IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 3, MAY zyxwvutsrq 1993 2095 zyx Eddy Current Losses at Cryogenic Temperatures V. Sokolovsky, V. Meerovich, and M. Slonim Absfrucf-The present paper analyses e5wt of thermal pro- cesses on eddy-current losses in construction elements of cryogenic and superconducting devices. Maxwell’s equations coupled with heat-conduction equation are solved with taking into account the dependence of resistivity, heat capacity and heat-transfer coefficient on temperature. Analysis of losses zyxwvut as a function of magnetic field, frequency and geometry factors is given for the case of thin strip in a uniform magnetic field. It is shown that losses calculated with taking into account the thermal processes may di5er from those obtained at constant temperature. INTRODUCTION NE of the problems arising at design of cryogenic 0 and superconducting devices is calculation of eddy current losses in constructional elements (CE), Le., in cryostat walls, radiation shields, etc. [1]-[4]. The accu- rate determination of these losses is required for the cool- ing system construction and for choosing the power of cryogenic refrigerators. The main feature which must be taken into account at cryogenic temperatures is the strong dependence of the metal’s resistivity zyxwvutsr p on temperature zyxwvutsr T. Thus, the resistiv- ity p may change by several times when the temperature varies by 20 K [5]. The temperature of CE may vary con- siderably even within the magnetic field period because the heat capacity of metals at 4.2 K is approximately two orders of magnitude lower than that at room temperature. Also, the boiling crisis in the coolant causes the reduction of the heat-transfer coefficient and the temperature in- crease [5]. Usually eddy current losses are calculated at constant resistivity corresponding to the selected temperature. The effect of temperature on resistivity is taken into account with the use of correction coefficients zyxwvuts [ 11, [6]. This work is devoted to the analysis of the effect of thermal processes on eddy-current losses in CE of devices operating at cryogenic temperatures. As is done in induc- tion heating problems [7], we will use an approach founded on the coupled solution of electromagnetic and thermal problems. STATEMENT OF THE PROBLEM Generally, eddy current losses, temperature and, con- sequently, resistivity are non-permanent in a volume of Manuscript received July 7, 1992; revised January 15, 1993. The authors are with the Department of Electrical and Computing En- IEEE Log Number 9208216. gineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel. CE. In metals whose resistivity p varies from point to point, free space charge u may exist and Maxwell’s equa- tion set takes the form: + + rotH= j; + div B = 0; (3) div E = U/E~ (4) + where zyxwvu 2, 2 ar: the elect@ and magnetic field inte!sities, respectively (E = pj ); i is the current density; B is the magnetic flux density (B = zyx ps); pa is the magnetic permeability, E, is the dielectric constant, t is the time. The space charge density u is described by the expres- sion: + u = E,j gradp. (5) From (5) it follows that u = 0 if grad p = 0. This as- sumption is used for most electromagnetic field problems. 5 should be noted that, even for great values of grad p, j grad p = 0 and hence u = 0 are satisfied when all the current lines are parallel or if one can neglect the propa- gation of heat in CE. The problem now is to solve (1)-(5) coupled with the heat-conduction equation: (6) aT C - at = div (X grad T) + W, where C is the specific heat capacity; X is+the thermal conductivity, T is the temperature; W = I E I2/p is the loss power per unit volume. In (1)-(6) C, X, p are functions of temperature and con- sequently of time and of point-to-point variation. The boundary conditions for Maxwell’s equations are usual conditions for the magnetic and electric fields at a conductor-dielectric interface. The type of boundary con- ditions for (6) is defined by the device construction, the cooling method and the form of the examined CE. Losses in a Thin Strip To analyze the influence of thermal processes on eddy current losses, we will consider, for example, a thin strip (Fig. 1) in a uniform magnetic field normal to its surface Hz = -H, sin at (a = 271-5 f is the frequency). This simple problem is a model for a number of practical cases. 0018-9464/93$03.00 0 1993 IEEE __- 7