ELECTRO MAGNETIC FIELD ANALYSIS 493 



Normalized Network Modki and Lagrange-Maxwell 

 Electrodynamical Equations 



Let us now return to the ideas of Lagrange as applied to electromagnetics.. 

 [n dynamics the Lagrange equations are formulated in terms of the kinetic 

 energy T expressed as a function of velocities, potential energy U expressed 

 as a function of coordinates, and a dissipation function F expressed as a 

 function of velocities. In network theory T is the magnetic energy ex- 

 pressed as a function of currents, U is the electric energy expressed in terms 

 of charges, and F is the dissipation function in terms of currents. Lagrange- 

 Maxwell equations are then written in the following form 



l[i'--"]-^'--«+a¥. = "-. » 



where In is the typical mesh current, ^„ is its time integral, and Vn is the 

 impressed electromotive force, that is, the electromotive force not accounted 

 for by the magnetic induction and the charges in the network. The various 

 functions in the equation are 



7^_>^vlr T T A'_vv ^"^ 9n 



^^mn (5 J. 



F = i:m^nhRmnImIn, 



where Lmn is the mutual inductance between two typical meshes (the self- 

 inductance if m = n), Cmn is the mutual capacitance and i?„,„ is the mutual 

 resistance. The mesh currents are introduced in order to insure that the 

 total current either entering or leaving a typical junction of the network 

 elements is zero. If we perform the diflferentiations indicated in equation 

 (4), we shall obtain the network equations in their usual form. 



Let us'now suppose that F = and F„ = 0. In higher algebra it is 

 shown that by a linear transformation two quadratic functions, T and U for 

 example, can be reduced to normal forms in which there are no mutual 

 terms 



T = i:nhLn / n , U = S„ fjlCn • (6) 



In this case equations (4) will assume the following simple form 



i„ ^" + 1^ = 0. (7) 



at Ln 



It is as if we had a certain number of isolated single-mesh circuits. Equa- 

 tions (7) represent the normal modes of oscillation of the network. 



