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# Emf Equation of BLPM SQW DC Motors

The basic torque emf equations of the brushless dc motor are quite simple and resemble those of the dc commutator motor.

EMF EQUATION OF BLPM SQW DC MOTORS

The basic torque emf equations of the brushless dc motor are quite simple and resemble those of the dc commutator motor.

The co-ordinate axis have been chosen so that the center of a north pole of the magnetic is aligned with the x-axis at ė© = 0 .the stator has 12 slots and a three phasing winding. Thus there are two slots per pole per phase.

v   Consider a BLPM SQW DC MOTOR

Let ŌĆśpŌĆśbe the number of poles (PM)

ŌĆśBgŌĆś be the flux density in the air gap in wb/m2.

Bk is assumed to be constant over the entire pole pitch in the air gap (180įÄ pole arc)

ŌĆśrŌĆś be the radius of the airgap in m.

ŌĆślŌĆś be the length of the armature in m.

ŌĆśTcŌĆś be the number of turns per coil.

ŌĆśŽēmŌĆś be the uniform angular velocity of the rotor in mechanical rad/sec.

Žēm=2ŽĆN/60 where N is the speed in rpm.

Flux density distribution in the air gap is as shown in fig 4.14.At t=0(it is assumed that the axis of the coil coincides with the axis of the permanent magnet at time t=0).

Let at Žēmt=0,the centre of N-pole magnet is aligned with x-axis.

At Žēmt=0,x-axis is along PM axis.

Therefore flux enclosed by the coli is

╬”max=B x 2ŽĆr/p x l                                          ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”...(4.1)

=flux/pole

╬”max=rlŌł½0ŽĆ B(╬Ė)d╬Ė

=Bg rl[╬Ė]0ŽĆ

=Bgrl[ŽĆ]

At Žēmt=0,the flux linkage of the coil is

╬ømax= (Bg x 2ŽĆr/p x l)Tc Žēb-T                                     ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”.(4.2) Let the rotor rotating in ccw direction and when Žēmt=ŽĆ/2, the flux enclosed by the coil ╬”, Therefore ╬╗=0.

The flux linkages of the coil vary with ╬Ė variation of the flux linkage is as shown above.

The flux linkages of the coil changes from BgrlTcŽĆ/p at Žēmt=0 (i.e) t= 0 t0 ╬Ė at t=ŽĆ/pŽēm.

Change of flux linkage of the coil (i.e) Ōłå╬╗ is

=0- (2BgrlTcŽĆ/p)/ (ŽĆ/pŽēm)

= -(2BgrlTcŽēm)                                          ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”...(4.3)

The emf induced in the coil ec= - d╬╗/dt

ec =2BgrlTcŽēm                                             ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”.(4.4)

Distribution of ec with respect to t is shown in fig 4.16 It is seen that the emf waveform is rectangular and it toggles between + ec to - ec. The period of the wave is 2ŽĆr/pŽēm sec and magnitude of ec is

ec =2BgrlTcŽēm  volts                             ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”...(4.5)

Consider two coils a1A1 and a2A2 as shown in fig 5.15.Coil a2A2 is adjacent to a1A1 is displaced from a1A1 by an angle 30įÄ(i.e.) slot angle ŽÆ .

The magnitude of emf induced in the coil a1A1

ec2 =BgrlTcŽēm volts                   ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”.(4.6)

The magnitude of emf induced in the coil a2A2

ec2 =BgrlTcŽēm volts                       ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”...(4.7)

Its emf waveform is also rectangular but displaced by the emf of waveform of coil ec1 by slot angle ŽÆ .

If the two coils are connected in series, the total phase voltage is the sum of the two separate coil voltages.

ec1      +ec2 =2BgrlTcŽēm   ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”..(4.8)

Let nc          be the number of coils that are connected in series per phase   ncTc  =Tph  be the

number of turns/phase.

eph= nc [2BgrlTcŽēm ]       ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”.(4.9)

eph= 2BgrlTphŽēm volts  ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”..(4.10)

eph=resultant emf when all nc coils are connected in series.

The waveforms are as shown in fig 4.17 The waveform of eph is stepped and its amplitude is 2BgrlTphŽēm volts.

At any instant 2-phase windings are connected in series across the supply terminals as shown in fig 4.18. Assumption

Armature winding is Y connected.

Electronic switches are so operated using rotor position sensor that the resultant emfs across the winding terminals is always = 2 eph.

Amplitude of back emf generated in Y connected armature winding E = 2 eph.

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