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
dθ
dt
(4.39)
that is,
Ω =
dθ
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
2π
=
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