Paul_C_Krause,_Oleg_Wasynczuk,_Scott_D_Sudhoff_Analysis_of_Electric

Machine Windings and Air-Gap MMF

Electromotive Force (EMF) and Torque

• Form of torque when currents i1 and i2 are constant:

  • Te = - K sin(θ) where K is a positive constant.

• Torque visualization: Interaction of magnetic poles from current in conductors.

  • Positive currents i1 and i2 create distinct polarities (N-S) as depicted in diagram.

• Flux behavior:

  • Flux from stator’s north pole (I-I' winding) enters air gap, while rotor's north pole (II-I' winding) does similarly.

• Stable operation occurs over the range of angle A: -π/250 < A < π/2.

Winding Configurations in Machines

• Types of machines:

  • 2-Pole, 3-Phase, Wye-Connected Salient-Pole Synchronous Machine as a basic model.

• Concepts extend to other induction and synchronous machines.

• Stator windings are embedded in slots along the stationary member’s circumference.

  • Each phase (as, bs, CS): Displaced by 120° with respect to others.

• Field winding (fd): Located on the rotor, generating flux as shown.

Symmetrical and Unsymmetrical Machines

• Symmetrical Induction Machine:

  • Identical multiphase stator windings and rotor windings.

• Unsymmetrical Induction Machine:

  • Non-identical multiphase stator windings.

• Coil Configuration: Each coil spans π/2 radians.

  • Coils defined by positive current direction indicated on diagrams.

• Importance of uniform distribution in windings leads to reduced harmonic generation.

Air-Gap Magnetomotive Force (MMF)

• Salient-pole machines typically consist of multiple poles with laminated steel and winding around poles.

• The developed diagram helps visualize and analyze air-gap MMF.

  • Displacements left of the origin are positive; angular velocities and displacements defined accordingly.

Flux Density and Magnetic Field Intensity

• Magnetic field intensity H and flux density B exist primarily in the air gap with relations depending on angular position.

• Ampere's Law helps determine air-gap MMF:

  • Application in described closed paths results in integrated expressions of MMF.

• Gauss's Law ensures net flux across the air gap is zero if MMF average values are satisfied.

Sinusoidal Approximations of MMF

• Air-gap MMF from stators is a coarse approximation of sinusoidal functions.

  • Essential to minimize voltage/current harmonics; distributed configurations lead to closer approximations.

• MMFs derived from balanced three-phase currents rotate in synchronism about the air gap aligned to the’s and s’ axes, represented as sinusoidal functions.

Inductance Relationships and Voltage Equations

• Self-inductance of windings is defined in relation to linked flux and operating current.

• Detailed inductance expressions calculated for both self and mutual inductances within the machine.

  • Example computations provided for both P-pole and 2-pole machines.

• Voltage equations exhibit complexity due to time-varying mutual inductances especially prominent within induction machines.

Summary

• The machine design significantly affects the performance characteristics, particularly air-gap MMF, inductances, and voltage outcomes.

• Further analysis can unveil behaviors under different configurations, leading to optimized designs for desired operational efficiencies.