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