Supercond. Sci. Technol. 10 (1997) 187–194. Printed in the UK PII: S0953-2048(97)79837-3 Calculations of hysteresis in a short cylindrical type II superconductor I Younas and J H P Watson Institute of Cryogenics and Energy Research, University of Southampton, Southampton SO17 1BJ, UK Received 26 November 1996, in final form 5 February 1997 Abstract. The phenomenon of hysteresis in magnetic materials has been widely studied for many decades and several models have been proposed to find good approximations for this phenomenon. The simulation presented in this paper describes hysteresis in a cylindrical type II superconductor of small aspect ratio. A fundamental approach, using energy minimization, has been applied by which the demagnetization effects are naturally included. A comparison has been made between the critical current derived from the magnetization using Bean’s formulation for the infinitely long cylinder and that for the short cylinder used in our calculations, and it indicates how the error in using the Bean model for a short cylinder can be minimized. 1. Introduction The distribution of magnetic vortices in a type II superconductor at finite temperature is governed mainly by the balance of electromagnetic driving forces, pinning forces of the material and flux creep. When an external applied magnetic field is varied between Hc1 and Hc2 the vortices begin to enter or leave the superconductor through its boundary. Whenever the driving forces overcome the pinning forces, the system of vortices arranges itself into another metastable state such that an equilibrium is achieved with the external magnetic field. Hence, a change in the applied magnetic field or temperature forces the unpinned vortices to move rapidly into another equilibrium state or to leave the superconductor so that a new quasistationary equilibrium of energy is obtained. Numerical calculations of magnetic field distribution in superconductors have been carried out by many authors. Frankel [1] calculated the current and field distribution in a thin superconducting disk in transverse fields by use of the critical-state model. Daümling and Larbalestier [2] extended the calculations by including the short-cylinder geometry, but they did not include the hysteresis effects. Fabbricatore et al [3] carried out hysteresis simulations on infinite slab and cylinder, but this is essentially a onedimensional case with no demagnetizing effect included. Zhu et al [4] looked at trapped flux, but their hysteresis calculations were limited to the thin-disk case. The present work is an extension of the earlier work by Watson and Younas [5] who looked at current and field profiles in short cylindrical superconductors in the remnant state using Kim’s relation [6] for the dependence of current on field. In the present work the Bean model [7] was used for the current flow in the superconductor, i.e. the current c 1997 IOP Publishing Ltd 0953-2048/97/040187+08$19.50 density never exceeded some critical value Jc which is independent of the magnetic field. This implies that for a moderate applied magnetic field (Hc1 < Ha < Hc2 ) the superconductor is divided into two regions: (i) the field-penetrated region where the current density is either +Jc or −Jc depending on the magnetic history and (ii) the field-free region where the current is identically zero. The boundary separating the above two regions is called the free boundary, which is highly sensitive to the magnitude and spatial dependence of the magnetic field, as well as the topology of the superconducting material. The free boundary is easily determined for a superconductor with a simple geometry, e.g. infinitely long cylinder or strip. For a geometry where the demagnetizing effect needs to be included, a determination of the free boundary can be non-trivial. Several front-tracking algorithms have been developed to determine the free boundary, e.g. [8, 9], some better suited to a particular problem than others, but there may be a topological form of the superconductor where the use of a particular methodology is not at all appropriate. In [10] the authors calculate the levitation force on a permanent magnet above a HTS material. Although they use the critical-state model and the current is dependent on the electric and magnetic fields it is not at all clear how they have included the demagnetizing effects. In [11] a vector potential notation for the current and a virtual conductivity together with a certain non-linear J – E constitutive relation are used to describe the critical state. The most comprehensive paper describing the field equations for the type II superconductor is given by [12], but unfortunately they only considered the infinitely long cylindrical conductor configuration. 187 I Younas and J H P Watson In the present work the problem of finding the free boundary has been avoided, although it can easily be found once a solution has been obtained. The problem is tackled by applying a fundamental physical approach using energy minimization of the system consisting of the superconductor and an external source giving a uniform magnetic field over the volume of the conductor. When the applied field varies with time the energy of the system is assumed to achieve a minimum configuration at each time step. 2. Theoretical background Consider a body of fixed currents situated in a magnetic medium of permeability µ and positioned at a finite distance from an arbitrary origin. The volume occupied by this body is denoted as V1 , and the magnetic energy T1 due to excitation currents in the body can be expressed as [13] Z 1 B1 · B1 dV (1) T1 = 2µ another formulation will be outlined which overcomes this difficulty. Assume that the body situated in V1 is a solenoid, and the body occupying the volume V2 is a type II superconductor. The field due to excitation currents in the solenoid gives rise to screening currents in the superconductor. The magnitude of the screening currents depends on the magnitude of the applied field of the solenoid. It is assumed that the field never exceeds the upper critical field Hc2 of the superconductor. Hence the conductor remains superconducting for the range of applied field. It will also be assumed that the screening currents are all volume currents, and that the Meissner state can be neglected (Hc1 ∼ = 0). In the following derivation the subindices 1 and 2 refer to the solenoid and the superconductor respectively. The first term of equation (3) can be rewritten as Z Z 1 1 B2 · H2 dV = ∇ × A2 · H2 dV (4) E1 = 2 2 V V V where the integration volume V is extended over all space. It is assumed that the relation between the magnetic field B and the field strength H is linear and the medium is isotropic. Now reduce the intensity of the excitation currents to zero, and introduce into a suitable cavity of the medium another body which is assumed initially to be unmagnetized, and the relation between B and H is linear. The volume occupied by the second body is denoted V2 . The excitation currents of the first body are subsequently restored to their original configuration. Since the energy of the system consisting of the two bodies is determined only by the initial and final values of the system variables, the total energy of the final state is expressed as: Z 1 (B1 + B2 ) · (B1 + B2 ) dV (2) T2 = 2µ V where B1 and B2 are magnetic fields due to excitations in the two bodies. The energy required to build up the currents of the first body to their initial state in the presence of the second body is the energy difference T = T2 − T1 , where T can be written as Z Z 1 B22 dV + 2 B1 · B2 dV . (3) T = 2µ V V The first term in equation (3) represents the energy due to the field of the second body alone, and the second term denotes the interaction energy between the two bodies. In the above discussion the magnetic response of the two bodies has been included in the expression for the field vectors. Hence, to calculate the energy at a given time the magnetic field must be integrated over all space. When a numerical solution is needed, and calculations are carried out, this becomes a cumbersome approach to determine the total energy of the system. In the following discussion 188 which when using the vector identity ∇ · (a × b) = b · (∇ × a) − a · (∇ × b) (5) (where a and b are two real arbitrary vectors) can be expressed as Z 1 [∇ · (A2 × H2 ) + A2 · (∇ × H2 )] dV (6) E1 = 2 V where the integration is over all space. The first term in equation (6) can be transformed into a surface integral by applying the divergence theorem. This surface needs to be extended to infinity to contain the whole space. Since A falls as r −1 and H as r −2 for increasing r, while the surface area increases as r 2 , this integral does not contribute to E1 . The second term in equation (6) will only contribute in the region V2 because curl H2 is zero outside V2 . The second term in equation (3) denotes the energy due to magnetic interaction between the fields of the solenoid and the superconductor. This term can be rewritten as Z Z (7) E2 = B1 · H2 dV = ∇ × A1 · H2 dV . V V Again, by applying identity (5) this expression can be reformulated as Z E2 = [∇ · (A1 × H2 ) + A1 · (∇ × H2 )] dV . (8) V By an argument similar to that for E1 the first term in equation (8) vanishes, and the second term contributes only in the region V2 . Hence the field formulation of equation (3) for the energy of the system can be recast in a current and vector formulation as Z 1 (9) T = (A2 · J + A1 · J ) dV 2 V2 Hysteresis in cylindrical type II superconductor field, and they screen as much of the conductor from the magnetic field as possible. It is assumed that for a given value of the applied field the screening currents adjust in such a way that the total energy of the system is minimized. Practically, this energy minimum was found by employing an iteration procedure using quasi-Newton and bisection methods in an intelligent way [14]. A conceptual algorithm for energy minimization is outlined below. Figure 1. A sketch of the geometry where the superconductor is immersed into the bore of the solenoid, which produces a uniform field over the volume of the conductor. Dimensions not to scale. where the current density J is given as: J = curl H2 . In this work the geometry of the superconductor is chosen as a short cylinder of known dimensions. The superconductor is positioned in the bore of the solenoid, which is configured such that it produces a uniform field over the volume of the superconductor (see figure 1). The simulation consists of two parts. (i) Minimize expression (3) or (9) for various values of the applied magnetic field. This will give different screening current patterns for the different values of applied magnetic field. (ii) Generate a complete hysteresis loop (M versus H ) curve of the trapped flux as a function of a timedependent uniform applied field. This is achieved by sweeping the applied magnetic field between −Bm and Bm where Bm is less than or equal to the full penetration field Bp . The relation between B and H is assumed linear and given as B = µ0 H where µ0 denotes the magnetic constant. 3. Implementation procedure The geometry of the superconductor was chosen as a short cylinder, with dimensions of radius R = 10 cm and height D = 20 cm. The dimensions of the solenoid were subsequently adjusted in such a way that it produces a uniform magnetic field over the volume of the conductor, parallel to the z-axis (see figure 1). Other ways of producing the uniform field are by means of a pair of Helmholtz coils, or by placing the conductor in the air gap between the poles of an electromagnet. It is assumed that the applied field is generated with a magnitude B0 by some means described above. The conductor is cooled to a temperature below Tc and the cylinder becomes superconducting and the applied field is subsequently switched on. Screening currents are generated in the superconductor in response to the applied (a) Choose the magnitude of the applied field B0 , and decide on the value of the critical current density Jc of the superconductor. This can be done by choosing experimentally determined values for a short cylindrical superconductor in the mixed state. B0 is chosen such that the penetration depth is small. (b) Add a current with density Jc in a very thin outer shell of the conductor as a starting guess for the current configuration. (c) Calculate the total energy of the system using equation (3) or (9). (d) Create another current shell next to the previous one, but where the current has only penetrated some finite distance towards the mid-plane, i.e. decreased in z with some step symmetrically from the outer surface for a constant value of r. (e) Determine the best configuration for z using bisection and Newton techniques which minimizes the total energy. (f) Increase the penetration depth radially towards the centre by a fixed step and repeat (d) and (e) again. (g) Repeat (d)–(f) until a global energy minimum is found for the chosen value of the applied field. The geometry of the concentric conductor shells was established by using a commercial software package called OPERA from Vector Fields Ltd in Oxford, UK [15]. 4. Results The results consist of two parts. The first task was to derive a useful minimization procedure by which the total energy of the system as given by equation (3) or (9) could be minimized. As an illustration the applied magnetic field B0 was chosen to be 3.0 T, and the critical current density Jc was chosen to be 108 A m−2 (which is close to experimentally measured values). The problem is now to arrange the screening currents of the superconductor such that the applied field is completely cancelled in the interior of the conductor, where B and hence J are zero. After applying the minimization procedure described in the previous section the following results were obtained. Figure 2 shows the screening current configuration of the superconductor to screen out the applied field of 3.0 T applied along the z-axis. The number of conductors N to model the screening currents was chosen to be 20, so that the computation effort was kept at a ‘manageable level’. An increase in N would increase the accuracy of the results obtained, but computation effort needed for the minimization procedure would also grow at least in the same proportion. 189 I Younas and J H P Watson Figure 2. A cross-section along the z -axis of the cylindrically shaped superconductor. The boundary between the screening currents and the interior of the conductor marks the border between the field-penetrated and field-free regions. Figure 3. (a ) Variation of the axial magnetic field (created by the screening currents alone) along the z -axis. (b ) Variation of the axial magnetic field (created by the screening currents alone) along the mid-plane (z = 0). Figure 3 displays field plots of the axial field component Bz (produced by the screening currents to cancel the applied field of 3.0 T) along the z-axis and along the mid-plane (z = 0). It can be seen from the figures that the axial component of the magnetic field produced by the screening currents is constant and equal to 3.0 T through most of the conductor. Hence it can be concluded that the screening currents have adjusted in such a way that the fields of these have cancelled the applied magnetic field (B0 = −3.0 T) in the current-free region. The radial field component of the screening currents is also zero (not shown) in the current-free region. The true field distribution for this configuration can be calculated by subtracting the applied magnetic field B0 = 3.0 T (applied axially, i.e. in the z-direction) from the field 190 generated by the screening currents in the superconductor. Figure 4 shows the variation of the resultant magnetic field across the superconductor. Since the applied magnetic field does not have a radial component, and the net field is zero in the interior of the conductor, it is concluded that the radial component of the field created by the screening currents is also zero in the current-free region. Hence the screening currents have adjusted in such a way that the demagnetizing effect of the geometry has been correctly accounted for. The error due to discretization of the screening currents and inaccuracy of the minimization procedure was estimated to be less than 3%. A set of simulations were carried out where the applied field was increased by 1 T in each simulation and the resulting screening current pattern was determined by the Hysteresis in cylindrical type II superconductor Figure 4. Variation of the net magnetic field (i.e. magnetic field generated by the screening currents and the applied field) across the superconductor. It can be seen that the magnetic field is very close to zero in the current-free region of the superconductor. energy minimization procedure described earlier. It was found that the superconductor was fully penetrated with field for a value of 10.8 T of the applied magnetic field. Figure 5 displays the different regions for various values of the applied field. Only one-fourth of the geometry needs to be shown, because of the axial and mirror symmetry of the magnetic field. When the applied magnetic field was decreased after increasing it to the full penetration value, some of the screening currents switched from +Jc to −Jc in the following way. The screening currents will try to prevent any changes in the established field pattern, which is also a lowest-energy configuration for the system. When the field is decreased to some value B2 after increasing it to a value B1 , the conductor senses a net field change of 1B = B1 − B2 (positive) which is opposite to the original field direction. To oppose the field changes some of the screening currents switch sign to −Jc starting from the edges and outer surface of the conductor (as this region of the conductor is most immediately affected by the field changes) such that a new energy minimum is achieved for the resulting field configuration. The average magnetization of the superconductor can be calculated by the following known formula: Z 1 (Btot − µ0 Ha ) dV (10) µ0 hMi = V2 V2 where V2 denotes the volume occupied by the superconductor. Btot is the total magnetic field in the superconductor Figure 5. Sketches for some free boundaries for various values of the applied magnetic field. and Ba = µ0 Ha is the applied magnetic field. When for a virgin superconductor the applied magnetic field is increased from zero to +Bm and then reduced to −Bm and lastly increased to +Bm again (i.e. a full field cycle) a hysteresis curve can be generated as shown in figure 6. The magnitude of the field sweep must be less than or equal to the full penetration field. Figure 6 shows a hysteresis curve 191 I Younas and J H P Watson Figure 7. Penetration depth as a function of the applied magnetic field for the infinitely long cylindrical superconductor in a parallel-field configuration. Figure 6. Magnetization versus applied field for the short cylindrical superconductor. The magnetization is calculated as µ0 M and the applied field as Ba = µ0 Ha , both having the dimension tesla. The maximum applied field is 10 T. (magnetization versus applied field) for Bm equal to 10 T. It is now interesting to explore the situation which arises experimentally. Usually magnetization curves have been measured on a short cylinder and the Bean model is subsequently applied to calculate Jc . We require therefore to calculate the errors produced by estimating Jc from magnetization of a short cylinder when using the Bean model for an infinitely long cylinder. Let us, as an illustration, calculate the critical current density Jc using five different magnetization curves. An example of a magnetization curve is shown in figure 6 with a maximum applied field of 10 T. In the calculations it will be assumed that the results obtained were derived using Bean formulae for an infinitely long cylindrical superconductor in a parallel-field configuration. The trapped field for the cylinder depends on the ratio of the magnitude of the applied field Ha to the full penetration field Hp (figure 7). The average field in the infinite cylinder can be calculated by the following formula µ0 hBi = πR 2 ZR Hs (r)2πr dr (11) 0 where Hs denotes the field distribution in the superconductor and R is the radius of the cylinder. When the applied field is increased to Hm ≤ Hp after which it is reduced to Ha ≥ 0 an expression for the average flux density in the superconductor can be derived: 1 1 1 2µ0 hBi = 2 − Hp Ha2 + Hp Ha Hm + Hp Hm2 Hp 4 2 4 1 3 1 1 1 − Ha − Ha Hm2 − Hm3 + Ha2 Hm . (12) 24 8 8 8 Equation (12) in the form shown is very general, and we shall consider two special cases as listed below. Assume that the applied field is increased to Hm ≤ Hp and kept at this value, i.e. Ha = Hm ; then the magnetization 192 of the superconductor is derived using equations (10) and (12) as 1 M = 2 Hp Ha2 − 13 Ha3 − Ha . (13) Hp This equation is a second-order polynomial equation in Hp = Jc R, when M and Ha are known. Equation (13) reduces to the well-known expression for M from the Bean model [7], i.e. 1 1 M = − Hp = − Jc R 3 3 (13a) when the applied field Ha is set to the full penetration field Hp . Now, if the applied field is reduced to zero after increasing it to Hm ≤ Hp , then an expression for the remnant magnetization is M= 1 Hp2 1 H H2 2 p m − 14 Hm3 (14) where Hm denotes the maximum applied field. Equation (14) is also a second-order polynomial equation in Hp = Jc R when M and Hm are known. It is interesting to note that (14) reduces to M= 1 1 Hp = Jc R 4 4 (14a) when the maximum applied field Hm equals the full penetration field Hp . Table 1 was generated using the calculated values for magnetization for the short cylinder using Jc = 108 A m−2 , and then applying the Bean equations (13) and (14) to find Jc . This is done in order to estimate the errors in Jc which arise when using the Bean formulae on a short cylinder where the demagnetizing field is high. The first column in table 1 contains values for the maximum applied field in the hysteresis cycle. The second column presents the calculated magnetization using equation (10) when the applied field is at the maximum value. In the third column the magnetization is calculated at zero field after the applied field is first increased to some maximum value and then subsequently reduced to zero. The fourth and fifth columns in table 1 contain values of Jc Hysteresis in cylindrical type II superconductor Table 1. Magnetization using equation (10) and derived values for Jc using equations (13) and (14) for different values for the sweep field. Maximum applied field Hm (A m−1 ) M using (10) at H = Hmax for short cylinder (A m−1 ) M using (10) at H = 0 for short cylinder (A m−1 ) Jc from (13) at H = Hmax using Bean formula (A m−2 ) Jc from (14) at H = 0 using Bean formula (A m−2 ) 1.5915×106 (2.0 T) −1.1331×106 2.3515×105 0.4931×108 50.7% error 0.4415×108 55.9% error 3.1831×106 (4.0 T) −1.7237×106 6.4768×105 0.5636×108 43.6% error 0.5598×108 44.0% error 4.7746×106 (6.0 T) −1.9934×106 1.1180×106 0.6035×108 39.7% error 0.6381×108 36.2% error 6.3662×106 (8.0 T) −2.1823×106 1.5580×106 0.6547×108 34.5% error 0.7448×108 25.5% error 7.9577×106 (10.0 T) −2.2150×106 1.8795×106 (0.6585×108 ) 34.2% error 1.0403×108 4.0% error which have been calculated using the values for M in the second and third columns and applying the Bean equations (13) and (14) respectively. The critical current density is found by first solving equations (13), (14) for Hp , i.e. the full penetration field, and then using Hp = Jc R for the infinitely long cylindrical superconductor in a parallel field. Since a second-order polynomial equation has always two solutions, two values for Jc are obtained. The Jc values that leads to values of Hp less than the maximum applied field (first column) are discarded. The derived values for Jc are similar in the two methods for small applied fields, but, as the magnitude of the maximum applied field is increased, the value of Jc obtained in the remnant state (i.e. Ha reduced to zero after some maximum value) becomes closer to the actual value used in the short cylinder, namely Jc = 108 A m−2 . The first column contains values for the maximum applied field. the second and third columns contain the magnetization of the superconductor at H = Hm and H = 0 in the short-cylinder configuration. The fourth and fifth columns contain derived values for Jc using Bean formulae. The fourth and fifth columns in table 1 contain values for Jc together with an estimated deviation from the actual value of Jc = 108 A m−2 . The last value for Jc in the fourth column is in parentheses because the corresponding Hp value is less than the maximum applied field (first column). The best value for Jc differs only by 4% from the value used in this work, i.e. Jc = 108 A m−2 . in earlier work in this context. The error arising from the discretization of the cylinder into finite concentric current segments was not greater than 3% in all calculations. A comparison has been made between the critical currents derived from the magnetization curves, assuming they were obtained for an infinitely long superconducting cylinder in a parallel magnetic field configuration (Bean model), with the value of Jc = 108 A m−2 used in this work. The results indicate that the demagnetizing effect of the cylinder is dominant for low values of the applied magnetic field. As the magnitude of the maximum applied field is increased, the discrepancy between the derived value for Jc and the reference value becomes smaller. For an applied field of 10 T the error in Jc is about 4% when the magnetization is measured in the remnant state, i.e. at H = 0. This indicates that the demagnetizing effect has been reduced to a large extent when the maximum applied field is 10 T in the current cylinder configuration. The real full penetration field in the current cylinder configuration was found to be 10.8 T. This value differs by 20% from the Bean value for the infinite cylinder configuration (13.07 T) when the maximum applied field is 10 T. The obtained results indicate that the best estimates for Jc , using the experimental values of magnetization for a short cylinder, are obtained when Hm ∼ = Hp and the remnant magnetization (i.e. H reduced to zero) is used to calculate Jc from the Bean formula. 5. Conclusion The authors acknowledge with gratitude support from the EPSRC for this project. We have shown that the demagnetizing field arising from the finite dimension of the type II superconductor can be accounted for and the flux penetration depth calculated for a given value of the applied field by an energy minimization approach. The hysteresis is then naturally derived by switching appropriate currents during the magnetic field sweep cycle. An enhancement of the minimization procedure together with a higher discretization will lead to higher accuracy of the energy minimization, but this has deliberately been avoided as the intention of this work was to describe a natural and obvious approach not used Acknowledgments References [1] Frankel D J 1979 Critical-state model for the determination of critical currents in disc-shaped superconductors J. Appl. Phys. 50 5402–7 [2] Daümling M and Larbalestier D C 1989 Critical state in disc-shaped superconductors Phys. Rev. B 40 9350–3 [3] Fabbricatore P, Gemme G, Musenich R, Occhetto M and Parodi R 1992 Simple numerical model to interpret the a.c. measurements on type II superconductors Cryogenics 32 559–68 193 I Younas and J H P Watson [4] Zhu J, Lockhart J and Turneaure J 1995 Field-dependent critical currents in thin Nb superconducting disks Physica C 241 17–24 [5] Watson J H P and Younas I 1995 Current and field distribution within short cylindrical superconductors Supercond. 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# Calculations of hysteresis in a short cylindrical type II superconductor