It is supposed that a reader is familiar with the basic knowledge of a synchronous machine theory, properties, and design configuration. The synchronous machine with salient poles on the rotor; symmetrical three-phase system a, b, c on the stator;
field winding f in the d-axis on the rotor; and damping winding, split into two parts perpendicular to each other (D and Q on the rotor), positioned in the d-axis and q-axis, as it is seen inFigure 34, is analyzed. The d-axis on the rotor is shifted about the angleϑrfrom the axis of the a-phase on the stator.
Basic equations for terminal voltage can be written for each of the winding separately, or briefly by one equation, in which the subscripts are gradually changed for each winding:
uj¼Rjijþdψj
dt (318)
where j = a, b, c, f, D, Q.
If symmetrical three-phase winding on the stator is supposed, then it can be supposed that their resistances are identical and can be marked by the subscript “s”:
Ra¼Rb¼Rc¼Rs (319)
Linkage magnetic flux can be also expressed briefly by the sum of all winding contributions:
ψj¼X
k
ψj,k¼X
k
Lj,kik (320)
Figure 32.
Simulations of the rotor wound asynchronous motor during the switching directly across the line. Time waveforms of the (a) speed, (b) phase current Ia, (c) developed electromagnetic torque, and (d) torque vs. speed at rated voltage.
Simulation waveforms are very similar with those of the squirrel cage rotor (high starting current and torques). But in the case of wound rotor, there is a possibility to add external resistors and to control the current and the torque (Figure 33).
PN= 4.4 kW Rs= 1.125Ω
UsN= 230 V , UrN= 64 V Ls= Lr= 0.1419 H
IsN= 9.4 A, IrN= 47 A Rr= 1.884Ω
fN= 50 Hz Lμ= 0.131 H
nN= 1370 min1 J = 0.04 kg m2
TN= 30 Nm p = 2
Tloss= 0.1 Nm
Table 4.
Nameplate and parameters of the investigated wound rotor asynchronous motor.
Figure 31.
Simulations of the asynchronous motor starting up by means of softstarter, time waveforms of (a) speed n, (c) phase current Ia, and (e) developed electromagnetic torque, and by means of a frequency converter, again in the same order: time waveforms of the (b) speed n, (d) phase current Ia, and (f) developed electromagnetic torque.
2.16 Synchronous machine and its inductances
It is supposed that a reader is familiar with the basic knowledge of a synchronous machine theory, properties, and design configuration. The synchronous machine with salient poles on the rotor; symmetrical three-phase system a, b, c on the stator;
field winding f in the d-axis on the rotor; and damping winding, split into two parts perpendicular to each other (D and Q on the rotor), positioned in the d-axis and q-axis, as it is seen inFigure 34, is analyzed. The d-axis on the rotor is shifted about the angleϑrfrom the axis of the a-phase on the stator.
Basic equations for terminal voltage can be written for each of the winding separately, or briefly by one equation, in which the subscripts are gradually changed for each winding:
uj¼Rjijþdψj
dt (318)
where j = a, b, c, f, D, Q.
If symmetrical three-phase winding on the stator is supposed, then it can be supposed that their resistances are identical and can be marked by the subscript “s”:
Ra¼Rb¼Rc¼Rs (319)
Linkage magnetic flux can be also expressed briefly by the sum of all winding contributions:
ψj¼X
k
ψj,k¼X
k
Lj,kik (320)
Figure 32.
Simulations of the rotor wound asynchronous motor during the switching directly across the line. Time waveforms of the (a) speed, (b) phase current Ia, (c) developed electromagnetic torque, and (d) torque vs. speed at rated voltage.
where j, k = a, b, c, f, D, Q. For a better review, here are all the equations with the sum of all members:
ψa¼LaaiaþLabibþLacicþLafifþLaDiDþLaQiQ, ψb¼LbaiaþLbbibþLbcicþLbfifþLbDiDþLbQiQ, ψc¼LcaiaþLcbibþLccicþLcfifþLcDiDþLcQiQ, ψf ¼LfaiaþLfbibþLfcicþLffifþLfDiDþLfQiQ, ψD¼LDaiaþLDbibþLDcicþLDfifþLDDiDþLDQiQ, ψQ ¼LQaiaþLQbibþLQcicþLQfifþLQDiDþLQQiQ:
(321)
Although it is known that mutual inductances of the windings that are perpen- dicular to each other are zero:
LfQ ¼LQf ¼LDQ ¼LQD¼0, (322)
Figure 33.
Time waveforms of the simulations during the starting up of the wound rotor asynchronous motor by means of rheostats added to the rotor circuits: (a) speed n, (c) phase current Ia, (e) developed electromagnetic torque, and time waveforms during the starting up by means of frequency converter, again in the same order: (b) speed n, (d) phase current Ia, and (f) developed electromagnetic torque.
for computer manipulation is more suitable if the original structure is kept and all inductances appear during the analysis:
Laa, Lbb, Lccare selfinductances of the stator windings: (323) Lff, LDD, LQQ are selfinductances of the rotor windings: (324) Lab, Lac, Lba, Lbc, Lca, Lcbare mutual inductances of the stator windings: (325)
LfD, LfQ, LDf, LDQ, LQf, LQDare mutual inductances of the rotor windings:
(326) The rest of the inductances are mutual inductances of the stator and rotor windings:
Laf, Lbf, Lcf, LaD, etc:
It is important to investigate if inductances depend on the rotor position or not.
2.16.1 Inductances that do not depend on the rotor position
Self- and mutual inductances of the rotor windings Lff, LQQ, LDD, LfDdo not depend on the rotor position because the stator is cylindrical, and if the stator slotting is neglected, then the air gap is for each winding constant. Thus, the
Figure 34.
Synchronous machine with salient poles on the rotor and three-phase winding a, b, c on the stator, field winding f, and damping winding split into two parts (D and Q ) perpendicular to each other, positioned in the d-axis and q-axis on the rotor. The d-axis on the rotor is shifted about the angleϑrfrom the axis of a-phase on the stator.
where j, k = a, b, c, f, D, Q. For a better review, here are all the equations with the sum of all members:
ψa¼LaaiaþLabibþLacicþLafifþLaDiDþLaQiQ, ψb¼LbaiaþLbbibþLbcicþLbfifþLbDiDþLbQiQ, ψc¼LcaiaþLcbibþLccicþLcfifþLcDiDþLcQiQ, ψf¼LfaiaþLfbibþLfcicþLffifþLfDiDþLfQiQ, ψD¼LDaiaþLDbibþLDcicþLDfifþLDDiDþLDQiQ, ψQ ¼LQaiaþLQbibþLQcicþLQfifþLQDiDþLQQiQ:
(321)
Although it is known that mutual inductances of the windings that are perpen- dicular to each other are zero:
LfQ ¼LQf¼LDQ ¼LQD¼0, (322)
Figure 33.
Time waveforms of the simulations during the starting up of the wound rotor asynchronous motor by means of rheostats added to the rotor circuits: (a) speed n, (c) phase current Ia, (e) developed electromagnetic torque, and time waveforms during the starting up by means of frequency converter, again in the same order: (b) speed n, (d) phase current Ia, and (f) developed electromagnetic torque.
for computer manipulation is more suitable if the original structure is kept and all inductances appear during the analysis:
Laa, Lbb, Lccare selfinductances of the stator windings: (323) Lff, LDD, LQQare selfinductances of the rotor windings: (324) Lab, Lac, Lba, Lbc, Lca, Lcbare mutual inductances of the stator windings: (325)
LfD, LfQ, LDf, LDQ, LQf, LQDare mutual inductances of the rotor windings:
(326) The rest of the inductances are mutual inductances of the stator and rotor windings:
Laf, Lbf, Lcf, LaD, etc:
It is important to investigate if inductances depend on the rotor position or not.
2.16.1 Inductances that do not depend on the rotor position
Self- and mutual inductances of the rotor windings Lff, LQQ, LDD, LfDdo not depend on the rotor position because the stator is cylindrical, and if the stator slotting is neglected, then the air gap is for each winding constant. Thus, the
Figure 34.
Synchronous machine with salient poles on the rotor and three-phase winding a, b, c on the stator, field winding f, and damping winding split into two parts (D and Q ) perpendicular to each other, positioned in the d-axis and q-axis on the rotor. The d-axis on the rotor is shifted about the angleϑrfrom the axis of a-phase on the stator.
magnetic permeance of the path of magnetic flux created by these windings does not change if the rotor rotates.
2.16.2 Inductances depending on the rotor position
2.16.2.1 Mutual inductances of the rotor and stator windings
Investigate, for example, a-phase winding on the stator and field winding f on the rotor, as it is shown inFigure 34.
When sinusoidally distributed windings are assumed, i.e., coefficients of higher harmonic components are zero, then the waveform of mutual inductance is cosinusoidal, if for the origin of the system such rotor position is chosen in which the a-phase axis and the axis of the field winding are identical (seeFigure 35).
Then the mutual inductances can be expressed as follows:
Laf¼Lfa¼Lafmaxcosϑr (327)
Lbf ¼Lfb¼Lafmaxcos ϑr�2π 3
(328) Lcf ¼Lfc¼Lafmaxcos ϑrþ2π
3
(329) similarly:
LaD¼LDa¼LaDmaxcosϑr (330)
LbD¼LDb¼LaDmaxcos ϑr�2π 3
(331) LcD¼LDc¼LaDmaxcos ϑrþ2π
3
(332)
Figure 35.
(a) Illustration to express mutual inductance of the a-phase on the stator and field winding f on the rotor, (b) waveform of the mutual inductance Lafversus rotor positionϑr:
Expressions for Q-winding positioned in the q-axis are written according to Figure 36a, where it is seen that the positive q-axis is ahead about 90° of the d-axis.
Hence if the d-axis is identified with the axis of the a-phase, the q-axis is perpen- dicular to it, and mutual inductance LaQis zero. To obtain a position in which LaQis maximal, it is necessary to go back about 90°, to identify q-axis with the a-phase axis. There the LaQreceives its magnitude. The magnitudes of the mutual induc- tances between Q-winding and b- and c-phases are shifted about 120°, as it is seen inFigure 36b.
LaQ ¼LQa¼LaQmaxcos ϑrþπ 2
¼ �LaQmaxsinϑr, (333) LbQ¼LQb¼bLaQmaxsin ϑr�2π
3
, (334)
LcQ ¼LQc¼cLaQmaxsin ϑrþ2π 3
(335)
2.16.2.2 Self-inductances of the stator
Self-inductances of the stator depend on the rotor position if there are salient poles. Self-inductance of the a-phase is maximal (Laamax), if its axis is identical with the axis of the pole. In this position the magnetic permeance is maximal. The minimal self-inductance of the a-phase (Laamin) occurs if the axis of the a-phase and axis of the pole are shifted aboutπ=2. Because the magnetic permeance is periodically changed for each pole, it means north and south, the cycle of the self inductance isπ, as it is seen inFigure 37.
Laa¼La0þL2cos 2ϑr, (336) Lbb¼La0þL2cos 2 ϑr�2π
3
, (337)
Figure 36.
(a) Illustration to express mutual inductance of the Q-winding on the rotor and a-phase on the stator and (b) waveform of the mutual inductances LaQ, LbQ, LcQ, versus rotor positionϑr.
magnetic permeance of the path of magnetic flux created by these windings does not change if the rotor rotates.
2.16.2 Inductances depending on the rotor position
2.16.2.1 Mutual inductances of the rotor and stator windings
Investigate, for example, a-phase winding on the stator and field winding f on the rotor, as it is shown inFigure 34.
When sinusoidally distributed windings are assumed, i.e., coefficients of higher harmonic components are zero, then the waveform of mutual inductance is cosinusoidal, if for the origin of the system such rotor position is chosen in which the a-phase axis and the axis of the field winding are identical (seeFigure 35).
Then the mutual inductances can be expressed as follows:
Laf ¼Lfa¼Lafmaxcosϑr (327)
Lbf¼Lfb¼Lafmaxcos ϑr�2π 3
(328) Lcf¼Lfc¼Lafmaxcos ϑrþ2π
3
(329) similarly:
LaD¼LDa¼LaDmaxcosϑr (330)
LbD¼LDb¼LaDmaxcos ϑr�2π 3
(331) LcD¼LDc¼LaDmaxcos ϑrþ2π
3
(332)
Figure 35.
(a) Illustration to express mutual inductance of the a-phase on the stator and field winding f on the rotor, (b) waveform of the mutual inductance Lafversus rotor positionϑr:
Expressions for Q-winding positioned in the q-axis are written according to Figure 36a, where it is seen that the positive q-axis is ahead about 90° of the d-axis.
Hence if the d-axis is identified with the axis of the a-phase, the q-axis is perpen- dicular to it, and mutual inductance LaQis zero. To obtain a position in which LaQis maximal, it is necessary to go back about 90°, to identify q-axis with the a-phase axis. There the LaQreceives its magnitude. The magnitudes of the mutual induc- tances between Q-winding and b- and c-phases are shifted about 120°, as it is seen inFigure 36b.
LaQ ¼LQa¼LaQmaxcos ϑrþπ 2
¼ �LaQmaxsinϑr, (333) LbQ ¼LQb¼bLaQmaxsin ϑr�2π
3
, (334)
LcQ ¼LQc¼cLaQmaxsin ϑrþ2π 3
(335)
2.16.2.2 Self-inductances of the stator
Self-inductances of the stator depend on the rotor position if there are salient poles. Self-inductance of the a-phase is maximal (Laamax), if its axis is identical with the axis of the pole. In this position the magnetic permeance is maximal. The minimal self-inductance of the a-phase (Laamin) occurs if the axis of the a-phase and axis of the pole are shifted aboutπ=2. Because the magnetic permeance is periodically changed for each pole, it means north and south, the cycle of the self inductance isπ, as it is seen inFigure 37.
Laa¼La0þL2cos 2ϑr, (336) Lbb¼La0þL2cos 2 ϑr�2π
3
, (337)
Figure 36.
(a) Illustration to express mutual inductance of the Q-winding on the rotor and a-phase on the stator and (b) waveform of the mutual inductances LaQ, LbQ, LcQ, versus rotor positionϑr.
Lcc¼La0þL2cos 2 ϑrþ2π 3
: (338)
The magnitude of the self-inductance Laamaxis obtained if the axis of the salient pole is identical with the axis of the stator a-phase; it meansϑr ¼0. Then:
Laamax ¼La0þL2: (339)
The minimal value of the self-inductance is obtained if the axis of the salient pole is perpendicular to the axis of the stator a-phase, i.e.,ϑr¼π=2. Then:
Laamin¼La0�L2: (340)
If the rotor rotates aboutϑr¼π, the self-inductance obtains again its maximal value, etc.; accordingly self-inductance does not obtain negative values, as it is seen inFigure 37b.
2.16.2.3 Mutual inductance of the stator windings
Mutual inductances of the stator windings depend on the rotor position only in the case of the salient poles on the rotor. These inductances are negative because they are shifted about 120° (see explanation inFigure 27b). The rotor is in a position where mutual inductance Lbcis maximal is shown inFigure 38a. Its waveform vs. rotor position is inFigure 38b.
It is possible to assume that for the sinusoidally distributed windings, the mag- nitudes of harmonic waveform L2are the same as in the case of the self-inductance of the stator windings. In the windings embedded in the slots, with a final number of the slots around the rotor periphery and the same number of the conductors in the slots, this assumption is not fulfilled; thus magnitudes of self and mutual waveforms can be different. Here a source of mistakes can be found and eventually
Figure 37.
(a) Illustration to express self-inductance Laaof the a-phase on the stator and (b) waveform of the self- inductance Laaversus rotor positionϑr:
discrepancies between the calculated and measured values. The waveforms in Figure 38bcan be written as follows:
�Lbc¼Lab0�L2cos 2ϑr (341)
�Lca¼Lab0�L2cos 2 ϑr�2π 3
¼Lab0�L2cos 2ϑrþ2π 3
(342)
�Lab¼Lab0�L2cos 2 ϑrþ2π 3
¼Lab0�L2cos 2ϑr�2π 3
(343) or
Lab¼ �Lab0þL2cos 2 ϑrþ2π 3
¼ �Lab0þL2cos 2ϑr�2π 3
(344) which better corresponds to the waveform inFigure 38.
Now all the expressions of these inductances are introduced into Eq. (65) and Eq. (318). They are equations with nonlinear periodically changed coefficients. To eliminate these coefficients, it is necessary to transform the currents, voltages, and linkage magnetic fluxes. The most suitable is Park linear transformation, which was explained in Section 4 and is applied again in the next chapter.