Lecture 6 EE743

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Lecture 6 - EE743
Single-Phase Reluctance Machine
Professor Ali Keyhani
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Single-Phase Reluctance Machine

• Consider a single-phase 1- ? reluctance machine
  shown below,

.
Te
?r
as
?r
as_axis
x
as
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Single-Phase Reluctance Machine

• The equations that describe a single-phase
  reluctance machine are,

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Single-Phase Reluctance Machine

• Note that when ? r 0, the reluctance is maximum
  along the magnetic as_axis.
• The torque expression - Te - is valid for the
  transient and steady-state conditions.

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Single-Phase Reluctance Machine

• Consider the case when iasconstant,

.
Reluctance Te ?r
rotor position
as
Maximum 0 0
as_axis (magnetic axis)
x
as
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Single-Phase Reluctance Machine
.
Reluctance Te ?r
rotor position
as
k
as_axis (magnetic axis)
x
as
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Single-Phase Reluctance Machine
.
Reluctance Te ?r
rotor position
as
0
as_axis (magnetic axis)
x
as
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Single-Phase Reluctance Machine
.
Reluctance Te ?r
rotor position
as
-k
as_axis (magnetic axis)
x
as
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Single-Phase Reluctance Machine
.
Reluctance Te ?r
rotor position
as
Maximum 0
as_axis (magnetic axis)
x
as
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Single-Phase Reluctance Machine

• Assume that there is no external torque on the
  shaft.
• There is a force created to minimize the
  reluctance of the magnetic system.
• For at the time when ias is
  increased from zero to a constant value, the
  torque Te would become .

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Single-Phase Reluctance Machine

• Thus, a positive Te would cause the rotor to
  rotate to . (Te gt 0 when ias ? 0)

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Single-Phase Reluctance Machine

• From the previous graph, we can deduce that
  is stable point of operation for the
  single-phase reluctance machine, that is, the Te
  0 at this position.
• Analysis
• If (decreases very slighty).
  Then, Te gt 0 . The Te increase back
  to .

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Single-Phase Reluctance Machine

• Analysis
• If (increases very slightly).
  Then, Te lt 0 . The Te increase back
  to .
• Therefore, are stable operating
  points.

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Single-Phase Reluctance Machine

• The previous analysis, will led us to ask what
  would happen if the rotor is initially positioned
  at when we increase ias from zero to a
  constant value.
• No external torque is applied to the rotor.
• Ias goes from zero to a constant value
• If

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Single-Phase Reluctance Machine

• Stability Test
• If ?r is increased slightly from zero, Te becomes
  positive, which would cause ?r to increase to a
  stable point ?/2.
• If ?r is decreased slightly from zero, Te lt 0
  (negative torque), which would cause ?r even
  further and the rotor will come to rest at a
  stable point -?/2.

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Single-Phase Reluctance Machine

• Therefore,
• If we set ?r ? ? and increase ias from 0 to a
  constant value.
• The rotor would theoretically remain at rest
  ?r ? ? .
• However, any disturbance, even a slightly one,
  will cause the rotor to rotate away from ?r ? ? .

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Single-Phase Reluctance Machine

• There is a 50-50 chance for the rotor to move
  counter-clockwise, or clockwise.

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Single-Phase Reluctance Machine

• Assume
• and

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Single-Phase Reluctance Machine

• For steady-state operation (?e constant),
• Recall that

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Single-Phase Reluctance Machine

• After some work, we will have,

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Single-Phase Reluctance Machine

• Term A is an average steady-state torque
  produced when ?r0.
• Term A or Term B will produce an average torque
  when ?r ?e or when ?r - ?e .
• ?r rotor or shaft speed
• ?e synchronous speed

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Single-Phase Reluctance Machine

• Let ?r ?e ,

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Single-Phase Reluctance Machine

• The steady-state torque will pulsate at 4 ?e and
  2 ?e , and the average torque is a double-angle
  sinusoidal function of

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Windings in relative motion

• Consider the following machine,

.
Te
?r
as_axis 2
1
?r
as_axis 1
x
1
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Windings in relative motion

• The equivalent magnetic circuit is shown below

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Windings in relative motion
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Windings in relative motion

• If we assume

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Windings in relative motion

• Note
• For the stator, the flux issues from a north pole
  of a magnet into the air toward the south pole
  (stator winding for a positive current).
• For the rotor, the flux produced by a positive
  current flowing in the rotor winding, enters the
  air gap from the north pole of the magnetic
  system of the rotor.

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Windings in relative motion
.
.
Sr
Nr
Ns Ss
Te
Ns Ss
Nr
Sr
s
x
s
x
.
.
Sr Nr
Ns Ss
Nr Sr
Sr
Nr
s
x
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Windings in relative motion

• An electromagnetic torque is produced in attempt
  to align the magnetic system established by the
  currents flowing in the stator and rotor windings
  to align the magnetic axes.
• The stable operation is reached when ?r 0
• The unstable operation is reached when ?r ?

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Windings in relative motion

• If we assumed that
• The, for the steady-state operation,

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Windings in relative motion
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Windings in relative motion

• An average torque is produced if ?e ?r or ?e
  - ?r
• If ?e ?r