Modeling of 3-Phase Induction Motor Including Slot Effects and Winding Harmonics A. K. Mishra and B. S. Rajpurohit School of Computing and Electrical Engineering, IIT Mandi, Mandi, H.P. 175001, India
[email protected],
[email protected] Abstract— In this paper induction motor model has been developed in three phase frame of reference. A qualitative and analytical approach has been adopted to consider the effects of rotor and stator slots with winding harmonics. Chronological inclusion of rotor slot effects and winding harmonics establishes the model. This order builds up the self and mutual inductance matrices which are the key variables used in voltage and current equations. An iterative algorithmic approach is used to solve these six voltage current differential equations and two motor dynamics equations which determines the response of the induction motor. Model has been excited with sinusoidal input only to demonstrate the effect of winding and slot harmonics. Index Terms— Non sinusoidal Flux density, air-gap, harmonics index, carter's factor, rotor slots, Modified winding function.
I. INTRODUCTION Drive design includes the power converter design and machine design. In machine design to achieve certain objectives a good model of that machine is needed so the each parameter effect on the output can be forecasted. Different models have been flourished for different applications. Mathematical modeling of a machine is nothing but to mimic the real machine through a set of mathematical equations subjected to some initial conditions. If the past is explored, we will find numerous models have been used. Applicability of models varies from speed estimation to design of the machines. A variety of application are such as speed estimation, fault identification, fault-level determination, eccentricity related issues, bearing faults analysis, mechanical to electrical instability, etc. Each application in this context can be analyzed properly if there is a model available of the motor for that kind of study. History of induction motor model developments shows a wide range of application specific models[1, 2]. Speed sensing, parameter estimation, condition monitoring, etc are the major application of this variety of models[3, 4]. Different Frame of references has been taken to model a machine in time domain. A majority of study is only concerned with twophase frame of references [5]. They proved it to be the best model according to their applications. Another community has worked in three-phase frame of reference to model the machine. Most of the models differ due to the self and mutual inductances of the rotor and stator. Authors are grateful to DST-FIST, for their financial support to carry out this research.
For a single loop winding in the circular periphery of an electrical rotating machine, the MMF waveform will be like square wave with respect to the space angle șr for direct current supply. In any electromechanical machine the desired constant MMF should be purely sinusoidally distributed in the space. When this machine is supplied with purely sinusoidal source the oscillatory MMF or the two way circulating flux will setup in the motor. In order to create stationary sinusoidal flux density in the air-gap the winding arrangement should be in distributive manner. Winding cannot be housed in continuous manner that's why flux path between the winding wires exists. Now the concept of purely sinusoidal MMF building becomes futile. So the harmonics occurred due to the winding placements in the air-gap are called the winding harmonics [6]. Another undesirable effect which introduces harmonics is the slots made for the winding housing. Slots reduce the permeability inside the slots and the conductors inside the slots offer the diamagnetic behavior which again detriment the main flux. So these harmonics called the space harmonics as they evict the space organization of the main flux in the airgap [7].
Fig. 1. Sources of space harmonics
Formation of these harmonics can be minimized or optimized in reference to the torque generation and the other figures of merit. Few space harmonics can give positive effect while they interact with the time harmonics of different order. Then the remaining is obviously adverse for the steady state operation which causes lot of vibrations in the shaft torque for the machine.[8] This doctrine give rise to the customize design of induction machines for the different application perspectives. Effect of the space harmonics in form of winding and slot harmonics is discussed in detail in next sections and
many application perspectives have been considered to exhibit the positive effect of few space harmonics. II. EFFECT OF WINDING HARMONICS As discussed above the winding harmonics are mainly due to winding distribution in the space. Variables those represent winding distributions are the pitch and belt factor. Process to analyze the winding harmonics starts with the analysis of single turn coil in the air-gap. In order to formulate mathematical model of electrical machine we need to write its voltage current equations. Voltage and currents acts as the input variables for the system and the output voltage variables are torque and speed. The analytical part is the finding of the mutual and self inductances. Mutual and self inductances can be considered the backbone for most of the model formulations of the induction motors. Self inductances and the mutual inductances can be directly connected with the findings of the winding formations. Different types of winding corresponds to different types of pitch and belt factors [9]. To calculate the inductance first we need to determine the spatial distribution of magneto motive force of single coil in the air-gap periphery. The spatial distribution for the MMF is calculated for any common phase "q" in the winding. Using this establishment, the harmonic magnetic flux linkages can be determined between any two generalized phases "q" and "r". These flux linkages are responsible for the harmonic inductances. To calculate the self inductances we will use "q=r" and for mutual inductances "qr". Starting with this assumption, the relation between the currents determined from differential equations and the torque output of the motor with the co-energy concept. MMF Placement in the Space Let us assume a coil of Tq turns for the qth phase. Due to the current iq in the coil the MMF distribution in the air-gap (d) can be presented by the equation (1). The MMF distribution with respect to the space angle is given by the following equation. ܶ ݅ ߛ ߨ ۍെ ՜ ߙ െ ߨ ߙ ൏ ߙ െ ʹ ʹ ێ ܶ ݅ ߨ ߨ ߛ ߛ (1) ܨሺߙሻ ൌ ێ ՜ ߙ െ ൏ ߙ ൏ ߙ ʹ ʹ ʹ ێ ߛ ߨ ܶ ێ ݅ ۏെ ʹ ՜ ߙ ʹ ൏ ߙ ߙ ߨ
Fig. 3. MMF Corresponding to Coil
After disintegrating the coil MMF into the Fourier series we reach at the following interpretation: ܨ ሺߙሻ ൌ
ʹ ܶ݅ ߨ
ஶ
ୀଵǡ ௗௗ
ͳ ߨ ቀ݇ߛ ቁ
ሾ݇ሺߙ െ ߙ ሻሿ ʹ ݇
(2)
Each coil in the air-gap periphery is responsible for the production of kth indexed space harmonic. In the mathematical form it can be represented as follows: ܨ ሺߙሻ ൌ
ʹ ͳ ߨ ܶ ݅ ቀ݇ߛ ቁ
ሾ݇ሺߙ െ ߙ ሻሿ ߨ ݇ ʹ
(3)
For all the Nq coils of a particular q phase the net MMF per phase can be determined from the superposition of likewise harmonics of all the phase coils. Hence the expression for MMF changes in the following manner ே
ʹ ͳ ߨ ܨ ሺߙሻ ൌ ܶ ݅ ቀ݇ߛ ቁ
ሾ݇ሺߙ െ ߙ ሻሿ ߨ ݇ ʹ
(4)
ୀଵ
Fig. 4. Coil Phase group in the space
It is assumed that the centre of phase q is at an angle Įq as shown in the figure, after solving the equation (2) with this assumption: ܨ ሺߙሻ ൌ
ʹ ͳ ܶ ܰ ݇ ܭௗ ݅
ൣ݇൫ߙ െ ߙ ൯൧ ߨ ݇
(5)
Ampere's law can be used to find out the magnetic flux density in the peripheral air-gap. We use the concept of negligible reluctance of iron path for the determination of flux. ܤ ሺߙሻ ൌ
ߤ ܨ ሺߙሻ ݀
(6)
After solving equation (6) with the help of equation (5) ܤ ሺߙሻ ൌ
Fig. 2. Coil Placement in the Machine
ߤ ʹ ͳ ܶ ܰ ݇
݅ ܭൣ݇൫ߙ െ ߙ ൯൧ ݀ ߨ ௗ ݇
(7)
To determine flux linkage mutually between phases q and r, magnetic flux of phase r included in phase q is required. When both the phases are same, i.e. q=r, phase q flux is known and when q r , mutual flux for phase q is known. Mutual harmonic component of magnetic flux density in c coil for any phase r is calculated by equation (7). This flux
density can be used to calculate the mutual and self fluxes of generic phase q with respect to any other generic phase r. ߣ ሺߙሻ ൌ ܶ න
ఊ ఈ ି ଶ
ఊ ఈ ି ଶ
ܤ ߙ݀ܦܮ
(8)
Let us assume the that a coil c of phase q is linking with the spatial flux of phase r. So kth order harmonics of this flux can be computed using equation (8). It is considered that any generic phase "q" is made up of Nq distributed coils. So the net mutual flux between phase q and r is determined by the equation (9). ߣ ሺߙሻ ൌ
ே
ୀଵ
ୀଵ
Limits for integration are given as
ቈන
ఈ̶
ఈᇱ
ܤ ߙ݀ܦܮ
(10)
ߙோ ߛ ߨ ʹ ʹ
(11)
ߙ̶ ൌ ߙ െ ൫ܰ െ ʹܿ ͳ൯
Equation (8) is the harmonic flux linkage between any two phases which includes the term of harmonic flux density, it is dependent on harmonic MMF. Using equations (4), (7) and (8), net mutual flux linkage of any two phases is given by ݇Ԣ ݇௪ ݇௪ ݅
ൣ݇൫ߙ െ ߙ ൯൧ ݇ଶ
(12)
Where the constant k'qr is given for: ݇Ԣ
݁ ൌ ܴ ݅ ܮ
݀݅ ݀ݐ
(16) ߷
and ݇௪ ൌ ݇ ݇ௗ is phase "q" winding factor ݇௪ ൌ ݇ ݇ௗ is phase "r" winding factor Angular position of "q" phase axis is abbreviated as ߙ kth order mutual harmonic flux linkage between any two generic phases q and r is given by equation(8), which can be abbreviated in form of equation (13) ߣ ݅
(13)
Inserting the relation (12) into the equation (13) gives the result as follows ߷ ൌ ܮ ݅
ൣ݇൫ߙ െ ߙ ൯൧
݅
݀߷ ൨ ݀ݐ
݀݅ ݀ݐ
In these equations the only variable term is the inductance which any time is dependent on the rotor position with respect to stator hence for the six voltage current variables we got six rotor and stator equations. But these equations have rotor angle with respect to stator which is in turn dependent on the torque developed. So it is necessary to include two torque and speed equations to solve the problem. From the electromechanical energy conversion energy principles it is clear that the torque is the differential product of co-energy function. The co-energy is given through currents as following ͳ ܹ ൌ ߷ ݅ ݅ ʹ
(17)
So the developed electromagnetic torque expression derived from
Ͷߤܦܮ ܰ ܰ ܶ ܶ ൌ ߨ݀
߷ ൌ
Hence the voltage-current relation is given as
(9)
ߙோ ߛ ߨ ߙᇱ ൌ ߙ െ ൫ܰ െ ʹܿ ͳ൯ െ ʹ ʹ
ߣ ൌ
ߣ ሺߙሻ ே
(15)
ߣ ൌ ܮ ݅ ߷ ݅
ൌ ܶ
and
In voltage current equations generic phase flux is ߣ (where q is generalization for q=A,B,C,a,b,c.). The generalized flux linkage ߣ is due the fundamental as well as harmonic components of the inductances. Hence general expression of flux linkage is transformed into the following form:
(14)
Where ܮ is the abbreviation for the expression ݇Ԣ ݇௪ ݇௪ ܮ ൌ ݇ଶ Ultimately we have arrived at the expression of the harmonic inductance which is given by equation (14). These harmonics in the inductance are due the space arrangement of the winding.
ܲ ߜܹ ȁ ʹ ߜߠ ௦௧௧௨௧
(18)
݀߷ ܲ ݅ ݅ ݀ߠ ͺ
(19)
ܶ ൌ
or ܶ ൌ
Equation (19) is solved along with motor dynamics equation and current and voltage equations in order to determine the current response and torque response. III. SLOT HARMONICS EFFECT In the previous subsection the effect of the winding placements were analyzed irrespective of the shape and the size of the slots. In order to analyze vibrations in the rotor the harmonics due to slots and due to slot sizing are been considered in this section. In the preceding sections its has been assumed that due to the harmonics of slots there may not be effect on the steady state toque or high loads. But in the smoothing applications of the induction motors, the small amount of torque pulsations can leave a lot of distortions on the surface hence cannot be neglected [10]. In literature the application of slot harmonics is mainly limited to the speed estimation of the electric device[11]. The method to analyze
the slots harmonics incorporates the estimation of air gap permeance due to the slotting in the air-gap periphery[12]. The tediousness is that slot harmonics are dependent on the harmonics in air-gap MMF of rotor and stator windings and the next is the permeance effect due to the slotting. Due to slotting some parts in the core saturates and in some parts the flux density is extremely low. Hence the net effect due to saturation may suppress some harmonics and some harmonics are increases in amplitude due to these harmonics effects. The frequency of the slot harmonics is given by ݂ ൌ ݂ ቈሺ݄ܴ േ ݊ௗ ሻ
ሺͳ െ ݏሻ േ ݇௧ ʹܲ
This expression contains the order of eccentricity and time harmonics. For the analysis of machine design time harmonic index kt=1, for normal eccentricity (at the centre) the constant nd =0 and multiplicity to rotor number can be considered as k=1. Modified winding function approach is used for the simulation purpose [13]. In this approach the effect of winding as well as slot harmonics is considered. As the winding distribution and pitching effect is already considered in the preceding section. So modified winding function is here mainly focused to incorporate the slot permeance effect. Specific slot permeance due to the stator and rotor slots can be given as ߩ௦ ൌ ߩ௦
ሺܵݕሻ ߩ
ሼܴሺ ݕെ ߱ ݐሻሽ ߩ௦ௗ
ሼሺܵ െ ܴሻ ݕ ܴ߱ ݐሽ ߩ௦ௗ
ሼሺܵ ܴሻ ݕെ ܴ߱ ݐሽ
(20)
This described permeance is dependent on the space parameter y and the speed of the rotor with respect to stationary frame of reference. Rotor and stator MMF will interact with given permeance and this gives rise to principle slot harmonics in the flux density. The flux density due to space harmonics can be considered as: ܤௗ ൌ ܤଵ ߩଶ
ቈሺ݄ܴ േ ʹܲ݇ሻݕ െ ቊ݄ܴ
(24)
ܽଵ ൌ ߪߟ
(25)
Another calculation for the inverse air-gap function is done for the slotted rotor with an assumption of smooth stator periphery ݀ିଵ ሺݕǡ ߠ ሻ ൌ
ͳ ͳ െ ܾଵ
ሼܴሺ ݕെ ߠ ሻሽ ݀ ߢଶ
(26)
െ ǥ ǥ ǥ ൨
Constants values for this equation are obtained same as for equation (23). After all the net air-gap function including stator and rotor slotting effects can be finalized as ିଵ ሺݕǡ ߠ ሻ ൌ ݀൫݀ିଵ ሺݕǡ ߠ ሻ݀ ିଵ ሺݕǡ ߠ ሻ൯ ݀௦ ௦
(27)
Till the analysis, the slots have been modeled into the airgap function. So these effects are still to be included into the main machine model. For this, following approach has been used. The turn functions are described for both the stator and the rotor and hence these turn functions are used to determine the modified winding functions. So the stator turn functions for a three phase induction motor is described as ஶ
ܶ ሺݕሻ ൌ
ܹ ሺ݅ሻ
ሺʹ݅ݕሻ ܰ
(28)
ୀଵǡଷǡହǥ ஶ
ܶ ሺݕሻ ൌ
ߨ ܹ ሺ݅ሻ
ቆʹ݅ ቀ ݕെ ቁቇ ܰ ͵
(29)
ஶ
(30)
ୀଵǡଷǡହǥ
ܶ ሺݕሻ ൌ
ܹ ሺ݅ሻ
ቆʹ݅ ൬ ݕെ ୀଵǡଷǡହǥ
where ܹ ሺ݅ሻ ൌ ܹ ሺ݅ሻ ൌ ܹ ሺ݅ሻ ൌ
(22)
ʹߨ ൰ቇ ܰ ͵
Ͷܰ ߨ݅ ͺܰ Ͷߨ݅ ݊݅ݏ ݊݅ݏ ͵ߨ݅ ͵ ͵ߨ݅ ͻ
Turn function for the lth rotor loop is given as ஶ
ܶ ሺݕǡ ߠ ሻ ൌ
ܹ ሺ݆ሻ
ቂ݆ሼ ݕെ ߠ ୀଵǡଷǡହǥ
െ ሺ݈ െ ͳሻߙሽ
And the߱ ݐof the equation (20) is replaced withߠ . Inverse air gap function for the machines stator slots with a consideration of smooth rotor has been calculated as ͳ ͳ െ ܽଵ
ሺܵݕሻ െ ǥ ǥ ǥ ൨ ݀ ߢଵ
݉ ݉ െ ͳǤߪ݉
And
ሺͳ െ ݏሻ േ ͳቋ ߱ ݐെ ߰ଵ ʹܲ
ͳ ͳ ܽଵ െ
ሺܵݕሻ ݀ ߢଵ ߢଶ ߢଶ ܾଵ
ሼܴሺ ݕെ ߠ ሻሽ െ ߢଵ ܽଵ ܾଵ
ሼሺܵ െ ܴሻ ݕ ܴߠ ሽ ʹ ܽଵ ܾଵ
ሼሺܵ ܴሻ ݕെ ܴߠ ሽ൨ ʹ
݀௦ିଵ ሺݕǡ ߠ ሻ ൌ
ߢଵ ൌ
(21)
The inverse air-gap function includes the effect of slot harmonics, is described as ିଵ ሺݕǡ ߠ ሻ ൌ ݀௦
Slot design factorsߪ, (dependent on the ratio of d and mp) and ߟ (dependent on the ratio of mo and mp) are used in computation of the design constants carter factor ߢଵ and ܽଵ
(23)
where ܹ ሺ݆ሻ ൌ ߙൌ
ߙ ቃ ʹߨ
Ͷ ݆ߙ ݆ߛ ൬ ൰ ൬ ൰ ߨ݆ ଶ ߛ ʹ ʹ
ʹߨ Ǣ ߛ ൌ ߙ כ ݎݐ݂ܿܽݓ݁݇ݏ ܴ
Skew factor is defined as the fraction of one rotor slot.
(31)
Modified winding function in presence of slot effects is as following ܯሺݕǡ ߠ ሻ ൌ ܶሺݕǡ ߠ ሻ െ ܯۃሺߠ ሻۄ
(32)
Final Inductance expression including winding harmonics and slot effects is derived from equation (15) and (34) and given as equation (44) ܮሺሻ ൌ
In equation (32) ܯۃሺߠ ሻۄ
ଶగ ͳ ሻ݀ ିଵ ሺݕǡ ߠ ሻ݀ݕ ൌ ିଵ ሺݕǡ ߠ ሻ ۄන ܶሺݕǡ ߠ ௦ ʹߨ݀ۃ௦
(33)
For two coils C1 and C2 the mutual inductance is the flux linkage of coil C1 due to the 1A current in coil C2, which determined in following fashion ଶగ
ିଵ ሺݕǡ ߠ ሻ݀ݕ ܮభమ ൌ ߤ ܴ ܮන ܶଵ ሺݕǡ ߠ ሻܯଶ ሺݕǡ ߠ ሻ݀௦
(34)
So this way the mutual inductance has been calculated which include the effect of slot harmonics. After the calculation of inductances the voltage current equations can be perfectly used to determine the exact machine behavior. Our main plant is to see the machine behavior in terms of the torque profile. So with these inductances the voltage and current equations are solved with Runge-Kutta differential equation solving numerical method along with the motor dynamics equation. Hence the final torque representation can be exhibited in the following format ߜܮ௦௦ ݅ۍ௦ ݅ ߜߠ ௦ ܶ ൌ ێ ܮߜ ݅ێ௦ ݅ ۏ ߜߠ ௦
ߜܮ௦ ݅௦ ݅ ې ߜߠ ۑ ߜܮ ۑ ݅ ݅ ߜߠ ے
(35)
IV. MOTOR DYNAMICS Three phase stator voltage equations are given for the induction motor as following ݀ߣ ݀ݐ ݀ߣ ݁ ൌ ܴ ݅ ݀ݐ ݀ߣ ݁ ൌ ܴ ݅ ݀ݐ ݁ ൌ ܴ ݅
(36) (37) (38)
Similarly for the rotor these are follows as ݀ߣ ݀ݐ ݀ߣ ݁ ൌ ܴ ݅ ݀ݐ ݀ߣ ݁ ൌ ܴ ݅ ݀ݐ
݁ ൌ ܴ ݅
(39) (40) (41)
These electrical equations can be solved together with motor dynamics equation. With this procedure motor response can be determined in terms of the torque and current waveforms. Motor dynamics equations are as follows ݀ଶ ߠ ܲ ൌ ሺܶ െ ܶ ሻ ݀ ݐଶ ʹ ܬ ݀ߠ ൌ ߱ ݀ݐ
(42) (43)
All Ȝ’s in voltage equations are determined collectively using equations (15) and (34). For the solution of these equations Runge-Kutta fourth order has been used.
ߣ ฬ ܮభ మ ݅ ஷଵ
(44)
V. MODELING AND RESULTS Motor parameters and other data have been procured from a 3-phase, 5H.P. , 1460 rpm, 440Volt, 50Hz motor. Machine for the study, was provided by electrical engineering lab, IIT Mandi as shown in Fig. 4. Motor has been disassembled to collect the design data in the laboratory with direct measurements. With these data and parameters in hand a software program was written to simulate the machine behavior. Programming for the modeling has been done using a three step algorithm. .
Fig. 5. Disassemble motor for measurements TABLE I. Parameter Stator Slot Opening Stator Slot pitch Stator Slot depth Carter Factor for Stator Slots Air Gap Diameter Machine Bore Length Air-gap
MOTOR DATA Value 2 mm 10mm 20mm 1.1143 11.18 cm 10.7 cm 0.38mm
First step of the algorithm is to model the machine with winding harmonics only using equations as described in section II. Second step is associated with the implementation of slot harmonics. Hence in the third step the net inductance due to all space harmonics effect was programmed using the equations described in section III. This inductance is used to find out the current and torque responses using equations (36)(41). Fan type load has been used in which torque is squarely proportional to speed. The reason behind the assumption of fan type load is that it is the most commonly used load for drive application. Either it could be in compressor or in water pumping applications. Steady state torque in fig. 6 could have followed a flat profile for continuous and current sheet edged air-gap periphery. Because of slotting effect in the air gap, lower value ripples in the torque can be seen in fig. 8. As slots in the stator deepens, the spike in the torque response at the lower limit of torque ripple will increase in magnitude. Major ripples in torque response are due to the winding distributions. An inference is drawn from the torque response that these ripples will decrease in magnitude if the each wire of every phase
group is distributed in space with minimum distribution angle. Fig. 9 demonstrates the current ripple in rotor. Ripples are visible in rotor current properly because current frequency to ripple frequency ratio is very high in rotor current. Current ripple visible in the waveforms are due to winding harmonics only as slot permeance is very less to demonstrate the effect. Inference of the fig. 6 and fig. 7 is that model is behaving similar to the real machine. Extension of this work is to validate these results with the real machine. Later this model will be optimized to reduce the ripples in the torque and current. This method of modeling and analysis is appropriate for low and medium power motor designs with high and medium speed applications. These type are suitable for smoothing application where high precision is obligatory. In smoothing application small value of torque ripple can be detrimental. Winding harmonics of 5th, 7th, and 11th order have been taken in account because for higher orders the magnitude decreases in mass, so those can be dropped in analysis.
VI. CONCLUSION Mathematical modeling for induction motor has been done in three phase frame of reference. Winding and slot harmonics effects have been programmed separately for the torque and current responses. On the next step of modeling both effects have been included simultaneously. Design data for programming the model has been taken from a real machine. After the modeling and its result analysis, large ripples have been observed in the torque waveform of the motor. Torque ripple has been calculated from the results and value found to be 11.12%. REFERENCES [1]
[2]
[3]
[4] [5] [6] Fig. 6. Torque Response
[7] [8] [9]
[10] Fig. 7. Current Response [11]
[12]
[13] Fig. 8. Torque Ripple due to Space Harmonics
Fig. 9. Current Ripple due to Space Harmonics
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