ELECTRO-MOTIVE FORCE IN ARMATURE.

79

also the number of wires on the armature so large, that their angular distance ▲ a is very small, in fact so small that the intensity of the field can be considered as constant within that angular distance. Since the electromotive force created in the wires is proportional to their speed, we can determine it for any convenient speed, and if it be required for a different speed, we can obtain it by multiplying the result first obtained with the ratio of the two speeds. In the present instance we fix as a convenient speed that which will bring each wire at the end of one second into the position occupied by its immediate neighbour at the beginning of the second, or

This is a very slow speed, and if we wish to know what will be the electro-motive force at the faster speed of n revolutions a minute, we shall have to multiply the electro-motive force at the low speed with the ratio of

n

60°

D and v.

Since A a Nt 2π we have also

π D and the ratio of the two speeds is

v =

n one-half of the circumference of the armature, then

the electro-motive force in these wires will be given by

The sum of all these forces gives the total electro-motive force created within the armature, which we denote in future by E..

But the expression F1 b v represents the number of lines contained between the first and second wire on the armature, since F, is the density, and bv the area of the space swept by the first wire in one second. Similarly F2 b v represents the number of lines between the second and third wire, and so on, the sum of all these expressions representing the total number of lines entering between the first and last wire on one-half circumference of the armature. Let z be that total number, and we find for the electro-motive force at the low speed,

precisely the same expression as already obtained above. If z be inserted in absolute measure, E. will also be obtained in absolute measure, and to obtain it in volts the right side of the equation must be multiplied with 10 -3. We can also write

ELECTRO-MOTIVE FORCE IN ARMATURE.

81

and if we measure the field intensity by means of a unit 6,000 times as great as the absolute unit, we can further simplify the equation to

Z being the total number of lines in the new system, which is related to the absolute system by the equation

The cross-sectional area of the armature core is 2 a b, and if we denote by m the average density of lines per square inch of armature core, we have,

Z = 2 a b m,

and inserting this value in 5), we find for the electromotive force also the expression,

Ea 2 a b m Nt n 10-6.

α

6).

This expression is sometimes more convenient than the former, because it enables us at once to see how the dimensions of the armature affect the electro-motive force. Experience has shown that the density of lines, m, in the core cannot exceed a certain limit, which is reached when the core is saturated with magnetism. This limit is m = 30, but in practical work a lower density is generally adopted, for reasons which will be explained in the following chapter. A fair average value in good modern dynamos and motors is m = 20, and the area, a b, must be taken as that actually filled by iron, and not the gross area of the core. To avoid waste of power and heating, the armature core of dynamos and motors must be subdivided into portions insulated from each other, the planes of division being parallel to the direction of the lines of force. The space wasted by such insulation must be deducted from the gross area of the core, and the remainder

G

-from 80 to 90 per cent. of it-is the portion actually carrying lines of force.

The electrical energy developed in the armature, if a current c be flowing through its coils, is E, c, and the horse-power represented by this energy is

The power to be applied must naturally be somewhat in excess of this in order to overcome mechanical resistances, as friction in the bearings and air resistance, and also the magnetic resistance due to imperfect subdivision of the core, and reaction of the armature on the magnets. In good dynamos these losses amount to about 10 per cent.

CHAPTER III.

Reversibility of Dynamo Machines-Different conditions in Dynamos and Motors-Theory of Motors-Horse-power of Motors-Losses due to Mechanical and Magnetic Friction-Efficiency of Conversion-Electrical Efficiency-Formulas for Dynamos and Motors.

AFTER what has been explained in the previous chapters it will be evident that dynamo machine and electro-motor are convertible terms. Any dynamo can be used practically as a motor, and in most cases any motor can be used to generate a current. On purely theoretical grounds this should be possible in all cases, but in practice it is found that the speed which is required to make some small motors act as self-exciting dynamos is so high as to render that application mechanically impossible. The reason for this is, that in small motors the polar surfaces are of very limited extent, and consequently the magnetic resistance of the path traversed by the lines of force is excessively high, requiring more electrical energy to excite the field magnets than the armature is capable of developing at a moderate and practical speed. This point will be more fully explained further on. For our present purpose it suffices to note that on purely theoretical grounds the same machine can act as a motor or as a dynamo. A separate investigation as to the theory of motors might, therefore, almost seem superfluous. But, on the other hand, experience has shown that although