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 Ó¨ = 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 â€˜pâ€˜be
the number of poles (PM)

â€˜B_{g}â€˜
be the flux density in the air gap in wb/m^{2}.

B_{k}
is assumed to be constant over the entire pole pitch in the air gap (180^{ÔŽ}
pole arc)

â€˜râ€˜ be
the radius of the airgap in m.

â€˜lâ€˜ be
the length of the armature in m.

â€˜T_{c}â€˜
be the number of turns per coil.

â€˜Ï‰_{m}â€˜
be the uniform angular velocity of the rotor in mechanical rad/sec.

Ï‰_{m}=2Ï€N/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 Ï‰_{mt}=0,the
centre of N-pole magnet is aligned with x-axis.

At Ï‰_{mt}=0,x-axis
is along PM axis.

Therefore
flux enclosed by the coli is

Î¦_{max}=B x 2Ï€r/p x l â€¦â€¦â€¦â€¦â€¦â€¦...(4.1)

=flux/pole

Î¦_{max}=rlâˆ«_{0}^{Ï€}
B(Î¸)dÎ¸

=B_{g}
rl[Î¸]_{0}^{Ï€}

=B_{g}rl[Ï€]

At Ï‰_{mt}=0,the
flux linkage of the coil is

Î›_{max}=
(B_{g} x 2Ï€r/p x l)T_{c} Ï‰b-T â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(4.2)

Let the
rotor rotating in ccw direction and when Ï‰_{mt}=Ï€/2, the flux enclosed
by the coil Î¦, Therefore Î»=0.

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

The flux
linkages of the coil changes from B_{g}rlTcÏ€/p at Ï‰_{mt}=0
(i.e) t= 0 t0 Î¸ at t=Ï€/pÏ‰_{m}.

Change of
flux linkage of the coil (i.e) âˆ†Î» is

âˆ†Î»/âˆ†t
=Final flux linkage â€“ Initial flux linkage/time.

=0- (2B_{g}rlTcÏ€/p)/
(Ï€/pÏ‰_{m})

= -(2B_{g}rlTcÏ‰_{m)} â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦...(4.3)

The emf
induced in the coil e_{c}= - dÎ»/dt

e_{c}
=2B_{g}rlTcÏ‰_{m} â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(4.4)

Distribution
of e_{c} with respect to t is shown in fig 4.16

It is
seen that the emf waveform is rectangular and it toggles between + e_{c}
to - e_{c}. The period of the wave is 2Ï€r/pÏ‰_{m} sec and
magnitude of e_{c} is

e_{c}
=2B_{g}rlTcÏ‰_{m} volts â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦...(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 30^{ÔŽ}(i.e.) slot angle Ï’ .

The
magnitude of emf induced in the coil a1A1

e_{c2} =B_{g}rlTcÏ‰_{m}
volts â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(4.6)

The
magnitude of emf induced in the coil a2A2

e_{c2} =B_{g}rlTcÏ‰_{m}
volts â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦...(4.7)

Its emf
waveform is also rectangular but displaced by the emf of waveform of coil e_{c1}
by slot angle Ï’ .

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

e_{c1} +e_{c2} =2B_{g}rlTcÏ‰_{m} â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(4.8)

Let nc be the number of coils that are
connected in series per phase n_{c}T_{c} =T_{ph} be the

number of
turns/phase.

e_{ph}=
n_{c} [2B_{g}rlTcÏ‰_{m} ] â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(4.9)

e_{ph}=
2B_{g}rlTphÏ‰m volts â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(4.10)

e_{ph}=resultant
emf when all nc coils are connected in series.

The
waveforms are as shown in fig 4.17

The
waveform of e_{ph} is stepped and its amplitude is 2B_{g}rlTphÏ‰_{m}
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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Special Electrical Machines : Permanent Magnet Brushless D.C. Motors : Emf Equation of BLPM SQW DC Motors |

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