Ohmic heating in a magnet coil by eddy currents
J.O. Rossi, J.J. Barroso, Y. Aso, and J.G. Ferreira Laboratório Associado de Plasma Instituto Nacional de Pesquisas Espaciais 12201-970 - São José dos Campos, SP, Brazil
1 Introduction
According to Faraday's law a voltage is induced in a massive conducting body if it is subject to a time-varying magnetic flux. The voltages induced at different points within the object give rise to internal currents circulating in appropriate paths that are referred to as eddy currents. In this paper, energy dissipation by eddy currents through ohmic losses is addressed, where the effective resistance of a magnet coil is examined on the basis of the experimental result of heat production. That coil is the single component of a water cooled magnet system that provides the required high magnetic field for operation of lhe 35 GHz, 100 kW gyrotron I1' under development at INPE. Each pancake coil is wound from an insulated copper tube of 25rn length which has a rectangular 'cross 2 - WICYI (9 x 9mm ) with a 5-mm-diamcter orifice leading the cooling water. As will be discussed, in conventional solenoids the action of eddy currents may be detrimental as the result of a nonuniform current distribution in the conductors which leads to higher ohmic losses.
2 Measurement of the energy loss in a pancake coil
To put into perspective the ohmic losses incurred by eddy currents, a test circuit was set up as in Fig. 1. The power loss in the pancake coil was then measured and compared with the calculated one. The power loss is given by Q = r'ncAT where m and AT are the mass flux and the temperature rise (AT = Tout — l\n) of the cooling water, respectively; c is the specific heat for water with value 4.18 J/gK. This power loss is equivalent to the ohmic loss in the pancake coil given by < I7 > R, where < I7 > is the mean square of the feeding current and R is the coil resistance. The current waveforms used in this experiment are shown in fig. 2 along with relevant parameters displayed in Tab. 1. The average < I > and the mean square values are expressed by 7 7 2 < / >= 2(/,T2 + l3T3ynT and < I >= {1 TX + /3 T3)27', where T = T, -I- T2 + T3 + T4.
47 The power loss measurement was made for four cases of 7=50,100,150, and 2D0 A, where 7 is the mean value of the coil current as indicated by a the DC power supply current meter (Tab. 2). It should be mentioned that the coil resistance of 7mfl is calculated from the resistivity of 1.72 x 10~8fim for copper, and confirmed by the measurement using a bridge circuit. However, the effective resistance, ranging from 95 to 48mfi, is 13.5 to 6.8 times larger than the expected 7mil value. This suggests that the observed higher oliuiic losses are due to the harmonic components of the applied current. In fact, by using another DC power supply with lower ripple rate, a similar experiment was carried out. The results presented in Tab. 3 indicate that the healing rate is in good agreement with the predicted value.
Accordingly, the increase in the effective resistance shown in Tab. 1 should be ascribed to the harmonic content of the feeding current. From the current waveform shown in Fig. 2, the frequency of first harmonic component is-180 Hz. Thus we may consider the effective coil resistance for the frequency range 0-lkIIz, which is discussed in the next section.
3 Eddy-current effects
Eddy currents manifest themselves by increasing the effective resistance and decreas- ing the effective inductance of the total impedance Z = Re/j + juiLe/j in such a coil (Fig. 3) according to the equations I1':
7 2 2 /?,// = Ko + pM^ /{p 4- /V), Leff = L0- lMW/(p + /V) (1)
where fi0 and Lo denote the resistance and the inductance at zero frequency, respectively; p and / are the resistance and the inductance for eddy currents, and M is the mutual inductance between the coil and the eddy current circuit. To estimate the inductance / we consider the configuration for eddy currents as shown in Fig.4, where L\ and Lj denote the self-inductances of the circular loop conductors of finite cross section. Thus, the inductance / is calculated as / = Lt + Lj — 2Mtj, where Ml2 is the mutual inductance between L\ and Lj. On the other hand, Rejj and Ltjj for a given frequency u> can be obtained experimentally from measurements of phase difference between the voltage and the current waveforms M. Such a measurement was made for a duplicate pancake coil and the results are given in Fig. 5. From this plot, the unknown parameters Lo and Ijp could be obtained. Using the calcualted value for the inductance /, the mutual inductance A/ was then estimated. It is apparent that the experimental data are in good agreement with the theoretical curves.
48 4 Power loss taking into account eddy-current effects
To account for the energy loss, the applied current waveform i(t) shown in Fig. 2 was expanded in Fourier series and the energy dissipation for each harmonic component was calculated using the effective resistance equation (1) for eddy-currcnl effects. Thus, the average dissipated power becomes
(2) n=l where c\ = a' + b\, with an = \ fj i(t)co*nu;tdt,bn = j. £ i(t)sinnu}tdt, and /?„ is the effective resistance in eqs. (1) corresponding to the n-th harmonic frequency. In Fig. 6, the average dissipated power < W >, which was calculated until the 10th harmonic component for the coil current displayed in Fig. 2, is shown along with the experimental data of Tabs. 1 and 2. The parameters p, A/, and / in cqs. (1) were determined so that the resulting < W > values might agree with measured data, thus giving 42mfi,35/iH and 10////, respectively, consistent with estimated values.
5 Conclusion
In a large magnet coil, it was found out that the action of a lime-varying current is to increase the effective coil resistance. This leads to higher ohmic losses which arc strongly dependent on the ripple rate of the applied current. Taking into account eddy-current effects, experimental data could be compared with calculations showing the effective coil resistance to be about one order of magnitude above the DC resistance value.
Finally, it should be pointed out that the experimental observations cannot be ac- counted for on the basis of the skin effect. By this effect, the current magnitude de- creases exponentially inward from the conductor surface to a nominal depth of penetra- tion 6 = J2t)/(iui, where t) and // are the resistivity (1.72 x 10~8ilm for copper) and the absolute magnetic permeability of the conductor, respectively. At higher frequencies, the current concentrates in a thin skin at the conductor surface, thus causing a higher resistance. To obtain the effective resistance given in Tab. 1, the skin depth needs to be nearly 0.3mm, which calls for an operaling frequency of 50 kHz well above the frequency range considered in the present study.
6 Reference
[1 ]] Y.Aso, J.G. Ferreira, and J.O. Rossi. "Electrical test of gyrotron coil". INPE Technical Report 4929/RPE/596. S.J.Campos, Sept. 1989.
49 COOLING WATER INLET
O.C. POWER SUPPLY
T2 T3
out I Fig. 2. Current waveform. OUTLET
Fig. 1. Test circuit of a pancake coil.
I 50 100 150 2UO A
<[> 52.8 94.3 A Ti I Tz T3 Tu h 146.9 188.6 f(A) 'm [IU 7.363 20, 385 Í 47.904 r A? 50 1.4 1.5 1.2 1.3 200 140 AT 8.4 18.1 29.7 41.1 K 1.4 * 100 1.8 1.2 1.0 320 160 702 1,510 ?,180 3,440 W
150 2.2 l.l 1.3 0.8 460 180 z
Tab. 1. Relevant parameters for Tab. 2. Power loss and effective the current waveform shown resistance. in Fig. 2.
7 (A) ripple rate aT (5) experiment calculation
50 6 0.2 0.2 •A/WW 1 25 0.5
100 15 1.1 0.8
150 12 2.3 1.9
200 9 3.7 3.3
Tab. 3. Temperature rise for the Fig. 3. Equivalent circuit of the cooling water in the lower magnet coii. ripple rate case.
50 Fig. 4. Circular loop circuits for eddy currents. Dimensions
area! = 5.0cm. bt = 5.2 cm. a, = 20.1 cm. 6j = 20.3 cm, and k = 1.9cm.
Fig. 5. R,n and L.fl for the ed- dy-current eíTects. o :ex- pcrimental results of R..IJ\ x:cxpcrimcnlal results of i"lt\ —:thcoretical curves from cqs. (1).
!0K
4T Fig. 6. Experimental and theo- retical dissipated powers as function of the effective cur- rent. 0 experimental val- ues; • :cstimatcd values from u •6 eq.(2) for p = 42mfl, / =
I- 7mil; xrexperimental values for the lower ripple rate case;
—theoretical curve for Rejj 100 200 300 = /?„ = Imil. Ieff (A)
51