A Field Construction Technique to Efficiently Model the Dynamic Vector Forces within Induction Machines Dezheng Wu, Steve Pekarek School of Electrical and Computer Engineering Purdue University May 20, 2008

Motivation for Research Fields-based modeling of machines valuable analysis tool –Investigate slot geometries, material properties –Calculate force vector (radial and tangential) –Readily model induced currents in magnetic material Limitation as a design tool –Numerical computation relatively expensive (compared to magnetic equivalent circuits) Field construction –Attempt to establish a fields-based model while minimizing computation requirements –Use fields-based models more extensively as an early design tool

Induction Machine Rotor conductors are made of solid aluminum or copper bars, connected using end- rings Rotor current is induced due to the change of magnetic field A 3-phase squirrel-cage induction machine

Finite Element Analysis (FEA) Maxwell’s equations. In 2-D FEA, the magnetic field calculate is obtained by solving the nonlinear Poisson’s equation

FEA of Induction Machines Two methods typically used to model transient response of induction machines. — Use static FE model coupled with lumped-parameter circuits. — The induced current in rotor conductors is included as part of a transient FEA model. To derive the transient FEA model, current density is composed of two parts Due to the voltage across the conductor Induced current (neglected in stator winding)

Field Construction – Basic Idea Use two FEA solutions to characterize machine behavior –Create basis functions for stator and rotor magnetic fields ‘Construct’ the magnetic field in the airgap using stator field and rotor basis functions. B n =B ns +B nr B t =B ts +B tr Calculate torque and radial force using the Maxwell Stress Tensor (MST) method under arbitrary stator excitation and rotor speed (assuming linear magnetics)

Assumptions for Field Construction Hysteresis, saturation in the iron is neglected Thermal conditions are assumed constant No deformation occurs in stator and rotor teeth

Stator Basis Function Derivation k ns k ts

Rotor Basis Function (k nr,k tr ) Derivation Impulse Response 1. Set a discrete-time impulse input to a transient FEA program. i as (t) = I 0 when t = t 0 i as (t) = 0 when t ≠ t 0 2. Record the flux density components (B nid, B tid ) for t ≥ t Subtract the stator magnetic field B nr = B nid – i as k ns, B tr = B tid – i as k ts 4. Divided by I 0. k nr = B nr / I 0, k tr = B tr / I 0

Complete Characterization Process

Magnetic Flux Density Due to Stator The flux density generated by arbitrary stator phase-a current is approximated as Due to symmetry, the total flux density generated by stator currents

Magnetic Flux Density Due to Rotor Obtain rotor magnetic field using the convolution of stator current signal and rotor basis function. where x can be ‘n’ or ‘t’. due to i as due to i bs due to i cs

Complete Field Construction Obtain the total flux density in the discrete-time form In the computer, the discrete convolution of the stator current and rotor basis function where x can be ‘n’ or ‘t’.

Example: Field-Oriented Control Simulated conditions: Current Used indirect field-oriented control Electrical rotor time constant τ r =0.226 s Moment of inertia J = 0.03 kg · m 2. Load torque is proportional to the square of rotor speed.

FEA vs FC Results Torque Radial force Speed Simulation Time: Commercial Transient FEA ~ 6 hours Field Construction ~ 3 min

Side Result of Research Analytical derivation of radial force over a pole as a function of rotor speed

Ongoing Research Voltage-Input Field Construction –Enables efficient coupling with drive circuits High Frequency Field Construction –Impulse response characterization requires high sampling rate, over relatively long time Core Loss Construction

The Induction Machine Studied 3-phase 4-pole squirrel-cage induction machine 36 stator slots, 45 rotor slots Rated power: 5 horsepower Rated speed: 1760 rpm Machine parametersValue Airgap1.42 mm Rotor outer diameter mm Stator outer diameter228.6 mm Stack length88.9 mm Shaft diameter39.4 mm Lamination materialM-19 Stator winding material Copper Rotor bar materialAluminum Number of turns per coil 22 Number of coils per phase 6 coils in series connection