . is an infinite Fourier’s series. The terms needed in this case are:
In the broadband case, we have: dFrdt q i Ž t . rCu s 0.
Ž 8.
If the coil losses are small, then the copper resistance rCu may be neglected, as discussed in Ref. w8x. The equation is then integrated to give:
F s Ž i ex q i Ž t . . L Ž t . s constant, or i Ž t . s yi ex q FrL Ž t . .
Ž 9.
The output current has no DC component, so by taking the time-average Žmarked by ² :. we get: 0 s yi ex q F ²1rL Ž t . : ´ F s i ex LG 0 ;
Ž 10 .
where LG0 is the geometric mean value of the pick-up coil induction 1rLG 0 s ²1rL Ž t . : .
Ž 11 .
Notice that LG0 differs from the arithmetic mean value L 0 s ² LŽ t .: from Eq. Ž6.. And from this we have the basic short-circuited fluxgate equation: i Ž t . s i ex
LG 0 LŽ t .
y1 .
Ž 12 .
The p y p value of the output current was derived in Ref. w9x: i py p s i ex
LG 0 Lmax
ž
Lmax Lmin
/
y1 .
Ž 13 .
The complete solution then follows from a Fourier’s analysis of Ž LG0 rLŽ t . y 1.. 2.2. The tuned secondary case The sensor secondary circuit may be parallel or series tuned to one of the even higher harmonics of the excitation frequency. This enhances the chosen harmonic signal and attenuates all other signals, in particular the omnipresent odd harmonics, which in some cases may be cumbersome. The parallel tuning was first described by Serson and Hannaford w2x and it is further discussed in Ref. w3x. Series tuning of the sensor secondary appears not to have been considered elsewhere, and it is briefly discussed theoretically in the following. In the second harmonic tuned case, the solution to iŽ t . is limited to the Serson–Hannaford approximation: i Ž t . ( i a cos Ž 2 v t . q i b sin Ž 2 v t .
163
Ž 14 .
because in the tuned circuit, the second harmonic is by far the dominant signal for any reasonable circuit Q-factor. In principle, the circuit could be tuned to any other even harmonic frequency to the same effect. The assumption of only second harmonic content in iŽ t . dramatically limits the number of harmonic components of LŽ t . entering the calculation, because the product iŽ t . P LŽ t .
L Ž t . ( Lo q L2 cos Ž 2 v t . q L 4 cos Ž 4v t . . Ž 15 . We substitute for iŽ t . and LŽ t . from Eqs. Ž14. and Ž15. into Eq. Ž5.: d Ž i ex q i a cos Ž 2 v t . q i b sin Ž 2 v t . .Ž Lo q L2 cos Ž 2 v t . dt qL4 cos Ž 4v t . . q rCu Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . 1
Ž 16 . Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . d t s 0. C Retaining only the terms leading to second harmonics, we get: d i L cos Ž 2 v t . q Lo Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . d t ex 2 q Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . L 4 cos Ž 4v t . q
H
qrCu Ž i a cos Ž 2 v t . q i b sin Ž 2 v t . . 1 q Ž 17 . Ž i sin Ž 2 v t . y i b cos Ž 2 v t . . s 0 2vC a applying the trigonometric product formulas: d i L cos Ž 2 v t . q i a Lo cos Ž 2 v t . q i b L o sin Ž 2 v t . d t ex 2 1 1 q i a L4 cos Ž 2 v t . y i b L 4 sin Ž 2 v t . 2 2 qi a rCu cos Ž 2 v t . q i b rCu sin Ž 2 v t . ia ib q sin Ž 2 v t . y cos Ž 2 v t . s 0 2vC 2vC and performing the differentiation:
Ž 18 .
y2 v i ex L 2 sin Ž 2 v t . y 2 v i a L o sin Ž 2 v t . q 2 v i b Lo cos Ž 2 v t . y v i a L 4 sin Ž 2 v t . y v i b L4 cos Ž 2 v t . q i a rCu cos Ž 2 v t . ia q i b rCu sin Ž 2 v t . q sin Ž 2 v t . 2vC ib y cos Ž 2 v t . s 0 Ž 19 . 2vC then, we divide through by 2 v L o and assume 2nd harmonic resonance: 1 Cs Ž 20 . 2 Ž 2 v . Lo . By considering the sine and cosine terms separately, we get the following matrix equation determining i a and i b : rCu
yL4
2 v Lo
2 Lo
yL4
rCu
2 Lo
2 v Lo
0 ia L 2 s i . ib Lo ex
Ž 21 .
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
164
The determinant is: 2
D s Ž rCu r2 v Lo . y Ž L4r2 Lo . s 0
for rCu s v L4.
For rCu - v L4 , i.e., for sufficiently small losses, the circuit becomes unstable as shown in Ref. w3x.
3. Measurements We performed our measurements on the standard ringcore fluxgate sensor similar to that described in Ref. w12x. The sensor core is made of 18r22 mm rings etched from 79Ni4Mn permalloy: 8 rings 50 mm thick form the core with cross-section of A Fe s 0.8 mm2 . The rings are stacked in the corundum ceramic holder. The sensor is excited by 4 kHz, 5 V rms sinewave from 50 V source. The excitation winding of N1 s 150 turns, f 0.3 mm is tightly wound on the ceramic core holder so it has the toroidal shape. Excitation winding is parallel tuned by C1 s 2.2 nF in order to increase the excitation field amplitude. The excitation winding resistance was R 1 s 0.7 V, its cross-section is A air s 16.2 mm2 . The pick-up coil has N2 s 100 turns of f 0.3 mm wire wound on glass-fiber bobbin. The coil has no split; its DC resistance is R 2 s 1.8 V, average cross-section is A pick s 196.3 mm2 and the coil length is l s 22 mm. The core holder with excitation winding is mounted in the middle of the pick-up coil bobbin. The pick-up coil was connected to the current-to-voltage converter with LT 1028 op amp, the feedback resistor value was R f s 5.1 k V. Our measurements were made using 1:10 voltage probe in order to fit the input range of the SR 830 lock-in amplifier. All the measurements were performed without feedback. The field sensitivity was measured at 1 and 5 mT for both untuned sensor and sensor tuned to 2nd harmonic by serial capacitor of C2 s 1.7 mF. The measured sensitivities for the first six even harmonics are shown in Table 1. Serial tuning increased the sensitivity at 2nd harmonic by the factor of 5, which is lower than the amplification factor achievable at voltage-output fluxgates. The signal at higher even harmonics is amplified approximately by the factor of 2, while in properly tuned voltage-mode sensors, the sensitivity at higher harmonics is decreased. Fig. 2 shows both the tuned and untuned current-output sensor for the measured field of 2 mT. Top trace is the Table 1 Field sensitivity of untuned and tuned sensor at first six even harmonics Žin mV, 1 mV corresponds to 2 mA. Harmonics
Untuned Tuned
1 mT 5 mT 1 mT 5 mT
2
4
6
8
10
12
9.1 43.4 47 219
6.7 38 11 69
2.6 30.5 7.3 50
4.2 22 7 36
0.6 13.5 3.5 25
1.1 6 2.3 15
Fig. 2. Ža,b. Excitation current Župper trace. and output current Žlower trace, 200 mArdiv.. of untuned Ža. and tuned Žb. fluxgate sensor. The measured field was 2 mT.
excitation current and bottom trace is the output current Žmeasured at the output of the IrU converter..
4. Calculated values With the ring-core completely magnetically saturated, the induction of the secondary pick-up coil is measured to be Lair ( Lsat s 0.1085 mH Ž0.1080 mH. at 1 kHz Ž10 kHz., and the maximum inductance for unsaturated core is L max s 0.2465 mH Ž0.2461 mH. at 1 kHz Ž10 kHz.. Let us approximate the LŽ t . waveform by a step function changing from Lmax to Lair w9x as shown in Fig. 3. The fraction of time spent in saturation by the core is estimated from the excitation currentroutput current traces shown in Fig. 1a,b to be d s 0.0686. The average induction L o is calculated from Eq. Ž8. as Lo s 0.22678 mH. 4.1. Broadband output current The current equivalent to the external field is i ex s Bex lr Ž mo N . s 9.151 mA for 50,000 nT external field along the sensor axis ŽFig. 4..
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
Fig. 3. The time variation of the fluxgate secondary pick-up coil inductance for two periods of the excitation frequency. During the times when the core is in magnetic saturation then the secondary inductance decreases to approximately the air-cored coil value. The average induction determining the resonance is L o .
An effective coil winding length l s 23 mm is adopted in order to match the calculated value to the measured secondary coil induction w3x. The waveform of the theoretical output current is also squarewave with a duty factor of d w9x. The Fourier’s analysis of a periodically repeated narrow square impulse of width d P T, period T and amplitude I yields the coefficients: i a Ž n . s 2 Ir Ž n P p . sin Ž n P p P d . i b Ž n. s 0
cosine terms
sine terms.
The six first even harmonic Fourier’s coefficients ŽFig. 5. of the computed broadband short-circuited output for 5000 nT are: 145.7, 142.3, 136.8, 129.3, 120.0 and 109.2 mA.For comparison with the measured numbers Žin mV. in Table 1, divide by 2.
165
Fig. 5. Broad-band even harmonics Fourier’s coefficients of the output currents. The external magnetic field is 50,000 nT.
4.2. The second harmonic tuned case This requires knowledge of the first few Fourier’s coefficients of the time-modulated secondary induction: L2 s 18.7874; L4 s 18.3528 mH. Lo s 0.22678 mH as used above. Using rCu s 1.8 V and 8 kHz as the second harmonic of the excitation frequency, we get: rCu s 0.15795 2 v Lo L4 2 L2
s 0.04048.
The circuit is stable because rCu ) v L4 . Solving the matrix Eq. Ž21. for i a,b gives: i a s 1.317 mA, i b s 5.139 mA and a total second harmonic current amplitude i 2 s 5.305 mA for Bex s 50,000 nT. Table 1 gives the measured second harmonic current in mA as twice the tabulated number in mV and for 5000 nT. The measured current i 2 s 0.438 mA and the calculated current i 2 s 0.531 mA are in a fair agreement. The tuned output current depends on the series resistance rCu . Any input impedance of the operational amplifier adds to rCu , and this may account for the slight difference between the measured and the calculated results.
5. Conclusion
Fig. 4. Calculated broad-band short-circuit output current from the sensor secondary pick-up coil. One excitation period is shown and the external magnetic field is 50,000 nT.
We have shown that the current-output fluxgate may be tuned by using a serial capacitor. Such tuning increases the sensor sensitivity in the situation when the pick-up coil has a low number of turns, and this may be advantageous in the process of the miniaturization of the fluxgate sensors.
166
P. Ripka, F. Primdahlr Sensors and Actuators 82 (2000) 161–166
We have increased the sensitivity of the ring-core shortcircuited fluxgate sensor five times while the level of the spurious feedthrough remained the same. Serial tuning is easily made only by decreasing the value of the input capacitor. The change of signal-to-noise ratio Žor effective noise in units of the measured field. with tuning is under investigation. The preliminary results are ambiguous: noise improvement with tuning was reported in Ref. w6x, while noise degradation by tuning was observed in Ref. w3x and recently in Ref. w13x.
The ring core fluxgate sensor null feed-through signal, Meas. Sci. Technol. 3 Ž1992. 1149–1154. w11x F. Primdahl, B. Hernando, O.V. Nielsen, J.R. Petersen, Demagnetising factor and noise in the fluxgate ring-core sensor, J. Phys. E: Sci. Instrum. 22 Ž1989. 1004–1008. w12x P. Ripka, P. Kaspar, Portable fluxgate magnetometer, Sensors and Actuators A 68 Ž1998. 286–289. w13x E.B. Pedersen, Digitalisation of Fluxgate Magnetometer, PhD Thesis, Academy of Technical Sciences, TERMA Elektronik and Department of Automation, Technical University of Denmark, DK-2800 Lyngby, Denmark, June 1999.
Biographies References w1x P. Ripka, Review of fluxgate sensors, Sensors and Actuators A 33 Ž1992. 129–141. w2x P.H. Serson, W.L.W. Hannaford, A portable electrical magnetometer, Can. J. Technol. 34 Ž1956. 232–243. w3x F. Primdahl, P. Anker Jensen, Noise in the tuned fluxgate, J. Phys. E: Sci. Instrum. 20 Ž1987. 637–642. w4x Z. Gao, R.D. Russell, Fluxgate sensor theory: sensitivity and phase plane analysis, IEEE Trans. Geosci. 25 Ž1987. 862–870. w5x M.A. Player, Parametric amplification in fluxgate sensors, J.Phys. D 21 Ž1988. 1473–1480. w6x P. Ripka, W. Billingsley, Fluxgate: tuned vs. untuned output, IEEE Trans. Magn. 34 Ž1998. 1303–1305. w7x A. Tipek, P. Ripka, P. Kaspar, Gradiometric sensor Žin Czech., Proc. ˇ of the Sensory-snimace-aplikace conf., Ostrava, 1998. w8x F. Primdahl, J.R. Petersen, C. Olin, K. Harbo Andersen, The shortcircuited fluxgate output current, J. Phys. E: Sci. Instrum. 22 Ž1989. 349–353. w9x F. Primdahl, P. Ripka, J.R. Petersen, O.V. Nielsen, The short-circuited fluxgate sensitivity parameters, Meas. Sci. Technol. 2 Ž1991. 1039–1045. w10x J.R. Petersen, F. Primdahl, B. Hernando, A. Fernandez, O.V. Nielsen,
PaÕel Ripka received an Ing. degree in 1984, a CSc Žequivalent to PhD. in 1989 and a doctoral degree in 1996 at the Czech Technical University, Prague, Czech Republic. He works at the Department of Measurement, Faculty of Electrical Engineering, Czech Technical University as a lecturer, teaching courses in Electrical Measurements and Instrumentation, Engineering Magnetism and Sensors. His main research interests are magnetic measurements and magnetic sensors, especially, fluxgate. He is a member of Elektra Society, Czech Metrological Society, Czech National IMEKO Committee and Eurosensors Steering Committee. Fritz Primdahl received an MSc in Electrical Engineering and Physics at the Technical University of Denmark in 1964. He worked as Research Scientist at the Department of Geophysics, Danish Meteorological Institute Ž1966–1968 and 1970–1980.. In 1968–1970, he was awarded a fellowship by the National Research Council of Canada. Since 1980, he has been with the Danish Space Research Institute, now on leave to lead the Ørsted satellite science instrumentation team at the Technical University of Denmark. He mainly works in space magnetometry and space plasma physics. He was a member or head of teams developing instrumentation for sounding rockets and satellites ŽCLUSTER, Oersted, ASTRID-2, CHAMP and SAC-C. and scientific teams planning the space experiments and analyzing the data. He is a member of Danish Geophysical Society.