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1 Design Principles of Superconducting Magnets Aki Korpela Tampere University of Technology Electromagnetics Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 2 DESIGN PRINCIPLES OF SUPERCONDUCTING MAGNETS Content of the presentation Background Short-sample measurement of a superconducting wire • Power law Critical current of a superconducting magnet • Design case of low temperature superconductor (NbTi) solenoidal coil • Design case of high temperature superconductor (Bi-2223/Ag) solenoidal coil Optimization of coil geometry • Design case of low temperature superconductor SMES Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 3 Background Critical quantities of superconductors are: • current density J, • magnetic flux density B, • temperature T. These three parameters constitute a critical surface, inside of which a material is in the superconducting state. • Piercing of this surface is called as a resistive transition. The maximum values of J, B and T that maintain superconductivity at certain operation conditions are called as: • critical current density Jc, • critical magnetic flux density Bc, • critical temperature Tc. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 4 Short-sample measurement of a superconducting wire In a typical short-sample measurement of a superconducting wire the critical current Ic is measured at constant temperature T and constant magnetic flux density B. As a result we get Ic(B)-dependence of the wire at a certain temperature, which is fundamental data concerning the superconducting magnet design. Short-sample data at 4.2 K for NbTi wire OK42 manufactured by Luvata. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 5 Short-sample measurement of a superconducting wire Critical current is determined with an electric field criterion Ec. • In LTS measurements the criterion is typically Ec = 0.1 μV/cm. • In HTS measurements the criterion is typically Ec = 1 μV/cm. Adjacent measurement is for Bi-2223/Ag tape (HTS material) at 77 K and self-field. In this case the distance between voltage taps is 2 cm, so Ic (≈ 120 A) is determined with the voltage criterion of 2 μV. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 6 Short-sample measurement of a superconducting wire: power law Voltage (V) - current (I) measurement of a superconducting wire can be modelled with so called power law, where critical current Ic is determined from the measured V(I)-curve with voltage criterion Vc, and exponent n describes steepness of the resistive transition: n ⎛I ⎞ V = Vc ⎜ ⎟ . ⎝ Ic ⎠ Adjacent figure shows power law fitting for measured data: • Vc = 2 μV, • Ic = 119.5 A, • n = 8 (+), n = 10 (o), n = 12 (∗). Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 7 Summary: short-sample measurement of a superconducting wire The idea is to find out Ic and n at certain temperature and magnetic flux density. • When Ic and n are known, V(I)-characteristics of a sample can be modelled with power law. Cool down the sample to a certain temperature. at first the magnetic flux density applied on the sample is zero, B = 0 while B ≤ Bmax measure the voltage - current curve of the sample determine Ic from measured V(I)-curve with electric field criterion Ec determine exponent n by fitting power law to measured V(I)-curve B = B + ΔB end Change the sample temperature and repeat the while loop. As a result we get Ic(B,T) and n(B,T) data of the superconducting sample, after which the V(I)-behaviour of the sample can be modelled with the power law. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 8 Critical current of a superconducting magnet Critical current of a superconducting magnet, Icm, is the maximum current that enables a stable operation at certain temperature. The computational determination of Icm for a certain coil geometry requires that: • short-sample data of the wire is available. • magnetic flux density distribution in the coil has to be computed. Basically the determination of Icm requires the solution of heat conduction equation inside a superconducting coil: ∂T ∇ ⋅ ⎡⎣λ (T ) ∇T ⎤⎦ + Q (T , I op ) = Cp (T ) , ∂t where λ and Cp are the thermal conductivity and the volumetric heat capacity of the coil, and Q is the heat generation inside the coil. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 9 Basic differences between LTS and HTS magnet design Resistive transition in LTS is much steeper than in HTS. • Typical value for exponent n is about 100 for LTS and about 10 for HTS. • In practice a steep transition means explosive heat generation with overcritical currents. Ic(B)-dependence of LTS wire is isotropic, but in HTS wires the situation is anisotropic. • In isotropic situation Ic depends only on the magnitude of B. • In anisotropic situation Ic depends on the magnitude and direction of B. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 10 Critical current of LTS magnet (1/3) The determination of Icm for LTS magnet simplifies to: • determine the critical current of the coil turn that is exposed to the highest magnetic flux density. We don't have to care about the solution of heat conduction equation inside the coil, because the resistive transition in LTS is so steep that any cooling cannot tolerate the explosive heat generation at overcritical currents. We don't have to care about the direction of magnetic flux density, because the Ic(B)dependence of LTS wires is isotropic. So, in order to determine Icm for LTS magnets, we have to compute the magnitude of B applied to each turn of the coil at certain operation current Iop. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 11 Critical current of LTS magnet (2/3) In solenoidal coil the maximum of magnetic flux density, Bmax, is located at the axial mid-plane of the coil's inner radius (red-coloured turns in the figure). If there is not any magnetic material in the vicinity of the coil, the dependence between Bmax and Iop is linear. • Iop(Bmax) is called a load line. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 12 Critical current of LTS magnet (3/3) The critical current of LTS magnet is found from the intersection of load line and shortsample based Ic(B)-dependence of the wire in the coil. • Typically 80-90% of the short-sample Ic is achived, when the wire is wound as a solenoidal coil. In adjacent figure the intersection of load line and Ic(B)-dependence of the wire in the coil is at the point (4.8 T, 264.4 A). In other words: • the operation current of 264.4 A is the critical current of the coil. • when the operation current of the coil is 264.4 A, the magnetic flux density in the mid-plane of the inner radius is 4.8 T. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 13 Critical current of HTS magnet (1/6) Preceding method to determine the critical current of LTS magnet cannot be applied to HTS magnet, because: • HTS materials have slanted resistive transition due to which HTS magnets can be momentarily operated with overcritical currents. • HTS materials have anisotropic Ic(B)-dependence due to which the critical current of the magnet cannot be determined by investigating a single turn of the coil. In LTS magnets, the loss of superconductivity typically occurs, when critical current is exceeded in the coil turn exposed to the highest magnetic flux density. In HTS magnets, the loss of superconductivity is viewed as a global temperature increase inside the coil rather than in a single turn. In the following we concentrate on determining the critical current of a solenoidal Bi-2223/Ag coil. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 14 Critical current of HTS magnet (2/6) Because Ic of HTS wire depends on the direction of B, the short-sample data of HTS wire is more complicated than the one of LTS. Adjacent figure presents Ic(B)- and n(B)-dependence at 20 K for Bi-2223/Ag wire manufactured by American Superconductor. • Curves from top to bottom represent the orientation of B from parallel (BII) with an interval of 10o to perpendicular (B⊥). Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 15 Critical current of HTS magnet (3/6) In solenoidal coil the maximum of magnetic flux density, Bmax, is located in the axial midplane of the coil's inner radius. • However, due to anisotropic Ic(B)-dependence, the point of Bmax is not the weak spot in HTS coils. • Typically the lowest Ic in HTS coils is found from the turn that is exposed to the highest perpendicular magnetic flux density B⊥ (in solenoidal coil near the coil flanges). Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 16 Critical current of HTS magnet (4/6) When the critical current and exponent n of each coil turn is known, the electric field E in n each coil turn can be computed from the power law: ⎛ I op ⎞ E = Ec ⎜ ⎟ . ⎝ Ic ⎠ When electric field E and current density J of each coil turn are known, the power loss density Q of each coil turn can be computed: Q = EJ. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 17 Critical current of HTS magnet (5/6) In order to accurately determine the critical current of HTS coil, the heat conduction equation inside the coil has to be solved: ∂T ∇ ⋅ ⎡⎣λ (T ) ∇T ⎤⎦ + Q (T , I op ) = Cp (T ) . ∂t The solution cannot be found analytically, but numerical approach is required. Finite Element Method (FEM) is one potential method to solve partial differential equations numerically. The idea is that the coil is operated with constant Iop, and then the temperature distribution inside the coil is computed by solving the heat conduction equation as a function of time. • If the minimum temperature inside the coil starts to increase ⇒ Iop > Icm. • If the maximum temperature inside the coil starts to decrease ⇒ Iop ≤ Icm. Thus, the magnet critical current, Icm, has to be searched iteratively in case of HTS. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 18 Critical current of HTS magnet (6/6) The accurate determination of HTS magnet critical current requires the solution of heat conduction equation inside the coil. This is a quite complicated computational task. However, the critical current of HTS magnet can be estimated by monitoring the average electric field, Eave, between the coil terminals: Eave = V/l, where V is the voltage between the coil terminals, and l is the length of the wire in the coil. These operation temperature dependent critical current estimates for Bi-2223/Ag coil are: • 20 K: Iop ≈ Icm, when Eave = 0.1 μV/cm. • 77 K: Iop ≈ Icm, when Eave = 0.3 μV/cm. • 4.2 K: Iop ≈ Icm, when Eave = 0.01 μV/cm. However, it has to be emphasized that these critical current criteria are only estimates and apply only for properly cooled Bi-2223/Ag coils. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 19 Summary: critical current of a superconducting magnet Generally speaking, the accurate determination of critical current of a superconducting magnet requires the solution of heat conduction equation inside the coil. However, this is unnecessarily complicated way to solve Icm in case of LTS, because: • LTS wires have very steep resistive transition and isotropic Ic(B)-dependence. ⇒ Icm of LTS coils can be determined as Ic of the coil turn exposed to Bmax. The accurate determination of Icm in case of HTS requires the solution of heat conduction equation, because: • HTS wires have slanted resistive transition and anisotropic Ic(B)-dependence. • Icm of HTS coil can be estimated with some voltage-based criteria, but these are always temperature and material dependent. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 20 Optimization of coil geometry (1/4) So far we have only been dealing with the determination of magnet critical current Icm, which is a fundamental parameter related to superconducting magnet design. Other, usually application-dependent, design parameters are: • coil volume (in order to save superconducting wire and hence money) • stored energy (for example is SMES), • geometry requirements (for example in MRI), • magnetic flux density distribution requirements (for example in MRI), • losses in AC use (for example in SMES), • etc... Generally speaking, the design of a superconducting magnet is a nonlinear optimization problem. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 21 Optimization of coil geometry (2/4) The aim of superconducting magnet design is to find an optimal coil geometry for a given application. • For example in solenoidal coil the task is to find optimal values for inner radius a, outer radius b and axial length d. The meaning of "optimal values" depends on the application. • If we for example want to minimize the coil volume and have the magnetic flux density of 5 T in the center of the coil bore, then we try to find such a, b and d that produce the required magnetic flux density with the smallest possible coil volume. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 22 Optimization of coil geometry (3/4) The design of a superconducting magnet can be formulated as a constrained nonlinear optimization problem. Let's consider the following task to design a solenoidal coil for superconducting magnetic energy storage (SMES) application: • the coil should be as small as possible. • the stored energy, W, should be at least 200 kJ when operated with Icm. • the maximum magnetic flux density, Bmax, in the coil is 5 T when operated with Icm. Coil design can be stated as a constrained optimization problem: minimize V(a,b,d) subject to g(a,b,d,Icm) ≤ 0, where V is the coil volume, and g is the vector of the constraint functions. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 23 Optimization of coil geometry (4/4) Thus, the design task is to minimize V(a,b,d) such that: • W(a,b,d,Icm) ≥ 2 × 105 J (constraint function g1), • Bmax(a,b,d,Icm) ≤ 5 T (constraint function g2). In other words the vector of constraint functions can be written as: ⎡ g1 ( a, b, d , I cm ) ⎤ ⎡ 2 × 105 − W ( a, b, d , I cm ) ⎤ g ( a, b, d , I cm ) ≤ 0 ⇔ ⎢ ⎥. ⎥=⎢ g a , b , d , I − B a , b , d , I 5 ( ) ( ) cm ⎦ cm ⎣ 2 ⎣ max ⎦ There are several algorithms to solve such optimization problems. One of the most effective ones is sequential quadratic programming (SQP) that is an iterative optimization algorithm for constrained nonlinear problems. • "Iterative" means that an initial value for the coil geometry has to be set in the beginning of iteration, after which the algorithm tries to find a feasible solution that minimizes the coil volume. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 24 Optimization of coil geometry: case of 200 MJ Nb3Sn SMES coil The task was to minimize the volume of a solenoidal coil such that: • the strored energy should be at least 200 kJ when operated with Icm. • the maximum magnetic flux density, Bmax, in the coil is 5 T when operated with Icm. • the maximum tolerable AC losses during charging the coil (30 s) is 1.5 W. • the maximum stress values are 100 MPa for hoop stress and 50 MPa for radial stress. Adjacent figures show the development of the solution during iteration with SQP. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 25 Summary: optimization of coil geometry Usually superconducting magnet design means that the coil geometry has to optimized for a certain application. Regardless of the application, the design process can be formulated as a nonlinear optimization problem, which can then be solved with some optimization algorithm. In order to build computer aided tools for the design of superconducting magnets, a software for solving partial differential equations by means of finite element method is practically required. • For my own PhD research the partial differential equations of field computations and thermal analysis were solved with Comsol Multiphysics. The optimization problem was solved with SQP, which was a built-in package in the Optimization toolbox of Matlab. Generally speaking, the design of a superconducting magnet is a nonlinear optimization task with some particular constraints set by the special features of superconductivity. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008 26 Thank you for your attention. Electromagnetics European Summer School on Superconductivity, 11-18 June 2008, Pori, Finland 25.6.2008