1903.] The " Hunting " of Alternating -Current Machines. 237 



clear, therefore, that in this case at any rate the hunting was not due 

 to resonance but to some essential instability in the motion of the motor 

 itself. 



V It is easy to see how such instability could arise. In the argument 

 given above it has been assumed that the torque is dependent only on 

 the relative position of motor and generator and not on their relative 

 velocity. As a matter of fact there is a term in the torque dependent 

 on the velocity, and the equation of motion is M + b + c = 0. 

 Higher differential coefficients than the second may and in fact do 

 come in, but these are the most important terms as a rule. The 

 solution of this equation is, if b is small, 



If b be positive, the amplitude of the oscillations continually 

 decreases. If, however, b were negative, even though very small, the 

 oscillations would continually increase and the motion be essentially 

 unstable. Most dynamical systems are affected with viscosity, in 

 which case b is positive, but systems are not unknown in which the 

 contrary is the case. Watt's Governor is such a system.* Its oscilla- 

 tions about steady motion are given by a cubic equation, the two 

 complex roots of which have positive real parts and correspond to 

 constantly increasing oscillations. There is no doubt that the motion 

 of a synchronous motor is under certain conditions another instance of 

 the same thing. 



Suppose for the present that the motor has a permanent magnet or 

 saturated field, and that it is working against a constant load. 



Let 6 be an angle defining the position of the armature in space, 

 I = A sin 6 the induction linked with the field coils and with the 

 armature when in position 6. In virtue of the above assumption, A is 

 constant. Let t be the time, and E cospt + F sin pt, the E.M.F. of the 

 source of supply ; L the self-induction and p the resistance of the 

 armature and any conductors in series with it. 



Assuming for the moment that the motor is moving with uniform 

 angular velocity, let u = a sin pt + /3 cos pt be the current in the 

 armature. The epoch of t is as yet un chosen ; choose it so that in the 

 steady motion 6 = pt. Now suppose that the state of steady motion is 

 slightly disturbed so that the motor oscillates about it. Then we have 

 in the disturbed motion : 



0=pt + and u = (a + a) sin pt + (/3 + /3) cos pt, 



where , a, /? are small quantities varying periodically with the time. 

 Experience shows that in all cases the period of the variation is long 



* See Routh's ' Rigid Dynamics,' TO!. 2 (1892), p. 74 ; also Maxwell's Collected 

 Papers, vol. 2, p. 105. 



