Synchronous Machines
The synchronous machine incorporates 3-phase
AC windings which are fed by 3-phase supply.
It
generates
3-phase
rotating
field
circumferentially moves along the stator. The
rotor speed is always the same as the rotating
field speed (so-called synchronous). The rotor
field is independently excited so that the
attraction between rotor field and stator rotating
field generate the desired torque.
(http://industrial-automation-systems.blogspot.com/2010/08/industrial-electric-ac-motor.html)
The rotor is usually wound with concentrated
DC windings and excited by external DC
current to form the desired number of poles via
slip rings and carbon brushes. It causes the
problems of maintenance requirement, bulky
accessories and safety issues.
The key is to make use of 3-phase AC windings
and 3-phase supply to generate the rotating
field.
(www.rfcafe.com/references/Electricity-Basic-Navy-Training-Courses/electricity-basic-navy-training-courses-chapter-16.htm)
Synchronous Machines - 1
Synchronous Machines - 2
A 3-phase winding consists of 3-phase coils
with axes physically separated by 120° elect.
deg. When it is energized by 3-phase currents,
the resultant flux pattern appears to move along
the winding. This travelling flux is called a
rotating magnetic field.
iA
iC
iB
Nomenclature:
A machine with p pole-pairs has 2pp electrical
radians or 360p electrical degrees in its periphery.
S total number of slots
C number of coils
Time
e emf of a conductor
et emf of a turn
p number of pole-pairs
t1 t2 t3
g = 2pp/S slot pitch in electrical radians
T
m number of phases
A ' A ' B B C ' C ' A A B' B' C C A ' A '
t = S/(2p) number of slots per pole (pole pitch)
At t1
N
y coil span (coil pitch)
S
At t2
N
S
s spread of a phase-group in electrical radians
g' = s/g number of slots per phase-group
b = 2p/m angle between emfs of successive phases
At t3
N
g = S/(2pm) number of slots per pole per phase
in electrical radians
S
Synchronous Machines - 3
Synchronous Machines - 4
In one cycle, the flux rotates through two polepitches. If the stator is wound for p pairs of
poles (2p pole-pitches per revolution), the flux
will rotate through 1/p revolution per cycle, or
(f / p) revolutions per second, where f is the
supply frequency. Hence, the synchronous
speed ns is defined as:
Rotational emf in a single turn:
a: The airgap flux density is sinusoidal.
b: The airgap flux density is rectangular.
ns = f / p
The AC windings of synchronous machines are
the same as that of induction machines. Both
operate based on the rotating magnetic field.
Their key difference is the rotor speed of
synchronous machines is always the same as
that of the rotating field but the rotor speed of
induction machines is always different from
that of the rotating field.
Types of synchronous machines:
w Synchronous generator (alternator)
w Synchronous motor
Synchronous Machines - 5
a
b
(Extracted from Say's AC Machines p.83)
Synchronous Machines - 6
Phase grouping:
In grouping c, it yields 6 phases with 60° spread.
An array of conductors in 12 slots is set out in a
diagrammatic development of a double pole-pitch.
Grouping b could not be used with a single-layer
winding for 3 phases, nor could grouping c be used
(a to d are extracted from Say's AC Machines p.84)
In grouping b, each band has a 120° spread, and the
summation of conductor emfs yielding 3 phase-emfs
with a time-phase displacement of 120°.
for 6 phases, because the subdivision of the winding
leaves no further free conductors a pole-pitch apart
to form the return conductors.
Grouping c could be used for 3 phases, if D, E and F
were employed as the completion of A, B and C.
Synchronous Machines - 7
Synchronous Machines - 8
Single-layer windings:
As shown in grouping d, where the phase-bands are
arranged in this way, and the phase sequence A, B,
The number of coils in a single-layer winding is one-
C obtained by re-lettering. The phase-bands follow
half the number of slots available, because each coil
in the order A, C', B, A', C, B'.
side completely occupies one slot: thus C=S/2.
Double-layer windings:
The armatures of nearly all synchronous generators
and motors, and of most induction motors above a
few kilowatts, are wound with double-layer
windings. The number of coils is equal to the
number of slots, i.e. C=S. In general, the windings
are with a whole number of slots per pole per phase
(g an integer) which are called integral-slot
windings.
Synchronous Machines - 9
Synchronous Machines - 10
Integral-slot windings:
The coils are chorded or short-pitched (y<t) to
reduce the end-connections and certain harmonics in
the phase-emfs. Certain slots hold coil-sides of
different phases. Chording reduces the phase-emf,
and the coil-span is rarely less than 2/3 pole-pitch
because additional turns become necessary which
offset the saving of overhang material.
Fundamental emf of integral-slot windings:
Consider L: axial core length, u: tangential speed, n:
rotational speed in rev/s, f1: flux per pole.
e$1 = B$1 Lu
where B$1 = ( p / 2 ) B1 , B1 = f1 / ( tL) and u = 2 ptn
Hence, e$1 = ppnf 1
Since f = pn and E t 1 = 2 e$1 / 2 , the emf per phase
is given by
E ph1 = 4.44f1 fN ph Kw1
a: m=3, t=9, y=9, g=3, g=20°, s=60°.
b: m=3, t=9, y=8, g=3, g=20°, s=60°.
where
Nph: no. of turns per phase,
Kw1: a correction factor so-called winding factor.
The winding factor Kw1 is defined as the product of
distribution factor Kd1 and pitch factor Kp1:
(Extracted from Say's AC Machines p.88)
Synchronous Machines - 11
Kw1 = Kd 1 K p1
Synchronous Machines - 12
Harmonic emfs of integral-slot windings:
Distribution factor:
Similar to the magnitude of fundamental emf, the
A phase comprises a number of coils connected in
nth harmonic emf is expressed as:
series and extended over a spread of s. The phase
E phn = 4.44 f nfn N ph Kwn
emf is the complex or vector (not scalar) summation
f n = nf
of the constituent individual coil emfs. Hence, the
Kwn = Kdn K pn
correction factor taking into account the effect of
distributed windings is expressed as:
Since f n = Bnt n L and t n = t / n , the normalised nth
harmonic flux is expressed as:
Kd 1 =
sin(s / 2)
g ¢ sin( g / 2)
fn 1 Bn 1 B$n
=
=
f1 n B1 n B$1
The percentage content of harmonic flux is 1/n times
the percentage content of harmonic flux density,
both normalised by the fundamentals.
(Extracted from Say's AC Machines p.99)
Synchronous Machines - 13
Synchronous Machines - 14
Pitch (coil-span, chording) factor:
Harmonic winding factor:
If a coil span is not a full pole-pitch, the conductor
Since the harmonic poles of the nth harmonic have a
emfs in the two coil-sides are not directly additive,
pitch only 1/n of the fundamental pole-pitch, the
but form an addition of vectors whose displacement
angles of the harmonic emf will be n times as large.
is e, the angle by which the span departs from its
Thus, the nth harmonic distribution and pitch factors
full-pitch value p. Hence, the correction factor
can simply be written as:
because of the effect of chording is expressed as:
K p1 = cos(e / 2)
Kdn =
sin( ns / 2 )
g ¢ sin( ng / 2 )
K pn = cos( ne / 2)
(Extracted from Say's AC Machines p.98)
Synchronous Machines - 15
(Extracted from Say's AC Machines p.98)
Synchronous Machines - 16
E = E12 + E32 + E52 + L
Eav = E1av +
e = eˆ1 sin wt + eˆ3 sin 3wt + eˆ5 sin 5wt + L
E º Average of e 2
e 2 = å eˆ 2p sin 2 ( pwt ) + å eˆ p eˆq sin( pwt ) sin( qwt )
p
p ¹q
Average of eˆ 2p sin 2 ( pwt )
2p
2
eˆ p
1
2
2
=
e
p
w
t
d
w
t
=
sin
(
)
(
)
ˆ
p
2p ò0
2
Average of eˆ p eˆq sin( pwt ) sin( qwt )
E3av E5 av
+
+L
3
5
Eav º Average of e
1 T /2
ò (eˆ1 sin wt + eˆ3 sin 3wt + eˆ5 sin 5wt + L)dt
T /2 0
2æ2
2
2
ö
= ç eˆ1 +
eˆ3 +
eˆ5 + L÷
3w
5w
T èw
ø
2æ
eˆ eˆ
ö
= ç eˆ1 + 3 + 5 + L÷
pè
3 5
ø
=
= E1av +
E3 av E5 av
+
+L
3
5
Form Factor :
2p
1
=
eˆ p eˆq sin( pwt ) sin( qwt )d (wt ) = 0
2p ò0
eˆ12 eˆ32 eˆ52
+ + +L =
\E=
2 2 2
E12 + E 32 + E 52 + L
Synchronous Machines - 17
E12 + E32 + E52 + L
E
Kf º
=
Eav E + E3av + E5 av + L
1av
3
5
ö
æ
2
2
2
ç
E1 + E3 + E5 + L ÷
p
÷
ç
=
2 2 ç E + E3 + E5 + L ÷
÷
ç 1
ø
è
3
5
Synchronous Machines - 18
Total rms emf:
Problems in AC Windings
(1)
slot stator carrying a 60o spread double-layer winding with 83.3% pitch coils. There are
10 conductors in each slot. If the maximum value of the sinusoidally distributed field
Due to the half-wave symmetry, there are no even
flux is 0.088 Wb, calculate the fundamental distribution and pitch factors and the phase
and line rms values of the generated emf. (0.958, 0.966, 1446V, 2505V)
harmonics. The total rms phase emf is given by
2
2
2
2
2
2
E ph = E ph
1 + E ph 3 + E ph 5 + E ph 7 + E ph 9 + E ph11L
A 3-phase, 4-pole, star-connected synchronous machine driven at 1500 rpm has a 48-
(2)
Determine the distributed and pitch factors for a 3-phase winding with 2 slots per pole
per phase. The coil-span is 5 slot pitches. How do these factors affect the output of the
machine? If the flux density wave in the gap consists of the fundamental and a 30%
3rd-harmonic, calculate the percentage increase in the rms value of the phase voltage
due to this harmonic. (1.28%)
The total rms line emf when in delta connection is
given by:
Eline
D
2
2
2
2
= E ph
1 + E ph 5 + E ph 7 + E ph11 L
(3)
The flux density distribution in the air-gap of a 50 Hz alternator is given by the
expression B = B1m sinq + (B1m/3) sin 3q. The armature winding, which can be
assumed to be uniformly distributed, is a 3-phase, narrow-spread, full-pitched winding
with 120 turns per phase. If the total flux per pole is 0.1 Wb, calculate the rms value of
the emf per phase and the form factor of the phase emf. (2345V, 1.06)
The total rms line emf when in star (wye) connection
is given by:
E line
Y
2
2
2
2
= 3 E ph
1 + E ph 5 + E ph 7 + E ph11L
Synchronous Machines - 19
Synchronous Machines - 20