“Permanent Magnet Synchronous Motor (PMSM) “;
5.1 Principle of operation
The Permanent Magnet Synchronous Motor (PMSM) is an AC synchronous motor whose field excitation is provided by permanent magnets but has a sinusoidal Back EMF wave form.
The PMSM is a close relative of the brushless DC (BLDC) motor. Both motors have a permanent magnet rotor and windings on the stator. However the PMSM motor is constructed such that the back EMF waveforms of the windings are sinusoidal.
The principle difference in controlling these two motors is the kind of drive signals that are supplied to the motor from the inverter.
A BLDC motor is controlled with trapezoidal waveforms, while a PMSM motor is controlled using sinusoid waveforms to match the back EMF waveform of each motor’s windings.
The operation of a PMSM is similar to that of a traditional synchronous motor excited by constant field current. When sinusoidally varying three-phase voltage is supplied to the armature winding, sinusoidally varying currents flow through the windings.
The armature conductors are distributed in such a way that the phase currents have an angular shift of 120° in space. The three-phase current produces three-phase mmfs which are also sinusoidally distributed with 120° phase differences from each other. The resultant mmfs have constant magnitude and rotate at constant speed in space.
The speed of the rotating armature field is given by 2f/P rps, where f = supply frequency and P = number of poles. This speed is known as synchronous speed.
Consider a three-phase, two-pole PMSM. The resultant mmf produced by the armature current can be represented by stator poles. At start, when the supply is given to the terminals of the PMSM, the rotor and stator magnetic poles are at the positions shown in figure(a).
At these positions, the rotor poles are attracted by the stator poles and the rotor has a tendency to move in the clockwise direction. During the next half cycle of the armature current (after 0.01 sec for a 50-Hz supply), the stator poles get interchanged and have the locations shown in figure(b). In this situation, the rotor and stator poles repel and the rotor tries to move in counterclockwise (CCW) direction.
During the next half cycle, the stator poles assume the position shown in figure(c) and produce magnetic force around the rotor poles. Thus for every half cycle, the torque changes its direction.
Due to inertia of the rotor and the fast reversal of the developed torque. The rotor cannot move and rests in the same position. Or in other words, the PMSM is not self-starting.
If we bring the motor speed to the speed of resultant armature (synchronous speed) by external means or by using damper winding, the rotor and stator poles get interlocked and rotor continues to run at synchronous speed in the direction of revolving magnetic field.
Relative position of the rotor and stator poles for half cycles
5.2 Ideal PMSM – EMF and Torque equations
Torque equation
The force on the elementary group of ampere-conductors is given by,
.
Together with the corresponding force on the opposite element, this force produces a couple 2Fr1 on the stator.
An equal and opposite couple acts on the rotor and the total electromagnetic torque on the rotor is the integral of the elementary contributions over the whole airgap periphery over p pole-pairs.
By integrating and substituting the rotating magnet flux distribution is
we get the torque equation as,
where,
The angle β is called the torque angle and is positive for motoring, it is measured in electrical radians or degrees.
Emf Equation
The e.m.f equation of the sinewave motor can be derived by considering the e.m.f induced in the elementary group of conductors in figure. Noting that figure is drawn for a two-pole machine (p= 1), for a machine with p pole-pairs in series this e.m.f. is,
By integrating the contributions of all the elementary groups of conductors, we get the instantaneous phase e.m.f as,
The phase e.m.f. is therefore,
and the line-line e.m.f. is E√3.
This equation is used to derive the fundamental e.m.f. for a practical winding in slots and practical expressions for NS are derived.
It is worth emphasizing again that the flux-density is the peak air gap flux-density produced by the magnet acting alone. In other words, it is the open-circuit value and does not include any contribution due to the m.m.f. of the stator currents.
The e.m.f. equation can also be derived From Faraday's law.
This alternative method is included here because it is the basis of the phasor diagram and provides the means for calculating the inductive volt drop due to armature reaction.
Faraday's law is more rigorous than the BLV formulation but it is useful to show that T or E or both methods gives the same result.
By Faraday's law, the instantaneous e.m.f. induced in the stationary phase winding of figure is given by,
.
where ψ is the instantaneous flux-linkage.
To calculate the flux-linkage, consider the coil formed by the elementary group of conductors within the angle dθ at angle θ and assume that the return conductors of this coil are located within the angle dθ at angle -θ.
The results are derived for p pole-pairs. On open-circuit there is no current in the coil and all the flux is due to the magnet.
The flux through the elementary coil is,
where D = 2rx is the stator bore.
The flux per pole can be extracted from this expression by setting θ = ∏/p and t = 0. Thus,
. This is a fixed flux that rotates with the rotor.
The flux-linkage of the elementary coil is,
The total flux-linkage of the winding is obtained by integrating the contributions of all the elementary coils with p pole-pairs then the result is,
The subscript 'M' has been added as a reminder that the flux is produced only by the magnet.
By Faraday's law, the instantaneous phase e.m.f. is given by,
5.3 Armature reaction MMF – Synchronous Reactance
Inductance of phase winding
Figure shows the single-phase, two-pole sine-distributed winding. The flux is now produced by the current in the stator winding and we assume that the magnet is unmagnetized while we calculate the inductance by determining the flux-linkage of the winding due to its own current, i.
If the steel in the rotor and stator is assumed to be infinitely permeable then the m.m.f. is concentrated entirely across the two air gaps.
Across each air gap, the m.m.f. drop is equal to one-half of the ampere conductors enclosed within an 'Ampere's law contour' or flux-line.
The flux-density across the gap and the magnet is assumed to be radial and the magnet is assumed to be equivalent to an air gap of length , as before. It gives an 'effective air gap'.
The subscript 'a' has been added to the peak air gap flux-density to denote that it is generated by armature current.
Calculation of armature reaction m.m.f. of sine wave winding.
By integrating the flux-density around the periphery of the air gap, the fundamental armature reaction flux per pole can be determined as,
where, is the stator bore.
This expression has exactly the same form as the flux per pole of the magnet and therefore it produces a flux-linkage given by,
The self-inductance is obtained as the flux-linkage per ampere. With NSturns in series per phase as,
The inductance is only half the value which would be obtained with the same number of turns concentrated into one pair of slots spanning 180 electrical degrees.
The inductance calculated above is the actual ‘air gap’ inductance, i.e. the value which would be measured with the rotor stationary and unmagnetized with the other phases open-circuited and with negligible leakage inductance from the slots or the end-turns.
The actual inductances of three simple full-pitch windings with one, two and three slots per pole per phase were calculated on the same basis.
The sine-distributed winding is a hypothetical case with an infinite number of slots per pole per phase, the number of conductors per slot being modulated sinusoidally to produce a sine-distributed airgap m.m.f.
Synchronous reactance
Three sine-distributed phase windings carrying balanced three-phase sinusoidal currents produce a sine-distributed ampere-conductor distribution represented by the expression.
This rotating flux wave established by armature reaction generates voltages in all three phases.
In each phase, the voltage is proportional to I and is therefore regarded as the voltage drop XSI across a fictitious reactance, the 'synchronous reactance' Xs.
By substituting the peak flux-density into the expression derived earlier for e.m.f., and dividing by I and we get,
This expression applies to an ideal two-pole sine-distributed three-phase winding with NS turns in series per phase and it neglects the leakage inductance of the slots and end-turns.
To obtain a practical formula for a real winding, we must first find an effective value for the sine-distributed turns. This is done by means of Fourier analysis and winding factors.