QuickField FEA Software Electrical Engineering Theoretical Foundations Additional Chapters S IMON DUBI TS KY H IGHER SCHOOL OF H IGH V OLTAGE E NERGY Autumn 2021 SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 1 Week 8: Static Magnetic Field 1. What is the source of magnetic field? 2.Which potential is used? 3.The governing equation in 2D symmetry. 4.Boundary conditions: Dirichlet, Neumann. 5.Calculated field quantities: locals and integrals 6.Example: applying external magnetic field 7. Example: DC magnetic shielding. SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 2 Source of magnetic field 1. Current 2. Permanent Magnet SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 3 Starting from Maxwell’s equations Integral Form Gauss’ theorem Magnetic flux leaving any closed surface S is zero for magnetic B ds = 0 field S Ampere’s Law Differential Form Conclusion Magnetic field has no sources nor sinks No magnetic charge exists. It allows as introducing of vector magnetic potential A. divB = 0 Magnetic field is created by H-field circulation around a closed Magnetic field is not currents. loop L is equal to free current conservative in areas through the surface S enclosed by L. where the current exists. Magnetic scalar potential Ψ only exists in current-free areas H dl = I rotH = J S Material Equation B = H = 0 (H + M ) SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM The state of material in magnetic field is determined by both external current and material magnetization 4 Which potential we use? Electric Field B rotE = − =0 t Magnetic Field in current-free region Electrostatic field is conservative Scalar potential φ exists: E=-grad φ rotH = J + D t Magnetic field is conservative in current-free area Scalar magnetic potential ψ only exists in current-free areas: H=-grad ψ Magnetic Field everywhere divB = 0 Magnetic field is solenoidal (divergent-free) Vector magnetic potential A always exists: B=rotA SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 5 What happens with vector potential in 2D ? Only out-of-plane component Magnetic Flux A=izAz J=izJz Stokes’ theorem Φ = B ds S = rotA ds = A dl S L B = rotA B=ixBx+ iyBy H=ixHx+ iyHy A Bx = y B = rotA B = − A y x Note! SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 6 Governing Equation of static magnetic field rot( 1 rotA) = J + rotHc ( = f rotA ) H cy H cx 1 A 1 A = − J + + − x y x y x y y x Poisson equation (coordinate independent form both 3D and 2D) 2D in Cartesian coordinates (x, y) (plane-parallel symmetry) 1 (rA) 1 (rA) H cr H cz 2D IN cylindrical coordinates (r, z) + = − J + − r r z r z rr z r (axial symmetry) z SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 7 Magnetic boundary conditions A = f ( x, y ) Plane rA = f ( x, y ) Axisymmetric Dirichlet Condition: the known magnetic potential A (or rA) (may depend on coordinates) 1 A Ht = = ( x, y ) n Neumann condition: the known tangential H-field, equal to surface current density A = const Bn = 0 Zero normal flux (superconductor): A special type of Dirichlet condition SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 8 Magnetic vs. electric boundary conditions E U lines SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM field lines B 9 Magnetic vs. electric boundary conditions Electric field Dirichlet Condition: Neumann condition (zero): Magnetic field Electric field lines are normal to a Dirichlet surface Magnetic field lines are tangential to a Dirichlet surface Electric field lines are tangential to a Neumann surface Magnetic field lines are normal to a Neumann surface SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 10 Calculation results 1. Locals Magnetic flux density (B-Field): B = rotA Magnetic field strength (H-Field): H = 1 B − H c Magnetic energy density: Linear material Nonlinear material BH w= 2 1B w = H ( B ) dB 20 SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 11 Calculation results 2. Integrals Mechanical force: F= 1 2 ( H ( B n ) + B ( H n) − n ( H B ))ds S Mechanical torque: 1 T = ((r H )(B n ) + (r B )(H n) − (r n )(H B ))ds 2S Magneto-motive force (MMF): MMF = H dl L Flux Linkage: 1 = A ds SL SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 12 Example: Lab 5 – DC Magnetic Shielding Steel spherical shield in uniform magnetic field B0 B0=12.6 mT SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 13 Sizing and material properties Sizing: External diameter 54 mm Internal diameter 44 mm Outer diameter 500 mm SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM B, T 0.030 0.206 0.325 0.445 0.535 0.600 0.650 0.685 0.720 0.745 0.770 0.800 0.820 0.840 0.860 H, A/m 50 150 200 250 300 350 400 450 500 550 600 650 700 750 800 14 Z-uniform field boundary condition 1 (rA) BZ = r r B = rotA = ( ) B = − 1 rA R r z Choose as: rA R Bz0=12.6 mT Air = 0.5r2BZ0 Check it by differentiating: 1 r 2 BZ 0 1 2 =B BZ = Z0 r r Axis of rotation Steel Z SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 15 Z-uniform field boundary condition Dirichlet boundary condition: rA = 0.5r2BZ0 SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 16 Shielding Factor Field outside B0=12.6 mT Air gap across the field Field inside shield Bi=0.807 mT The shielding factor: k = 12.6/0.807 = 15.6 Air gap along the field Field inside shield Bi=0.158 mT The shielding factor: k = 12.6/0.158 = 79.7 SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 17 Field created by a permanent magnet Permanent Magnet SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 18 Material Magnetic Properties Magnetic Permeability Examples Free Space: μr = 1 Paramagnetic: μr > 1 Diamagnetic: μr < 1 Tungsten, Cesium, Aluminum, Magnesium, Sodium Soft magnetic: B = f(H), Hc ≈ 10…103 A/m Hard magnetic: B = f(H), Hc ≈ 104 …106 A/m Bismuth, Mercury, Silver, Lead, Graphite, Copper, Water Supermalloy (16Fe:79Ni:5Mo) Permalloy (Fe:4Ni) Elecrtic steel (96Fe:4Si) Ferrite (Zn, Fe, Ni, O) Disk drive recording media (Cr, Co, Pt) Neodimium magnets (Nd, Fe, B) Samaruim Cobalt magnets (SmCo5) SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 19 Typical hysteresis loop SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 20 Hard and Soft Magnetic Materials Hard Soft SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 21 Magnetic properties of Hard Magnetics Definitions by IEC Magnetic remanence Br Coercive (magnetic) field strength Hc Remanent magnetic flux density in a substance, when it departs from magnetic saturation by monotonic reduction to zero of the applied magnetic field strength Magnetic field strength to be applied to a magnetic substance to bring the magnetic flux density to zero. (any intersection with H-axis of the magnetization curve (for flux density, polarization or magnetization) SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 22 Material Magnetic Properties in QuickField Non magnetized state (HC=0) : B=f(H) Magnetized up to maximum (HC=147.3 kA/m) B = f(H+M) SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 23 Material Magnetic Properties in QuickField The modern magnets, both Neodymium and Samarium-Cobalt, are almost linear: The constant permeability μr = 1.05 = const The magnitude and direction of initial magnetization SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 24 Examples of Magnetization Direction Polar Cartesian M α=0 Horizontal M α=45° α=0 Radial α=90° Tangential Sloped 2.38125 M α=90° α=180° Vertical M Horizontal SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 25 Example: Pulling Force of Disc Magnet Manufacturer’s Data: δ Neodymium Disc Magnet N40 3/16"x1/32" Gauss Rating: 12800 gauss Pulling Force: 0.49 lbs Price: 0.06 $ t Ød Geometry d 4.7625 mm (R = 2.38125 mm) t 0.7937 mm δ h/20 =0.04 mm D 100 mm Magnet Exercise: Find pulling force with δ=0.04 mm δ=0.08 mm δ=0.12 mm SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM μr 1.05 Br 12800 Gs = 1.28 T Hc Br / (μ0μr) = 970 kA/m F 0.49 lbs = 1.8 N 26 One Side Magnet: The Halbach Array SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 27 Example 2: Refrigerator Magnet (Triangle Halbach Array) Given: D = 1 mm, H = 1 mm, B = 20 mm, Magnet material - N35, Steel fridge door thickness 1 mm, relative magnetic permeability 1000. Coercive HC=100 000 A/m SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 28 Refrigerator Magnet (Halbach Array) SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 29 Comparing pulled force Manufacturer Data QuickField Calculation Film size = 2cm * 0.5cm = 1cm2 SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 30 How to magnetize the fridge magnet? SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 31 How the magnetizer looks like? SEMINAR 08. MAGNETOSTATICS 1: INTRODUCTION, EXTERNAL FIELD, PM 32