Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych
Nr 62                        Politechniki Wrocławskiej                    Nr 62
Studia i Materiały                              Nr 28                                            2008

                                                electrotechnology, synchronous motors, FEM modeling



    This paper deals with finite element calculation of magnetic forces inside two-speed, large power,
synchronous motor. Prediction of such forces are very important in understanding vibration phenomena
of electrical machines. In configuration for lower rotational speed pole numbers of magnetic field and
numbers of excited poles of field-winding are not equal. Because of non symmetrical armature and field
windings only one way of investigation is finite element (FE) modeling. The two-dimensional filed-
circuit model is elaborated and the time-stepping approach in motor type GAe1510/12p is used.
Simulation are done for nominal load operation point, for two different rotational rotor speeds: n =
500 rpm (2p = 12) and n = 600 rpm (2p = 10), and corresponding nominal active powers: P = 600 kW
and 1050 kW. As a results of calculations are: results of magnetic field, magnetic pressure inside the
machine and magnetic forces acting on stator yoke of investigated machine.

                                      1. INTRODUCTION
   Two-speed synchronous, large power motors are nowadays used to drive main fans
in polish coal and copper mains of the exhaust system. Positive aspects of using such
motors are:
   – two-step control of rotational speed,
   – two-step start up of the motor, which limits maximum values of armature
currents and the voltage drop,
   – savings in energy consumption,
   – less investments in comparison to alternatives solution,
   Negative aspects of presented motor are:
   – magnetic asymmetry,
   – distortion of the torque characteristic,
   – different forces values acting on stator yoke.
Two-speed synchronous motor are examples of no symmetrical machines (asymmetric
armature and field winding). Therefore investigation can be done only with help of
    * Politechnika Wrocławska, Instytut Maszyn, Napędów i Pomiarów Elektrycznych, ul. Smoluchowskiego
19, 50-372 Wrocław,,

finite element modeling (FEM) methods [1, 2]. In this paper calculation results of the
two–speed synchronous motor type GAe1510/12p are presented. This motor has two
different speeds: n = 500 rpm (2p = 12) and n = 600 rpm (2p = 10) and corresponding
nominal powers P = 600 kW and 1050 kW. Determination of magnetic forces acting
in mentioned motor is the goal of this paper.

                          2. MAGNETIC CALCULATION
In all simulation, elaborated 2D field-circuit model is used. Details of the model can
be found in [1, 2]. In Fig. 1 numeric model (field part) of investigated motor, together
with cylindrical coordinate system (situated in air-gap, 0.3 mm below stator inner
surface), and part of finite element mesh is shown. Second order approximation of the
magnetic vector potential is used. To calculate the magnetic forces the magnetic flux
density for next 100 (for higher rotational speed; 120 for lower rotational speed) time
steps, for nominal load is determined.

                         Fig. 1. Numeric model of investigated motor

Magnetic field is sampled in 1024 equidistance points in air-gap. Examples of a space
magnetic flux density (radial component) are shown in Fig. 2. For the picture clarity
only first three space distribution are show. Results are valid for both rotational
speeds. Finally, collecting all calculation results (for all time steps), matrix of flux
density B(m, n) is obtained, where M is a number of space samples, and N – time

samples. By means of 2DFT, matrix B(m, n) can be converted into spectral domain
B(µ, ν) [5]. In all further analysis the RMS value are taken on, both in space and time.
   Figure 3 and Figure 4. shows time/space distribution and modal/frequency
spectrum of radial component of flux density. Result are presented for both rotational
speeds and for field winding current If = 200 A.

a)                                                                b)
 1,2       Bn [T]                                                  1,2       Bn [T]

 0,8                                                               0,8

 0,4                                                               0,4

 0,0                                                               0,0

-0,4                                                              -0,4

-0,8                                                              -0,8

-1,2                                                              -1,2
       0              60            120       τ [o ]   180               0            60    120      τ [o ]   180

 Fig. 2. Instantaneous flux density distribution in air-gap – nominal load, for 600 rpm (a) and for 500 rpm
                         (b); distribution valid only for half machine circumference

a)                                                           b)

            Fig. 3. Radial component of flux density in air-gap for 600 rpm: time/space distribution (a)
                             modal/frequency spectrum (centered) (b) – nominal load

a)                                               b)

      Fig. 4. Radial component of flux density in air-gap for 500 rpm: time/space distribution (a)
                       modal/frequency spectrum (centered) (b) – nominal load

Maxell stress tensor components in air-gap, in 2D calculations, in cylindrical
coordinate system, can be calculated from the well know equations [3]. The 2DFT
approach to magnetic stresses in air-gap can be used as well. In the Fig. 5 an example
for time/space distributions of the magnetic stress (normal component), valid for both
rotational speed are shown. In addition in Fig.6 the modal/frequency spectrum is
presented. All results are valid for nominal load point of motor. According to [5]
modal/frequency spectrum of magnetic pressure should be limited only to lowest
harmonic in space, because from vibroacoustic point of view only longest waves are
important. In case of large power, two-speed, silent pole, synchronous motors such
approach can not be used. For all rotational speeds in modal/frequency spectrums of
magnetic stresses (radial component), harmonics close to 1 kHz are observed. These
harmonics are connected with numbers of stator slots (108) and numbers of pole pairs
(n = 500 rpm, p = 6 and n = 600 rpm, p = 5). For lower rotational rotor speed the
harmonics number are 102 and 114, and for higher speed – harmonics number 103
and 113. Magnitudes of these harmonics are similar to harmonic’s amplitudes of
small order. In addition these harmonics are very close to natural frequencies of
a mechanical construction of two-speed synchronous motor [3]. Therefore vibration
of such structure with big amplitudes can be expected.

a)                                                   b)

         Fig. 5. Radial component of magnetic pressure in air-gap (time/space distribution) for:
                                     600 rpm (a) and 500 rpm (b)

 a)                                                    b)

 Fig. 6. Modal/frequency spectrum of radial component of magnetic pressure in air-gap for nominal load:
                                  a) for n = 600 rpm; b) n = 500 rpm

Currently are existing only two commercial programs, which allows to solve vibration
problems with help of one finite element mesh – Ansys [7] and J-Mag [4]. In case for
lack of these programs user is forced to look for alternative solution, which allow to
transfer magnetic load onto mechanical structure. Two methods exist to deal with the
problem [5]:

    – calculation of magnetic forces which are applied to nodes of finite element mesh
of structural model,
    – structural load are applied as a magnetic pressure, applied to appropriate areas of
structural model.
Method of force calculation, which are applied to nodes of finite element structural
mesh, is chosen. Reaction of magnetic field with mechanical structure of two-speed
synchronous motor are represented with help of two force component: radial and
tangential, which are applied to stator-iron teeth (actually to nodes of finite element
structural mesh in stator teeth area). Magnetic force acting on one single stator tooth
can be calculated in according to [6]. Method is explained in Fig. 7.

             Stator                                                  Armature
              Iron                                                    winding

                                                                     1            4

                                 iron                                      Damper cage

 Fig. 7. Application of Maxwell stress tensor to calculation of magnetic forces acting on stator’s yoke
   teeth. To every stator tooth a closed contour (consist 4 arcs) is assigned (in 3D model this will be
  4 surfaces) where the appropriate magnetic stress components are determined. After integration of
magnetic pressure components (radial or tangential) along defined contour, the magnetic force acting on
                                    one single stator tooth is obtained

   In Fig.8 the time distribution of magnetic force, acting on three chosen stator tooth
are presented (only radial component). The Fourier analysis is used to obtain the
harmonic spectrum of magnetic forces (Fig. 9). According to these results, more dense
vibration spectrum with higher amplitudes for lower rotational speed (n = 500 rpm)
are expected (harmonics are closer to the natural frequencies [3] of the investigated
motor – possible resonance phenomena can occur). In addition in Fig. 10 time
distribution of tangential component of magnetic forces acting on teeth of stator yoke
are presented.

  a)                                                                    b)
4000                                                                    6000                                                          Ząb 10
           F n [N]                                         Ząb 10                  F n [N]
                                                           Ząb 20                                                                     Ząb 20
3500                                                       Ząb 30                                                                     Ząb 30

2000                                                                    3000

                                                       Czas [s]                                                                  Czas [s]
  0                                                                        0
       0                0,01    0,02    0,03    0,04         0,05              0             0,01    0,02   0,03   0,04   0,05          0,06

            Fig. 8.Time distribution of magnetic forces (radial component) acting on teeth of stator-yoke:
                                        a) for n=600 rpm, b) for n = 500 rpm

  a)                                                                     b)
 2000            Fn[N]                                                  2000           Fn[N]

 1500                                                                   1500

 1000                                                                   1000

  500                                                                    500
                                                         f [Hz]                                                                    f [Hz]
       0                                                                     0
            0             200     400     600    800             1000              0           167    333    500   667    833          1000

           Fig. 9. Harmonic spectrum of magnetic forces (radial component) acting on teeth of stator-yoke:
                                      a) for n = 600 rpm, b) for n = 500 rpm

  a)                                                                     b)
                                                           Tooth 10
  200          Fτ [N]                                                   400        F τ [N]                                            Ząb 10
                                                           Tooth 20                                                                   Ząb 20
                                                           Tooth 30                                                                   Ząb 30


                                                         t [s]                                                                   Czas [s]
 -800                                                                   -800
           0             0,01    0,02    0,03   0,04             0,05          0             0,01    0,02   0,03   0,04   0,05          0,06

   Fig. 10. Time distribution of magnetic forces (tangential component) acting on teeth of stator-yoke:
                                  a) for n = 600 rpm, b) for n = 500 rpm

                                         5. CONCLUSIONS
   In this paper the detailed approach to calculate magnetic forces in two-speed, large
power, salient pole, synchronous motors, using finite element model is presented.
This approach in combination with time stepping analysis seems to be the only way to
analyze vibration problems in unsymmetrical machines. According to calculation
results more dense vibration spectrum on lower rotational speed n = 500 rpm can be


[1] BIALIK J., ZAWILAK J., ANTAL L., Polowo-obwodowy model dwubiegowego silnika
    synchronicznego, Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych
    Politechniki Wrocławskiej, Nr 56, Studia i Materiały, Nr 24, Wrocław, 2004, pp. 43–54.
[2] BIALIK J., ZAWILAK J., Drgania oraz siły pochodzenia elektromagnetycznego w dwubiegowych
    silnikach synchronicznych du ej mocy, Proceedings of XLI International Symposium on Electrical
    Machines SME 2005, Jarnołtówek, 2005, pp. 55–64.
[3] BIALIK J., ZAWILAK J., Vibration modeling of the two-speed, large Power, synchronous motor, 6th
    IEEE International Symposium on Diagnostic for Electric Machines, Power Electronics and Drives,
    Cracow 6-8 September 2007, pp. 173–177.
[4] GIERAS J. F., WANG CH., CHO LAI J., Noise of polyphase electric machines, CRS Press Taylor &
    Francis Group USA 2006.
[5] WITCZAK P., WAWRZYNIAK B., Analiza magnetycznych sił wibracyjnych w maszynach
    z magnesami trwałymi w ujęciu modalno-częstotliwościowym, Proceedings of XLI International
    Symposium on Electrical Machines SME 2005, Jarnołtówek, 2005, pp. 214–219.
[6] WITCZAK P., Magnetic force and stress analysis in electric machinery, International XIII
    Symposium on Micromachines and Servodrives Mis 2002, Krasiczyn 2002, pp. 169–176.
[7] ANSYS Help,


     W artykule zaprezentowano sposób obliczania sił magnetycznych działających w dwubiegowym silniku
synchronicznym du ej mocy z zastosowaniem techniki MES. Wiedza o tych siłach jest pomocna w zrozumieniu
zjawisk drgań w silnikach elektrycznych. W silnikach dwubiegowych w przypadku pracy na mniejszej prędkości
obrotowej liczba biegunów magnetycznych i biegunów mechanicznych wirnika jest ró na. Asymetria uzwojeń
stojana oraz wirnika jest przesłanką do zastosowania metody elementów skończonych w analizach tego typu
silników. Opracowany dwuwymiarowy polowo-obwodowy model obliczeń wraz z techniką kroków czasowych
zostały zastosowane w obliczeniach sinika typu GAe1510/12p. Symulacje przeprowadzono dla stanu obcią enia
znamionowego dla dwóch ró nych prędkości obrotowych: n =500 obr/min (2p = 12) i n = 600 obr/min
(2p = 10), oraz mocach znamionowych: P = 600 kW i 1050 kW. Jako wyniki obliczeń przedstawiono obliczenia
pól magnetycznych w silniku, ciśnienia magnetycznego oraz sił magnetycznych działających na strukturę stojana
badanego silnika.

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