Hardware Reference
In-Depth Information
then the currents i a , i b and i c form a 3-phase sinusoidal current set, which is
symmetric in the time domain. In this case, the airgap MMF generated by the
windings changes to
F g (θ,t)=F a1 (θ,t)+F b1 (θ,t)+F c1 (θ,t)
(4.37)
3KI m
2
=
cos(ωt −pθ)=F m1 cos(ωt −pθ)
Equation 4.37 is a typical expression of traveling wave, and its phase angle
τ is determined by
τ = ωt −pθ.
(4.38)
Therefore, if an MMF value or a phase angle τ is given at certain time
instant t, the position of the traveling wave θ at that phase angle can be
determined using equation 4.38. It is clear that this position θ is a function of
time.
For a gi ven τ,differentiating both sides of equation 4.38 yields
0=ω−p
dt
(4.39)
that is,
Ω =
dt = ω
p = 2πf
,
(4.40)
p
where Ω is the angular speed of the traveling wave whose unit is radian per
second, and f the frequency of the current in Hz (Hertz).
Conventional practice in engineering expresses rotational speed in units of
revolution per minute or RPM and uses the symbol n to describe it. From
4.40, it can be shown that,
n = 60Ω
= 60f
.
(4.41)
p
It shows that, the rotational speed of the MMF or of the airgap fl ux density
is proportional to the frequency of the current, and inversely proportional to
the pole-pair of the magnetic fi eld. In other words, a rotating magnetic fi eld
can be generated by inputting the 3-phase time-symmetric alternating currents
into the 3-phase space-symmetric windings. This is an electrical method to
realize the rotational fi eld. The fi eld speed can be adjusted by changing the
frequency of the current. Otherwise, for a given frequency of the current in
the winding, the speed of the rotating fi eld can be designed with a suitable
magnetic pole-pair in the motor design stage.
We can also know from equation 4.31 that, besides the fundamental MMF,
the winding-A, winding-B and winding-C also contain harmonic fi elds. It is
not diﬃcult to prove that, when the input currents are symmetric in the time
domain, the m th order harmonic fi elds produced by the 3-phase winding can
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