Download Starting Performance of Synchronous Motors

Starting Performance of Synchronous Motors
Windings
Fig. 1
The figure shows half a pole of a salient pole synchronous machine. A damper (amortisseur) winding
consisting of round bars in the pole face are connected through endrings on both sides, like the squirrel
cage winding of an induction motor. During starting the field winding is short circuited through a discharge resistor, and both the damper winding and the field winding contribute to the starting torque.
The discharge resistor is chosen large enough to give significant torque, most of it near synchronous
speed. If the resistance is too large, the induced voltage will be too high near standstill. In the calculations of starting performance the stator is a smooth circular arc, and the air gap is enlarged with the
Carter’s coefficient to account for slots and ducts. The stator winding is modeled as a sinusoidal current
sheet, as it will be shown later.Per unit d- and q-axis circuits
2
The d-axis has armature mmf centered on the pole axis, q-axis between the poles. The armature resistance ra is often added to the circuits in series with the stator leakage reactance jxl. The admittance
seen by Id is then Yd and by Iq it is Yq. Voltage sources to the left in the circuits are one per unit at rated
voltage. Subscripts kd and kq refer to the damper winding, f to the field winding. The slip is one at
standstill and zero at synchronous speed. An average per unit armature (inrush) current during starting
can be calculated as:
Ia = ( Id + Iq )/2
and the per unit torque often as (U is per unit voltage):
T = U2 Real ( Yd + Yq )/2 - Ia 2 ra
where ra is included in Yd and Yq.
This method of calculation, where the torques in the two axes are calculated independent of each other,
is only accurate enough when Yd and Yq are of about the same order of magnitude. In practical terms,
this means that the damper winding must have complete endrings with connections between the poles.
Sometimes damper windings are made without such connections, and Yq becomes very much smaller
than Yd. The approximate solution gives a smooth torque-speed curve also in this case, whereas in actual fact the torque exhibits a sharp dip around half speed. The more accurate solution was too complicated to be used before the age of computers, but it is used here. In the equations which follow admittances Yd and Yq are without ra in series with jxl in the axis circuits. D, Yd and Yq are complex numbers.
 = 2 -1
D =  + ra [ ra Yd Yq + ( Yd +Yq ) ]
T = (U2 /2 ǀDǀ2) Real [ ( Yd + Yq ) + ra ( 2 Yd Yq + ǀYd ǀ2 + ǀYq ǀ2 ) + ra2 ( Yd ǀYq ǀ2 + Yq ǀYd ǀ2 ) ]
Per unit damper impedances are very hard to calculate with sufficient accuracy, because of the saliency
of the poles and uncertain current distribution between bars. It is particularly problematic when the bars
are made of different materials, which they often are. The method described here solves that problem. It
even makes it possible to calculate individual bar currents at all speeds during starting.
Individual bar impedance
Fig 2 shows a damper bar which gets induced current from a magnetic field set up by a current in the
red region to the right. The induced current is subject to skin effect, which makes the current density
higher towards the top of the bar. The higher the frequency is, the more pronounced is the skin effect.
3
Fig. 2
Calculations are made with FLD8 initially for one meter length. The bar is short circuited with zero
voltage.
0 = jZ1I1 + jX12I2
Index 1 refers to the bar, 2 to the red current region. Mutual reactance
X12 is calculated from Fig. 3.
Fig. 3
4
Vector potential zero is at the left boundary. The flux responsible for the mutual reactance is equal to
the vector potential at the point in the stator just above the damper bar. FLD8 calculates the induced
current I1 in the bar. I2 is always 100A. That gives the bar impedance:
Z1 = -X12 I2 /I1
Bar resistance is the real component, reactance the imaginary component of Z1. The reactance must
now be divided up in two parts, that due to flux crossing the air gap and that due to flux crossing the
bar and the slot above it. Flux crossing the air gap is equal to the vector potential at the uppermost right
corner in figures 2 and 3. The difference in flux is due to the induced bar current. Reactance due to flux
crossing the bar and the slot is considered to be constant for all the bars, but changes with motor speed
due to skin effect. So does the resistance.
There is a simpler way of calculating impedance of a single bar, Fig.4. FLD8 calculates the impressed
voltage for a given current in that case. Voltage divided by current gives exactly the same Z1.
Fig. 4
The problem is that the simpler method cannot be used for double bars, which are used in some motors.
In those cases the upper bar is made of brass or bronze with higher resistivity than the lower bar, which
is made of copper. The arrangement is shown on Fig. 5. At start the upper bar carries most of the current. Torque becomes high and inrush current low because of the high resistance. Near synchronous
speed the lower bar carries more of the current and produces most of the torque.
5
Fig. 5
Basic terms and equations
,B
↑
●
A1
●
A2
A=vector potential
=flux
B=flux density
/m= A2 - A1
U=2 f (rms values)
B=µH
ʃ Hdl=I (line integral)
6
Magnetic field calculations are essential here. They consist basically of calculating vector potentials.
Calculations are two dimensional. A flux line or a point with zero potential is always specified. The
flux in webers per meter length between two points is simply the difference between the vector potentials. They can be calculated as scalars with program FLD11 if phase angles are of no importance, as
complex numbers with FLD8 if they are. Induced rms voltage E is minus 2times frequency times rms
flux. Impressed voltage U is -E. Flux density in tesla is permeability times magnetic field strength H,
which is ampereturns divided by length in meters. A line integral of field strength times length around a
closed path is equal to the current inside that path, and this can be used to derive finite element equations. In a triangular grid covering the region where the magnetic field is to be calculated, a path around
a point goes through centroids of surrounding triangles. That gives an equation for the vector potential
at the center as a function of those for the surrounding points. Triangles are considered to have uniform
flux and current densities. There may be thousands of points in a calculation, and the large number of
equations with as many unknowns are solved directly with Gaussian elimination.
Current in field winding
Fig. 6
In what follows, symbols have subscripts which refer to:
1:
2:
3:
4:
5:
Armature winding (not shown on Fig. 6)
Field winding
Damper bar at or closest to pole centerline
Next damper bar (not shown, but the location at the pole face is known to the program)
Last outermost damper bar
A flux line with reference vector potential zero is specified at the pole centerline. The average vector
potential A2 in the field winding determines the self reactance X2. The flux A4 crossing the air gap at
damper bar 4 is calculated the same now with current in the field winding as if bar 4 carried the same
current. The reason for that has been explained earlier in connection with figures 3 and 4. The self reactance X4 of bar 4 is also due to flux crossing the bar and the slot above it, which is calculated earlier
7
and must be added. A4 is also used to calculate mutual reactances X24 and X45. Vector potential A5 is
used to calculate self reactance X5 and mutual reactance X25.
Fig. 7 shows the finite element grid, with 2201 nodes.
Fig. 7
Current in armature winding, d-axis
Fig. 8
The armature winding is modeled as a sinusoidal current sheet with uniform current density. The average vector potential of the armature winding is used to determine a self reactance X1 due to flux crossing the air gap. It corresponds to Xad in the per unit d-axis circuit. Very little flux crosses the field winding. Mutual reactance X12 can be determined from a vector potential A2 inside of it. Mutual reactances
between the armature winding and the two outermost damper bars X14 and X15 are calclated by the vector potentials at their locations.
Current in armature winding, q-axis
8
Fig. 9
A flux line with reference vector potential zero is now specified at the centerline between poles. Self
reactance X1 of the armature winding is determined by average vector potential and corresponds to Xaq.
Mutual reactances X13, X14 and X15 between the armature winding and the damper bars are calculated
from the vector potentials A3, A4 and A5 at their locations.
Current in damper bar at or nearest the pole centerline
Fig. 10
Total flux crossing the air gap is equal to the vector potential at the lower left corner. The calculation of
self reactance X3 must also take into account the flux crossing the bar and the slot above it, which is
calculated more accurately earlier. Vector potentials A4 and A5 are used to calculate self reactances X4
and X5, and mutual reactances X34, X35 and X45.Per unit field impedance
9
Armature reactance X1, field reactance X2 and mutual reactance X12 between them are determined in
the half pole field calculations. The field winding is short circuited with zero voltage across it during
starting. If damper impedances and field resistance are ignored,
0 = X2 I2 - X12 I1
That gives current through the field winding if the armature current is chosen as I1 = 1:
I2 = X12 / X2
Armature voltage becomes:
U1hp = X1 I1 - X12 I2 = X1 - X122 / X2
At open circuit with I1 = 1, the armature voltage is X1 in the half pole calculations, Xad in per unit.
The ratio Xad / X1 between the two armature voltages also applies during starting.
U1pu = U1hp Xad / X1
Per unit armature voltage with I1 = 1 under the same conditions as for the half pole calculations is current times impedance of the parallel connection of Xf and Xad (see per unit d-axis circuit):
U1pu = Xf Xad / (Xf + Xad ) = U1hp Xad / X1 = Xad ( 1 - X122 / X1 X2 )
Per unit field current is armature current one minus current through Xad :
If = 1 - U1pu / Xad
Per unit field reactance is per unit voltage divided by per unit current:
Xf = U1pu / (1 - U1pu / Xad ) = 1 / [(1/U1pu) - (1/ Xad)]
Per unit field resistance is calculated conventionally, with the discharge resistor included:
Rf = RΩ ( ARAT kd / Nf )2 10-3 / KVA
ARAT: Armature reaction ampereturns
Nf :
Number of field turns in series per pole
Pole shape factor kd refers armature ampereturns to field ampereturns. It is the ratio between fundamental flux densities d-axis armature excited / field winding excited when ampereturns are the same in both
cases. It is one of several pole shape factors also calculated by the program.Per unit damper impedance
10
Reactances and resistances for the damper winding have been calculated earlier. They are now used in
circuit calculations with program ACCAN (Ac circuit analysis) to produce damper impedances for the
per unit axis circuits. The ACCAN-circuit for the q-axis with endrings connected between poles is
shown in Fig. 11. Other circuits are similar, with minor modifications. Resistances of endring segments
are to the right. The lowest one is for the connection between poles in the q-axis. That branch is simply
removed if the connection is not there. Bar currents within half the pole must add up to zero in that
case. With five bars the half pole also has only two endring segments in the d-axis. The upper Fig. 11
segment resistance represents the connection between the two half center bars. The resistance becomes
zero, but the connection remains. Since only damper impedance is to be calculated, the field winding is
not there.
Fig. 11
With a current source one ampere in the armature winding, an armature voltage U1hp is calculated for
the d-axis. It is converted to a per unit voltage in the same way as for the field winding:
U1pu = U1hp Xad / X1
Per unit current in the damper winding is also found in the same way as for the field winding, equal to
armature current one minus current through Xad. It is now necessary to use complex numbers in the
calculations.
Ikd = 1 - U1pu / j Xad
The per unit damper impedance is:
Zkd = U1pu / Ikd
Per unit damper resistance is the real component of Zkd , reactance the imaginary component.
For the q-axis calculations are done in exactly the same way as for the d-axis if endrings are connected
between the poles, only with subscript d replaced by q in the equations. If endrings are not connected
between poles, Zkq will be much lower than Zkd. Such motors can only be used when there is no need
for a large starting torque near half speed. Open endrings are more often used in generators, for which
damper impedances can also be calculated by the program.