Accurate Estimation of a Coil Magnetic Dipole Moment

Accurate Estimation of a Coil Magnetic Dipole Moment Antonio Mosch itta, Alessio D e Angelis, Francesc o Santoni, Marco Dio nigi, Paolo Ca rbone Department of Engineerin g University of Perugia Perugia, Italy {antonio.moschitta,ales sio.deangelis,francesco.sa nto ni, marco.dionigi, paolo.ca rbone }@unipg.it Guido De Angel is Regione Umbria Perugia, Ital y ing.guidodeangelis.gmail.c om Abstract — In th is pap er, a technique for a ccurate estimation of the moment of magnetic dipole is proposed. The achievable accuracy is investigated, as a fu nction of measurement noise affecting estimation of magnetic field cartesian components. The proposed technique is validated both via si mulations and experimentally. Keywords — magnetic dipole moment ; estimation ; measurement; coils; AC magnetic field; positioning I. I NTRODU CTION The Internet- of - Things (Io T) paradigm is b ased o n the developm ent of intelligen t systems , capable of collecting , aggregatin g, an d processin g in formation originat ing in th e real world. As such, positioning systems (P Ss) are a strong enabler for I oT bas ed appli cations , th at inc lude L ocation Based Services , Domotics, Wireless Sensor Netw orks, and production line traceability . Solutions proposed in the lite rature are based on vari ous measurement principles and processing techn iqu es [1- 13 ] . Apart from solutions b ased o n image process ing, PSs are usually base d on the transm ission o f kn own signals between a m obile n ode and a set of beacons. Then, by measurin g a set o f physical quantities that d epend on th e transmitte d signal, ranging and positioning can be perform ed using fitti ng techni ques on a know n propagati on model . Among PSs , those bas ed on measurement of AC mag netic fields, generat ed by either mobile nodes or fixed b eacons are often mentioned in the recent litera ture, because this appr oach is b oth easily im plemented and robust to m ost envir onmen tal factors , such as the presence of obstacles and the static geomagn etic fiel d [7]- 12 ]. The accuracy of positioning systems is lim ited by the accurate know ledge of the references and of the mobile node character istics. For AC Magnetic P Ss (MPSs) the references are often a set of fixe d coils acting as beacons, described by their geometric al character istics, position, and bearing, while the mobile node is realized by an additional coil. Hence, accurate AC MPSs require accu rate knowledge of coils properties , that can be obtaine d either by careful manuf acturing, lea ding to stri ct toleranc e requir ements , or by accurate coils’ characte rizatio n. In short-range MPS systems applicati ons, estimatin g the position with sub-centimeter accuracy and the attitu de with 1° accuracy requir es at least an equally accurate know ledge of the bea cons ’ ones [14]. Moreover, active coils used in MPSs are often described by an approxim ated model, su mmarizing coils ’ knowledge by their magn etic dipole moments. I n fact , the magnetic field induced in a giv en positi on by an active coil can be expressed easily, as a functi on of the coil’s magnetic d ipole m oment and the vector that describes the distance between the coil and the position of interest [9][10][14 ] . The voltag e appearing at the output of a probe coil can be expressed in a sim ilar way. T hus, by assuming that voltage or magnetic field m easurements are collecte d between the mobile node and a set of b eacons, the position and the attitu de of a m obile coil can be estim ated using numerical fittin g. T his leads to computationa lly light positionin g algorithm s, suita ble for real time applicati ons , especially with respect to models that estim ate the magnetic field usin g finite element an aly sis [15]. Mo reover, th e appr oach based o n m agnetic dipole moment can be indifferently applied to MPSs featur ing active beacon and m obile nodes equi pped with a passive probe or to MPSs with a dual architectu re, featuring an activ e mo bile node an d a set of passive beacons. Consequently , this pa per is focuse d on a simple character ization technique , aim ed at accurate ly estimating the magn etic dip ole moment of a coil. While magnetic dipole measurem ent is mentione d in the literature, m ost works do not target accu rate charact erizati on of active coils, being mostly focused on approxim ate characterizati on of ele ctric appliances [ 16 - 17 ], on cha racteriz ation of perm anent m agnets [ 18 ], on magn etostatic characte rization of space equipment [ 19 ] , or are strongly focused o n modeling [ 20 ]. The proposed approach is simple and computationa lly light, and was validated both by simu lations and expe rimentall y , using small coils compatible with short range MPSs. It is show n that the magnitu de and the directi on of an active coil m agn etic dipole mom ent can be estimated w ith an accu racy of less th an 4 %. © 2017 IEEE. Pe rsonal use of th is material is per mitted. Permiss ion from I EEE must be o btained for all other uses, in a ny cur rent or future media, including reprinting/re publishing this ma terial for advertising or promotional pur poses, creating new col lective wor ks, for resale or redistribution to serv ers or l ists, or reuse of any copyrig hted component of this w ork in other w orks Preprint ve rsion. Presented at 2017 I EEE I nternational Wor kshop on Measureme nt and Networking ( M&N) , Napl es, 2017. II. M EASUREMENT M ODEL AND P ROCEDURE A. Mea surement model Let us a ssume to operate unde r sinusoidal steady state, at a frequency f 0 , and that the magnetic dipo le to be measured is described by its pha sor m   ( m x , m y , m z ) . T his magnetic dipole moment is generated by a p lanar and circular transmi tting co il, so its magnitude m is gi ven by m = N t S t I t , where N t is the number of coil windings, S t is the area of a coil winding, and I t is the phasor of the current stimulating the co il. Let us also assume that the tran smitting co il is placed i n the known position P m  ( x m , y m , z m ). The measurement proced ure can be developed by recalling that, in a given position P  ( x , y , z ), the phasor B  of the AC magnetic field i nduced by m  is given by     5 2 0 3 4 ) , , ( r r m r r m z y x B           , (1) where µ 0 is the vacuu m permeab ility, r  is the d istance vecto r between the magnetic dip ole applicatio n point P m and the position P, given b y ) , , ( ) , , ( m m m z y x z z y y x x r r r r       , (2) and r= || r  || is the Euclidean nor m of r  , that is the Euclidean distance between P and P m . Let us also a ssume t hat a planar coil with radius R p , N p windings, and attitude de scribed by i.e. unit vect or p n  =( n px , n py , n pz ), is placed in P, acting as probe . By assumin g that the magnetic field is co nstant across the coil sectio n, t he phasor V descr ibing the probe coil output voltage is given by           0 0 2 0 0 5 2 5 2 0 2 , , 4 , ) ( 3 ) ( 3 , , , , 2 f R S N S K n r r m r r m r m r m K n r r m r r m K n z y x B K n z y x B N S f V p p p p p p z z y y x x p p p p p p p p                                      (3) where S p is the probe co il area . B. Mea surement proced ure Provided that the current feeding t he transmittin g coil a nd the voltage at t he output of the coil ca n be simultaneously measured, (3) can be used to develop a simple measurement procedure. In particular, let us assume t hat, without loss of generality, the transmittin g c oil is placed in the ori gin of a Cartesian coord inate system, i.e. P m  (0,0,0). For instance, if the prob e coil is placed on the x axis, then r  =( r x ,0,0)= r x x n  , where x n  is the unit vector desc ribing t he direction of t he x axis, and (3) r educes to       . ) ( 3 3 ) ( 3 3 2 2 5 2 5 pz z py y px x px x x p p x x x x x p p x x x x x x p n m n m n m n m r K n r m n r m r K n r m n r r m r K V                 (4) Moreover, if the probe coil is alig ned to th e x axis, that is x p n n    , (4) further reduces to   3 3 2 3 x x p x x x p r m K m m r K V    . (5) Note that the probe place ment and orientation leading to (5) deco uples the measureme nt of m x from the measurement of the components m y and m z . Using (5), m x is obtained as x p x x V K r m 2 3  , (6) where V x indicates the voltage phasor when t he probe co il is placed o n the x axis and aligned with it. No te that the si gn of m x is given by t he sig n of V x , that in turn d epends o n th e relationship bet ween the pha se lag bet ween the c urrent feeding the trans mitting coil and the pr obe coil output voltage. In particular, since the sys tem op erates at very low frequencies, by assuming that the I t phasor is positive and real, V x can be either real p ositive or real nega tive. B y placing the prob e coil T ABLE I – P ARAMETERS USED IN THE NUMERICAL SIMULATI ONS . Number of Mo nte Carlo iterat ions 10 3 Number of orie ntations 126 Number of noise level s 11 Coil under test: ra dius 5 mm Coil under test: number of turns 20 Coil under test: driving curre nt (amplitude) 0.28 A Coil under test: driving curre nt (freque ncy) 184 kHz Probe coil: radius 19 mm Probe coil: number of turns 5 Sampling fre quency 3 MSa/s Number of sampl es 10 3 Fig. 1 – Proposed measureme nt setup. The magnetic dipole moment generated by a coi l placed in the origin of a Cartesian coordinate system is assessed by measuring voltage at the output of a probe coil, sequentially placed on each cartesian axis and aligned to the axis itself. Phase between the phasor I t of the curre nt feeding t he active coil and each coll ected voltage is measured as wel l. on the y and z axe s, each time aligned with the correspo nding axis, m y and m z can be estim ated as well. The measureme nt setup is summarized in Fig. 1. Additional p henomena affecting the m easurement result, such as gain of th e measurement c hain and coil resonance, may be kept into account by first cal ibrating the s ystem. III. S IMULATI ONS RESULTS To investigate the effect o f noise and orientation on t he performance o f the propo sed method, Mo nte Carlo numerical simulations were perfo rmed. T he coil under test was assumed to be centered at the origi n of the coord inate syste m with an arbitrary orienta tion, and to be generating a sinusoidal time- varying magnetic field. T he simulations were repeated for 126 different orientatio ns of the c oil under test, with an azimuthal angle from 0° to 360° and an elevation angle from 22 .5° to 90° . Some o f the unit vec tors describing t he 126 orientatio ns are shown in Fi g. 2. T hree probe coils were simulated, ea ch centered o n one of the axes, at a distance of 30 cm from the center of the coil under test, and oriented along the corresponding axis. The voltage induced at ea ch probe coil was si mulat ed according to the model (1)-( 5). Subsequently, additive white gaussian noise ( AWGN) was ad ded to the samples o f t he induced voltage and t he amplitude a nd phase of the induced voltage were estimated based on the noisy samples using a 3 - parameters sinefit al gorithm [ 20 ]. Finally, the resulting V x , V y , and V z estimates were used to calculate the m x , m y , and m z components of the magnetic dipole vector, respectively, according to the prop osed method as derived in Sect ion II. The simulations were repeated for 11 different values o f the no ise standard deviation σ AWGN , correspo nding to an SNR ranging from -10 d B to 40 d B. For each value of σ AWGN a nd for each value of the o rientation, 1 0 3 Monte Carlo iterations were performed. The numerical simulation para meters are summarized in Tab le I. Fig. 3 – Numerical simulation results. Each curve represents one orientation of the coil under test. Fig. 2 – 3D represe ntation of some of the 126 unit vectors, describing simulated orien tations of the coil u nder te st. Fig. 4 – Empiric al cumulative distribution function ( E CD F ) of the angular error. Fig. 5 – Emp irical cumulative distribution function ( ECDF) of the relative magnitude err or We define the angular error as the angle between the direction of the true magnetic dipole moment and the directio n of the estimated magnetic dipole moment. Moreo ver, the magnitude error is defined as t he difference bet ween the true magnitude of the magnetic dipo le moment of the coil under tes t and its estimated value. Fi nally, we evaluate the relative magnitude error as the absolute value of the magnitude erro r divided by the true magnitude of the magnetic dipole moment of the coil under test. Results f or varying sign al- to -noise r atio and orienta tions are show n in Figs. 2 -5. It can be noticed that, if the SNR at the probe coil is 5 dB or greater, the propos ed metho d allow s for estimatin g the magnetic dipole momen t vector with an average angular error o f less than 1° an d an average ma gnitude error of less than 0.1%. IV. M EASUREMENT R ESULTS In order to validate the proposed measurem ent method, experim ents were performed by means of the experim ental setup shown in Fig. 6. The coil under test was driven w ith a sinusoidal v oltage s ignal a t 184 kHz, 20 V pp . The driv ing sig nal was provided by an Agilent 332 20a fu nction generator. A probe coil was placed at fixed and known positions . Test coil and probe c oil pa rameters are sh own in Table II. The probe coil was co nnected in paralle l w ith a 330 nF capacitor (LC circuit) and the voltage gain (Q) due to th e resonant circuit w as measured to be 5.57. The voltage induced on the probe coil was amplified by an instrum entati on amplifier (INA), AD8421 by Analog Devices, with a gain of 100. T he o utput of the I NA was co nnecte d to a Fluke 8845A voltm eter. Furtherm ore, the driving signal from the function generat or was co nnecte d to channel 1 o f an oscillos cope, while the output of the INA was connected to channel 2 of the o scil loscope. This allow ed for identify ing whether the drivin g signal and the receiv ed signal were in phase or in anti phase. The coil under test was placed in the center of a 3D -printed holder realized as an icosahe dron. The icosahedron shape was used because it allow ed for placing the test coil in a set of known and contro lled orientati ons. The center o f the coil under test coincided with the orig in of th e local coordinate fram e considere d in the ex perimen ts. The probe c oil was pl aced in t he follow ing three positions (coordin ates express ed in meters) , P x =(0.194 , 0, 0), P y =(0, 0.19 4, 0), and P z =(0, 0, 0.214). This correspon ds to placing the coil center on the x , y , and z axes respective ly. Each time, the coil axis w as aligned to the correspon ding ca rtesian axis. The measurements were perform ed for a set of orienta tions of the coil under test. For each or ientat ion and probe positi on, the measured voltages are shown in Table II. Subsequ ently, the measured voltage was used to estimate the coil dipole moment , obtaining the results show n in Table II I. The m easured magn etic dipole intensity is compared with the the oretical value 4.33x10-4 Am 2 in order to obtain the relative error reporte d in the rightm ost colu mn of Ta ble III. Note that the large azimuth error affectin g the first measure (-55.95° against a true value of 0°) does not represent a measurem ent problem. In deed in this case the test coil magn etic dipole is parallel to the z axis (elevation 90°), hence any azimuth defines the same directi on becau se of the cylindri cal symm etry of the problem. Thus, even in this case, the measured direction deviates from the verti cal direction by an angle of only 0 .81°. In all conside red cases a small angular error w as obtain ed, up per bou nded by 3.4°. T ABLE II – E XPERIMENT SETUP : TE ST COIL AND PROBE COIL PARAMETERS Coil under test: ra dius 5 mm Coil under test: number of turns 20 Coil under test: driving curre nt (amplitude) 0.28 A Coil under test: driving curre nt (freque ncy) 184 kHz Probe coil: radius 19 mm Probe coil: number of turns 5 T ABLE III – E XPERIMENTAL R ESULTS Test coil directi on (deg) Measured v oltage (m V) Magnetic dip ole (10 -4 Am 2 ) Azimut h Elevation Angular error Probe 1 Probe 2 Probe 3 Measured Error % Real Measured Real Measured 0.00 -55.95 -90.00 -89.19 0.81 -0.35 0.52 33.14 4.45 1.9 -108.00 -109.65 -26.56 -26.08 1.56 13.07 36.60 14.17 4.33 0.9 . 72.00 70.02 26.56 26.42 1.78 -13.21 -36.33 -14.31 4.32 1.0 180.00 177.92 -26.56 23.73 3.40 38.54 -1.40 12.63 4.21 3.5 0.00 -2.21 26.56 25.19 2.42 -38.16 1.47 -13.38 4.22 3.4 Fig. 6 – The e xperimental setup. V. C ONCLUSIONS A method for estim ating the magn etic d ipole moment of a coil fed by a sinusoidal current was presented, and validated both by means of sim ulations and experim entally . The proposed approach is efficient, requiring only 3 measurem ents, and achieves a good accuracy , with a rela tive error of less than 4% on th e magnitude o f the m agnetic dipole mom ent and less than 4° on its b earing. A CKNOWL EDGMENT Thi s res ea rch activ ity was fun ded throug h grant PRIN 2015C 37 B25 by the Ital ian Minis try of Inst ructi on , Univ ers ity an d Re sear ch (MIUR) , wh ose su pp ort th e au th ors g rat eful ly ack now led ge . R EFERENCES . [1] A. Mancini, E. F rontoni and P. Zingarett i, "Embedded Mu ltisensor Syste m for Safe Point- to -Point Navigation of Impaired Use rs," in IEEE Transactions on Int elligent Transpor tation S ystems , vol. 16, no. 6, pp. 3543-3555, Dec. 2015.doi: 10.1 109/TI TS.2015.2489261. [2] C. M edina, J. C. Segura, A. 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