A simplified approach to choosing number and diameter of strands in litz wire is presented. Compared to previous analyses, the method is easier to use. The parameters needed are only the skin depth at the frequency of operation, the number of turns, the breadth of the core window, and a constant from a table provided in the paper. In addition, guidance is provided on litz wire construction-how many strands or sub-bundles to combine at each twisting operation. The maximum number of strands to combine in the first twisting operation is given by a simple formula requiring only the skin depth and strand diameter. Different constructions are compared experimentally.

Abstract-The losses of realistic litz wires are characterized while explicitly accounting for their construction, using a procedure that computes the current-driven and magnetic-field-driven copper losses using fast numerical simulations. We present a case study that examines loss variation in one-and two-level litz wires as a function of twisting pitch, over a wide range of values and in small increments. Experimental confirmation is presented for predictions made by numerical simulations. Results confirm the capability and efficiency of numerical methods to provide valuable insights into the realistic construction of litz wire.

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Semidefinite programs (SDPs) are powerful theoretical tools that have been studied for over two decades, but their practical use remains limited due to computational difficulties in solving large-scale, realistic-sized problems. In this paper, we describe a modified interior-point method for the efficient solution of large-and-sparse low-rank SDPs, which finds applications in graph theory, approximation theory, control theory, sum-of-squares, etc. Given that the problem data is large-and-sparse, conjugate gradients (CG) can be used to avoid forming, storing, and factoring the large and fully-dense interior-point Hessian matrix, but the resulting convergence rate is usually slow due to ill-conditioning. Our central insight is that, for a rank-k, size-n SDP, the Hessian matrix is ill-conditioned only due to a rank-nk perturbation, which can be explicitly computed using a size-n eigendecomposition. We construct a preconditioner to "correct" the low-rank perturbation, thereby allowing preconditioned CG to solve the Hessian equation in a few tens of iterations. This modification is incorporated within SeDuMi, and used to reduce the solution time and memory requirements of large-scale matrix-completion problems by several orders of magnitude.Assumption 1 (Nondegeneracy). We assume:1) (Slater's condition) There exist X 0, y, and S 0, such that A i • X = b i and i y i A i + S = C. 2) (Strict complementarity) rank (X ) + rank (S ) = n.These are generic properties of SDPs, and are satisfied by almost all instances [10]. Note that Slater's condition is satisfied in solvers like SeDuMi [11] and MOSEK [12] using the homogenous self-dual embedding technique [13].We further assume that the data matrices A 1 , . . . , A m are structured in a way that allow certain matrix-implicit operations to be efficiently performed.Assumption 2 (Sparsity). Define the matrix A [vec A 1 , . . . , vec A m ]. We assume that matrix-vector products with A, A T and (A T A) −1 may each be applied in O(m) flops and memory.

Clique tree conversion solves large-scale semidefinite programs by splitting an n × n matrix variable into up to n smaller matrix variables, each representing a principal submatrix. Its fundamental weakness is the need to introduce overlap constraints that enforce agreement between different matrix variables, because these can result in dense coupling. In this paper, we show that by dualizing the clique tree conversion, the coupling due to the overlap constraints is guaranteed to be sparse over dense blocks, with a block sparsity pattern that coincides with the adjacency matrix of a tree. In two classes of semidefinite programs with favorable sparsity patterns that encompass the MAXCUT and MAX k-CUT relaxations, the Lovasz Theta problem, and the AC optimal power flow relaxation, we prove that the per-iteration cost of an interior-point method is linear O(n) time and memory, so an -accurate and -feasible iterate is obtained after O( √ n log(1/ )) iterations in near-linear O(n 1.5 log(1/ )) time. We confirm our theoretical insights with numerical results on semidefinite programs as large as n = 13659.

A fast numerical method is presented for the simulation of complicated 3D structures, such as inductors constructed from litz or stranded wires, above or sandwiched between planar lossy magnetic media. Basing upon its smoothness, the quasistatic multilayer Green's function is numerically computed using finite differences, and its source height dependence is computed using an optimal Toeplitz-plus-Hankel decomposition. We show that a modified precorrected FFT method can be applied to reduce the dense linear algebra problem to near-linear time, and that frequency-dependent setups can be avoided to result in a considerable speed-up. Experimental verifications are made for a 16-strand litz wire coil, realistically modeled down to each individual strand. Results are obtained in 2-3 hours, showing an excellent agreement to measurements, and can be used to study the impact of transposition patterns in litz wire construction.

The definition of treatment 'response' in a progressive neurodegenerative disease can encompass a variety of outcomes, including short-term improvement, longer-term stabilization and a slowed decline in one or more clinically relevant symptoms or symptom domains. The ability to identify groups of people who respond to donepezil underscores the clinical utility of the medication and may contribute to more focused assessments of the cost effectiveness of cholinesterase inhibitors.

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