112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



both X and (p. Thus equation (2) is a differential equation in which (p, x, 

 and d(p/dt are the variables. With the armature at rest, x is constant, 

 and (2) may be solved to give the initial field development in operate, 

 or decay in release. When the armature is moving, the variation of <p and 

 X is governed jointly by (2) and the mechanical force equation. 



THE MECHANICAL EQUATION 



The forces acting to accelerate the armature and associated moving 

 parts, of effective mass m, are the magnetic pull F and the forces exerted 

 by the contact and other springs. The total spring force may be written 

 as dV/dx, where V is the potential energy stored in the springs, expressed 

 as a function of x. The force equation may be written in the form: 



As X measures the gap opening, the velocity dx/dt is positive as the 

 gap opens, and negative as it closes. The pull is directed toward the 

 closed position and therefore tends to algebraically decrease the velocity. 

 The spring force tends to open the gap and to algebraically increase the 

 velocity, as T" decreases with .r, and dV/dx is negative. 



On the assumption that the dynamic and static field patterns are the 

 same, the pull F can be evaluated from the static magnetization rela- 

 tions. F is a function of (p and .r, the variables of the flux development 

 equation (2). Thus the dynamic performance is governed jointly by (2) 

 and (3), to which the concurrent variation in (p and x with time must 

 conform. To complete the formulation there are required explicit ex- 

 pressions for (R and F in terms of (p and x. 



RELUCTANCE AND PULL 



As shown in a companion article,^ the magnetization relations for 

 most electromagnets conform to the simple magnetic circuit of Fig. 1. 

 Through the region of linear magnetization, in which the flux density is 

 below that producing incipient saturation, the reluctances (Ro and (Rl 

 and the area A are constants. If the magnetization relations conform 

 to this schematic, the total reluctance (R is given bj^: 



(Rl((Ro + ^ 

 (R = ^ ^ . (4) 



(Rz. + (Ro + J 

 The constants (Ro , (Rl , and A of Fig. 1 and equation (4) are called 



