a positive potential to one segment and the two ends which are at a negative potential to the other segment, both coils are made to deliver their currents in the same direction to the external circuit.

Fig. 151 illustrates the arrangement for employing two such coils; they are similarly wound (right-handedly in this case), and their adjacent ends are joined to the same section of the commutator. Now, as they are at opposite extremities of a diameter, they pass at every moment through parts of the field where they act with equal effect, and therefore, as already pointed out, the E.M.F. will be the same at the extremities of each coil. Since the ends of the two coils, which are at the same E. M. F., are joined to the same segment of the commutator, the E.M.F. due to both coils is only the same as that produced by one of them, and the current will rise and fall in precisely the same manner as with a single coil. It is, in fact, an exactly analogous case to that of joining two primary cells of equal E.M.F. in parallel. There is also the corresponding advantage here that because the coils are joined in parallel the internal resistance between the two segments is only half that of one coil, and, as we have seen, any arrangement that so reduces the internal resistance of a current generator is sometimes very valuable. By increasing the number of turns in the coils we can increase the F.M.F., because a greater number of conductors in series, round the periphery, are then usefully cutting lines of force; but, of course, the number must be exactly the same in each coil. In figs. 149 and 151 it will be observed that there are two active conductors to each coil.

We are now in a position to proceed with the consideration of a method for making the short fluctuating currents depicted in fig. 146 approach more nearly to a continuous steady current. These short currents are at a minimum when the coils are at right angles to the lines of force, or at that point where the reversal of the induced current takes place, and it is evident that if a second pair of coils be placed at right angles to this existing pair, as in fig. 152, they will always lie parallel to the lines of force, or be in the position of best action, just at the moment when the first pair are almost idle. But it now becomes necessary to divide the commutator into four parts, all the coils being, of course, similarly

wound, and the adjacent ends of adjacent pairs connected to the same segment of the commutator. When only two segments are employed, the brushes, as we have observed, are placed so that the divisions of the commutator pass them just at the moment when the coils are at right angles to the lines of force, and when they are almost idle. In the present case, with four coils, the brushes must also be placed so that the division between each pair of segments on the commutator passes a brush when the coil connected to that pair of segments is in the position of least activity, viz. with its plane at right angles to the lines of force. We shall see presently that there are several causes which combine

to slightly vary this particular position, although it can always be found.

In fig. 152 the brushes are indicated by dotted lines, and are shown slightly out of what we have hitherto considered to be their correct position. Supposing the lines of force to pass straight across from one pole-piece to the other, the currents in the various coils would flow in the direction indicated by the arrows, and the resulting current could be led from the armature to the external circuit by the upper brush B,, entering the armature again by the lower brush B. The two horizontal coils D and E are in the position of greatest activity, while the vertical coils a and c are almost idle, and merely serve to conduct the current generated by the

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active coils to that segment of the commutator which the brush is touching. A moment later a and c will each begin to generate a current in the opposite direction to the one now flowing in them, but as by that time they will have passed the brushes, their opposite ends will now be in contact with these same brushes, and the direction of the current in the external circuit will remain unaltered. When the plane of each of the four coils makes an angle of 45° with the lines of force, they are all equally active (although the activity of no one coil is so great as that of the coils E and D while they are in the best position, as shown in fig. 152), and the E.M.F. at the brushes is twice that which is at that moment being developed by one coil.

The resulting E.M.F., due to the joint effect of the double instead of the single pair of coils, is still far from constant, and, as

before, we must determine at what positions of the coils this resulting E.M.F. is at a maximum and where it becomes a minimum. The curve (fig. 146) illustrates the variation of the E.M.F. due to one coil or one pair of coils, and as, when this E.M.F. is highest, that of the second pair of coils is lowest, and vice versa, the relative magnitude of the E.M.F. generated by two pairs of coils at different positions may be indicated by the overlapping curves in fig. 153. From this we wish to construct a curve which shall show how the E.M.F. at the brushes due to the effect of both pairs of coils varies. Now, twice during each revolution one of the two pairs is for the moment acting alone, and, consequently, the E.M.F. at the brushes is simply that due to this pair, and is proportional to the length of the perpendicular line o A. At this moment the E.M.F. at the brushes is at its lowest value, and the length of this line o A determines the lowest point on the curve which we desire to construct. Immediately after this point is passed both pairs are acting together, the activity of one increasing and that

of the other decreasing. At a certain stage they will be acting with exactly equal effect, and this stage is indicated by the intersection of the two curves in B; it occurs, as we have seen, at the moment when each coil makes an angle of 45° with the lines of force. To obtain, therefore, the resulting E.M.F. at the brushes, we must add together these two equal E.M.F.'s; consequently, twice the length of the line C B must be taken as the height of this the highest point in the new curve. When the coils have rotated through another 45°, one pair is again idle and the other at its maximum activity, so that we again reach the lowest point of the curve. The curve so constructed is shown in fig. 154, and it indicates the manner in which the total E.M. F. at the commutator brushes fluctuates when the armature consists of two pairs of coils arranged as in fig. 152. The resulting current will also fluctuate

similarly, depending in strength upon the gross resistance in the circuit.

It is obvious that the variation in the E.M.F. can be further diminished by the employment of a yet greater number of pairs of coils in the armature, provided that they are placed so that each pair comes into the position of best action at the moment when the resulting E.M.F., without their individual aid, would be at a minimum.

For instance, a coil might be placed exactly midway between each of those wound on the armature shown in fig. 152; that armature would then consist of eight coils in four pairs, and the commutator of eight bars or segments (fig. 155). The black portions of the circle represent the metallic segments, the white spaces between them indicating the insulating material. The current from such an armature would be far more steady than one

from the four-coil armature; in fact, it may be stated generally that the greater the number of coils composing the armature, the less the fluctuation of the current. Of course there is a practical limit to the number of coils; for instance, the commutator with this kind of armature must have as many segments as the armature has single coils or sections, and its construction and the making of the necessary connections would be difficult and expensive if the number were excessively increased.

It will be observed that in fig. 155 the whole armature conductor is wound continuously round the core; it is divided into sections having four convolutions each, and a connecting wire is led from the junction of every two adjacent sections to the proper segment of the commutator. The result is of course the same as if the ends of each section were brought direct to the commutator segment, while the actual length of the armature conductor, and therefore the resistance, is slightly reduced. In the following chapter several illustrations will be given of the way in which these commutator connections are effected in ordinary practice.

In order to increase the E.M.F. developed in a given field at a given speed, we must increase the number of conductors on the outer periphery of the armature, which can be done by adding to the number of convolutions, although this also increases the internal resistance. In the armature illustrated there are thirty-two active portions of the wire round the whole external periphery, but as they are joined up in two sets in parallel, the total E.M.F. is only sixteen times that of one active portion.

If we know the number of active conductors joined in series and the number of lines of force which they cut per second, it is easy to calculate the resulting E.M.F. The E.M.F. developed at any moment by any particular conductor moving circularly in a uniform field varies with its position, and is, as we have seen (Chapter VIII.), proportional to the cosine of the angle which the plane of the coil of which it forms a part makes with the lines of force; or to the sine of the angle through which the coil has turned from its position at right angles to the lines of force. But we need not now trouble ourselves with this consideration, for, in a symmetrically constructed armature of many convolutions, the place of each conductor as it moves to a position of greater or less activity is