FEM ANALYSIS OF ROGOWSKI COILS COUPLED WITH BAR CONDUCTORS

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1 XIX IMEKO orld Congress Fundmentl nd Applied Metrology September 6 11, 2009, Lisbon, Portugl FEM ANALYSIS OF ROGOSKI COILS COUPLED ITH BAR CONDUCTORS Mirko Mrrcci, Bernrdo Tellini, Crmine Zppcost University of Pis (Deprtment of Electricl Systems nd Automtion), Pis, Itly, bernrdo.tellini@dse.unipi.it Abstrct e study the behvior of Rogowski coils coupled with br conductors under qusi-sttic conditions. Through finite element (FEM) nlysis, we estimte the current distribution cross the br nd the flux linked by the trnsducer for vrious positions of the primry conductor nd for vrious operting frequencies. Simultion nd experimentl results re reported in the text. Keywords: Rogowski coil, FEM nlysis, qusi-sttic conditions. 2. BASIC CONFIGURATIONS In Fig. 1, we show schemtic of circulr Rogowski coil with 8 turns crossed by primry br conductor. The internl nd externl rdii of the torus re = 3 cm nd b = 4,5 cm, respectively, while its height is w = 3,2 cm. Dimensions of the br re width c = 3 cm nd thickness d = 0,5 cm. The primry conductor is ssumed to be infinite in the xil direction. 1. INTRODUCTION Rogowski coils re often employed for mesuring high lternting currents or high current pulses [1], [2]. The bsic principle of opertion is bsed on the mutul coupling between the current trnsducer nd the primry conductor. Aprt from the idel cse, the mutul inductnce M depends on the current distribution cross the primry conductor nd on the relevnt positions between the ctive edges of the winding nd the primry conductor itself. In mny ctul configurtions, the ccurte estimtion of M should require full electromgnetic nlysis. Once we know the current distribution cross the primry conductor, the estimtion of M cn be obtined from the geometry of the system by mens of nlyticl or qusi-nlyticl methods. Biot-Svrt s lw, prtil inductnce technique, or, more in generl, mgnetic energy method cn be dopted to solve the problem [3], [4]. On the other hnd, the current distribution cross mssive conductor obeys to diffusion eqution nd, except for prticulr geometries nd pths of the primry conductor numericl solution must be performed. Through frequency nlysis crried out by commercil FEM code we estimte the vrition of the flux linked by circulr Rogowski coils with rectngulr cross sections coupled with n infinitely long br conductor. Our reserch objective consists in improving the design technique nd possibly the clibrtion of such current trnsducers. In prticulr, we focus on Rogowski coils with smll number of turns being the number of turns very criticl in terms of vrition of M [3]. d c b Fig. 1. Schemtic representtion of the reference configurtion position 1. As to be expected the vrition of M reduces significntly with the number of turns N. Indeed, s known from the theory, Rogowski coil with very smll pitch, i.e., very high number of turns, cn be used to show experimentlly the Ampère s low. In Fig. 2, we show the primry conductor shifted of 2,25 cm long the y-direction, while in Fig. 3 the br is shifted of 1,45 cm long the x-xis. b-

2 b Fig. 2. Second configurtion with the primry br conductor shifted long the y-direction of 2,25 cm. Position 2. b Fig. 3. Third configurtion with primry br conductor shifted long the x-direction of 1,45 cm. Position FEM ANALYSIS e dopted time-hrmonic qusi-sttic formultion to solve the field distribution of our problems. Thus, strting from the two Mxwell s equtions : E = iω A V (1) B = A (2) introducing two new potentils: A ' = A + ψ (3) nd fixing the guge b- b- V ' = V iωψ (4) V ψ = (5) i ω we obtin the driving eqution: r e µ 1 0 A + iωa = J (6) e strted our simultions solving 2D problem representing n indefinite conducting cylinder. Simultion results hve been compred with the nlyticl clcultion nd dt were in complete greement with respect to our purposes. Percentge differences were less thn one prt per ten thousnd. In prticulr, the flux linked by the modelled Rogowski hs been clculted through the circultion of the mgnetic vector potentil long its ctive edges nd compred with the direct ppliction of the Ampère s lw. On the other hnd, the current distribution cross the round wire hs been compred with the nlyticl solution expressed in terms of Bessel functions. In Fig. 4, we show the colour mp of the current density distribution through the br t 200 Hz, 1000 Hz, nd 2000 Hz. For ll the simultions the clcultion of the flux linked by the Rogowski is done vi the circultion of the mgnetic vector potentil A. The 2D computtion domin is represented by circulegion of rdius r = 0,15 m. The number of tringulr-type elements is bout corresponding to number of degrees of freedom of bout Simultions were performed on 64-bit 16 GB Rm pltform nd the computtionl time for ech simultion ws bout 1 minute. It is pprent how skin effects re prcticlly negligible t 200 Hz, while they re significnt t 2000 Hz. A rough estimtion of the eddy current effect cn be evluted for copper lmintion by the clcultion of the penetrtion depth δ: 1 δ = (7) π f µσ where σ is the electricl conductivity nd µ the mgnetic permebility. From (7) we obtin penetrtion depth of bout 4,7 mm, 2,1 mm nd 1,5 mm t 200 Hz, 1000 Hz nd 2000 Hz, respectively. In more detil, the hell blue color (upper imge 200 Hz nd centre imge 1000 Hz) corresponds to current density of 0,06 A/mm 2, while the red colour (lower plot 2000 Hz) indictes current density vlue of bout 0,35 A/mm 2. In Fig. 5, we show the colour mp of the mgnetic field distribution in the computtion domin. Dt re evluted t 2000 Hz. 4. EXPERIMENTAL SETUP In Fig. 6, it is shown prt of the dopted experimentl setup. It is possible to recognize the br, the Rogowski coil nd the output signl cble. A one meter copper br (thickness 0.5 cm, width 3 cm) is closed to form rectngulr loop of 4 m 3 m to void proximity effects between the Rogowski coil nd the rest of the circuit.

3 Fig. 6. Prt of the dopted experimentl setup. Fig. 4. Current density distribution through the br conductor t 200 Hz, 1000 Hz nd 2000 Hz. e wound two times eight turns on two plexigls dielectric supports of thickness 8 mm nd plced t distnce of 16 mm between the internl fces, obtining totl height of 32 mm of the trnsducer. The internl nd externl dimeters re 30 mm nd 45 mm, respectively. In Fig. 7, it is shown the dopted Rogowski coil nd segment of the primry br conductor plced s shown in Fig. 1. Fig. 5. Mgnetic field (H) in the computtion domin t 2000 Hz An rbitrry function genertor (Agilent 33220A) is connected to the loop vi trnsconductive power mplifier (not shown in the figure), so tht it works s current genertor. A Lbview pltform drives ll the system through the IEEE 488 protocol. A sinusoidl current with mplitude 10 A is utomticlly set t ech investigted frequency. The current through the circuit is mesured by mens of hll probe. Another DMM Agilent 34401A mesures the induced voltge t the remote ends of the Rogowski coil. Fig. 7. Rogowski coil coupled with the br conductor in the position RESULTS AND DISCUSSION In Tble I, we report the M vlues for the described configurtions provided by the FEM simultion t 200 Hz, 1000 Hz, nd 2000 Hz. For ll the simultions nd mesurements, position 1 refers to Fig. 1, position 2 to Fig. 2 nd position 3 to Fig. 3, nd position 1 is the benchmrk.

4 TABLE I. Mutul inductnce vlues for the investigted configurtions. (Vlues re in µh). Position 1 43,49 43,49 43,49 Position 2 42,12 41,80 41,81 error (%) Position 3 45,41 46,30 46,55 In Tble II, the percentge vritions with respect to the centred position re reported. TABLE II. Percentge vrition of M for the investigted configurtions (simulted dt). Position Position 2-3,2-3,9-3,9 Position 3 +4,4 +6,4 +7,0 In Tble III, we report the mesured percentge vritions of M for the investigted configurtions. TABLE III. Percentge vrition of M for the investigted configurtions (mesured dt) frequency (Hz) Fig. 8. Mesured (circles) nd computed (crosses) vritions of the induced voltge with respect to the benchmrk for position 2 (lower plot) nd position 3 (upper plot). In prticulr, for our implemented geometries the mximum percentge vrition of M hs been observed t 2000 Hz from 3,9 % up to + 7,0 % with respect to the benchmrk. This cn mke the Rogowski n inccurte trnsducer if such prmeters re not tken into ccount. It is pprent how numericl simultion cn be useful tool in the designing phse of ir-cored current trnsducer. In ddition, n ccurte evlution of the prmeter M over the frequency rnge of interest cn be very importnt for correct reconstruction of distorted primry currents. Indeed, the intrinsic linerity of the Rogowski coil mkes possible to evlute the spectrum of the output voltge nd then vi the M(f) reltion to obtin the nonsinusoidl primry current. Preliminry results pper to be promising for extending the FEM nlysis to more complex geometries s tht shown in Fig. 9. Further investigtions re currently in progress to improve the design of such current trnsducers. Position Position 2-3,0-3,6-3,8 Position 3 +3,9 +5,1 +5,6 The obtined results re summrized in Fig. 8 tht reports the mesured (circle symbols) nd computed (cross symbols) percentge vritions of the mutul inductnce M for the investigted configurtions. At ech frequency, lower nd upper plots refer to position 2 nd 3, respectively. Results show how the frequency nd position ply n importnt role nd how it is importnt n ccurte knowledge of the relevnt positions mong the ctive elements nd clcultion of the current density distribution for correct reconstruction of the primry current. Fig. 9. 3D problem, rectngulr C-shped primry conductor.

5 6. CONCLUSIONS Our min purpose hs been to develop FEM nlysis for investigting how the current density distribution cn ffect the estimtion of the mutul coupling between Rogowski coils nd br conductors. Results hve shown how frequency nd position re criticl prmeters for such trnsducers, especilly for Rogowski coil hving smll number of turns. The FEM nlysis ppers to be n effective tool to improve the design of such current trnsducers. REFERENCES [1] M. Rezee, nd H. Heydri, Mutul Inductnces Comprison in Rogowski Coil with Circulr nd Rectngulr Cross-Sections nd Its Improvements, proceedings of IEEE ICEA 2008, 3-5 June, pp [2] L. Ferkovic, D. Ilic, R. Mlric, Mutul Inductnce of Precise Rogowski Coil in Dependence of the Position of Primry Conductor, IEEE Trns. Instrum. Mes., vol 58, no. 1, Jn. 2009, p [3] G. Becherini, S. Di Fri, M. Mrrcci, B. Tellini, C. Zppcost, G. Robles, Criticl Prmters for Mutul Inductnce between Rogowski Coil nd Primry Conductor, Proc. I2MTC 2009, My 5-7, Singpore, p [4] F.. Grover, Inductnce Clcultions: orking Formuls nd Tbles, Dover Publictions Inc., 1973.