lecture 21 synchronous machines (1) 2… · lecture 21 synchronous machines (1)...
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Copyright © 2017 Hassan Sowidan
ECE 330
POWER CIRCUITS AND ELECTROMECHANICS
LECTURE 21
SYNCHRONOUS MACHINES (1)
Acknowledgment-These handouts and lecture notes given in class are based on material from Prof. Peter
Sauer’s ECE 330 lecture notes. Some slides are taken from Ali Bazi’s presentations
Disclaimer- These handouts only provide highlights and should not be used to replace the course textbook.
11/26/2017
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Copyright © 2017 Hassan Sowidan
SYNCHRONOUS MACHINES
• The main element in terms of generation of power.
• Range all the way from a few MVA to 1100 MVA.
• Can be operated as either a generator or motor.
• The large majority of applications are as generators.
• The three-phase generators have an AC winding on
the stator with a wye-connection.
• The rotor is excited by a DC field winding
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Copyright © 2017 Hassan Sowidan
SYNCHRONOUS MACHINES
• It is connected to the prime movers, such as steam or
hydro-turbine.
• As motors, synchronous machines are less used
except at low power levels such as permanent
magnet synchronous motors (PMSM).
• In certain cases, synchronous machines at a high
rating are operated to act as power factor correcting
devices. 11/26/2017 3
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Copyright © 2017 Hassan Sowidan
SYNCHRONOUS MACHINES
• We will discuss the fundamental concepts of
deriving torque expressions and only the sinusoidal
steady-state operation using the equivalent circuit.
• For a proper understanding of a three-phase
machine, we will motivate it via the single- and two-
phase machines.
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SYNCHRONOUS MACHINES
Source: emadrlc.blogspot.com
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SYNCHRONOUS MACHINES
Source: pelectric.blogsky.com
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SYNCHRONOUS MACHINES
SALIENT POLE
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SYNCHRONOUS MACHINES
ROUND ROTOR
• Source: en.partzsch.de
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SINGLE-PHASE ROTATING MACHINE
The fundamental component of mutual inductance
will be .
In practical machines, the number of turns are
so positioned on stator and rotor that the higher
harmonics are minimized and the mutual inductance is
largely due to this fundamental component.
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0 1 2 /s rN N L cosM
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Copyright © 2017 Hassan Sowidan
SINGLE-PHASE ROTATING MACHINE
The winding, instead of being concentrated, is
distributed.
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Copyright © 2017 Hassan Sowidan
SINGLE-PHASE ROTATING MACHINE
• We have already seen flux linkage derivation of
single-phase machines:
2
0 0
2
0 0
21 ( )
21 ( )
( ) ( ) cos( )
s s s s r r s s sr r
r r r s r s r r rs s
sr rs
N L i N N L i L i L i
N L i N N L i L i L i
L L M
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Copyright © 2017 Hassan Sowidan
SINGLE-PHASE ROTATING MACHINE
• The co-energy and torque are:
' ' ' '
0 0
2 2
' ( ,0, ) ( , , )
1 1( )
2 2
( )'sin( )
s ri i
m s s s r s r r
s s r r sr s r
e srms r s r
W i di i i d i
L i L i L i i
LWT i i i i M
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Copyright © 2017 Hassan Sowidan
SINGLE-PHASE ROTATING MACHINE
• The electrical differential equations are:
ss s s
rr r r
dv i R
dt
dv i R
dt
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Copyright © 2017 Hassan Sowidan
SINGLE-PHASE ROTATING MACHINE
• The mechanical differential equation is:
2
2
e
m
d dJ K B T T
dt dt
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Copyright © 2017 Hassan Sowidan
SINGLE-PHASE ROTATING MACHINE
Under sinusoidal excitation, the power becomes:
cos( )
cos( )
cos( )cos( )sin( )
( .)
cos( )cos( )sin( )
s s s
r r s
e e
m m m s r s r
m
m m s r s r m
i I t
i I t
dP T T I I M t t
dt
t is some arbitary const
P I I M t t t
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Copyright © 2017 Hassan Sowidan
SINGLE-PHASE ROTATING MACHINE
• Power can also be expressed as:
where:
1 2
3 4
sin ( ) sin ( )4
sin ( ) sin ( )m m s r
t tP I I M
t t
1 2
3 4
,
,
m s r m s r
m s r m s r
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SINGLE-PHASE ROTATING MACHINE
Since a sinusoidal function can have no average value
Pm can have an average value only if for
, or 4, i.e.,
• If ,
A necessary condition for average power is that one of
the ’s is zero, and a sufficient condition is that
.
m s r 2 0
( )
sin( )
4
m s rm av
I I MP
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= 0i
= 1,2,3i =m s r
i sin 0
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Copyright © 2017 Hassan Sowidan
TWO-PHASE ROTATING MACHINE
There is average power when there is pulsating torque
due to other ’ s.
To eliminate this, we can have a two-phase machine.
In the two-phase machine, there is an additional
winding on both the stator and the rotor. The two-
phase machine creates a rotating magnetic field.
i
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Copyright © 2017 Hassan Sowidan
TWO-PHASE ROTATING MACHINE
• We look at the stator magnetic field of one- and two-
phase machines.
• With sinusoidal excitation, no rotating field as it is
• always maximum at ψ = 90o:
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π
H
ψ2π-π-2π 0
Nsis/(2g)
-Nsis/(2g)
Ns, is
Hψ
sin( )2
s sN iH
g
sin( )cos( )2
s ss
N IH t
g
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Copyright © 2017 Hassan Sowidan
TWO-PHASE ROTATING MACHINE
• Two-phase machine: Rotating magnetic field!
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π
Hb
ψ2π-π-2π 0
Nsibs/(2g)
-Nsibs/(2g)
π
Ha
ψ3π-π 0
Nsias/(2g)
-Nsias/(2g)
2 223π2
Ns, ibs
Ha,Hbψ
Ns, ias
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TWO-PHASE ROTATING MACHINE
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cos ( ) sin ( )2 2
: cos , sin
cos( )cos( ) sin( )sin( )2
s as bs
s as s bss
as s s bs s s
s ss s
H H H
N i N iH
g g
Assume i I t i I t
N IH t t
g
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TWO-PHASE ROTATING MACHINE
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cos( )2
Revolving magnetic field
at 0, 0
,
Re
s ss
s
N IH t
g
t peak is at
t t peak is at t
volves continue clockwise