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Origin of Parasitic Time-Periodic Torques and Forces in Electrical Machines
Periodic Control Systems — PSYCO 2010 Antalya, Turkey, August 26-28, 2010
Origin of Parasitic Time-Periodic Torques and Forces in Electrical Machines A. Arkkio, K. Zenger Aalto University, School of Science and Technology, P.O. Box 13000, FI-00076 Aalto, Finland (e-mail: [email protected], [email protected])
Abstract: The origins of the time-periodic torques and forces acting on the rotor of an electrical machine are searched for. The main task of an electrical machine operated in a steady state is to produce a constant torque. In an ideal case, this would be the only magnetic force on the rotor. In practice, the electromagnetic torque usually has relatively large ripple and significant lateral forces may also occur. The examples discussed are from radial flux machines, especially, from cage induction motors. As the electromagnetic torque and forces can be calculated from the air-gap magnetic field, the air-gap flux density and its harmonics are under special examination. Understanding the magnetic interactions may provide means to actively control the air-gap flux density and eliminate the parasitic effects. Keywords: electrical machines, time-periodic, magnetic field, forces, torque, finite element analysis.
A rotating electrical machine converts electrical energy to mechanical work, or vice versa. Electromagnetic forces acting over the air gap of the machine enable the energy conversion. In an ideal machine in steady state, the torque would be constant. In practice, there is relatively large time variation in the torque. In addition to the circumferential force needed for the energy conversion, an electrical machine also generates time-varying radial or lateral forces that generate harmful vibration in the machine structures. The sources of these parasitic, typically, time-periodic force and torque variations are studied. The paper focuses on the timeperiodic components in the torque and magnetic forces acting on the rotor. The time-varying torque may excite torsion vibrations in the electrical machine – load machine system. The lateral magnetic forces excite stator and rotor vibrations and may even lead to rotor-dynamic instability. Figure 1 shows the steady-state starting torque of a 37 kW cage induction motor. The machine is supplied from a threephase 50 Hz voltage source. The rotor is locked. The motor produces a relatively large constant torque but there are also significant harmonics. The main harmonic varies at the 100 Hz frequency. The simulation result is obtained from finiteelement analysis described in Section 3. A space-vector equivalent circuit of Figure 2, which could be used for online control of the machine, predicts a constant torque when the machine is supplied from a balanced three-phase voltage source.
high-frequency counter-rotating force acting on the rotor. A negative frequency in the force spectrum means that a force component is rotating in a direction opposite the direction of rotation of the rotor. The harmonic effects become more dominant if we stop the whirling motion but keep the rotor eccentric as shown in Figure 4. This state of rotor motion is called static eccentricity. Compared to the case of dynamic eccentricity in the previous figure, the main component of the force is reduced, the high-frequency component is approximately the same and a new force component appears at twice the supply frequency. 1200 1000 Torque [Nm]
1. INTRODUCTION
800 600 400 200 0 0
20
40
60
80
100
Time [ms]
Figure 1. Starting torque of a 37 kW cage induction motor.
i ks
Rs
jZ k\ ks
LsV
d dt
~
LrV
d dt
j Z k Z \ kr
~
Rr
i kr
Figure 3 shows the magnetic forces acting on the rotor of a 2.6 MW cage induction motor when the rotor is in cylindrical whirling motion at the rotation frequency of the rotor. The motor is loaded by the rated torque and supplied from a 60 Hz balanced voltage source. This state of rotor motion is often called dynamic eccentricity. The main component of the force rotates at the whirling frequency but there is also a
Figure 2. Equivalent circuit of a space-vector model of an induction motor.
ISBN 978-3-902661-86-9/11/$20.00 © 2010 IFAC
1
u ks
d\ ks dt
Lm
d dt
d\ kr dt
u kr
The simple electromagnetic force model below (Arkkio et al. 2000) predicts the behavior of the fundamental component of the magnetic force but the higher harmonics are neglected
2500
1250
Fy [N]
F s 0
-2500 -1250
0
1250
2500
From the examples above it is clear that electrical machines present many phenomena that need time-variant or timeperiodic models for their description. Typically, these phenomena can be modeled by numerical methods based on, for instance, finite element analysis. However, finite-element models are still too heavy to be used for on-line control algorithms. For the control purposes, something more comprehensive than the models in Figure 2 and Equation 1 but simpler than the finite element analysis are needed. Understanding the physics behind the parasitic time-periodic phenomena will help us to construct the time-variant models needed.
Fx [N]
2500
Force [N]
2000
1500
1000
500
0 -3000
-2000
-1000
0
1000
2000
(1)
The force F and displacement u are expressed as complexvalued variables on the x,y plane. K is a second-order transfer function from the rotor displacement to the force. The realvalued parameters ci and complex-valued ai must be estimated by finite-element simulations.
-1250
-2500
§ c p 1 c p1 · K(s )u s ¨ c0 ¸ u s ¨ s ap 1 s ap 1 ¸¹ ©
3000
Frequency [Hz]
The analysis below is focused on radial flux electrical machines. Somewhat similar equations could be derived for axial or transversal flux machines but as the radial flux machines are the most common ones the analysis is restricted to them.
Figure 3. Lateral force on the rotor of the 2.6 MW machine. The dynamic eccentricity is 10% of the radial air-gap length. The upper figure shows the trace of the force vector, the bottom one gives the frequency spectrum of the force. 800
2. ANALYTICAL MODELS The magnetic force and torque on the rotor can be calculated from Maxwell’s stress
Fy [N]
400
0
F
1 ª
1
2
º n » dS ¼
1
2
v³ P0 «¬ B n B 2 B S
-400
T
1 ª
v³ r u P0 «¬ B n B 2 B S
º n » dS ¼
(2)
(3)
-800 0
400
800
1200
where P0 is the permeability of free space, B is the magnetic flux density vector, S is a surface in the air gap enclosing the rotor and n is the unit normal vector of this surface.
1600
Fx [N]
1000
The flux density in the air gap of the machine should be known for the force and torque calculation. The magnetic vector potential A (i.e. B u A ) combined with Maxwell’s equations leads to a partial differential equation
Force [N]
750
500
250
0 -3000
§1 · u¨ u A¸ ©P ¹ -2000
-1000
0
1000
2000
3000
J
(4)
where J is the electric current density. There is no current density in the air gap, and the material is linear. The problem reduces to the solution of Laplace equation. The solution for the 2D problem, presented in cylindrical coordinates, is
Frequency [Hz]
Figure 4. Lateral force on the rotor of the 2.6 MW machine. The rotor has been displaced in the x-direction 10% of the air gap. The upper figure shows the trace of the force vector, the bottom figure gives the frequency spectrum of the force.
2
To keep the practical application in mind, the possible harmonics behind the torque variation of Figure 1 are briefly reviewed. Figure 5 shows the cross section and magnetic field of the 37 kW cage induction motor obtained from finite element analysis. The rotor is locked and the machine supplied from the rated voltage source.
ªC t r n D n t r n º sin nI ½ f ¼ ° ° B r , I , t n °¬ n ¦ ® ¾ ° r n n n 1 r ° ª G n t r Hn t r º cos nI ° ° ¼ ¯ ¬ ¿ ° (5) ® ° n n º ª ½ f ° n °¬C n t r D n t r ¼ sin nI ° °BI r , I , t ¦ ® ¾ n n ° n 1 r ° ª G n t r Hn t r º cos nI ° ¼ ¯ ¬ ¿ ¯
The air-gap flux-density distribution of the locked-rotor machine at one instant of time is shown in Figure 6. The harmonics in the air-gap magnetic field are of the same order of magnitude as the fundamental harmonic producing most of the constant torque component of the machine.
The time dependence has been included in the coefficients Cn, Dn, Gn and Hn, which should be defined from the boundary conditions. The angular dependence follows the harmonic sine and cosine functions. In an ideal electrical machine, there would be only one harmonic of wave number p in the air-gap flux density, where p is the number of pole pairs of the machine. However, due to the slotting, winding construction, magnetic saturation and different types of asymmetry, many other harmonics also exist in the air gap, see Equations 9–15. Equation (5) also shows that the radial and circumferential flux-density components are coupled. If the radial flux density has some harmonic component n, the circumferential flux density also has a harmonic of the same wave number n, and vice versa. The dependency on the radial coordinate is often neglected from the flux-density equations. The radial flux density, as an example, is expressed
Figure 5. Flux of the 37 kW motor in locked-rotor condition.
f
Br I , t
Br 0 t ¦ ª¬an t sin nI bn t cos nI º¼
(6)
n 1
1.50 Radial flux density [T]
Concerning the neglected radial dependence, one can think that the integration surface in Equation (2) is fixed to a constant radius, for instance, in the middle of the air gap and the flux density is studied only on this surface. In ac machines, the time dependence cannot be neglected. Typically, some fundamental frequency Z0 can be assumed so that the time dependence can be expressed as a Fourierseries. This leads to a double series in space and time
Br I , t
f
ª fnQ cos nI QZ0 t D nQ
¦ ¦ « g
n 0Q
0« ¬
nQ
0.00 -0.50 -1.00
0
60
120
0
60
120
180
240
300
360
180
240
300
360
1.50
(7)
The sine and cosine functions can be combined to f
0.50
-1.50
Tangential flux density [T]
Br I , t
ªanQ cos nI cos QZ0 t º « » f f b cos nI sin QZ t « nQ 0 » ¦ ¦« » n 0 Q 0 «+cnQ sin nI cos QZ0 t » « dnQ sin nI sin QZ0 t » ¬ ¼
1.00
cos nI QZ0 t E nQ
º » »¼
1.00 0.50 0.00 -0.50 -1.00 -1.50
(8)
Polar angle [°]
In this expression, the radial flux density is given as superposition of flux-density waves having different wave numbers and moving at different velocities both to the positive (minus sign in the argument of the cosine term) and negative directions.
Figure 6. Radial and circumferential components of flux density in the air-gap of the four-pole 37 kW cage induction at one instant of time. The rotor is locked. The fundamental component of the radial flux density is also shown.
3
In the following analytical equations, the radial component of air-gap flux density is given, only. However as already discussed, the conservation of magnetic flux requires that if there is a radial flux-density component, there also is a corresponding circumferential component in the air-gap flux density.
Equation (8) includes all the potential air-gap harmonics. The symmetry of the machine eliminates many of them (Heller & Hamata 1977). A symmetric three-phase stator winding in a symmetric geometry can only produce harmonics of the form Bˆ q cos qpI B Zs t Ms
Bsq
Bs
(9)
Bˆ p cos pI Zs t D p
(11)
where Qs is the numbers of stator slots. The superscript s indicates that these harmonics are presented in the stator frame of reference. The stator-slot harmonics partly result from the modulation of the fundamental harmonic by the stator slotting. Partly they originate from the stator currents distributed in the discrete slots.
cos ª Qs r p 1 I rZs Zw t DQs r p D w º ¬ ¼
Qs r p
BˆQr r p cos ª¬(Qr r p)I QrZr r Zs t DQr r p º¼
(15)
Bˆ Qr r p cos ª Qr r p 1 I QrZr r Zs Zw t DQr r p D w º ¬ ¼ Bˆ Qr r p cos ª Qr r p 1 I QrZr r Zs Zw t DQr r p D w º ¬ ¼
The method presented by Coulomb (1982) was used for computing the electromagnetic forces. It is based on the principle of virtual work, and the force is obtained as a volume integral computed in an air layer surrounding the rotor. In the two-dimensional formulation, the calculation reduces to a surface integration over the finite elements in the air gap.
In a similar manner, the rotor slotting produces its own harmonics into the air-gap flux density
The motion of the rotor is obtained by changing the finiteelement mesh in the air gap. The centre point of the rotor was forced to move along a circular path to model the whirling motion. In addition, the rotor was rotated at the mechanical angular frequency. Triangular, second-order, elements were used. A typical finite-element mesh for the cross section of the motor contained about 15 000 nodes. One period of supply frequency was divided into 600 time steps.
Bˆ Qr r p cos ª¬ Qr r p I QrZr r Zs t DQr r p º¼ (12)
s BQ r rp
H ˆ B
2G 0
The calculation of the magnetic field and forces is based on time-discretized finite-element analysis. The magnetic field in the core region of the motor is assumed to be twodimensional. End-winding inductances are used in circuit equations of the windings to model approximately the end effects. The field and circuit equations are discretized and solved together. The time-dependence of the variables is modeled by the Crank-Nicolson method. The magnetic field, currents and potential differences of the windings are obtained directly in the solution of the coupled equations.
(10)
Bˆ Qs r p cos ª¬ Qs r p I B Zs t DQs r p º¼
cos ª Qs r p 1 I rZs Zw t DQs r p D w º ¬ ¼
Qs r p
3. NUMERICAL ANALYSIS
and the stator-slot harmonics s BQ s rp
(14)
H ˆ B
2G 0
H 2G 0 H 2G 0
where p is the number of pole pairs of the machine, Zs is its supply frequency and q 6n r 1; n 0, 1, 2,... is an integer. The largest of the harmonics produced by the stator winding typically are the fundamental harmonic Bps
Bˆ Qs r p cos ª¬(Qs r p )I B Zs t DQs r p º¼
Bs
where Qr is the number of rotor slots and Zr the mechanical rotation frequency of the rotor. An eccentric rotor modulates the air-gap flux-density harmonics. For the fundamental harmonic (Heller & Hamata 1977)
4. ORIGINS OF THE PARASITIC EFFECTS B I , t
Bˆ p cos pI Zs t D p
H 2G 0 H 2G 0
Bˆ p cos ª p 1 I Zs Zw t D p D w º ¬ ¼
The main parameters of the 37 kW and 2.6 MW cage induction motors are given in Table 1. The cross-sectional geometry of the smaller motor was shown in Figure 5.
(13)
Bˆ p cos ª p 1 I Zs Zw t D p D w º ¬ ¼
When assuming a two-dimensional magnetic field and expressing it in the cylindrical coordinates, the equation of the electromagnetic torque (3) is simplified to
The flux density of the deformed air gap includes the fundamental harmonic but there are also two additional eccentricity harmonics at wave numbers p ± .
T
When the radial air-gap length is distorted by the eccentricity, the stator- and rotor-slot harmonics are also split into three components, i.e. the original parent harmonic plus two new eccentricity harmonics with wave numbers on both sides of the parent wave number.
l
P0
2S
³ Br BI r
2
dI
(16)
0
where l is the length of the machine and Br and BI are the radial and circumferential components of the air-gap flux density. The integration radius r is within the air gap of the machine.
4
The wave numbers predicted by Equation (12) are 38 and 42. Equation (9) predicts wave numbers 2, 10, 14, 22, 26, 34, 38, 46, 50, … Wave number 38 is the common one for the two windings. It is obtained by using the upper ones of the plusminus signs in Equation (9) and the bottom ones in Equation (12). The corresponding frequencies are Zs and QrZr Zs . By requiring these to be equal for the synchronization, we get the rotation speed for a possible synchronous torque dip
Table 1. Main parameters of the two induction motors. Parameter
37 kW motor
2.6 MW motor
4 2 250 200 48 40 0.70 Star 50 37
2 1 768 480 48 40 10 Star 60 2600
Number of poles Number of parallel paths Core length [mm] Air-gap diameter [mm] Number of stator slots Number of rotor slots Weight of the rotor [kN] Connection Rated supply frequency [Hz] Rated power [kW]
Zr
(17)
This is exactly the measured synchronization frequency at 150 rpm. To explain the smaller synchronous torque dip in Figure 7, the second-order rotor-slot harmonics have to be included in the study.
It is interesting to see from Equation (16) that both a radial and circumferential flux density is needed to produce torque from an electrical machine. It can further be shown by substituting two harmonics from the general equation of airgap flux density (5) in Equation (16) that only such radial and circumferential flux-density harmonics that have equal wave numbers can interact and produce a non-zero torque.
Thus, the harmonics of wave number 38 probably produce the larger torque dip of Figure 7. By substituting the corresponding flux-density harmonics in the torque equation (16), the frequency of torque variation as function of the mechanical rotation frequency is obtained
The search of the origin of the large torque ripple in Figure 1 could be started by studying the harmonics of the air-gap flux obtained from finite-element analysis. However, we choose an alternative approach and start from the torque versus speed curve measured for this motor, Figure 7. The torque versus speed curve has a significant synchronous torque dip at 150 rpm rotation speed.
ZT
QrZr 2Zs
ª Qr 1 s º 2 » Zs « p ¬ ¼
(18)
where s is the slip of the rotor. At the locked rotor condition, s = 1, this frequency is 100 Hz. The same two harmonics of wave number 38 that produce the large synchronous torque dip at 150 rpm also produce main part of the 100 Hz torque ripple at the locked rotor condition.
A synchronous torque dip occurs when the stator winding and rotor winding produce independently two flux-density harmonics with equal wave numbers. Furthermore, the two harmonics can synchronize only if their frequencies are equal. From this we get two conditions to search for a pair of synchronizing harmonics. The harmonics produced by a symmetric three-phase stator winding are given by Equations (9), those of the rotor winding by Equation (12). The machine studied has four poles and 40 rotor slots (p = 2, Qr = 40).
Finally, Figure 8 shows the torque computed for the 37 kW motor at the rated operation point, s = 0.016. There are several harmonics in the torque, but also at the rated operation point, the flux-density harmonics of wave number 38 seem to produce the largest ripple. 350 300 Torque [Nm]
600 500 Torque [Nm]
2Zs Qr
400
250 200 150 100 50
300
0
200
0
20
30
40
Time [ms]
100 0 -500 -250
10
Figure 8. Torque of the 37 kW machine at the rated operation point. 0
250
500
750 1000 1250 1500
Speed [rpm]
Holopainen & Arkkio (2008) have discussed the forces acting on whirling rotors of induction motors in detail. Here, we briefly study the origins of the harmonic forces shown in Figures 3 and 4. The force equation (2) is expressed in cylindrical coordinates
Figure 7. Torque versus speed curve measured for the 37 kW cage induction motor at a reduced 320 V supply voltage. There are two synchronous torque dips present, a large one at 150 rpm and a smaller one at –300 rpm.
5
F
l
P0
2S
³ ª¬ 21 Br 0
2
BI2 er Br BI eI º rdI ¼
manner but the results will be somewhat different. The main reason for this is that the p–1 harmonic can flow relatively freely in machines with a pole-pair number larger than 1. In the two-pole machine as already discussed, the p–1 harmonic becomes a homopolar flux the flow of which is more or less restricted.
(19)
From the orthogonality of the sine and cosine functions, we get the condition that two harmonics in the air-gap flux density can produce a non-zero lateral force on the rotor only if the wave numbers n and k of the two harmonics fulfill the condition nk
1
5. CONCLUSIONS The magnetic field in the air-gap of a radial flux electrical machine was written as a sine and cosine series expressed in the cylindrical coordinates. The equations for the forces and torque acting on the rotor were derived from Maxwell’s stress and expressed in the same coordinates. In this way, it is relatively easy to follow how the different harmonic components interact and produce torque and forces on the rotor. The torque ripple is typically caused by two fluxdensity harmonics that are produced independently by the stator and rotor windings or constructions and that have equal wave lengths. In a cage induction motor, the same harmonics produce synchronous torque dips.
(20)
If a force is produced, it rotates at the frequency
Z = Zn Zk
(21)
where Zn and Zk are the frequencies of the two interacting harmonic waves. Condition (20) specifies that an air-gap harmonic cannot produce a lateral force by itself. It also eliminates most pairs of air-gap harmonics from producing a net force. However, when studying the eccentricity harmonics of Equations (13), (14) and (15), one can see that many possible pairs of harmonics still remain to participate in the lateral force production.
A lateral force results from the interaction of two air-gap harmonics the wave numbers of which differ by one. Such harmonics are typically produced by rotor eccentricity or some other asymmetries in the machine structures. The eccentricity harmonics of the fundamental flux-density as well as those of slot harmonics produce force at the whirling frequency. In addition, the stator-slot and rotor-slot harmonics produce force at a frequency of |2Zs – Zw|. Interactions of the rotor-slot eccentricity harmonics with the non-fundamental harmonics produced by the stator winding can lead to forces oscillating within the kilohertz range.
The force components at the whirling frequency in Figures (3) and (4) probably originate from the fundamental harmonic interacting with its eccentricity harmonic p+1. The p–1 harmonic of a two-pole machine is a homopolar flux and its magnitude is zero in a 2D finite element model. In addition to the force from the fundamental flux density, the stator- and rotor-slot harmonics also produce force at the whirling frequency.
Acknowledgement
In a two-pole machine, the stator- and rotor-slot harmonics may also produce force at a frequency
This research was supported by the Academy of Finland. REFERENCES
Z = 2Zs Zw
Arkkio A., Antila M., Pokki K., Simon A., Lantto E., 2000, Electromagnetic force on a whirling cage rotor. IEE Proceedings – Electric Power Applications, Vol. 147 Iss. 5, pp. 353–360. Coulomb J.L., 1982, A methodology for the determination of global electromechanical quantities from the finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness. IEEE Transactions on Magnetics, Vol. 19, Iss. 6, pp. 2514– 2519. Heller B., Hamata V., 1977, Harmonic field effects in induction motors. Elsevier Scientific Publishing Company, Oxford. 330 p. Holopainen T., Arkkio A., 2008, Electromechanical interaction in rotordynamics of electrical machines – an overview. 9th International Conference on Vibrations in Rotating Machinery. 8–10 September 2008, Exeter, Devon, UK. pp. 423–436.
where Zs is the supply frequency and Zw the whirling frequency. The double-frequency force in Figure 4 results from this interaction between a slot harmonic and its eccentricity harmonic. The first-order stator-slot eccentricity harmonics by themselves or first-order rotor-slot eccentricity harmonics by themselves do not produce high-frequency components on the lateral force. If the stator and rotor slot numbers are close to each other, interactions with the stator-slot and rotor-slot harmonics may occur. However, this is not the case in the 2.6 MW motor. To explain the high-frequency lateral forces in Figures (3) and (4), the harmonics of Equation (9) and their interactions with the rotor-slot harmonics have to be studied. As an example, by choosing q = 7 and p = 1, we get a force at frequency QrZr, which corresponds to the high frequency variation in Figure 4. The discussion above is valid for two-pole machines. For other pole numbers, the analysis can be repeated in a similar
6
Origin of Parasitic Time-Periodic Torques and Forces in Electrical Machines A. Arkkio, K. Zenger Aalto University, School of Science and Technology, P.O. Box 13000, FI-00076 Aalto, Finland (e-mail: [email protected], [email protected])
Abstract: The origins of the time-periodic torques and forces acting on the rotor of an electrical machine are searched for. The main task of an electrical machine operated in a steady state is to produce a constant torque. In an ideal case, this would be the only magnetic force on the rotor. In practice, the electromagnetic torque usually has relatively large ripple and significant lateral forces may also occur. The examples discussed are from radial flux machines, especially, from cage induction motors. As the electromagnetic torque and forces can be calculated from the air-gap magnetic field, the air-gap flux density and its harmonics are under special examination. Understanding the magnetic interactions may provide means to actively control the air-gap flux density and eliminate the parasitic effects. Keywords: electrical machines, time-periodic, magnetic field, forces, torque, finite element analysis.
A rotating electrical machine converts electrical energy to mechanical work, or vice versa. Electromagnetic forces acting over the air gap of the machine enable the energy conversion. In an ideal machine in steady state, the torque would be constant. In practice, there is relatively large time variation in the torque. In addition to the circumferential force needed for the energy conversion, an electrical machine also generates time-varying radial or lateral forces that generate harmful vibration in the machine structures. The sources of these parasitic, typically, time-periodic force and torque variations are studied. The paper focuses on the timeperiodic components in the torque and magnetic forces acting on the rotor. The time-varying torque may excite torsion vibrations in the electrical machine – load machine system. The lateral magnetic forces excite stator and rotor vibrations and may even lead to rotor-dynamic instability. Figure 1 shows the steady-state starting torque of a 37 kW cage induction motor. The machine is supplied from a threephase 50 Hz voltage source. The rotor is locked. The motor produces a relatively large constant torque but there are also significant harmonics. The main harmonic varies at the 100 Hz frequency. The simulation result is obtained from finiteelement analysis described in Section 3. A space-vector equivalent circuit of Figure 2, which could be used for online control of the machine, predicts a constant torque when the machine is supplied from a balanced three-phase voltage source.
high-frequency counter-rotating force acting on the rotor. A negative frequency in the force spectrum means that a force component is rotating in a direction opposite the direction of rotation of the rotor. The harmonic effects become more dominant if we stop the whirling motion but keep the rotor eccentric as shown in Figure 4. This state of rotor motion is called static eccentricity. Compared to the case of dynamic eccentricity in the previous figure, the main component of the force is reduced, the high-frequency component is approximately the same and a new force component appears at twice the supply frequency. 1200 1000 Torque [Nm]
1. INTRODUCTION
800 600 400 200 0 0
20
40
60
80
100
Time [ms]
Figure 1. Starting torque of a 37 kW cage induction motor.
i ks
Rs
jZ k\ ks
LsV
d dt
~
LrV
d dt
j Z k Z \ kr
~
Rr
i kr
Figure 3 shows the magnetic forces acting on the rotor of a 2.6 MW cage induction motor when the rotor is in cylindrical whirling motion at the rotation frequency of the rotor. The motor is loaded by the rated torque and supplied from a 60 Hz balanced voltage source. This state of rotor motion is often called dynamic eccentricity. The main component of the force rotates at the whirling frequency but there is also a
Figure 2. Equivalent circuit of a space-vector model of an induction motor.
ISBN 978-3-902661-86-9/11/$20.00 © 2010 IFAC
1
u ks
d\ ks dt
Lm
d dt
d\ kr dt
u kr
The simple electromagnetic force model below (Arkkio et al. 2000) predicts the behavior of the fundamental component of the magnetic force but the higher harmonics are neglected
2500
1250
Fy [N]
F s 0
-2500 -1250
0
1250
2500
From the examples above it is clear that electrical machines present many phenomena that need time-variant or timeperiodic models for their description. Typically, these phenomena can be modeled by numerical methods based on, for instance, finite element analysis. However, finite-element models are still too heavy to be used for on-line control algorithms. For the control purposes, something more comprehensive than the models in Figure 2 and Equation 1 but simpler than the finite element analysis are needed. Understanding the physics behind the parasitic time-periodic phenomena will help us to construct the time-variant models needed.
Fx [N]
2500
Force [N]
2000
1500
1000
500
0 -3000
-2000
-1000
0
1000
2000
(1)
The force F and displacement u are expressed as complexvalued variables on the x,y plane. K is a second-order transfer function from the rotor displacement to the force. The realvalued parameters ci and complex-valued ai must be estimated by finite-element simulations.
-1250
-2500
§ c p 1 c p1 · K(s )u s ¨ c0 ¸ u s ¨ s ap 1 s ap 1 ¸¹ ©
3000
Frequency [Hz]
The analysis below is focused on radial flux electrical machines. Somewhat similar equations could be derived for axial or transversal flux machines but as the radial flux machines are the most common ones the analysis is restricted to them.
Figure 3. Lateral force on the rotor of the 2.6 MW machine. The dynamic eccentricity is 10% of the radial air-gap length. The upper figure shows the trace of the force vector, the bottom one gives the frequency spectrum of the force. 800
2. ANALYTICAL MODELS The magnetic force and torque on the rotor can be calculated from Maxwell’s stress
Fy [N]
400
0
F
1 ª
1
2
º n » dS ¼
1
2
v³ P0 «¬ B n B 2 B S
-400
T
1 ª
v³ r u P0 «¬ B n B 2 B S
º n » dS ¼
(2)
(3)
-800 0
400
800
1200
where P0 is the permeability of free space, B is the magnetic flux density vector, S is a surface in the air gap enclosing the rotor and n is the unit normal vector of this surface.
1600
Fx [N]
1000
The flux density in the air gap of the machine should be known for the force and torque calculation. The magnetic vector potential A (i.e. B u A ) combined with Maxwell’s equations leads to a partial differential equation
Force [N]
750
500
250
0 -3000
§1 · u¨ u A¸ ©P ¹ -2000
-1000
0
1000
2000
3000
J
(4)
where J is the electric current density. There is no current density in the air gap, and the material is linear. The problem reduces to the solution of Laplace equation. The solution for the 2D problem, presented in cylindrical coordinates, is
Frequency [Hz]
Figure 4. Lateral force on the rotor of the 2.6 MW machine. The rotor has been displaced in the x-direction 10% of the air gap. The upper figure shows the trace of the force vector, the bottom figure gives the frequency spectrum of the force.
2
To keep the practical application in mind, the possible harmonics behind the torque variation of Figure 1 are briefly reviewed. Figure 5 shows the cross section and magnetic field of the 37 kW cage induction motor obtained from finite element analysis. The rotor is locked and the machine supplied from the rated voltage source.
ªC t r n D n t r n º sin nI ½ f ¼ ° ° B r , I , t n °¬ n ¦ ® ¾ ° r n n n 1 r ° ª G n t r Hn t r º cos nI ° ° ¼ ¯ ¬ ¿ ° (5) ® ° n n º ª ½ f ° n °¬C n t r D n t r ¼ sin nI ° °BI r , I , t ¦ ® ¾ n n ° n 1 r ° ª G n t r Hn t r º cos nI ° ¼ ¯ ¬ ¿ ¯
The air-gap flux-density distribution of the locked-rotor machine at one instant of time is shown in Figure 6. The harmonics in the air-gap magnetic field are of the same order of magnitude as the fundamental harmonic producing most of the constant torque component of the machine.
The time dependence has been included in the coefficients Cn, Dn, Gn and Hn, which should be defined from the boundary conditions. The angular dependence follows the harmonic sine and cosine functions. In an ideal electrical machine, there would be only one harmonic of wave number p in the air-gap flux density, where p is the number of pole pairs of the machine. However, due to the slotting, winding construction, magnetic saturation and different types of asymmetry, many other harmonics also exist in the air gap, see Equations 9–15. Equation (5) also shows that the radial and circumferential flux-density components are coupled. If the radial flux density has some harmonic component n, the circumferential flux density also has a harmonic of the same wave number n, and vice versa. The dependency on the radial coordinate is often neglected from the flux-density equations. The radial flux density, as an example, is expressed
Figure 5. Flux of the 37 kW motor in locked-rotor condition.
f
Br I , t
Br 0 t ¦ ª¬an t sin nI bn t cos nI º¼
(6)
n 1
1.50 Radial flux density [T]
Concerning the neglected radial dependence, one can think that the integration surface in Equation (2) is fixed to a constant radius, for instance, in the middle of the air gap and the flux density is studied only on this surface. In ac machines, the time dependence cannot be neglected. Typically, some fundamental frequency Z0 can be assumed so that the time dependence can be expressed as a Fourierseries. This leads to a double series in space and time
Br I , t
f
ª fnQ cos nI QZ0 t D nQ
¦ ¦ « g
n 0Q
0« ¬
nQ
0.00 -0.50 -1.00
0
60
120
0
60
120
180
240
300
360
180
240
300
360
1.50
(7)
The sine and cosine functions can be combined to f
0.50
-1.50
Tangential flux density [T]
Br I , t
ªanQ cos nI cos QZ0 t º « » f f b cos nI sin QZ t « nQ 0 » ¦ ¦« » n 0 Q 0 «+cnQ sin nI cos QZ0 t » « dnQ sin nI sin QZ0 t » ¬ ¼
1.00
cos nI QZ0 t E nQ
º » »¼
1.00 0.50 0.00 -0.50 -1.00 -1.50
(8)
Polar angle [°]
In this expression, the radial flux density is given as superposition of flux-density waves having different wave numbers and moving at different velocities both to the positive (minus sign in the argument of the cosine term) and negative directions.
Figure 6. Radial and circumferential components of flux density in the air-gap of the four-pole 37 kW cage induction at one instant of time. The rotor is locked. The fundamental component of the radial flux density is also shown.
3
In the following analytical equations, the radial component of air-gap flux density is given, only. However as already discussed, the conservation of magnetic flux requires that if there is a radial flux-density component, there also is a corresponding circumferential component in the air-gap flux density.
Equation (8) includes all the potential air-gap harmonics. The symmetry of the machine eliminates many of them (Heller & Hamata 1977). A symmetric three-phase stator winding in a symmetric geometry can only produce harmonics of the form Bˆ q cos qpI B Zs t Ms
Bsq
Bs
(9)
Bˆ p cos pI Zs t D p
(11)
where Qs is the numbers of stator slots. The superscript s indicates that these harmonics are presented in the stator frame of reference. The stator-slot harmonics partly result from the modulation of the fundamental harmonic by the stator slotting. Partly they originate from the stator currents distributed in the discrete slots.
cos ª Qs r p 1 I rZs Zw t DQs r p D w º ¬ ¼
Qs r p
BˆQr r p cos ª¬(Qr r p)I QrZr r Zs t DQr r p º¼
(15)
Bˆ Qr r p cos ª Qr r p 1 I QrZr r Zs Zw t DQr r p D w º ¬ ¼ Bˆ Qr r p cos ª Qr r p 1 I QrZr r Zs Zw t DQr r p D w º ¬ ¼
The method presented by Coulomb (1982) was used for computing the electromagnetic forces. It is based on the principle of virtual work, and the force is obtained as a volume integral computed in an air layer surrounding the rotor. In the two-dimensional formulation, the calculation reduces to a surface integration over the finite elements in the air gap.
In a similar manner, the rotor slotting produces its own harmonics into the air-gap flux density
The motion of the rotor is obtained by changing the finiteelement mesh in the air gap. The centre point of the rotor was forced to move along a circular path to model the whirling motion. In addition, the rotor was rotated at the mechanical angular frequency. Triangular, second-order, elements were used. A typical finite-element mesh for the cross section of the motor contained about 15 000 nodes. One period of supply frequency was divided into 600 time steps.
Bˆ Qr r p cos ª¬ Qr r p I QrZr r Zs t DQr r p º¼ (12)
s BQ r rp
H ˆ B
2G 0
The calculation of the magnetic field and forces is based on time-discretized finite-element analysis. The magnetic field in the core region of the motor is assumed to be twodimensional. End-winding inductances are used in circuit equations of the windings to model approximately the end effects. The field and circuit equations are discretized and solved together. The time-dependence of the variables is modeled by the Crank-Nicolson method. The magnetic field, currents and potential differences of the windings are obtained directly in the solution of the coupled equations.
(10)
Bˆ Qs r p cos ª¬ Qs r p I B Zs t DQs r p º¼
cos ª Qs r p 1 I rZs Zw t DQs r p D w º ¬ ¼
Qs r p
3. NUMERICAL ANALYSIS
and the stator-slot harmonics s BQ s rp
(14)
H ˆ B
2G 0
H 2G 0 H 2G 0
where p is the number of pole pairs of the machine, Zs is its supply frequency and q 6n r 1; n 0, 1, 2,... is an integer. The largest of the harmonics produced by the stator winding typically are the fundamental harmonic Bps
Bˆ Qs r p cos ª¬(Qs r p )I B Zs t DQs r p º¼
Bs
where Qr is the number of rotor slots and Zr the mechanical rotation frequency of the rotor. An eccentric rotor modulates the air-gap flux-density harmonics. For the fundamental harmonic (Heller & Hamata 1977)
4. ORIGINS OF THE PARASITIC EFFECTS B I , t
Bˆ p cos pI Zs t D p
H 2G 0 H 2G 0
Bˆ p cos ª p 1 I Zs Zw t D p D w º ¬ ¼
The main parameters of the 37 kW and 2.6 MW cage induction motors are given in Table 1. The cross-sectional geometry of the smaller motor was shown in Figure 5.
(13)
Bˆ p cos ª p 1 I Zs Zw t D p D w º ¬ ¼
When assuming a two-dimensional magnetic field and expressing it in the cylindrical coordinates, the equation of the electromagnetic torque (3) is simplified to
The flux density of the deformed air gap includes the fundamental harmonic but there are also two additional eccentricity harmonics at wave numbers p ± .
T
When the radial air-gap length is distorted by the eccentricity, the stator- and rotor-slot harmonics are also split into three components, i.e. the original parent harmonic plus two new eccentricity harmonics with wave numbers on both sides of the parent wave number.
l
P0
2S
³ Br BI r
2
dI
(16)
0
where l is the length of the machine and Br and BI are the radial and circumferential components of the air-gap flux density. The integration radius r is within the air gap of the machine.
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The wave numbers predicted by Equation (12) are 38 and 42. Equation (9) predicts wave numbers 2, 10, 14, 22, 26, 34, 38, 46, 50, … Wave number 38 is the common one for the two windings. It is obtained by using the upper ones of the plusminus signs in Equation (9) and the bottom ones in Equation (12). The corresponding frequencies are Zs and QrZr Zs . By requiring these to be equal for the synchronization, we get the rotation speed for a possible synchronous torque dip
Table 1. Main parameters of the two induction motors. Parameter
37 kW motor
2.6 MW motor
4 2 250 200 48 40 0.70 Star 50 37
2 1 768 480 48 40 10 Star 60 2600
Number of poles Number of parallel paths Core length [mm] Air-gap diameter [mm] Number of stator slots Number of rotor slots Weight of the rotor [kN] Connection Rated supply frequency [Hz] Rated power [kW]
Zr
(17)
This is exactly the measured synchronization frequency at 150 rpm. To explain the smaller synchronous torque dip in Figure 7, the second-order rotor-slot harmonics have to be included in the study.
It is interesting to see from Equation (16) that both a radial and circumferential flux density is needed to produce torque from an electrical machine. It can further be shown by substituting two harmonics from the general equation of airgap flux density (5) in Equation (16) that only such radial and circumferential flux-density harmonics that have equal wave numbers can interact and produce a non-zero torque.
Thus, the harmonics of wave number 38 probably produce the larger torque dip of Figure 7. By substituting the corresponding flux-density harmonics in the torque equation (16), the frequency of torque variation as function of the mechanical rotation frequency is obtained
The search of the origin of the large torque ripple in Figure 1 could be started by studying the harmonics of the air-gap flux obtained from finite-element analysis. However, we choose an alternative approach and start from the torque versus speed curve measured for this motor, Figure 7. The torque versus speed curve has a significant synchronous torque dip at 150 rpm rotation speed.
ZT
QrZr 2Zs
ª Qr 1 s º 2 » Zs « p ¬ ¼
(18)
where s is the slip of the rotor. At the locked rotor condition, s = 1, this frequency is 100 Hz. The same two harmonics of wave number 38 that produce the large synchronous torque dip at 150 rpm also produce main part of the 100 Hz torque ripple at the locked rotor condition.
A synchronous torque dip occurs when the stator winding and rotor winding produce independently two flux-density harmonics with equal wave numbers. Furthermore, the two harmonics can synchronize only if their frequencies are equal. From this we get two conditions to search for a pair of synchronizing harmonics. The harmonics produced by a symmetric three-phase stator winding are given by Equations (9), those of the rotor winding by Equation (12). The machine studied has four poles and 40 rotor slots (p = 2, Qr = 40).
Finally, Figure 8 shows the torque computed for the 37 kW motor at the rated operation point, s = 0.016. There are several harmonics in the torque, but also at the rated operation point, the flux-density harmonics of wave number 38 seem to produce the largest ripple. 350 300 Torque [Nm]
600 500 Torque [Nm]
2Zs Qr
400
250 200 150 100 50
300
0
200
0
20
30
40
Time [ms]
100 0 -500 -250
10
Figure 8. Torque of the 37 kW machine at the rated operation point. 0
250
500
750 1000 1250 1500
Speed [rpm]
Holopainen & Arkkio (2008) have discussed the forces acting on whirling rotors of induction motors in detail. Here, we briefly study the origins of the harmonic forces shown in Figures 3 and 4. The force equation (2) is expressed in cylindrical coordinates
Figure 7. Torque versus speed curve measured for the 37 kW cage induction motor at a reduced 320 V supply voltage. There are two synchronous torque dips present, a large one at 150 rpm and a smaller one at –300 rpm.
5
F
l
P0
2S
³ ª¬ 21 Br 0
2
BI2 er Br BI eI º rdI ¼
manner but the results will be somewhat different. The main reason for this is that the p–1 harmonic can flow relatively freely in machines with a pole-pair number larger than 1. In the two-pole machine as already discussed, the p–1 harmonic becomes a homopolar flux the flow of which is more or less restricted.
(19)
From the orthogonality of the sine and cosine functions, we get the condition that two harmonics in the air-gap flux density can produce a non-zero lateral force on the rotor only if the wave numbers n and k of the two harmonics fulfill the condition nk
1
5. CONCLUSIONS The magnetic field in the air-gap of a radial flux electrical machine was written as a sine and cosine series expressed in the cylindrical coordinates. The equations for the forces and torque acting on the rotor were derived from Maxwell’s stress and expressed in the same coordinates. In this way, it is relatively easy to follow how the different harmonic components interact and produce torque and forces on the rotor. The torque ripple is typically caused by two fluxdensity harmonics that are produced independently by the stator and rotor windings or constructions and that have equal wave lengths. In a cage induction motor, the same harmonics produce synchronous torque dips.
(20)
If a force is produced, it rotates at the frequency
Z = Zn Zk
(21)
where Zn and Zk are the frequencies of the two interacting harmonic waves. Condition (20) specifies that an air-gap harmonic cannot produce a lateral force by itself. It also eliminates most pairs of air-gap harmonics from producing a net force. However, when studying the eccentricity harmonics of Equations (13), (14) and (15), one can see that many possible pairs of harmonics still remain to participate in the lateral force production.
A lateral force results from the interaction of two air-gap harmonics the wave numbers of which differ by one. Such harmonics are typically produced by rotor eccentricity or some other asymmetries in the machine structures. The eccentricity harmonics of the fundamental flux-density as well as those of slot harmonics produce force at the whirling frequency. In addition, the stator-slot and rotor-slot harmonics produce force at a frequency of |2Zs – Zw|. Interactions of the rotor-slot eccentricity harmonics with the non-fundamental harmonics produced by the stator winding can lead to forces oscillating within the kilohertz range.
The force components at the whirling frequency in Figures (3) and (4) probably originate from the fundamental harmonic interacting with its eccentricity harmonic p+1. The p–1 harmonic of a two-pole machine is a homopolar flux and its magnitude is zero in a 2D finite element model. In addition to the force from the fundamental flux density, the stator- and rotor-slot harmonics also produce force at the whirling frequency.
Acknowledgement
In a two-pole machine, the stator- and rotor-slot harmonics may also produce force at a frequency
This research was supported by the Academy of Finland. REFERENCES
Z = 2Zs Zw
Arkkio A., Antila M., Pokki K., Simon A., Lantto E., 2000, Electromagnetic force on a whirling cage rotor. IEE Proceedings – Electric Power Applications, Vol. 147 Iss. 5, pp. 353–360. Coulomb J.L., 1982, A methodology for the determination of global electromechanical quantities from the finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness. IEEE Transactions on Magnetics, Vol. 19, Iss. 6, pp. 2514– 2519. Heller B., Hamata V., 1977, Harmonic field effects in induction motors. Elsevier Scientific Publishing Company, Oxford. 330 p. Holopainen T., Arkkio A., 2008, Electromechanical interaction in rotordynamics of electrical machines – an overview. 9th International Conference on Vibrations in Rotating Machinery. 8–10 September 2008, Exeter, Devon, UK. pp. 423–436.
where Zs is the supply frequency and Zw the whirling frequency. The double-frequency force in Figure 4 results from this interaction between a slot harmonic and its eccentricity harmonic. The first-order stator-slot eccentricity harmonics by themselves or first-order rotor-slot eccentricity harmonics by themselves do not produce high-frequency components on the lateral force. If the stator and rotor slot numbers are close to each other, interactions with the stator-slot and rotor-slot harmonics may occur. However, this is not the case in the 2.6 MW motor. To explain the high-frequency lateral forces in Figures (3) and (4), the harmonics of Equation (9) and their interactions with the rotor-slot harmonics have to be studied. As an example, by choosing q = 7 and p = 1, we get a force at frequency QrZr, which corresponds to the high frequency variation in Figure 4. The discussion above is valid for two-pole machines. For other pole numbers, the analysis can be repeated in a similar
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