E-mail: [email protected]. Abstract-This paper presents the design optimization of linear induction motor. Application of linear induction motor includes various ...
Design Optimization of Linear Induction Motor Abhay Kumarl M.A.Hasan2 I
Md. Junaid Akhta?
S.K.Parida2
RK.Behera2
Department of Electrical Engineering, Indian Institute of Technology, llT Guwahati, Guwahati, India 2
Department of Electrical Engineering, Indian Institute of Technology, ITT Patna, Patna, India E-mail: skparida@ iitp.ac.in
Abstract-This paper presents the design optimization of linear induction motor. Application of linear induction motor includes various industrial processes like conveyor belt, vertical movement, steel power plant, induction heating and automated material handling. All these applications require an efficient and high torque providing linear induction motor. Objective function of the optimization problem discussed in this paper includes efficiency, output thrust and machine weight. Various machine design parameters have been used as constraint variables. Optimization done with Quasi Newton process shows significant improvement in machine efficiency and output torque compared to the results reported in literature. Machine design and optimization is carried out using RMxprt software. Keywords-Linear Induction Motor, Design Optimization, Quassi-Newton Technique
I.
INTRODUCTION
Certain merits of linear induction motor like high starting torque, simple structure, reduction in mechanical losses and smaller size make it an attractive choice as motor in motor driven systems [1]. There are various applications in industries where vertical motion of the machine is required. For example, robotic machines made for building constructions, rope-less lifts in super-skyscrapers, material transport for underground constructions etc. Linear induction motor can be used for all these purposes being able to facilitate linear upward and downward motion. One of the advantages of linear induction motor is its non contact motion and quick response [2]. In steel power plants, for high quality and high productivity, linear induction motor has been researched and practically used for transporting thin steel plates during reheating and galvanization process [3]. Basic principle of operation of linear induction motor also facilitates eddy current generation by travelling magnetic field. This can help in induction heating [4]. In coal plants, conveyor belt is used to carry the coal blocks from one place to other for further processing. Conventional induction motors used for rotating conveyor belts provide forces at relatively smaller area. Due to stretching or mismatch of forces at two ends, belt faces danger of getting slipped. Linear induction motor can provide such a provision where a belt of conducting material passes between a pair of linear stator blocks carrying poly-phase system of coils [5]. Design of linear induction motor demands preparation of data sheet for stator and rotor dimensions. Number of stator and rotor slots, inner and outer dimensions, tooth and slot dimensions and conductor size requirement are some of the design parameters require to manufacture a linear induction motor [6]. RMxpert software provide
platform to design various parameters of the electrical machine by providing performance characteristics of motor for a set of stator and rotor dimensions. By varying these dimensions, a suitable design can be obtained. Optimization provides an optimum solution for design problem. An optimized design of the motor gives stator and rotor dimension values for which machine gives optimum performance. Various optimization techniques are available in the literature [7]. This paper uses Quasi Newton optimization process to perform the design optimization. Paper is organized as follows. Section 2 presents linear induction motor modeling. Section 3 discusses Quasi Newton optimization technique used in this paper. Results obtained for optimized machine is presented in section 4. Finally conclusion discusses the significance of presented work and results obtained. II. LINEAR INDUCTION MOTOR MODELING Fig. 1 presents the cross-sectional view of the rotor and stator section of the LIM. Non-uniform airgap causes an effective airgap ge different from physical airgap gm Relation between physical and effective airgap is given in (1-2). (1)
go =gm+d
(2)
where d is the thickness of the conducting layer and kc is Carter's coefficient given by (3),
(3)
Parameter A is the slot pitch which is the distance between the centers of the two consecutive teeth. Stator slot depth hs can be calculated from (4). h,
A, W,
(4)
where As is the cross sectional area of the slot and Ws is the slot width. A relation for the slot cross sectional area with number of turns per slot and conductor cross section area is given in (5). A,
(5)
Here Nc is the number of turns per slot. Yoke height of the stator core hy is the portion of the core below teeth. If it assumed that the flux in the yoke is one half of the flux in air gap, it can be expressed as,
(13)
(14)
III. DESIGN OPTIMIZATION r--
I
--
�
I
I I I I I
I I I I I
Solution of an optimization problem follows an iterative process which starts with an initial point xc, producing a sequence of points Xk that converges at an optimum point x*. If x is a vector of design variables with constraints Xmin< X < Xmax and f is a function to be optimized, then second order Taylor expansion around Xk is given by (15)
(15) Fig. 1: Cross section of linear induction motor
(6)
A linear induction motor of the specification given in table I is designed using RMxpert software. Complete datasheet is presented in table II. Mathematical model is extended to obtain the electrical equivalent circuit model of LIM. Electrical equivalent circuit of LIM consists of series elements Rl and Xl, core reactance component Xm in parallel with a variable resistance Rs/s representing mechanical load. Equations representing equivalent circuit components are given in (7-11).
(7)
2flo1if Xl
[( ( !J ) ;: } A, 1 +
+
A,,
+
AJ,
=
P
Xm
(8)
_Xm 2-
(10)
ge
2J.1o Jr2 =
nge
Objective function for the optimization used in this paper includes efficiency, electromagnetic torque and weight of the machine. Objective function is built so as to maximize efficiency and torque and minimize weight of the machine. Objective function is given by (16). (16)
=
(9)
G
Slot width: 10 mm < Ws < 25 mm Slot height: 20 mm < hs < 35 mm Primary height: 45 mm < hy < 60 mm Secondary sheet thickness: 2.5 mm < d < 7.5 mm Air gap: 0 mm < gm < 7 mm Current density: 3 A/mm2 < J 1 < 6 A/mm2
J(x) (T(x)*ry(x))/W(x)
,'
24 J.1o JifW,.ekwNi2r J[ 2 pge R
Where P=X-Xk, and Bk denotes the Hessian matrix (second order partial derivatives of x). In this paper following design variable constraints have been used.
(11)
where Pw, lw and AWl are volume resistivity, length and cross sectional area of stator winding copper wire. Kp is pitch factor, Kw is winding factor and ge is equivalent air gap. Based on the electrical equivalent circuit, performance parameters of LIM like electromagnetic torque, power output and efficiency can be obtained as given in (12-14). (12)
Where T(x), �(x) and W(x) represent torque, efficiency and weight function respectively. Table 1: Linear induction motor specification
Specification
Value
Rated Power Rated Voltage Number of poles Rated frequency Rated speed
3.73 kW 440V 4 50 Hz 1428
Table 2: Complete Datasheet
Design Parameter
Design value
Number of stator slots Stator outer diameter Stator inner diameter Number of rotor slots Inner diameter of rotor
36 213 mm 119 mm 30 6l.8 mm
IV. SIMULATION AND RESULTS Electrical machine design is an iterative process. Search of a machine design with desired output require repetitive calculations to satisty all the constraints. Help of software based platform in this work is highly appreciated. ANSYS RMxprt is a design tool which calculates the performance of a hypothetical machine based on electrical equivalent circuit. It helps in making decision regarding machine dimensions and material selection. User provides a set of initial machine dimensions. Software does rigorous electromagnetic transient analysis to produce possible performance of the machine. Table 3: Design of LIM
Design parameter
Design value (in mm)
Stator top tooth width Stator bottom tooth width Length of the stator core Diameter of the conductor Stator resistance Air gap Length of the rotor Rotor stacking factor
5.98 5.96 140 1.151
As given in table 3 and 4, a theoretical design of Linear Induction motor of efficiency 67 percent, power factor 0.87 and electromagnetic torque of 24 N-m is produced. Flux density and magnetic field distribution shows the end effect in LIM. Red color closed loop lines at extreme left end shows that due to end effect, flux density is more than elsewhere. As we increase the load, the load current changes. Corresponding change in speed and input power is given in fig. 4-5. For full load condition, full load current of approx. 10 A is drawn at rated speed of 1428 rpm. As load is increased, output power requirement increases and hence secondary current increases. Since magnetizing reactance remains constant, primary side current increases. This characteristic is shown in fig. 4-5. As we increase speed of the motor, input power has to be increased if constant torque operation is desired. This is presented by fig. 6. In low speed range, torque is directly proportional to the speed. As speed increases, slip decreases. When speed is increased beyond full load speed, voltage drop across magnetizing reactance becomes significant. If the load is increased beyond breakdown point, decrease in rotor power factor becomes significant and torque decreases. This is presented by fig. 7-8.
4.12 ohms 0.3 140 0.96
Table 4: Optimum Design of LIM
Design parameter
Primary current density Primary width Primary height Secondary sheet thickness Air gap Motor length Tooth width Slot width Efficiency Weight Output torque
Design value (in mm)
Fig. 2: Flux density distribution
5.6 140 49.5 5 5 1090 12 18 71.47 50 kg 43.74
Fig. 3: Magnetic field distribution
I
15.00 12.50
Based on iterative process, a machine design for the specification given in table I and 2 is prepared. Table 3 presents complete stator and rotor dimensions for the LIM design. Performance of this machine is given in Table 4. ANSYS RMxprt also generates the magnetic field and flux density distribution in the air gap. Magnitude of air gap flux and magnetic field is shown through different color lines. Fig. 2-3 gives magnetic field and flux representation.
g10.00
�
£
-
5.00 2.50 0.00
ClJ..,. lofo
- Input Currentt
I
�-
u 7.50
�
:
,::::::j
1 ------0.00
2000.00
4000. 00
I 6000.00
Qrtput Paw.r (W)
I 8000. 00
Fig. 4: Current Vs output power
I
10000. 00
12000. 00
0.0000317 30.00 25.00
�"
I�
'---�---:----""--;::: ---:------:::-;:; : ::-=l I I I �":t' ,."t. I ·
r--:==I===::::::=;=-o--._
I
I
0.0000313 1n
o
uO.0000307
0.0000302
I
5.00 0.00
�eration
� ::-----:: � 500 .00 :----:: 7':':' 50.":: :!"-:: :!'C ::--""'1-::i 500.00 .00 1000 0.00 .00::----=:!':: 250 .00--1-:: 250 00---::
Speedl'pm)
Fig. 5: Current Vs speed
----:----..,--;:::::-;:;:---"'1 .. -----....,.----:-
3750.00 ,-
2500.00
1250.00
1000.00
2000.00
Speedi'pm)
3000.00
4000.00
4500.00
Fig. 6: Power Vs speed
87.50
,-----:----....,---;;::::;:;:1
67.50
47.50
27.50
7.50
·12.50
---
-- -t--
-
1-
- -
---
±---=:---"":'::'-::----:::r: ---:-::r.::-:-c::r-::--:::\ 0.00 250.00 500.00 750.00 1000.00 1250.00 1 500 .00 Speedi,pm) Fig. 7: Torque Vs speed
0.80
5
0.60
Above discussed theoretical machine has a very high efficiency but lower electromagnetic torque. For applications discussed in this paper where LIM's are employed, there is need of high starting and full load torque. Also a low weight machine is preferred for dynamic operation. In order to achieve these requirements, optimization is done. An initial setting of efficiency, weight and torque is fed to Quasi-Newton algorithm. Based on results obtained and cost Vs iteration relation, initial settings may be required to be changed. Finally an optimized design is obtained. Accuracy and closeness of desired output in optimization varies with number of iterations. Fig. 9 presents the cost Vs iteration curve. Cost signifies the closeness of result to the desired value. For higher accuracy, more number of iteration is required. Optimization presented in this paper use 50 iterations. Table 4 presents the optimized machine design. [t was found that different design constraints affect motor performance in different ways. Study of independent effect of each constraint is analyzed by varying a particular constraint keeping others constant. An increase in secondary sheet thickness increases the primary weight and decreases the torque. Power factor and efficiency increases first then starts decreasing. Thus design optimization requires a compromise. In this work power factor is compromised which allows a higher value of air gap. An increase in current density keeping output torque constant decreases conductor wire diameter and increases the winding resistance. This results in reduction in weight and efficiency. Since both weight and efficiency optimization is desired, so the results may vary with different number of iterations. This work has presented the result obtained for 50 iterations. Objective function is optimized for a value of 3.1 *l0-5. V. CONCLUSION
...�
�040
�
0.20
0.00
Fig. 9: Cost Vs iteration
� ::---....,:: r:: : .00:----:c50 :!C 0.OO ::-- --:::b---:: :: :!".::-00 2 50 ....,::1 12 :-:c50.00:---�1500 .00 1000 0.00
Fig. 8: Power factor Vs speed
This paper has presented an optimized design of linear induction motor for better efficiency, weight and torque. A conventional design based on electrical equivalent circuit is prepared in RMxprt software. Optimization of the machine design has been carried out using Quasi-Newton optimization technique. All design parameters affect performance of the machine in a different way. Their independent effects have been discussed in this paper.
Optimized machine achieves better desired output as compared to the conventional design
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