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Flux www.cedrat.com Example Creation date Characterization of a ferromagnetic material 2009 Author : Pascal Ferran - Université Claude Bernard Lyon Ref. FLU2_MH_MAG_01 Program Dimension Version Physics Application Work area Flux 2D 10.3 Magnetic Steady state Magnetic FRAMEWORK Presentation General remarks In this example, we determine the characterization of materials with the method of the torus. To do so, we’ll consider a coil torus with 2 (primary and secondary) coils. The torus is made of a material characterized by a model defined by a combination of a straight line and a curve in tangent arc. Js and µr are the parameters of this model. A sinusoidal current is injected in the primary coil and an induced voltage is measured at secondary coil. The magnetic field H corresponds to the current, the magnetic induction field B corresponds to the voltage. With the resulting B(H) curve, we can characterize the material. We will compare the original B(H) curve with the one “measured” with the torus. Objective - Computation of the magnetic induction in the torus from the voltage at the terminals of the secondary coil - Computation of the magnetic field H in the ferromagnetic torus The parameters the user can change are : Average relative permeability of the magnetic material (MUR) Magnetic polarization at material saturation (JS) Maximum current (I_MAX) injected in the primary coil The number of spires of the primary and secondary coils (NP, NS) Theoretical reminders See Annex CEDRAT S.A. 15, Chemin de Malacher Inovallée – 38246 MEYLAN Cedex (France) – Tél : +33 (0)4 76 90 50 45 – Email : [email protected] Flux Properties - Average rated radius of the torus: R_MOY = 28.75 mm - Rated number of spires of the primary and secondary coild: Np = 4000 spires and Ns = 1000 spires - Rated characteristics of the torus material : Js = 1.2 T and µr = 500 - Maximal peak value of the current injected in the primary coil : I_MAX = 1 A - Current source frequency = 1 Hz Illustration Main characteristics Some results … Material B(H) curve (all the parameters are rated) PAGE 2 Title Flux FRAMEWORK Evolution of the Vs (V) voltage according to the inducting current (A) (all the parameters are rated) To go further … - Title Impact of an airgap in a magnetic circuit with a toric shape Study of current sensors Characterization of materials with the method of the torus in 3D … PAGE 3 MODEL IN FLUX Flux MODEL IN FLUX Domain Dimension 2D Depth DEPTH Infinite Box Disk Length unit. mm Angle unit. degrees Size Out. Radius : 150mm Periodicity In. radius : 100mm Symmetry Characteristics none Repetition number : Offset angle : Even/odd periodicity Application Steady magnetic Properties Geometry / Mesh Full model in the FLUX environment 2nd order type Mesh Mesh Number of nodes 10055 Input Parameters Name DEPTH ALPHA I_MAX Js MUR Np Ns FREQ PAGE 4 Type Geometrical Physical Physical Physical Physical Physical Physical Physical Description Problem’s depth Current variation coefficient Current maximale intensity Magnetic polarization at material saturation Material Average relative permeability Number of spires of the primary coil Number of spires of the secondary coil Injected current frequency Rated value 7 mm 1 1A 1.2 T 500 4000 spires 1000 spires 1 Hz Title Flux MODEL IN FLUX Material Base NAME B(H) model Magnetic property J(H) model Electrical property D(E) model Dielectric property K(T) model K(T) characteristics RCP(T) model RCP(T) characteristics MATERIAL Analytical isotropic saturation MUR - Js Regions NAME Nature AIR Surface region Air region or vacuum - INFINITE Surface region Air region or vacuum - COMPONENT Surface region Non conductive magnetic region MATERIAL - PN Surface region Coil conductor type region - - - - COILCONDUCTOR_1 Electrical characteristics - - - Current source - - - Np spires – Negative current orientation - Thermal characteristics - - - - Possible thermal source - - - - NAME Nature PP Surface region SN Line region Type Coil conductor type region Coil conductor type region Material Mechanical Set Corresponding circuit component - - SP Line region Coil conductor type region - COILCONDUCTOR_1 COILCONDUCTOR_2 COILCONDUCTOR_2 Current source Np spires – Positive current orientation - Ns spires – Negative current orientation - Ns – Positive current orientation - Thermal characteristics - - - Possible thermal source - - - Type Material Mechanical Set Corresponding circuit component Electrical characteristics Title PAGE 5 MODEL IN FLUX Flux Mechanical Set Fixed part : Compressible part : Type Characteristics Miscellaneous Mobile part : Type of kinematics Internal characteristics: External characteristics : Mechanical stops Electrical circuit Component Type COILCONDUCTOR_1 Coil conductor COILCONDUCTOR_2 Coil conductor RESISTOR Resistance Characteristics Imposed current : I_MAX x ALPHA Coil conductor associated to a circuit 1 M Ohms Associated Region PN PP SN SP - Electric scheme Solving process options Type of linear system solver Type of non-linear system solver Automatically chosen Parameters Precision Newton Raphson Automatically defined 0.0001 Method for computing the relaxation factor Nb iterations 100 Automatically defined Thermal coupling Advanced characteristics Solving PAGE 6 Title Flux Scenario SCENARIO_1 MODEL IN FLUX Name of parameter ALPHA Duration of the solving Title Controllable parameter Physical 1 min 30s Variation method Steps list Interval definition Step selection 0.0 to 1.0 0.000 0.020 0.075 0.200 0.400 0.600 0.800 1.000 Operating System - 0.010 0.050 0.100 0.300 0.500 0.700 0.900 - Windows XP 32 bits PAGE 7 ANNEX Flux ANNEX Theoretical reminders Magnetic induction measurement In the magnetic circuit, the voltage at the terminals of the secondary coil can be measured. Then we can deduct the average flux in the magnetic circuit from the below relation: vs NS d dt avec B S Then : B vs H computation 1 NS 2 f S Let’s calculate H with the Amoere theorem formula : H dl j Nj ij c In this case, the current circulating in the secondary coil is 0 and we integrate H along the average contour of the torus (average radius = R_MOY). Obtained result: ALPHA I _ MAX NP H 2 R _ MOY From which we get : H Magnetic material characterization ALPHA I _ MAX NP 2 R _ MOY In Flux, a material can be characterized by the following magnetic property: « Analytical isotropic saturation (arctg, 2 coef.) ». The corresponding mathematical expression of the B(H) is : PAGE 8 Title Flux Notations and symbols ANNEX symbol description unit Voltage at the secondray coil terminals Average Flux in the magnetic circuit Magnetic induction field Torus section Inductive current signal frequency Magnetic field Peak value of the current circulating in the primary coil Average radius of the torus Number of spires of the primary coil Number of spires of the secondary coil Coefficient to adapt the current value to its maximal value vs B S f H I_MAX R_MOY NP NS ALPHA V Wb T m² Hz A/m A m Numerical applications Presentation Analytical computation of different magnitudes seeked for the following working point : - Determination of H Js = 1.2 T - µr = 500 R_MOY = 28.75 mm Np = 4000 spires – Np = 1000 spires I_MAX = 1 A ALPHA = 0.05 It is now possible to determine the value of H for the working point defined by applying the Ampere theorem. H Determination of B (Flux formula) Magnetic material property : Torus average radius : Number of spires : Maximale intensity : Adjustment coefficient : ALPHA I _ MAX NP 0.05 1 4000 1107 A / m 2 R _ MOY 2 28.75 10 3 From the H value, it is possible to determine the corresponding value of B : ( µr 1 ) µ0 H arctg 2 Js (500 1) 4 10 7 1107 2 1.2 7 B 4 10 1107 arctg 2 1 . 2 B µ0 H 2 Js B 0.56 T Title PAGE 9