Topic 2: Fundamentals of Electric Motors Spring 2004 ECE Electric Drives

Generic Electric Motor Below figure shows cartoon of induction motor. Most (but not all) machines have this structure. Ref: J.L. Kirtley, Jr. MIT Course 6.11s (June 2003)Course Notes

Generic Electric Motor (cont’d)  Rotor - mounted on shaft supported by bearings (usually rotor on inside - but not always) - shown with conductors but may have permanent magnets instead - sometimes just an oddly shaped piece of steel (variable reluctance machines)

 Stator - armature (electrical input) on stator (opposite to DC brush motor) - on outside with windings  In most electric motors, rotor and stator are made of highly magnetically permeable materials - steel or iron.  In many common motors, rotor and stator are made of thin sheets of silicon steel (laminations). Punched into these sheets are slots which contain rotor and stator conductors. Generic Electric Motor (cont’d)

 Time-varying magnetic fields passing through ferromagnetic materials (iron/steel) -> eddy currents to flow -> energy loss (power dissipation). Laminations (thin sheets of steel) are used to minimize eddy current losses.  Windings - many turns of Al/Cu conductor concentrically wound about a common axis.  Field winding carries excitation flux.  Armature winding carries electrical power.

Basic Principles of Operation of Electric Motors  Changes in flux linkage between rotor and stator creates torque and therefore relative motion between rotor and stator.  F=q(vxB)  F= l(ixB)

Basic Principles of Operation of Electric Motors (cont’d)

Electrical Radians and Synchronous Speed

Electrical Radians and Synchronous Speed (cont’d) frequency of induced voltage where P= # of poles p=# of pole pairs and N=synchronous speed of rotor (rpm) electrical rads. electrical rads./sec. Hz

Flux per Pole Consider a sinusoidally distributed flux density, B( e )=B pk cos  e. The flux per pole is given by:

Induced Voltage Full-pitched coil w/N turns moving laterally w.r.t. sinusoidal flux density.

Induced Voltage (cont’d) At t=0 coil’s axis coincides w/flux density wave peak. Thus, at time t, flux linked by coil is given by:  induced voltage in full-pitch coil is given by: transformer voltage speed voltage

RMS Value of Induced Voltage RMS value of sinusoidally varying speed voltage term is: In high power ac machines may have distributed or short-pitch windings. Use distribution and pitch factors (k d and k p respectively) to account for these designs. The rms value of the induced voltage under these conditions becomes: where k w =k d k p is the winding factor.

Distribution Factor Phase windings may have series/parallel coils under a different pole-pair. Within each pole-pair region, the coils of a distributed winding are spread out over several pairs of slots.

Distribution Factor (cont’d) The voltages induced in component coils for a single phase winding occupying adjacent slots will be separated by the slot angle separating them  s e (electrical angle subtended by arc between two adjacent slots.)

Distribution Factor (cont’d) The distribution factor can be defined as the ratio of the resultant voltage with coils distributed to resultant voltage if coils were in one location, i.e. k d = Resultant voltage of coils under one pole-pair |E pole | Arithmetic sum of coil voltages  i |E ci | If a phase winding has q coils/phase/pole, |E pole | = 2R E sin(q s e /2) and |E ci |=2R E sin( s e /2), and

Pitch Factor Short-pitching is when coils with less than one pole-pitch are used.

Pitch Factor (cont’d) Short-pitching is used in machines with fractional-slot windings (non-integral slots/pole or slots/pole/phase) in a double- layer winding arrangement. Allows for a finite set of stampings with a fixed number of slots to be used for different speed machines. Also, short-pitching can be used to suppress certain harmonics in the phase emfs. Although short-pitching also offers shorter end connections, the resultant fundamental phase emf is reduced.

Pitch Factor (cont’d) The pitch factor k p is defined by: k p = Resultant voltage in short-pitch coil Arithmetic sum of voltages induced in full coil With sinusoidal voltages, each coil voltage is the phasor sum of its two coil-side voltages. Thus, for coil a, E ca = E a +E -a =>

Spatial MMF Distribution of a Winding A current i flowing through a single coil of n c turns creates a quasi-square wave mmf of amplitude F 1 given by F 1 = n c i/2.

Spatial MMF Distribution of a Winding (cont’d) The fundamental component of this quasi- square wave mmf distribution is given by: The fundamental component of the airgap flux density in a uniform airgap machine is: where g is the airgap.

Effect of Distributing Phase Coils Consider a three-phase distributed winding with 4 coils each per phase in a 2-layer arrangement in a 2-pole stator.

Effect of Distributing Phase Coils (cont’d) If each coil has n c turns, the sum of the fundamental mmf components produced by coils a1 and a2 is given by: where  e is the angle measured from the a-phase winding and  s is the angle between the center lines of adjacent slots. Similarly for coils a3 and a4,

Effect of Distributing Phase Coils (cont’d) Therefore distribution factor is:

Effect of Short-Pitching Consider a layout of windings that are short- pitched by one slot angle. Let’s consider this to be made of four fictitious full-pitch coils: (a1,-a3), (a4,-a2), (a2,-a1) and (a3,-a4).

Effect of Short-Pitching (cont’d) The fundamental mmf component from these 4 fictitious full-pitch coils is given by:

Effect of Short-Pitching (cont’d) Comparing this expression to: and allowing for the distribution of the four full-pitch coils by a k d of cos( s /2), the factor due to the short-pitching by one slot angle is given by:

Effect of Short-Pitching (cont’d) In general, the expression for the fundamental mmf component of a distributed winding with winding factor k w and a total of n pole turns over a two- pole region is given by:

Effect of Short-Pitching (cont’d) Assuming total phase turns N ph in the P- pole machine are divided equally among P/2 pole-pair regions, number of turns per pole-pair = N ph /P. In terms of Nph, fundamental mmf is: The effective number of full-pitch concentric coils per pole-pair to achieve this same fundamental mmf is:

Winding Inductances Here we derive expressions for self- and mutual winding inductances for the elementary machine shown below.

Winding Inductances (cont’d) Self-inductance of the stator winding,L ss, with N effs turns per pole-pair linking  pole (ignoring leakage inductances) is given by: Similarly, the self-inductance of the rotor winding, L rr, with N effr turns per pole-pair is given by:

Winding Inductances (cont’d) An expression for the mutual inductance between the stator and rotor windings can be obtained by considering the flux linking the windings, rs which is given by: 

Rotating Fields The fundamental component of space mmf for a single-phase winding carrying a sinusoidal current i=I a cost is given by: where is the peak value of the fundamental mmf and  a is the electrical angle measured in the counter- clockwise direction from the winding axis.

Rotating Fields (cont’d) This equation may be rewritten as: Two interpretations: 1) pulsating standing wave 2) two counter-revolving mmf waves of half the amplitude of the resultant; forward component rotates counter- clockwise, reverse component rotates clockwise.

Rotating Fields (cont’d)

In a three-phase machine the axes of the windings are spaced 2/3 apart. Assuming balanced operation (phase currents are of same magnitude) the currents are given by:

Rotating Fields (cont’d) The fundamental airgap mmfs of the three phases are given (in terms of  a ) by:

Rotating Fields (cont’d) The sum of these three winding mmfs is: Therefore the resulting airgap mmfs is a constant amplitude sinusoidal wave rotating wave whose peak coincides with the magnetic axis of the a-phase winding at t=0 and rotates with a speed  in a direction corresponding to the sequence of peaking of the phase currents.

Torque in a Uniform Airgap Machine From basic energy conversion principles, the torque developed in a machine is given by: The co-energy is the complement of the field energy: W fld ’ = i - W fld

Torque in a Uniform Airgap Machine (cont’d) Example of three-phase machine (see text for other approaches/results)