+Model ARTICLE IN PRESS SNA 6110 1–8 Available online at www.sciencedirect.com Sensors and Actuators A xxx (2007) xxx–xxx Racetrack fluxgate sensor core demagnetization factor 3 J. Kubı́k a,∗ , P. Ripka b 4 a 6 b Q1 Tyndall National Institute, Lee Maltings, Prospect Row, Cork, Ireland Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 16627 Praha 6, Czech Republic OF 5 Received 23 August 2007; received in revised form 19 October 2007; accepted 19 October 2007 7 RO Abstract 8 17 Keywords: Fluxgate; Racetrack; Demagnetization factor 11 12 13 14 15 DP 16 The demagnetization factor of a fluxgate core plays an important role in the resulting sensor sensitivity and noise. The global (magnetometric) demagnetization factor of fluxgate sensor ring cores was evaluated using finite element modelling. This new method was verified using measured effective demagnetization factors for ring cores and subsequently used in the modelling of the global demagnetization factor of racetrack fluxgate cores. The results of modelling and measurements were compared and show a very close match. Based on modelling, an empirical formula is presented to quickly evaluate the fluxgate racetrack core global demagnetization factor based on core geometry and material permeability. Finally, the modelling, measurement and empirical estimation were compared showing a good agreement on five measured cores. The derived empirical formula can be easily utilized in designing the sensor racetrack core shape. © 2007 Elsevier B.V. All rights reserved. 9 10 TE 18 1. Introduction 1 10 The demagnetization factors of ferromagnetic objects have been studied both analytically and experimentally in many publications. When the ferromagnetic object is exposed to a homogeneous magnetic field Hext , the magnetic field in the ferromagnetic object HC is smaller than the external field Hext . (In general, also the direction of the HC vector may be different from Hext , but we will not consider this case here.) The difference is characterized by the demagnetization field DMC , acting against the external field: 11 HC = Hext − DMC . 2 4 5 6 7 8 9 RR EC 3 (1) 16 HC = 14 ∗ Hext Hext = 1 + Dκ 1 + D(μ − 1) UN 13 CO 15 where D is a dimensionless demagnetization factor (in the general case it is a tensor and in our simplified case it is a number between 0 and 1), and MC = κHC is the magnetization. From that we can write: 12 Corresponding author. E-mail addresses: [email protected] (J. Kubı́k), [email protected] (P. Ripka). 1 2 (2) An important fact is that the demagnetizing field is homogeneous only in the case of ellipsoidal objects. In that case, D is a constant for the whole object and it is independent of the permeability and depends only on the object geometry (supposing that the object is made of isotropic and linear magnetically soft material. The tables and formulas for the calculation of demagnetization factors for ellipsoidal shapes were proposed by Osborn [1]. In all other cases than the ellipsoid, the local demagnetization field is a function of position in the volume of the object, but the local demagnetization factor is still independent of μr by definition. Therefore, it is impossible to generalize the demagnetization factor to such objects without some sort of averaging. The literature distinguishes between two types of averaging [2]: ballistic (or local, fluxmetric) and magnetometric (or global) demagnetization factors. The magnetometric (global) demagnetization factor concerns the whole average magnetization of the object (corresponding to a “magnetometric” measurement with a large search coil compared to the object’s dimensions) and the ballistic demagnetization factor corresponds to the average magnetization in the object’s midplane (corresponding to measurement with a very short search coil at the object’s midplane). Notice the meaning of the “large search coil” for the traditional magnetometric insertion measurement: the magnetization of the object is measured by inserting the object inside such a coil or turning it by 180◦ . The voltage induced in the coil is then integrated. 0924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2007.10.066 Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007), doi:10.1016/j.sna.2007.10.066 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 +Model ARTICLE IN PRESS SNA 6110 1–8 J. Kubı́k, P. Ripka / Sensors and Actuators A xxx (2007) xxx–xxx 59 Vi = −N2 μ0 HC S 45 46 47 48 49 50 51 52 53 54 55 56 57 60 61 62 63 64 65 66 dμr (t) dt (3) The longitudinal in-axis magnetic field HC in such a core is equal to the longitudinal external magnetic field Hext . In case of a different core shape, the demagnetization factor has to be taken into account. This can be rewritten as the decrease of the material relative permeability μr to the apparent permeability μA [5]: μr μA = 1 + D(μr − 1) (4) 69 Vi = −N2 μ0 HS 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 RR 72 Eq. (5) clearly indicates that the demagnetization factor D of the sensor core is one of the key factors in the fluxgate sensor sensitivity. In general, the highest sensor output would be obtained for a demagnetization factor D equal to zero. Demagnetization factor of 1 would yield a zero output. Furthermore, the demagnetization factor plays an important role in the sensor noise performance. The proportionality of the external noise field to the demagnetization factor was shown in Ref. [6]. One of the first attempts to estimate the demagnetization factor of the common fluxgate sensor core shape – the ring core – was made in 1972 by Burger [10]. The effective demagnetization factor (the demagnetization factor with respect to a given core and measurement coil geometry) was calculated from the measured pick-up coil inductance with (Lcore ) and without (Lair ) inserted core [6]. The apparent core permeability μA and the effective demagnetization factor D for a given sensor core with cross-section area Acore and given coil with cross-section area Acoil can be calculated using equations (from Ref. [6]): CO 71 (5) UN 70 dμr (t) 1−D [1 + D(μr (t) − 1)]2 dt EC 68 Substituting the apparent permeability for the relative permeability, and differentiating, the fluxgate output voltage becomes: 67 Notice that both μA and D are in this case not exclusive properties of the core, but slightly depend also on the coil geometry. For coils wound around the core and having a similar length as the core, the effective demagnetization factor is not very different from the global demagnetization factor. This may not be true for coils much shorter or longer. For completeness, notice that the pick-up coil can also be located inside the core [8]. Experimentally measured data from other articles were collected and an empirical equation for the local demagnetization factor Dlocal of ring core with cross-sectional area A and diameter d was derived (7) based on the experimental data covering a large range of fluxgate ring-core dimensions [5]. Dlocal = 1.83 Lcore −Lair Acoil × +1 μA = Lair Acore (μr /μA )−1 and D= μr −1 (6) A d2 (7) The relationship between local and global demagnetization factors Dlocal and Dglobal was determined for a ring core in Ref. [5]: Dglobal = 2Dlocal + DP 44 TE 43 OF 58 Analytical solutions of demagnetization factors were presented for several special cases of objects such as cylinders [2] or rectangular prisms [3,4]. When a square cross-section prism with the same length to width ratio is compared to an elongated spheroid of the same length/width ratio, the demagnetization factor of a rectangular prism is much higher. Demagnetization factor plays an important role in fluxgate sensor core design. The most interesting property for a fluxgate sensor designer is the demagnetization factor in the direction of the sensor sensitivity axis. Thus, from now on, the demagnetization factor in the direction of measured field will be discussed (unless otherwise noted). The fluxgate sensor output formula (3) derived from Faraday’s law of electric induction applies only to sensors with infinitely long cores (with zero demagnetization factors). Vi denotes voltage induced to pick-up coil with N2 turns, μr is the relative permeability of the sensor core with cross-section area S. 42 RO 2 1 1 = 2Dlocal + κ μr − 1 (8) Finally, the analytical approach to ring core geometry demagnetization has been recently published [11] showing a good agreement with measured values. The racetrack geometry is another common fluxgate sensor core shape, but to the best of authors’ knowledge neither analytical calculations nor empirical estimations of demagnetization factor have been published so far. 2. Modelling and veriﬁcation on ring cores 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 FEM (finite element method) models of the demagnetization field can be used for the calculation of global demagnetization factors. We will first verify this method on a ring-core geometry and later use it for racetrack sensor cores. The ring cores data collected in Ref. [5] will be used for verification of the models. FEM modelling can provide data on magnetic field vector HC and magnetic flux density vector BC at each point inside the ferromagnetic core. This data can be used to determine the magnetometric demagnetization factor of any core shape. The FEM model consists of a core with defined permeability exposed to an external homogeneous magnetic field Hext in the direction of intended demagnetization factor determination. When considering the demagnetization factor in one direction (z-axis in our case) then only collinear components of HC (x,y,z) and BC (x,y,z) vectors are used for the calculation of the magnetometric demagnetization factor Dm . The relation (9) between external field Hext , the average of core internal field collinear component HC (z) and core average magnetization collinear component MC (z) over the whole core volume is the basis of Dm calculation. HC (z) = Hext − Dm MC (z) 90 (9) Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007), doi:10.1016/j.sna.2007.10.066 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 +Model ARTICLE IN PRESS SNA 6110 1–8 J. Kubı́k, P. Ripka / Sensors and Actuators A xxx (2007) xxx–xxx 137 The substitution of the average magnetization (10) into (9) yields (11). 138 MC (z) = BC (z) − HC (z) μ0 (10) 141 This approach is not limited to linear materials and thus the modelled material may have a permeability defined by a BH curve. 142 Dm = 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 OF 147 Fig. 2. Global demagnetization factors of selected ring cores with cross-section area A and diameter d: comparison of measured values, empirical equation and FEM model. core used by Clarke [9]. The RMS errors of FEM modelled and empirically estimated ring core magnetometric demagnetization factors compared in Table 1 are 0.53 × 10−3 and 0.67 × 10−3 , respectively. The FEM models provide magnetometric demagnetization factor values for ring cores which show a strong correlation with the measured values and also with the values obtained from empirical formulas (7) and (8) by Primdahl et al. [5]. The FEM models will be used when modelling the global demagnetization factor of racetrack cores to provide data on a wide range of cores to create an empirical estimation formula for this type of core without actually physically creating cores. Real cores then will be used for verification of this formula. DP 146 TE 145 The principle of the calculation of the global demagnetization factor from FEM models was verified using alreadyTable 1 measured fluxgate sensor cores demagnetization factors. Seven of twenty-six ring cores presented in Ref. [5] with known permeability values covering a wide spectrum of dimensions were chosen and FEM modelled. A linear ferromagnetic material was used for modelling (constant relative permeability). The cores were exposed to an external homogeneous field Bext = 50 T (corresponding to Hext ≈ 40 A/m). An example of the magnetic field flux density (collinear to the applied field in z-axis direction) distribution for the thickest ring core measured by Primdahl et al. [5] is shown in Fig. 1. An external magnetic flux density Bext = 50 T applied in the z-axis direction creates the maximum core flux density Bmax = 15 mT. The magnetometric demagnetization factor was calculated from the results of the FEM analysis using (11). The results of the measurements (from Ref. [5] and references therein) recalculated to the global demagnetization factors, FEM models and an empirical estimation using (7) and (8) are compared in and Fig. 2 with the dimension notation of Fig. 3. The maximum deviation of the FEM model from the measured value was +18% for a ring core used by Ripka and Primdahl [7] formed out of a stack of 8 rings etched from an amorphous magnetic material which was a different construction than other thin-tape wound cores. The deviations from other measured values are below 10%. Interestingly, a large deviation of the empirical estimation from the measured demagnetization factor (+16%) also occurs for the ring core used by Ripka and Primdahl [7] and a deviation of +18% occurs for a ring EC 144 (11) RR 143 HC (z) − Hext HC (z) − (BC (z)/μ0 ) 3. Racetrack core models The modelled racetrack cores will be described using the dimensions notation shown in Fig. 4. Two basic geometries are modelled: the “regular” racetrack core with constant track width and the asymmetrical core with a slight asymmetry in track width (originally used for sensor feedthrough adjusting [12]). The modelled geometries cover a wide spectrum of racetrack core shapes with lengths l from 18 to 140 mm, core widths d ranging from 6.2 to 12 mm, track widths T ranging from 1.2 to 2.4 mm CO 140 UN 139 RO 136 3 Fig. 1. 17 mm diameter, 0.5 mm thick ring core surface magnetic flux density in z-direction in [T] (Hext = 50 T applied in the z-axis). Fig. 3. Ring-core dimensions. Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007), doi:10.1016/j.sna.2007.10.066 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 +Model ARTICLE IN PRESS SNA 6110 1–8 4 J. Kubı́k, P. Ripka / Sensors and Actuators A xxx (2007) xxx–xxx Table 1 Global demagnetization factor of ring cores selected from Ref. [5] 35,800 33,000 31,000 28,700 40,000 1,871 2,500 d [mm] T [mm] h [mm] 17.0 17.0 17.0 17.0 20.0 16.4 12.7 0.125 0.250 0.375 0.500 2.000 0.264 0.254 1 1 1 1 0.4 1 1.6 Q4 d, diameter; T, thickness, h, height. 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 1.87 3.56 5.16 6.92 7.47 4.29 8.58 1.61 3.20 4.78 6.37 7.35 4.14 9.62 track core with a track width, T, equal to the asymmetrical core average track width. 4. Racetrack cores measurement A method for the measurement of an effective demagnetization factor of fluxgate sensor cores was proposed in Ref. [6] and was used for this measurement. The plastic coil bobbin was made with an inner hollow aperture slightly wider than the tested racetrack cores (Fig. 6). Using 0.2 mm diameter copper wire, 131 turns were wound in a single layer around the bobbin to form a 30 mm long coil equal to the length of the tested racetrack cores. Thus it can be considered as “a large pick-up coil” and the measured demagnetization factor as the global demagnetization factor of the sensor core. The core longitudinal axis (or the sensitivity axes of the whole sensor) was placed horizontally and oriented in the East-West direction during all demagnetization factor measurements to avoid longitudinal magnetization of the core by the horizontal component of the geomagnetic field. A Hewlett-Packard 4284A precision LCR meter with Kelvin clips was used to measure the air coil inductance at frequencies ranging from 90 to 100 kHz. The self-inductance of the air coil was (within 5%) constant in the whole frequency range indicating only a minor influence of the parasitic capacitance. Fig. 4. Racetrack dimensions: regular (left) and asymmetrical core (right). Fig. 5. 30 mm long, 8 mm wide, 1.55 mm track wide, 25 m thick racetrack core surface magnetic flux density in z-direction in [T] (Hext = 50 T applied in the z-axis direction). DP 199 TE 198 1.70 3.28 4.88 6.38 6.34 4.01 8.18 Empirical EC 197 RR 196 0.0004 0.0009 0.0013 0.0017 0.0020 0.0010 0.0025 FEM CO 195 and core thicknesses t ranging from 25 to 125 m. 23 geometries were modelled, in total. The 30 mm long asymmetrical cores with Tmax = 1.7 mm and Tmin = 1.4 mm were also modelled and compared to a core with a track thickness T = 1.55 mm equal to the average track thickness of the asymmetrical core. At first, the linear magnetic material with relative permeability μr = 25,000 was used, corresponding to a measured permeability μ4 (measured at Hmax = 0.4 A/m) of the Vitrovac 6025X amorphous alloy [13] (the maximum measured amplitude permeability was μr = 63,000). An example of the distribution of magnetic flux density in the racetrack core is shown in Fig. 5 for 30 mm long racetrack core. With a relative permeability of 25,000, a maximum flux density of BC = 0.24 T occurs in the core when an external magnetic flux density Bext = 50 T is applied in the longitudinal direction (the model’s z-axis). Table 2 shows the modelled global demagnetization factors of fluxgate racetrack cores either for linear magnetic material with μr = 25,000 and for nonlinear magnetic material characterized by a BH curve based on amplitude permeability measurement of Vitrovac 6025X [13]. It may be seen that the differences are negligible and thus the modelling with nonlinear material, which takes much more computer time, is not necessary in this case. The largest difference in the modelled magnetometric demagnetization factor occurs for very long and thin cores (such as 140 mm long, 25 m thick: 0.03 × 10−3 vs. 0.05 × 10−3 ) which could saturate even for applied field of Bext = 50 T when a BH curve is used instead of a constant permeability. When comparing a core with a thickness t = 25 m to the thicker core with a thickness t = 50 m, the global demagnetization factor is approximately two times higher for the thicker core, which corresponds to the behaviour of global demagnetization factors in ring cores. Note that the magnetometric demagnetization factor Dglobal = 0.00027 for an asymmetrical core is equal to the magnetometric demagnetization factor for a regular race- Measured UN 193 194 Dglobal × 103 OF Primdahl et al. [6] Primdahl et al. [6] Primdahl et al. [6] Primdahl et al. [6] Ripka and Primdahl [7] Nielsen et al. [8] Clarke [9] A/d2 Dimensions μr RO Sample Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007), doi:10.1016/j.sna.2007.10.066 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 +Model ARTICLE IN PRESS SNA 6110 1–8 J. Kubı́k, P. Ripka / Sensors and Actuators A xxx (2007) xxx–xxx 5 Table 2 Magnetometric demagnetization factor of racetrack cores modelled with constant relative permeability and nonlinear BH curve Dimensions 20 mm 20 18.2 6.4 6.2 1.2 1.8 25 25 75 136 0.45 0.70 0.47 0.69 30 mm 30 29.8 30 7.8 7.8 9 1.2 1.8 2.4 25 25 25 33 51 67 0.23 0.33 0.38 0.22 0.32 0.37 2 2 25 25 10 3 0.09 0.03 0.10 0.05 70 140 12 12 t [m] 20 18.2 6.4 6.2 1.2 1.8 50 50 150 272 30 mm 30 30 29.8 30 7.8 8 7.8 9 1.2 1.55 1.8 2.4 50 25 50 50 67 43 101 133 2 2 50 50 70 140 12 12 29.8 30 30 7.8 9 9 1.8 2.4 2.4 75 75 100 As. 30 mm As. 30 mm As. 30 mm As. 30 mm As. 30 mm 30 30 30 30 30 8 8 8 8 8 1.55 1.55 1.55 1.55 1.55 25 50 75 100 125 TE 254 255 256 257 260 261 262 263 264 Q2 CO 259 266 0.45 0.28 0.64 0.75 0.46 0.27 0.63 0.74 20 5 0.17 0.06 0.16 0.07 152 200 267 0.94 1.11 1.48 – – – 43 86 129 172 215 0.28 0.55 0.81 1.07 1.33 0.27 0.55 0.80 1.06 1.32 5. Empirical formula for racetrack core demagnetization factor 267 268 5.1. Influence of shape 269 The empirical estimation of a ring core local demagnetization factor and the relationship between the local and the global demagnetization factor including the influence of relative permeability has been published in Ref. [5]. A racetrack core is a more complicated shape and thus estimating the global demagnetization factor will require more input parameters representing the geometrical dimensions. The FEM racetrack models with a relative permeability μr = 25,000 will be used for the estimation. Some similarity of racetrack core physical dimensions influence on the core global demagnetization factor will probably be similar to a ring core (thickness, core width), others will be new. In RR 258 265 The measuring current in a constant current mode was 5 mA corresponding to a maximum axial field of approximately 20.6 A/m in the centre and 10.4 A/m at the end of the coil. The field values ratio corresponds to a long thin solenoid having at the end 50% of the field in the centre [14]. The coil selfinductance at 10 kHz was then measured for five thicknesses of the as-cast Vitrovac 6025X racetrack core (for dimensions see the last five rows of Table 2), i.e. one to five etched cores placed on each other yielding 25–125 m thickness range. The core permeability used for the global demagnetization factor calculation (6) was μr = 63,000. This value was measured as the maximum amplitude permeability at 10 kHz. Measurements with the external wire-wound coil produced repeatable and consistent results on the five cores. The global demagnetization factor of a 30 mm long, 8 mm wide and 25 m thick racetrack core is 0.00030—see Table 3, Col. 5 for the complete results. EC 253 0.86 1.23 DP 30 mm 0.88 1.38 RO 20 mm OF T [mm] Q5 “As.” denotes the asymmetrical core. 252 Dglobal × 103 BH curve d [mm] 70 mm 140 mm 250 Dglobal × 103 μr = 25,000 l [mm] 70 mm 140 mm 251 A/l2 × 106 UN Table 3 The global demagnetization factor of five racetrack cores, l = 30 mm, d = 8 mm, T = 1.55 mm Thickness Dglobal × 103 t [m] FEM μr = 25,000 FEM BH curve (nonlinear) Empirically estimated μr = 25,000 Empirically estimated μr = 63,000 Measured Vitrovac 6025X 25 50 75 100 125 0.28 0.55 0.81 1.07 1.33 0.27 0.55 0.80 1.06 1.32 0.28 0.54 0.80 1.06 1.32 0.28 0.54 0.79 1.05 1.31 0.30 0.55 0.82 1.03 1.30 Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007), doi:10.1016/j.sna.2007.10.066 270 271 272 273 274 275 276 277 278 279 280 +Model ARTICLE IN PRESS SNA 6110 1–8 J. Kubı́k, P. Ripka / Sensors and Actuators A xxx (2007) xxx–xxx OF 6 Fig. 7. Global demagnetization factor as a function of physical dimensions of racetrack core. Comparison of FEM models and estimation. 283 284 285 286 287 • thicker cores mean higher demagnetization factor (t parameter), • wider tracks mean higher demagnetization factor (T parameter) and • longer cores mean lower demagnetization factor (l parameter). 291 Dglobal (l, T, t) ≈ k1G1 289 EC 290 Starting with these assumptions, an analogy to ring-core formulas (7) and (8) can be utilized with arbitrary parameters k1G and k2G : 288 2Tt + k2G1 l2 (12) 301 Dglobal (l, d, T, t) ≈ k1G2 295 296 297 298 299 302 303 304 305 306 307 308 CO 294 2Tt + k2G2 (l + k3G2 d)2 Dglobal (μr ) ≈ 0.2 + Dmin μr (14) The RMS errors are 2.1 × 10−6 for 30 mm core and 6.7 × 10−6 for 70 mm core. 5.3. Combining geometry and permeability influence Comparing the empirical Eqs. (13) and (14) we get the empirical estimation of the global demagnetization factor for fluxgate racetrack cores as a function of core geometry and permeability: Dglobal (l, d, T, t, μr ) ≈ k1 2Tt 0.2 + + k2 2 μr (l + k3 d) (15) (13) Minimizing the RMS error yielded the k1G , k2G and k3G constants k1G2 = 6.58, k2G2 = 23 × 10−6 and k3G2 = 1.8 with the RMS error of eG2 = 1.82 × 10−5 . Note that the RMS error here is almost three times lower than for the previous estimation. Fig. 7 shows the comparison of the global demagnetization factor calculated by FEM to the global demagnetization factor estimated using (13). 310 311 312 313 314 315 316 317 318 319 320 UN 293 RR 300 Based on the global demagnetization factors modelled on 23 various racetracks, k1G1 and k2G1 were acquired by the least squares method: k1G1 = 2.70 and k2G1 = 55 × 10−6 . The RMS error of this empirical formula is eG1 = 5.2 × 10−5 . The increasing racetrack width d for the racetrack constant length l increases the distance between the two tracks thus decreasing their interaction. Therefore, we can expect that d decreases the global demagnetization factor in a similar fashion to l. This is reflected by an arbitrary constant k3G2 in the proposed formula (13). 292 309 The dependence of the global demagnetization factor on the relative permeability of two selected cores’ FEM models (30 mm long, track 1.55 mm wide and 25 m thick and 70 mm long, track 2 mm wide and 50 m thick) is displayed in Fig. 8 for a given core. This relation can be expressed as (14), where Dmin is the global demagnetization factor for a given core shape with very high relative permeability. DP 282 general, these rules will apply: TE 281 5.2. Influence of permeability RO Fig. 6. Coil dimensions (in mm) of wire-wound coil for global demagnetization factor measurement of 30 mm long racetrack fluxgate sensor cores. Fig. 8. Global demagnetization factor as a function of relative permeability for 30 mm (Dmin = 0.271 × 10−3 ) and 70 mm (Dmin = 0.161 × 10−3 ) long core. Comparison of discrete points of FEM models and solid lines of estimation (14). Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007), doi:10.1016/j.sna.2007.10.066 321 322 323 324 +Model ARTICLE IN PRESS SNA 6110 1–8 J. Kubı́k, P. Ripka / Sensors and Actuators A xxx (2007) xxx–xxx 7 lower relative permeability as indicated by the models (Fig. 8). Fig. 9 further illustrates the close match of FEM models, empirical estimations and measurement of the global demagnetization factors of racetrack cores. The comparison with ring core fluxgate cores is very interesting: the demagnetizing factor of racetrack fluxgate cores is lower compared to all the ring cores in Table 1. This fact indicates that long, thin racetrack fluxgate cores may be an advisable option for highly sensitive sensors and magnetometers. where 325 k1 = 6.58, 326 k2 = 23 × 10 −6 0.2 − = 15 × 10−6 , 25, 000 k3 = 1.8. 327 (16) 332 Eq. (15) with corresponding parameters (16) provides a quick estimation of the fluxgate racetrack core global demagnetization factor without the need for FEM modelling. The input parameters are only the basic geometrical dimensions defined in Fig. 4 and the relative permeability of the core material. 333 6. Comparison of models, estimation and measurement 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 Q3 The comparison of the global demagnetization factors of five racetrack cores is shown in Table 3. The table contains the global demagnetization factors acquired by linear and nonlinear FEM models (Cols. 2 and 3), the empirical estimation using Eq. (15) with two relative permeability values corresponding to μ4 and μmax (Cols. 4 and 5) and the global demagnetization factors measured on real sensor cores using the method with external pick-up coil described above (Col. 6). The Col. 2 global demagnetization factors were modelled with μr = 25,000 corresponding to the measured μ4 of the used material. The Col. 3 global demagnetization factors were modelled with nonlinear hysteresis-free BH curve (hysteresis-free BH curve was required by the FEM software). The BH curve was constructed by connecting the endpoints of measured BH loops with gradually increasing H field. The RMS errors of the FEM models compared with measured values are 2.6 × 10−5 and 2.1 × 10−5 using linear and nonlinear magnetic material model, respectively. The RMS errors of empirically estimated values compared with measured values are 2.0 × 10−5 and 2.1 × 10−5 using relative permeability values of μr = 25,000 and μr = 63,000 for the estimation, respectively. The modelled and estimated values of the global demagnetization factor show a strong correlation with the measured values with a maximum error of 9%. The nonlinear FEM model shows only slightly better results than the linear model with μr = 25,000. The influence of the relative permeability on the empirically estimated global demagnetization factors is negligible due to the use of high-permeability material. Much more caution is necessary when using a core material with a EC 334 RR 331 CO 330 UN 329 TE DP 328 The fluxgate sensor racetrack core magnetometric (global) demagnetization factor was modelled by FEM and an empirical estimation relation has been fitted (Eq. (15) with constants (16)). This empirical estimation is based on the geometrical dimensions of the racetrack core and the core relative permeability. The results of the measurements on the racetrack core samples are in agreement with the models and the empirical estimations. The empirical formula provides a ready-to-use solution for a quick estimation of the global demagnetization factor of a given fluxgate sensor racetrack core prior to FEM modelling, manufacturing and measurement of the actual core. RO Fig. 9. The global demagnetization factor for 30 mm long racetrack fluxgate core. Comparison of FEM model, empirical estimation and measured values. OF 7. Conclusions References [1] J.A. Osborn, Demagnetizing factors of the general ellipsoid, Phys. Rev. 67 (11–12) (1945) 351–357. [2] D.-X. Chen, J.A. Brug, R.B. Goldfarb, Demagnetizing factors for cylinders, IEEE Trans. Magn. 27 (4) (1991) 3601–3619. [3] A. Aharoni, Demagnetizing factors for rectangular ferromagnetic prisms, J. Appl. Phys. 83 (6) (1998) 3432–3434. [4] A. Aharoni, “Local” demagnetization in a rectangular ferromagnetic prism, Phys. Status Solidi (b) 229 (3) (2002) 1413–1416. [5] F. Primdahl, P. Brauer, J.M.G. Merayo, O.V. Nielsen, The fluxgate ring-core internal field, Meas. Sci. Technol. (13) (2002) 1248–1258. [6] 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) (1989) 1004–1008. [7] P. Ripka, F. Primdahl, Tuned current-output fluxgate, Sens. Act A 82 (2000) 161–166. [8] O.V. Nielsen, J.R. Petersen, F. Primdahl, P. Brauer, B. Hernando, A. Fernandez, J.M.G. Merayo, P. Ripka, Development, construction and analysis of the ‘Ørsted’ fluxgate magnetometer, Meas. Sci. Technol. 6 (1995) 1099–1115. [9] D.B. Clarke, Demagnetization factors of ringcores, IEEE Trans. Magn. 35 (1999) 4440–4444. [10] J.R. Burger, The theoretical output of a ring core fluxgate sensor, IEEE Trans. Magn. MAG-8 (4) (1972) 791–796. [11] M. De Graef, M. Beleggia, The fluxgate ring-core demagnetization field, J. Magn. Magn. Mater. 305 (2006) 403–409. [12] P. Ripka, Race-track fluxgate with adjustable feedthrough, Sens. Act A 85 (2000) 227–231. [13] J. Kubı́k, PCB fluxgate sensors, Dissertation thesis, CTU in Prague, 2006, pp. 56–58. [14] D. Jiles, Introduction to magnetism and magnetic materials, Chapman & Hall, London, 1996, pp. 19, ISBN 0 412 38640 2. Biographies Jan Kubı́k received his Ing. degree (M. Eng. equivalent) in the field of computercontrolled measurement systems in March 2003 at the Faculty of Electrical Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007), doi:10.1016/j.sna.2007.10.066 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 +Model ARTICLE IN PRESS SNA 6110 1–8 8 427 428 Pavel Ripka received an Ing. degree in 1984, a CSc (equivalent to PhD) in 1989 and Prof. degree in 2001 at the Czech Technical University, Prague, Czech OF 426 RO 425 DP 424 TE 423 EC 422 RR 421 Republic. He works at the Department of Measurement, Faculty of Electrical Engineering, Czech Technical University as a full professor, teaching courses in electrical measurements and instrumentation, engineering magnetism and sensors. He also worked as visiting scientist at Danish Technical University (1990–1993), National University of Ireland (2001) and in the Institute for the Protection and the Security of the Citizen, European Commission Joint Research Centre in Italy (2005/6). His main research interests are magnetic measurements and magnetic sensors, especially fluxgate. He is an author of >50 SCI journal papers, 2 books and 5 patents. He is a member of IEEE, Elektra society, Czech Metrological Society, Czech National IMEKO Committee and Eurosensors Steering Committee. He served as an associate editor of the IEEE Sensors Journal. He was a General Chairman of Eurosensors 2002 conference. CO 420 Engineering of Czech Technical University in Prague. He successfully defended his PhD thesis “PCB Fluxgate Sensors” under supervision of Prof. Pavel Ripka in December 2006 at the same university. He also worked at the National University of Ireland in Galway (2001 and 2002) and at the Lappeenranta University of Technology in Lappeenranta, Finland (2003). He has joined the Tyndall National Institute in Cork in September 2006 as a researcher. His research interests range from magnetic field sensors design and characterization to computer-controlled measurement systems development, FEM modelling of magnetic systems and microcontroller applications. He is the main author or co-author of more than 35 journal and conference papers. UN 418 419 J. Kubı́k, P. Ripka / Sensors and Actuators A xxx (2007) xxx–xxx Please cite this article in press as: J. Kubı́k, P. Ripka, Racetrack fluxgate sensor core demagnetization factor, Sens. Actuators A: Phys. (2007), doi:10.1016/j.sna.2007.10.066 429 430 431 432 433 434 435 436 437 438 439 440

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