Reference has already been made to the importance in dynamo design of the predetermination of the flux due to a given number of ampere-turns wound on the field-magnet, or, conversely, of the number of ampere-turns which must The magnetic circuit. be furnished by the exciting coils in order that a certain flux corresponding to one field may flow through the armature core from each pole. An equally important problem is the correct proportioning of the field-magnet, so that the useful flux Za may be obtained with the greatest economy in materials and exciting energy. The key to the two problems is to be found in the concept of a magnetic circuit as originated by H.A. Rowland and R.H.M. Bosanquet;[16] and the full solution of both may be especially connected with the name of Dr J. Hopkinson, from his practical application of the concept in his design of the Edison-Hopkinson machine, and in his paper on “Dynamo-Electric Machinery.”[17] The publication of this paper in 1886 begins the second era in the history of the dynamo; it at once raised its design from the level of empirical rules-of-thumb to a science, and is thus worthy to be ranked as the necessary supplement of the original discoveries of Faraday. The process of predetermining the necessary ampere-turns is described in a simple case under [Electromagnetism]. In its extension to the complete dynamo, it consists merely in the division of the magnetic circuit into such portions as have the same sectional area and permeability and carry approximately the same total flux; the difference of magnetic potential that must exist between the ends of each section of the magnet in order that the flux may pass through it is then calculated seriatim for the several portions into which the magnetic circuit is divided, and the separate items are summed up into one magnetomotive force that must be furnished by the exciting coils.
| Fig. 29. |
The chief sections of the magnetic circuit are (1) the air-gaps, (2) the armature core, and (3) the iron magnet.
The air-gap of a dynamo with smooth-core armature is partly filled with copper and partly with the cotton, mica, or other materials used to insulate the core and wires; all these substances are, however, sensibly non-magnetic, so that the whole interferric gap between the iron of the pole-pieces and the iron of the armature may be treated as an air-space, of which the permeability is constant for all values of the flux density, and in the C.G.S. system is unity. Hence if lg and Ag be the length and area of the single air-gap in cm. and sq. cm., the reluctance of the double air-gap is 2lg / Ag, and the difference of magnetic potential required to pass Za lines over this reluctance is Za·2lg / Ag = Bg·2lg; or, since one ampere-turn gives 1.257 C.G.S. units of magnetomotive force, the exciting power in ampere-turns required over the two air-gaps is Xg = Bg·2lg / 1.257 = 0.8Bg·2lg. In the determination of the area Ag small allowance must be made for the fringe of lines which extend beyond the actual polar face. In the toothed armature with open slots, the lines are no longer uniformly distributed over the air-gap area, but are graduated into alternate bands of dense and weak induction corresponding to the teeth and slots. Further, the lines curve round into the sides of the teeth, so that their average length of path in the air and the air-gap reluctance is not so easily calculated. Allowance must be made for this by taking an increased length of air-gap = mlg, where m is the ratio maximum density/mean density, of which the value is chiefly determined by the ratios of the width of tooth to width of slot and of the width of slot to the air-gap between pole-face and surface of the armature core.
The armature core must be divided into the teeth and the core proper below the teeth. Owing to the tapering section of the teeth, the density rises towards their root, and when this reaches a high value, such as 18,000 or more lines per sq. cm., the saturation of the iron again forces an increasing proportion of the lines outwards into the slot. A distinction must then be drawn between the “apparent” induction which would hold if all the lines were concentrated in the teeth, and the “real” induction. The area of the iron is obtained by multiplying the number of teeth under the pole-face by their width and by the net length of the iron core parallel to the axis of rotation. The latter is the gross length of the armature less the space lost through the insulating varnish or paper between the disks or through the presence of ventilating ducts, which are introduced at intervals along the length of the core. The former deduction averages about 7 to 10% of the gross length, while the latter, especially in large multipolar machines, is an even more important item. Alter calculating the density at different sections of the teeth, reference has now to be made to a (B, H) or flux-density curve, from which may be found the number of ampere-turns required per cm. length of path. This number may be expressed as a function of the density in the teeth, and ƒ(Bt) be its average value over the length of a tooth, the ampere-turns of excitation required over the teeth on either side of the core as the lines of one field enter or leave the armature is Xt = ƒ(Bt)·2lt, where lt is the length of a single tooth in cm.
In the core proper below the teeth the length of path continually shortens as we pass from the middle of the pole towards the centre line of symmetry. On the other hand, as the lines gradually accumulate in the core, their density increases from zero midway under the poles until it reaches a maximum on the line of symmetry. The two effects partially counteract one another, and tend to equalize the difference of magnetic potential required over the paths of varying lengths; but since the reluctivity of the iron increases more rapidly than the density of the lines, we may approximately take for the length of path (la) the minimum peripheral distance between the edges of adjacent pole-faces, and then assume the maximum value of the density of the lines as holding throughout this entire path. In ring and drum machines the flux issuing from one pole divides into two halves in the armature core, so that the maximum density of lines in the armature is Ba = Za / 2ab, where a = the radial depth of the disks in centimetres and b = the net length of iron core. The total exciting power required between the pole-pieces is therefore, at no load, Xp = Xg + Xt + Xa, where Xa = ƒ(Ba)·la; in order, however, to allow for the effect of the armature current, which increases with the load, a further term Xb, must be added.
| Fig. 30. |
In the continuous-current dynamo it may be, and usually is, necessary to move the brushes forward from the interpolar line of symmetry through a small angle in the direction of rotation, in order to avoid sparking between the brushes and the commutator (vide infra). When the dynamo is giving current, the wires on either side of the diameter of commutation form a current-sheet flowing along the surface of the armature from end to end, and whatever the actual end-connexions of the wires, the wires may be imagined to be joined together into a system of loops such that the two sides of each loop are carrying current in opposite directions. Thus a number of armature ampere-turns are formed, and their effect on the entire system of magnet and armature must be taken into account. So long as the diameter of commutation coincides with the line of symmetry, the armature may be regarded as a cylindrical electromagnet producing a flux of lines, as shown in fig. 30. The direction of the self-induced flux in the air-gaps is the same as that of the lines of the external field in one quadrant on one side of DC, but opposed to it in the other quadrant on the same side of DC; hence in the resultant field due to the combined action of the field-magnet and armature ampere-turns, the flux is as much strengthened over the one half of each polar face as it is weakened over the other, and the total number of lines is unaffected, although their distribution is altered. The armature ampere-turns are then called cross-turns, since they produce a cross-field, which, when combined with the symmetrical field, causes the leading pole-corners ll to be weakened and the trailing pole-corners tt to be strengthened, the neutral line of zero field being thus twisted forwards in the direction of rotation. But when the brushes and diameter of commutation are shifted forward, as shown in fig. 31, it will be seen that a number of ampere-turns, forming a zone between the lines Dn and mC, are in effect wound immediately on the magnetic circuit proper, and this belt of ampere-turns is in direct opposition to the ampere-turns of the field, as shown by the dotted and crossed wires on the pole-pieces. The armature ampere-turns are then divisible into the two bands, the back-turns, included within twice the angle of lead λ, weakening the field, and the cross-turns, bounded by the lines Dm, nC, again producing distortion of the weakened symmetrical field. If, therefore, a certain flux is to be passed through the armature core in opposition to the demagnetizing turns, the difference of magnetic potential between the pole-faces must include not only Xa, Xt, and Xg, but also an item Xb, in order to balance the “back” ampere-turns of the armature. The amount by which the brushes must be shifted forward increases with the armature current, and in corresponding proportion the back ampere-turns are also increased, their value being cτ2λ / 360°, where c = the current carried by each of the τ active wires. Thus the term Xb, takes into account the effect of the armature reaction on the total flux; it varies as the armature current and angle of lead required to avoid sparking are increased; and the reason for its introduction in the fourth place (Xp = Xg + Xt + Xa + Xb), is that it increases the magnetic difference of potential which must exist between the poles of the dynamo, and to which the greater part of the leakage is due. The leakage paths which are in parallel with the armature across the poles must now be estimated, and so a new value be derived for the flux at the commencement of the iron-magnet path. If P = their joint permeance, the leakage flux due to the difference of potential at the poles is zl = 1.257Xp × P, and this must be added to the useful flux Za, or Zp = Za + Zl. There are also certain leakage paths in parallel with the magnet cores, and upon the permeance of these a varying number of ampere-turns is acting as we proceed along the magnet coils; the magnet flux therefore increases by the addition of leakage along the length of the limbs, and finally reaches a maximum near the yoke. Either, then, the density in the magnet Bm = Zm / Am will vary if the same sectional area be retained throughout, or the sectional area of the magnet must itself be progressively increased. In general, sufficient accuracy will be obtained by assuming a certain number of additional leakage lines zn as traversing the entire length of magnet limbs and yoke (= lm), so that the density in the magnet has the uniform value Bm = (Zp + zn) / Am. The leakage flux added on actually within the length of the magnet core or zn will be approximately equal to half the total M.M.F. of the coils multiplied by the permeance of the leakage paths around one coil. The corresponding value of H can then be obtained from the (B, H) curve of the material of which the magnet is composed, and the ampere-turns thus determined must be added to Xp, or X = Xp + Xm, where Xm = ƒ(Bm)lm. The final equation for the exciting power required on a magnetic circuit as a whole will therefore take the form
X = AT = 0.8Bg·2lg + ƒ(Bt) 2lt + ƒ(Ba) la + Xb + ƒ(Bm) lm.