Estimation of optimum values for the main dimensions of a.d.c. machine by graphical method.

Easwarlal, C. ; Palanisamy, V. ; Sanavullah, M. Y. 等

Introduction

[1] Design may be defined as a creative physical realization of theoretical concepts. Engineering design is application of science, technology and invention to produce machines to perform specified task with optimum economy and efficiency. Engineering is the economical application of scientific principles to practical design problems. If the items of cost and durability are omitted from a problem, the results obtained have no engineering value. The problem of design and manufacture of electric machinery is to build, as economically as possible, a machine which fulfills a certain set of specifications and guarantees. The design is subordinated to the question of economic manufacture. The major considerations to evolve a good design are:

I. Cost

II. Durability

III. Compliance with performance criteria as laid down in specifications.

In most of the situations, it becomes difficult to design a machine which meets all the performance indices and also satisfies the cost and durability criteria because these requirements are usually conflicting. It is impossible to design a machine which is cheap and is also durable at the same time. This is because a machine which is to have a long life span must use high quality materials and advanced manufacturing techniques which obviously make it costly. The performance indices have to be met for certain. However, a compromise between cost and durability can be had. In order to achieve optimality of an objective, the values that are used for the variables should be optimum besides assuming which of the variables are free and fixed variables suitably.

The main dimensions of a d.c. machine can be expressed in terms of the rated power output, desired magnetic and electric loadings and speed of the machine in the form of an equation, which is known as output equation of the machine. The choice of specific magnetic and electric loadings is made based on performance factors, such as:

I. Maximum flux density in iron parts of the machine

II. Magnetizing current

III. Core losses

for specific magnetic loading and

I. Permissible temperature rise

II. Voltage

III. Size of machine

IV. Current density

for specific electric loading.

It is clear that these specific loadings are design variable. Similarly the main dimensions are also design variables. Speed of the machine may be treated as fixed variable or constant. Hence the power output for a constant or fixed speed machine can be expressed in terms of the above four variables. The general procedure in design is to assume some variables as known and fixed and some variables as independent or free variables and evolve. Since the specific loadings are dependent on the main dimensions, the specific loadings are treated as fixed variables and the main dimensions are treated as independent or free variables.

To find optimum values for the free design variables, an objective function and constraint, if any is required in terms of the design variables. The above equations are written using only two variables namely the armature diameter and armature core length, and optimum values are obtained by the proposed graphical method. Referring to ref. [2], for a 750 W, 350 rpm, ceiling fan motor, efficiency is considered as the objective function whereas temperature rise and motor weight are the constraints and the slot electric loading, magnet-fraction, slot-fraction, air gap and air gap flux density are the design variables. [3] Messine, F (et. al) showed the advantage of a deterministic global-optimization method in the optimal design of electromechanical actuators. The numerical methods classically used are founded either on nonlinear programming techniques (i.e., augmented Lagrangian, sequential quadratic programming) or on stochastic approaches which are more satisfactorily adopted to global optimum research (i.e., genetic algorithm, simulated annealing). However, the later methods only guarantee reaching this global optimum with some probability. In this paper, the design variables are actually non-linear in nature and limited to two in number. Further the power output is taken as constraint. The constraint equation also contains the same design variables, which helps to propose [4], [5], [6] a simple graphical method for getting optimum values for the design variables. The values are obtained simultaneously, which is a special feature of this method and satisfy the objective.

Symbols

[P.sub.a] = Power developed by armature, kW

L = Armature core length, m

[B.sub.av]= Specific magnetic loading, Wb/m2 A

P = Rated power output, kW

[PSI] = Ratio pole arc to pole pitch

[C.sub.o] = Output coefficient

[b.sub.p] = Width of pole body

Lf = Lagrangian function

p = Number of poles

D = Armature diameter, m

n = speed in rps

ac = Specific electric loading,

[eta] = Efficiency

[tau] = Pole pitch = [pi]D/p

b = Pole arc

[lambda] = Lagrange multiplier

X = A constant

Description of Design Problem

[1] To find the optimum values for the main dimensions of a 5 kW, 250 V, 4 pole, 1500 rpm simplex lap wound d.c. shunt generator. The loadings are: Average flux density in the gap = 0.42 Wb/m2 and ampere conductors per metre = 15,000 A. Assume full load efficiency = 0.87.

Solution by Conventional Method

Design makes use of assumptions, empirical relations derived out of experience, approximations, etc., Neglecting friction, windage and iron losses, the power developed by the armature in general can be written as:

[P.sub.a] = P/[eta] = 5/0.87 = 5.75 kW

Speed, n = 1500/60 = 25 r.p.s

output equation, [P.sub.a] = [[pi].sup.2][B.sub.av] ac [D.sup.2] Ln x [10.sup.- 3] (1)

The output coefficient

[C.sub.o] = [[pi].sup.2][B.sub.av] ac x [10.sup.-3] = [[pi].sup.2] x 0.42 x 15000 x [10.sup.-3]

[C.sub.o] = 62.178 [approximately equal to] 62.1 (2)

and [D.sup.2] L = [P.sub.a]/([[pi].sup.2][B.sub.av] ac x [10.sup.-3]) n = [P.sub.a]/[C.sub.o] n

[D.sup.2] L = 5.75/(62.1 x 25) = 0.0037037 [m.sup.3] (3)

[D.sup.2] L [approximately equal to] 0.0037 [m.sup.3] (4)

The value as a product in terms of D2 and L is obtained for D and L. In order to separate D and L, the following techniques depending upon the requirement of appearance of the machine are adopted.

i. Square section (L = [b.sub.p])

ii. Length equaling twice the width of pole body (L = 2 [b.sub.p])

iii. Square pole face (L = b)

The above indicates only pole proportion such that the machine appears nice and beautiful. Let us take the case of square pole face such that the armature core length equals pole arc.

For a square pole face, core length/pole arc = L/b = 1 Or L/[PSI] [tau] = 1

Taking, [PSI] = pole arc, b/pole pitch, [tau] = 0.64,

L = [PSI] x [pi]D/p = 0.64 [pi]D/4 = 0.50265D

L [approximately equal to] 0.503D (5)

[FIGURE 1 OMITTED]

Substituting eqn. (5) in eqn. (4)

0.503 [D.sup.3] = 0.0037

D = 0.19448 [approximately equal to] 0.194 m

and L [approximately equal to] 0.1 m

Thus the values for the main dimensions are found analytically one after another.

Optimum Value by Graphical Method

[2] Optimization is a process of finding the conditions that give the maximum or minimum value of a function. For any optimization problem, an objective is compulsorily required along with constraint, if any is to be specified. Here the conditions refer to the values for the main dimensions, namely D and L such that the objective function is made maximum or minimum. The function refers only to the objective function. The aim now is to bring in an objective function. The armature of a d.c.machine labeling D and L is as shown in Fig.1. Whenever we refer armature diameter, D it remainds only a circle, whereas armature core length, L may be thought of a linear distance. The circle (i.e. armature) diameter is also considered as a linear distance with a distance of D. Hence the objective function may be written in terms of D and L, which are variables, as

f(D,L) (6)

The objective is to minimize the sum of linear distances of D and L and hence the objective function equation is written as

D + L = X (7)

Where X is a constant, which must be minimum.

The design problem is now stated as to find the values for D and L such that total linear distance in the machine is optimum (i.e. minimum) and at the same time delivering the rated power. Therefore the output equation is the constraint equation, which is already in terms of D and L. The corresponding equations are eqn.7 and eqn.4, which are rewritten as

D + L = X (objective function)

and [D.sup.2] L = 0.0037 [m.sup.3] (constraint equation).

The above equations contain only two variables. The constraint is an equality constraint type. Since there are two variables, a simple graphical method is suggested.

The graph for constraint equation is drawn first. As the machine has to deliver its rated power output, the output equation of the d.c. machine is taken and values for D are obtained for different assumed values of L and a graph between D and L is drawn. The shape of the graph is obviously non-linear and takes the form of a parabola as shown in Figure 2. The readings for the graph are as shown in Table 1.

To draw the objective function graph, a constant value for X is assumed. Then for different assumed values for L, the corresponding values for D are obtained. It is a straight line graph. This process is repeated for different X. The objective function is thus represented as a series of straight line graphs as shown in Fig.2. It may be noted or seen that some graphs may be above the constraint graph and some may be below it. But at the same time, one straight line graph may become tangent to the constraint graph. The points for objective function graph for different value of X are given in Table 2, Table 3 and Table 4 as examples.

[FIGURE 2 OMITTED]

The constraint graph and the objective function graphs are shown in Fig.2. From the tangent point, the values for D and L are obtained as

D = 0.192 m

L = 0.1m

Which are optimum values for D and L.

Optimality by Lagrangian Method

The objective function is D + L = X

and constraint equation is [D.sup.2] L - 0.0037 = 0

The Lagrangian function [L.sub.f] can be written as

[L.sub.f] (D,L,[lambda]) = D + L + [lambda]([D.sup.2] L - 0.0037)

The necessary conditions for optimum are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[partial derivative][L.sub.f/[partial derivative][lambda] = 0 (10)

On solving, the optimum values are

D = 0.1948 m

and L = 0.0974 m

Comparison of Results

The values for armature diameter and core length obtained by all methods are given in Table 5 for comparison.

It is seen that the sum of armature diameter and core length is less in graphical method than that obtained by conventional method which indicate that the values for D and L are optimum. It is verified with the help of Lagrangian method.

Conclusion

A d.c. machine design problem is solved by conventional method, which employs assumptions, approximations, etc., and the values for the main dimensions are calculated. The values are not optimum. The same design problem is converted as an optimization problem. The objective function equation and constraint equation are carefully introduced. The optimum values are estimated from the proposed graphical method. The values are verified with the value obtained by popularly available optimizing method. The values are agreeable. The speciality of this method is that both values are obtained simultaneously. Such method can be used in intermediate steps in the process of design of an electric machine, provided the constraint equation and objective function contain only two same variables. The performance of the d.c. machine would improve because the optimum values are estimated at the first step itself which will be used in further calculations. Hence such graphical method may be adopted for obtaining optimum values for the two variables at the time of design of an electric machine.

References

[1] A. K. Sawhney " A course in Electrical machine Design", Dhanpat Rai & Co (p) Ltd., Delhi-110006, 2001.

[2] "Genetic algorithm based design optimization of a permanent magnet brushless dc motor", J.Appl.phys.97, 10Q516(2005); D.O.I.: 10.1063/1.1860891.

[3] Messine F., Nogarrede. B., Lagowanelle. J. L., " Optimal design of electromechanical actuators: a new method based on global optimization," IEEE Transactions on Magnetics, Vol.34, Issue 1, Jan 1998, PP. 299-308, D.O.I.:10.1109/20.650361.

[4] Singirasu S. Rao, "Engineering optimization theory and practice," 3rd Ed., New Age international (p) Ltd., New Delhi-110002, 1999, pp. 11-15.

[5] Easwarlal C., Palanisamy V., Sanavullah M. Y., Gopila, M., "Graphical Estimation of optimum weights of iron and copper of a transformer", ISBN: 0-7803-9772-X/06/$20.00 [C] 2006 IEEE.

[6] C. Easwarlal, V. Palanisamy and M. Y. Sanavullah, "Optimim full load losses of a Transformer by graphical method," International Journal of Electrical and Power Engineering, (3):359-362, 2007, [C]Medwell journals, 2007.

C. Easwarlal (1), V. Palanisamy (3), and M.Y. Sanavullah (4)

(1) Prof/EEE, Lecturer/EEE (2), Sona College of Technology, Salem--636005 E-mail: easwarlalc@yahoo.co.in

(3) Principal, Info Institute of Engineering, N.H.209, Sathy Road Kovilpalayam, Coimbatore--641107.

(4) Dean/EEE, K.S.R. college of Technology

Table 1: Readings for L and D (output equation)

L, m             0        0.01     0.02    0.05    0.07     0.09

[D.sup.2],   [infinity]   0.37     0.185   0.074   0.053    0.041
[m.sup.2]
D, m         [infinity]   0.608    0.43    0.272   0.23     0.202

L, m            0.1       0.12     0.14    0.16    0.2      0.25

[D.sup.2],     0.037      0.0308   0.026   0.023   0.0185   0.0148
[m.sup.2]
D, m           0.192      0.1756   0.162   0.152   0.136    0.121

L, m           0.3      [infinity]

[D.sup.2],     0.0123       0
[m.sup.2]
D, m           0.111        0

Table 2: D + L = 0.2

D,m        0       0.2
L,m       0.2       0

Table 3: D + L = 0.3

D,m        0       0.3
L,m       0.3       0

Table 4: D + L = 0.4

D,m        0       0.4
L,m       0.4       0

Table 5: Comparison of values

                                    Methods
Design variables
                   Conventional    Graphical    Lagrangian

D, m                  0.194          0.192        0.1948
L, m                  0.100          0.100        0.0974
(D+L), m              0.294          0.292        0.2922