Theoretical Analysis of AC Resistance of Coil Made by Copper Clad Aluminum Wires

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730 PIERS Proceedings, Stockholm, Sweden, Aug. 2 5, 203 Theoretical Analysis of AC Resistance of Coil Made by Copper Clad Aluminum Wires C. Kamidaki and N.

1 730 PIERS Proceedings, Stockholm, Sweden, Aug. 2 5, 203 Theoretical Analysis of AC Resistance of Coil Made by Copper Clad Aluminum Wires C. Kamidaki and N. Guan Power and Telecommunication Cable System R&D Department, Fujikura Ltd., Japan Abstract In this paper, we propose coils made by copper clad aluminum (CCA) in wireless power transfer (WPT) systems, which show lower AC resistance than Cu ones in a certain range of frequencies. Resistance of CCA or Cu coil is formulated by analysis of the skin effect, proximity effect and a shape factor of the coil which describes intensity of magnetic fields created by applied current in the coil itself. Boundary frequencies of the range where CCA coils are superior to Cu ones and corresponding resistances are quantitatively analyzed. The condition where CCA coils show lower resistance compared to Cu ones is clarified as configuration of coils and operation frequency.. INTRODUCTION In a wireless power transfer system using inductive coupling through magnetic fields, power transfer efficiency is significantly influenced by the quality factor Q = ωl/r ac of its coil, where ω, L and R ac are the angular frequency, inductance and AC resistance, respectively ]. In order to obtain a higher Q, higher frequency and lower R ac are desirable, but R ac increases quickly with frequency due to the skin effect as well as the proximity effect which comes from the eddy current induced by current flowing in neighbor wires. We have proposed copper clad aluminum (CCA) wires which are aluminum (Al) wires coated with thin Cu layer via metallic bond, as shown in Fig. 2]. The CCA coils are not only costeffective, light-weight and solderable as Cu, but also show lower R ac than Cu under certain circumstance. In the study, both the skin and proximity effects for a round metallic wire with multiple layers are theoretically analyzed and the AC resistance is formulated as a summation of an AC resistance caused by the skin effect and a product of a loss caused by the proximity effect and a shape factor of the coil. Figure : Copper clad aluminum wire. Figure 2: Analysis model of CCA wire. In this paper, the shape factor of a coil, which was obtained by curve-fitting from measurement, is numerically formulated. Frequencies f and f 2 between which CCA coils have lower R ac than Cu ones and corresponding resistance R and R 2 are quantitatively analyzed. It is demonstrated that our theoretical analysis agrees with measurement very well. 2. FORMULATION OF AC RESISTANCE OF CCA COILS A CCA wire is modeled as a round wire uniformly distributed along z-direction with two-layers where the i-th layer has a radius of r i, conductivity of σ i and relative permeability of µ i, as shown in Fig. 2. Assuming a time factor of e jωt, the z-component of electric field E z at the i-th layer induced by a current in z-direction satisfies the following equation: 2 E z r 2 + r E z r jωµ iµ 0 σ i E z = 0 ()

2 Progress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 2-5, which has a solution of E z = { A J 0 (k r) (r r ) A 2 J 0 (k 2 r) + B 2 Y 0 (k 2 r) (r < r r 2 ) (2) where A i and B i are constants, J ν and Y ν are the Bessel and Neumann functions of the ν-th order, respectively, and ki 2 = jωµ iµ 0 σ i. The energy consumption in the wire is equal to the power flow entering the wire from surface and is expressed by surface integration of Poynting vector on the wire. Therefore, when AC current flows in the wire, resistance caused by the skin effect per unit length is given by jωµ2 µ 0 R s = R A2J ] 0 (ξ) + B 2 Y 0 (ξ) 2πξ A 2 J 0 (ξ) + B 2Y 0 (ξ) (3) where ξ = k 2 r 2 and R denotes real part. Assume that an AC magnetic field with intensity of H 0 is applied to the wire along x-direction as shown in Fig. 2, the z-component of magnetic potential A z satisfies the following equation: 2 A z r 2 + r A z r + r 2 2 A z θ 2 + k2 i A z = 0 (4) which has a solution of C J (k r) (r r ) A z = sin θ C 2 J (k 2 r) + D 2 Y (k 2 r) (r < r r 2 ) C 3 r + D 3 r (r2 < r) where C i and D i are constants. Then the loss due to eddy current in the wire per unit length is calculated by the power flow passing through the surface of the wire and is given by (5) P p = 2π ξ 2 H 0 2 σ 2 ξxy Z 2 (6) where X = C 2 J (ξ) + D 2 Y (ξ) Y = C 2 J (ξ) + D 2 Y (ξ) Z = (µ 2 )X + ξ C 2 J 0 (ξ) + D 2 Y 0 (ξ)] Noting that magnetic field is generated by current flowing in the wire for a coil, then the magnetic field is proportional to the magnitude of the current, i.e., H 0 = α I (7) Since eddy current loss is expressed as a product of the AC resistance due to the proximity effect and a square of the applied current, AC resistance of coils wound by litz wire with N wires and length of l is expressed by R ac = ( R s + α 2 ) l D p (8) N where D p is associated with the loss caused by the proximity effect per unit length and is given by 4π ξ 2 D p = R σ 2 (ξ XY Z 2 ). (9) Assume that a coil wound with T turns is N T concentric circle wires as shown in Fig. 3, the intensity of applied magnetic field to the i-th wire H i is obtained as a summation of magnetic fields from circular current flowing in all other wires in the case of air core coils, and can be expressed

3 732 PIERS Proceedings, Stockholm, Sweden, Aug. 2 5, 203 Figure 3: Analysis model of coil. Figure 4: Picture of measured coil. by 3] H 2 i = N T j i H rij 2 N T + j i H zij H rij = i j z i z j 2π r j (ri + r j ) 2 + (z i z j ) 2 H zij = i j 2π k c = 2 K(k c ) + r2 i + r2 j + (z ] i z j ) 2 (r i r j ) 2 + (z i z j ) 2 E(k c) K(k c ) r2 i r2 j + (z i z j ) 2 (ri + r j ) 2 + (z i z j ) 2 (r i r j ) 2 + (z i z j ) 2 E(k c) ] (0) () (2) 4r i r j (r i + r j ) 2 + (z i z j ) 2 (3) where i j is the current flowing in the j-th wire, K and E are complete elliptic integrals of the first and second kinds, respectively. Then α is calculated by N T i= 2πr i Hi 2 α = N T (4) i= 2πr i i 2 i 3. NUMERICAL RESULTS For a coil shown in Fig. 4, where cables are stranded with 4 wires of Φ0.40 mm for 8 layers of 0 turns on a bobbin of Φ20 mm, Fig. 5 shows the measured and calculated R ac of the coil wound by Cu or CCA wire which consists of 5% Cu and 95% Al in area ratio. In the measurement, R ac increases with frequency and R ac of CCA coil is lower than Cu one from 5 to 450 khz. In the calculation, the shape factor α is obtained as 3.6 mm by Eq. (4). This value makes the calculation agree with the measurement very well. Figure 5: Measured and calculated R ac for 4 Φ0.40 mm coils. Figure 6: Calculated R ac for different coils.

4 Progress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 2-5, Figure 6 shows the calculated R ac of coils which are wound by wires with the same sectional area and configuration but different number and thickness of wires. R ac for CCA or Cu coils wound by a litz wire with 56 wires of Φ0.20 mm and one with 6 wires of Φ0.6 mm are added to the calculated result in Fig. 5. Although f and f 2 shift to higher frequency as the wires get thinner, the corresponding R and R 2 are independent on the thickness of wires. This phenomenon is explained as follows. At a low frequency at which wire radius is smaller enough than the skin depth δ = 2/ωσµ, R s and D p are approximated by 4]: Then, f and R are easily obtained to R s πσ 2 (r2 2 r2 ) + πσ r 2, (5) D p π (ωµ 0) 2 ( σ2 r r 4 ) + πσ r 4 ]. (6) f = παr fcu f CCA (7) R = R Cu + r2 2 r 2 R CCA (8) where R CCA and R Cu are DC resistances of CCA and Cu coil, f Cu is defined as a frequency at which the radius of wire is equal to the skin depth of Cu, and f CCA a frequency at which the radius of wire is equal to the skin depth of an uniform material with the same DC resistance of CCA wire. They are expressed by f Cu = f CCA = πµ 0 σ 2 r 2, (9) r 2 2 πµ 0 σ2 (r2 2 r2 ) + σ ] r 2. (20) r 2 Equation (8) leads to an interesting result that R depends on DC resistance of these coils and sectional area ratio of Al in CCA. Furthermore, R approaches to the summation of DC resistances of wires as the outer layer gets thinner. In the measurement in Fig. 5, R CCA, R Cu and R are 20, 7 and 97 mω, respectively. According to Eq. (8), R is 97 mω and coincides with the measurement. Frequency f obtained by Eq. (7) is 6.8 khz which differs a little with the measurement of 5 khz as shown in Fig. 5. Since the proximity effect is prominent in R ac at high frequency, f 2 is obtained as the frequency at which D p s of CCA and Cu show a same value. 4. CONCLUSION We have proposed an analytical expression of AC resistance of CCA coil by developing an analysis formulation for the skin effect, proximity effect and a shape factor of a coil. The theoretical and experimental results agreed well with each other and both demonstrated the superiority of CCA coil over Cu one regarding on AC resistance in a certain condition. In real applications, it should be easily confirmed whether it is effective or not to make replacement of Cu by CCA for not only saving weight but also decreasing energy loss, by using estimated f and f 2. It will take advantage of these features by applying this phenomenon to WPT systems to save total energy consumption, especially in power charging for electrical vehicles. REFERENCES. Yamanaka, T., Y. Kaneko, S. Abe, and T. Yasuda, 0 kw contactless power transfer system for rapid charger of electric vehicle, Int. Battery, Hybrid and Fuel Cell Electric Vehicle Symp., 9, Los Angeles, USA, May Guan, N., C. Kamidaki, T. Shinmoto, and K. Yashiro, AC resistance of copper clad aluminum wires, Proc. of the 202 Int. Symp. on Antennas and Propagation, , Nagoya, Japan, Oct. 202.

5 734 PIERS Proceedings, Stockholm, Sweden, Aug. 2 5, Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, Sullivan, C. R., Aluminum windings and other strategies for high-frequency magnetics design in an era of high copper and energy costs, IEEE Applied Power Electronics Conf., 78 84, Anaheim, USA, Feb

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