344 
DRS. J. AND E. HOPKINSON ON DYNAMO-ELECTRIC MACHINERY. 
recovered whilst the brush is in contact with only one bar of the commutator. The 
field thus oscillates slightly, owing to the disturbance caused by reversing the 
direction of the current in the successive sections of the armature. The number of 
oscillations in a Gramme armature or in a Siemens’ armature with an even number of 
sections will be pm, where p is the number of revolutions per second, but in a 
Siemens’ armature with an odd number of sections it will be 2 pm* This oscillation 
of the field is only another way of expressing the effect of the self-induction of the 
section, but it must be remembered that if the self-induction, multiplied by change of 
current, is expressed as a change in the field we must omit self-induction as a separate 
term in our electrical equations. The precise lead to be given to the brushes in order 
to avoid sparking in any given case depends on many circumstances—the form and 
extent of the pole-pieces, the number of sections in the armature, and the duration of 
the short circuit which the brushes cause in any section of the armature. The 
adjustment of the position of the collecting brushes is generally made by hand at the 
discretion of the attendant, and is in some cases fixed once for all to suit an average 
condition of the machine. We shall, therefore, treat X the lead as an independent 
variable, controlled by the attendant. 
Let I be total induction through the armature, I + T total induction through the 
magnets, I' being the waste field. Let C be current in armature, c in the magnets. 
Let gV be the line integral of magnetic force from a point on one pole-piece to a point 
on the other, the line being drawn external to the armature, g will be approximately 
constant. Omitting as comparatively unimportant the magnetising force in the pole- 
pieces and iron core of the armature we have the following equations :— 
4\mC+ gY=0 
Ao 
= 47mc. 
4 hnC + 2 4 —+ l s f( ^ 
When C=0 we observed 
whence 
9= 
1 2 i 3 
c-1 A s 
[* Added Aug. 17.—Armatures with an odd number of convolutions are open to one theoretical 
objection, which would be a practical one if the number of convolutions were very small. The 2?n-f-l 
convolutions constitute in themselves a closed circuit, having a resistance four times the mean actual 
resistance of the armature measured between the collecting brushes. When any one convolution is 
exactly in the middle of the field the E.M.E. of the other 2m convolutions exactly balance, so that there 
is upon the closed circuit an E.M.F. due to the single convolution somewhat in excess of ~th part of 
the actual E.M.F. of the machine. Thus there will be an alternating E.M.F. around the closed circuit 
of the armature capable of causing a considerable waste of power. This waste is materially checked by 
the self-induction of the circuit.] 
