Equations from Section 21 are used for transient simulations of a concrete syn- chronous motor with field winding. The nameplate and parameters of this motor are shown inTable 5.
Seeing that the simulation model is created in the d, q-system, linked with the rotor position, also the terminal voltages must be given in this system. It is made by Eqs. (419) and (420). The field winding voltage is a DC value and is constant during the whole simulation.
As it is known from the theory of synchronous motor, the starting up of syn- chronous motor is usually not possible by directly switching it across the line. If the synchronous motor has a damping winding, which is originally dedicated for damping of the oscillating process during motor operation, this winding can act as a squirrel cage and develop an asynchronous starting torque sufficient to get started.
After the motor achieves the speed close to synchronous speed, it falls spontane- ously into synchronism.
The damping winding is no longer active in torque development. The investi- gated motor has no damping winding; therefore a frequency starting up is carried out, which means continuously increasing voltage magnitude and frequency in such a way that their ratio is constant. InFigure 40, simulation waveforms of frequency starting up of this motor are shown, which means time dependence of variables n =
SN= 56 kVA Rs= 0.0694Ω
UN= 231 V Lσs= 0.391 mH
IN= 80.81 A Lμd= 10.269 mH
fN= 50 Hz Lμq= 10.05 mH
Uf= 171.71 V Rf’
´= 0.061035Ω
If= 11.07 A Lσf= 0.773 mH
nN= 1500 min1 J = 0.475 kg m2
TN= 285 Nm p = 2
Tloss= 1 Nm
Table 5.
Nameplate and parameters of the investigated synchronous motor with field winding.
Figure 40.
Simulation time waveforms of the synchronous motor with field winding starting up: (a) speed n, (b) current of the a-phase ia= is, (c) developed electromagnetic torque te, and (d) load angleϑL.
At first it is necessary to determine linkage magnetic flux of permanent magnets ψPM, by which an electrical voltage is induced in the stator winding. Obviously, it is measured on a real machine at no load condition in generating operation. From Eq. (73) it can be derived that in a general form, the PM linkage magnetic flux is given by equation:
ψPM¼Ui
ω:
In Eq. (428), there is instead the expression with a field current directlyψPM. As it is supposed that this magnetic flux is constant, its derivation is zero, and Eq. (437) is in the form:
ud¼RsidþLddid
dt �ωrLqiq, (444)
uq¼RsiqþLqdiq
dt þωrLdidþωrψPMdq: (445) In that equation the PM linkage magnetic flux is transformed into the dq0 system, because also the other variables are in this system. To distinguish it from the measured value, here a subscript “dq” is added. It can be determined as follows: In no load condition at the rated frequency, the currents id, iqare zero; therefore also the voltage udis zero according to Eq. (444), and the voltage in q-axis, according to Eq. (445), is:
uq¼ωrNψPMdq (446)
and at the same time according to Eq. (418), in absolute value, is:
uq¼ ffiffiffi3 2 r
Umax, , (447)
because also load angle is, in the no-load condition, zero. Then:
ψPMdq¼ ffiffiffi3 2 r Umax
ωrN : (448)
This value is introduced into Eq. (445), to calculate the currents id, iq. The real currents in the phase windings are obtained by an inverse transformation according to Eq. (443). Examples of these motor simulations are in Section 23.2.
2.23 Transients of a concrete synchronous motor
2.23.1 Synchronous motor with field windingEquations from Section 21 are used for transient simulations of a concrete syn- chronous motor with field winding. The nameplate and parameters of this motor are shown inTable 5.
Seeing that the simulation model is created in the d, q-system, linked with the rotor position, also the terminal voltages must be given in this system. It is made by Eqs. (419) and (420). The field winding voltage is a DC value and is constant during the whole simulation.
As it is known from the theory of synchronous motor, the starting up of syn- chronous motor is usually not possible by directly switching it across the line. If the synchronous motor has a damping winding, which is originally dedicated for damping of the oscillating process during motor operation, this winding can act as a squirrel cage and develop an asynchronous starting torque sufficient to get started.
After the motor achieves the speed close to synchronous speed, it falls spontane- ously into synchronism.
The damping winding is no longer active in torque development. The investi- gated motor has no damping winding; therefore a frequency starting up is carried out, which means continuously increasing voltage magnitude and frequency in such a way that their ratio is constant. InFigure 40, simulation waveforms of frequency starting up of this motor are shown, which means time dependence of variables n =
SN= 56 kVA Rs= 0.0694Ω
UN= 231 V Lσs= 0.391 mH
IN= 80.81 A Lμd= 10.269 mH
fN= 50 Hz Lμq= 10.05 mH
Uf= 171.71 V Rf’
´= 0.061035Ω
If= 11.07 A Lσf= 0.773 mH
nN= 1500 min1 J = 0.475 kg m2
TN= 285 Nm p = 2
Tloss= 1 Nm
Table 5.
Nameplate and parameters of the investigated synchronous motor with field winding.
Figure 40.
Simulation time waveforms of the synchronous motor with field winding starting up: (a) speed n, (b) current of the a-phase ia= is, (c) developed electromagnetic torque te, and (d) load angleϑL.
f(t), ia= f(t), te= f(t), andϑL= f(t). Motor is at the instant t = 3 s loaded by rated torque TN= 285 Nm.
The mechanical part of the model is linked with Eq. (423); however, it had to be spread by the damping coefficient of the mechanical movement of the rotor. This coefficient enabled implementation of the real damping, which resulted in a more stable operation of the rotor.
2.23.2 Synchronous motor with PM
Equations derived in Section 22 are employed in transient simulation of the synchronous motor with PM. The equation for the developed electromagnetic torque (221) is modified to the form:
te¼p Lð didþψPMÞiq�Lqiqid: (449)
Figure 41.
Simulated time waveforms of the synchronous motor with PM: (a) speed n, (b) current of the a-phase ia, (c) electromagnetic developed torque te, and (d) load angleϑL.
PN= 2 kW Rs= 3.826Ω
UphN= 230 V, star connection Y Ld= 0.07902 H
IN= 10 A Lq= 0.16315 H
fN= 36 Hz ψPM= 0.8363 Wb
nN= 360 min�1 J = 0.02 kg m2
TN= 54 Nm p = 6
Tloss= 0.5 Nm
Table 6.
Nameplate and parameters of the investigated synchronous motor with PM (SMPM).
Also this simulation model is created in the system d, q linked with the rotor position; therefore, it is necessary to adjust the terminal voltage to this system. The nameplate and parameters of the investigated motor are inTable6.
InFigure 41, there are simulation waveforms of the simulated variables during the frequency starting up of the synchronous motor with PM, if stator voltage and its frequency are continuously increasing in such a way that their ratio is constant. It is true that this way of the starting up is not typical for this kind of the motors. Such motors are usually controlled by field-oriented control (FOC), but its explaining and application exceed the scope of this textbook.
Simulation waveforms show time dependence of the n = f(t), ia= f(t), te= f(t), andϑL= f(t). The motor is at the instant t = 3 s loaded by rated torque TN= 54 Nm.
The mechanical part is widened by damping of the mechanical movement of the rotor.
f(t), ia= f(t), te= f(t), andϑL= f(t). Motor is at the instant t = 3 s loaded by rated torque TN= 285 Nm.
The mechanical part of the model is linked with Eq. (423); however, it had to be spread by the damping coefficient of the mechanical movement of the rotor. This coefficient enabled implementation of the real damping, which resulted in a more stable operation of the rotor.
2.23.2 Synchronous motor with PM
Equations derived in Section 22 are employed in transient simulation of the synchronous motor with PM. The equation for the developed electromagnetic torque (221) is modified to the form:
te¼p Lð didþψPMÞiq�Lqiqid: (449)
Figure 41.
Simulated time waveforms of the synchronous motor with PM: (a) speed n, (b) current of the a-phase ia, (c) electromagnetic developed torque te, and (d) load angleϑL.
PN= 2 kW Rs= 3.826Ω
UphN= 230 V, star connection Y Ld= 0.07902 H
IN= 10 A Lq= 0.16315 H
fN= 36 Hz ψPM= 0.8363 Wb
nN= 360 min�1 J = 0.02 kg m2
TN= 54 Nm p = 6
Tloss= 0.5 Nm
Table 6.
Nameplate and parameters of the investigated synchronous motor with PM (SMPM).
Also this simulation model is created in the system d, q linked with the rotor position; therefore, it is necessary to adjust the terminal voltage to this system. The nameplate and parameters of the investigated motor are inTable6.
InFigure 41, there are simulation waveforms of the simulated variables during the frequency starting up of the synchronous motor with PM, if stator voltage and its frequency are continuously increasing in such a way that their ratio is constant. It is true that this way of the starting up is not typical for this kind of the motors. Such motors are usually controlled by field-oriented control (FOC), but its explaining and application exceed the scope of this textbook.
Simulation waveforms show time dependence of the n = f(t), ia= f(t), te= f(t), andϑL= f(t). The motor is at the instant t = 3 s loaded by rated torque TN= 54 Nm.
The mechanical part is widened by damping of the mechanical movement of the rotor.