271 



Dividing by B'^lCh; 



(ChlBg)c>z/dt-{8/STr) sin (a^+^)±[sin (at+(3)^zY=0. 



From equation (153), paragraph 244: 



ChlBg= (8/3x) tan 0/a. 

 Giving: 



(S/Stt) tan 0d2/da^-(8/37r) sin (ai+/3)±[sin {at+^)+zf=0. (28A) 



Expanding the last term of equation (28A), the differential equation 

 for z becomes: 



(8/37r) tan <^d2/dai— (8/37r) sin {at+^)±sva? (a^+/3) 



±22sin (ai+/3)±22=0. (29A) 



The correction factor, z=i/B, is therefore a function of the angular 

 lag, (f), and the phase, at-\-(3, of the primary current. 



21. Since z is relatively small, a first approximation to its value 

 for given values of and a/+/3 may be derived by dropping its square 

 from equation (29A) and neglecting its effect upon the sign of the 

 velocity. 



Rearranging, equation (29A) then becomes: 



(8/3x) tan cl>()z/()ai+2z[±sm (,at-\-^)] 



- (8/3 tt) sin (a^+ /3) isin^ (at+ 13) =0 (30 A) 



in which the positive sign is to be applied when sin (at-\-l3) is positive 

 and the negative sign when it is negative. Angles are in radians. 



Equation (30A) does not appear integrable, but the values of z for 

 a given value of 0, and for successive values of a/+/S increasing by 

 sufficiently small increments may be derived by a somewhat laborious 

 arithmetical solution. The increment selected, in degrees, will be 

 designated Aat°. Its value in radians is then TrAat°ll80. Since 

 differential equations remain approximately true when small finite 

 increments are substituted for the differentials, the first term may be 

 written: 



(8/3 x) tan 0A2/(7rAai°/18O) 



in whichA2 is the increase in z due to an increment of Aat° degrees in 

 the phase of the primary current. If Aat° is sufficiently small, the 

 value of Az does not differ materially from the mcrease in z during 

 the preceding increment in the phase. Designating the preceding 

 value of z as Zo, the first term of equation (30A) then becomes: 



(480/^2 AaO tan 0(2-2o) = K2-2o) 



