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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 Ө = 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 ‘p‘be the number of poles (PM)

‘Bg‘ 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 (180Ԏ pole arc)

‘r‘ be the radius of the airgap in m.

‘l‘ be the length of the armature in m.

‘Tc‘ be the number of turns per coil.

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

ωm=2πN/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 ωmt=0,the centre of N-pole magnet is aligned with x-axis.

At ωmt=0,x-axis is along PM axis.

Therefore flux enclosed by the coli is

Φmax=B x 2πr/p x l                                          ………………...(4.1)

=flux/pole

Φmax=rl∫0π B(θ)dθ

=Bg rl[θ]0π

=Bgrl[π]

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

Λmax= (Bg x 2πr/p x l)Tc ωb-T                                     …………………….(4.2)

Let the rotor rotating in ccw direction and when ωmt=π/2, the flux enclosed by the coil Φ, Therefore λ=0.

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

The flux linkages of the coil changes from BgrlTcπ/p at ωmt=0 (i.e) t= 0 t0 θ at t=π/pωm.

Change of flux linkage of the coil (i.e) ∆λ is

∆λ/∆t =Final flux linkage – Initial flux linkage/time.

=0- (2BgrlTcπ/p)/ (π/pωm)

= -(2BgrlTcωm)                                          …………………………...(4.3)

The emf induced in the coil ec= - dλ/dt

ec =2BgrlTcωm                                             …………………………….(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 2πr/pωm sec and magnitude of ec is

ec =2BgrlTcωm  volts                             ………………………………...(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 30Ԏ(i.e.) slot angle ϒ .

The magnitude of emf induced in the coil a1A1

ec2 =BgrlTcωm volts                   …………………………….(4.6)

The magnitude of emf induced in the coil a2A2

ec2 =BgrlTcωm volts                       …………………………...(4.7)

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

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

ec1      +ec2 =2BgrlTcωm   ………………………………..(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 [2BgrlTcωm ]       ……………………………….(4.9)

eph= 2BgrlTphωm volts  ………………………………..(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 2BgrlTphωm 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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