1
ECSE 462
PRACTICE SESSION IV
Trigonometric Identities
cos(A+B)=cosAcosBsinAsinB
cos(AB)=cosAcosB+sinAsinB
sin(A+B)=sinAcosB+cosAsinB
sin(AB)=sinAcosBcosAsinB
In Lecture 6, a current i
a
of the aphase winding
having an airgap Bfield
cos
aa
m
Bi
b
θ
=
as shown in Fig. 1
90
°
180
°
270
°
360
°
am
ib
B
Fig. 1
can be represented by space vector
a
Φ
.
The space vector has a magnitude proportional to the current i
a
and is orientated in the
direction of the winding axis. The space vectors of aphase and bphase can be summed
vectorially as illustrated in Fig. 2.
Fig. 2
a
Φ
a
i
N
ϕ
b
i
N
b
Φ
N
S
T
Φ
+=
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a
Φ
JG
represents
cos
aa
m
Bi
b
θ
=
b
Φ
represents
cos(
)
m
b
ϕ
=−
T
Φ
represents
cos(
)
TT
BB
ξ
Question 1
Fig. 3
Fig. 3 shows the stator and rotor windings in the
α
β
−
frame. The windings have 2poles.
The stator iron is stationary. The stator currents are:
cos2 .60
sin 2 .60
s
S
s
i
t
I
i
t
π
⎡⎤
=
⎢⎥
⎣⎦
Represent the Bfield by a resultant stator flux vector
S
Φ
. Show that the speed of rotation
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 Spring '09
 BonTekOoi
 Vector Space, Rotation, Euclidean vector, Clockwise, bm cos

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