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MAGNETIC RESONANT COUPLINGS USED IN WIRELESS POWER TRANSFER TO CHARGE THE ELECTRIC VEHICLE BATTERIES BATTERIES Dragoş NICULAE ,,POLITEHNICA”Technical University of Bucureşti Lucia DUMITRIU Mihai IORDACHE Andrei ILIE ,,POLITEHNICA”Technical University of Bucureşti Lucian MANDACHE University of Craiova REZUMAT. În prezent aplicaţii precum reîncărcarea acumulatorilor autovehicolelor electrice au focalizat interesul cercetătorilor spre transferul de putere prin inducţie. Transmisia fără conductoa conductoare (wireless) a energiei electromagnetice este utilă în cazurile în care reţeaua reţeaua de conductoare este dificil sau chiar imposibil de realizat. realizat. Transferul este eficient la o distanţă la care câmpul electromagnetic este suficient de puternic pentru a asigura o putere putere rezonabil rezonabilă ilă în circuitul de sarcină, sarcină, ceea ce este posibil dacă ambele circuite – emiţătorul şi receptorul - funcţionează la rezonanţă. Cei doi parametri care controlează valoarea puterii transmise prin cuplaj magnetic şi randamentul transmisiei sunt frecvenţa şi coeficientul coeficientul de cuplaj. Lucrarea este focalizată pe calculul inductivităţii mutuale al sistemului wireless de care depinde ultimul parametru. Cuvinte cheie: cheie transfer de putere prin inducţie, reîncărcarea acumulatoarelor, vehicole electrice, inductivitate mutuală. ABSTRACT. Nowadays applications applications like recharging electric car batteries have focused the research on the wireless power transfer. Wireless transmission is useful in cases when when instantaneous or continuous energy is needed but interconnecting wires wires are inconvenient even impossible. The transfer is efficient over a distance at which the electromagnetic field is strong enough to to allow a reasonable power transfer, that is possible if both the emitter and the receiver achieve magnetic resonance. The two parameters which control the amount of the transferred power by magnetic coupling and the transfer efficiency as well are the frequency and the coupling coefficient. The paper is focused on the mutual inductance computation of the wireless system which control the last parameter. parameter. Keywords: Keywords wireless power transfer, battery recharging, electric vehicles, mutual inductance. 1. INTRODUCTION One of the main issues regarding electric vehicles is the process of automatically charging. The technology of wireless power transfer (Witricity – from WIreless elecTRICITY) provides large air gaps and large amounts of power. The antenna radiation technology is not suitable for this application. The main reason is that the radiated electromagnetic power is small, making this technology more suitable to transfer information than power. Also, in order to increase efficiency the antenna system emission pattern must be oriented towards the receiver, otherwise most of the power is radiated away and does not reach the receiver antenna system. This technology is affected by nearby or metal objects placed between the emitter and receiver, which absorb the radiated energy. Because the physical principle behind the Witricity concept relies on the fact that resonant objects exchange energy efficiently, the simplest Witricity system for power transfer consists in two circuits (resonators) – Source (emitter) and Load (receiver) – inductively coupled (Fig. 1). The source (emitter) resonator emits a non-radiative magnetic field oscillating at high frequencies, which mediates an efficient power transfer to the load (receiver) resonator. In applications like light propagation in waveguides or optical cavities, the Witricity concept is usualy described by the coupled mode theory [1-3]. _____________________________________________________________________________________________ Buletinul AGIR nr. 4/2011 ● octombrie-decembrie 155 An alternative theory is using of numerical methods in order to solve the Maxwell equations which arise in the study. For the bigger power applications, like recharging of the electric vehicle battery, because to perform wireless power transfer RLC resonators are used, the circuit theory can be applied as well [4]. 2 2 M = 4π Rr − k F − k k E (1) in which R and r are the radiuses of the two circles, k= 2 Rr (2) (R + r ) 2 + h 2 and h is the distance between their centers. F and E are the complete elliptic integrals of the first and the second kind. In general, elliptic integrals cannot be expressed in terms of elementary functions. It can be expressed as a power series. Weinstein gives an expression for mutual inductance of two coaxial circles in terms of the complementary modulus k’: Fig. 1. Wireless system foo recharging the electric vehicle battery. From the reported researches’ results some remarks have to be made: The interaction between the source and device is strong enough so that the interactions with nonresonant objects can be neglected, and an efficient wireless channel for power transmission is built. Magnetic resonance is particularly suitable for applications because, in general, the common materials do not interact with magnetic fields. It seems [5] that the power transfer is not visibly affected when humans and various objects, such as metals, wood, electronic devices, are placed between the two coils at more then few centimetres from each of them, even in cases where they completely obstruct the line of sight between source and device. Some materials (such as aluminium foil and humans) just shift the resonant frequency, which can in principle be easily corrected with a feedback circuit. The two parameters which control the amount of the transferred power by magnetic coupling and the transfer efficiency as well are the frequency and the coupling coefficient. In the following we shall develop a computing procedure of the mutual inductance in wireless systems which control the last parameter. 2. MUTUAL INDUCTANCE COMPUTATION In order to compute the mutual inductance of the various types of electric circuits occurring in practice, many formulas have been developed. The first and the most important formula to compute the mutual inductance between two coaxial coils is the elliptic integral formula given by Maxwell [6]: 3 2 33 4 107 6 k' + k ' +... ⋅ 1 + k ' + 4 64 256 15 M = 4π Rr k '4 + 1 + 4 ⋅ log − 1 − 128 k ' 185 6 k ' +... + 1536 (3) where k ' = 1 − k 2 . Nagaoka has given formulas for the calculation of the mutual inductance of coaxial circles, without using the elliptic integral tables [7]. These formulas make use of Jacobi’s q-series. The first formula is to be used when the circles are not near each other and has the following expression: 3 M = 16π 2 Rr q 2 (1 + ε ) , (4) where ε = 3q 4 − 4q 6 + 9q 8 − 12q 10 + ... 5 q= l= 9 l l l + 2 + 15 + ... 2 2 2 (5) 1− k' 1+ k' Nagaoka’s second formula is for coils which are near each other and the expression is: M = 4π Rr 1 2 2(1 − 2q1 ) 1 2 1 + 8q1 − 8q1 + ε1 log − 4 q1 [ ] (6) where _____________________________________________________________________________________________ 156 Buletinul AGIR nr. 4/2011 ● octombrie-decembrie 5 q1 = l1 = 9 l1 l l + 2 1 + 15 1 + ... 2 2 2 1− k (7) 1+ k ε1 = 32q13 − 40q14 + 48q15 − ... Other formulas based on certain definite integrals of Bessel functions or on elliptic integrals of the third kind have been proposed by many authors. To facilitate calculations in such problems, we have implemented a procedure in Matlab based on Neumann formula [8] : M = Φ 21 µ0 = i1 4π ∫∫ C1C 2 dl1dl 2 R12 (8) We have implemented the above expression in Matlab using two grids for each coil – one of them circumscribed and the other inscribed (Fig. 2); in this way we can limited the values between a maximum and a minimum, using a minimum mesh density. Case 3 Case 4 To compute the integral in (8) we have used 50 points in the mesh construction for each coil. The results of the analysis are presented in Table 2 together with the values taken from the literature, for comparison. In this table Mp – is the value of the mutual inductance, according to our procedure; Mt – is the value of the mutual inductance, according to the tables for elliptic integrals. Using two identic coils (one connected to an autotransformer at 50 Hz, the other connected two an oscilloscope in order to measure the induced voltage) we obtain the values presented in Table 3. By means of simple formulas, we then calculate the mutual inductance, having the current from primary circuit and the voltage from the second one. Mp – the value according to our procedure; Mm – the value obtained by measurements; Fig. 2. The two grids for each coil. 3. EXAMPLES For comparison we have chosen from the literature [6] four cases of mutual inductance, analytically computed by means of various formulas, some of them discussed in the precedent section. In Table 1 these cases are presented. Table 1. The theoretical cases Case 1 Case 2 Fig. 3. The device for measurements and the electrical scheme. _____________________________________________________________________________________________ Buletinul AGIR nr. 4/2011 ● octombrie-decembrie 157 Table 2. The comparison of the results x = 80 mm R1 = 250 mm R2 = 200 mm x = 40 mm R = 100 mm Mp=1,352e-7H Mt=1,351e-7H Mp=2,893e-7H Mt=2,9e-7H Case 1 Case 2 r = 80 mm R1 = 100 mm R2 = 60 mm cos θ = 0,7 r = 400 mm R = 50 mm cos θ = 0,4 Mp=-4,896e-11H Mt=-4,9e-11H Mp=2,44e-8H Mt=2,32e-8H Case 3 Case 4 Table 3. The values obtained by our computation procedure and the measurement results x = 30 mm R = 110 mm x =30 mm R = 110 mm θ = 450 Mm=51,97e-6H Mp=50,33e-6H Case 1 Mm=17,61e-6H Mp=17,01e-6H x =100 mm R = 110 mm x =100 mm R = 110 mm θ =450 Case 3 4. CONCLUSIONS Safe and accurate wireless power transfer remains a challenge and a current research theme. We have implemented a procedure to compute the mutual inductance between two circular coils. The computation results are in very good agreement with the values reported in the technical literature. We have built a device with two copper coils and a system that allow us to place the two coils in various positions and to make the measurements. The differences between measured values and the values obtained from software simulations could be explained by means of apparatus tolerances and by the fact that in our procedure we neglect the repartition of coil turns, considering them concentrated in one turn. As future papers and as a part of optimization transfer process, we will enhance our procedure in order to calculate the capacitance, allowing us to bring the benefits of resonance in the process. Mm=16,19e-6H Mp=15,92e-6H Case 2 Mm=14,73e-6H Mp=16,52e-6H Case 4 REFERENCE [1] H.A. Haus and W. Huang, “Coupled-Mode Theory”, Proc. of the IEEE, Vol. 79, No. 10, October 1991, pp. 1505-1518. [2] R. A. Moffatt, Wireless Transfer of electric power, thesis for Bachelor of Science in Physics - superviser M. Soljačić, June 2009. [3] T. Imura, H. Okabe, and Y. Hori, “Basic Experimental Study on Helical Antennas of Wireless Power Transfer for Electric Vehicles by using Magnetic Resonant Couplings”, Proc. of Vehicle Power and Propulsion Conf., Sept. 2009, IEEE Xplore, 4/10/2010, pp. 936-940. [4] M. Iordache, Lucia Dumitriu, Teoria modernă a circuitelor electrice, Vol.2, Editura ALL, 2000, ISBN 973-684-337-8. [5] A. Kurs, Power transfer through strongly coupled resonances, thesis for Master of Science in Physics superviser M. Soljačić, Sept. 2007. [6] P.L. Kalantarov, L.A. Teitlin, Calculul inductantelor, Editura Tehnica Bucuresti, 1958. [7] E.B. Rosa, F.W. Grover, Formulas and tables for the calculation of mutual and self-inductance, US Government Printing Office Washington 1948. [8] A. Timotin, V. Hortopan, A. Ifrim, M. Preda, Lectii de Bazele Electrotehnicii, Editura Didactica si Pedagogica, Bucuresti, 1970. _____________________________________________________________________________________________ 158 Buletinul AGIR nr. 4/2011 ● octombrie-decembrie