CIRCUIT DESCRIBED IN TERMS OF NORMAL MODES 243 



By using (6.16), we obtain 



Pn = AnAt 0~\ (r; + /35) ff (wnY dx dj (6.26) 



It is also of interest to express the z component of the nth mode, Ezn , 

 expHcitly. For the wave traveling to the right we have, from (6.18), 



£.„ = An (^j (r'„ + /3n)7^„(.^^ y) (6.27) 



Let the field at some particular position, say, x = y = 0, be E^no ■ Then 



^"- (rl + /3^)x„(0,0) ^^-2^^ 



and from (6.26) 



^•' = (^'""^"'"*^ 2,l(oro)(rl'+ gg ) // l*"(^' ^)'' '"" ''y ('^■2") 



We can rewrite this 



E^noE^nO* 27^l(0, 0)(rl + /3o) 



{-Tl)Pn . ^ , ^2^ /Tr. / M2, , (6.30) 



jwer„( — r^„) // [7r„(:r, y)f dx dy 



For an active mode in a lossless circuit, r„ is a pure imaginary, and the 

 negative of its square is the square of the phase constant. Thus, for a par- 

 ticular mode of propagation we can identify (6.30) with the circuit parame- 

 ter E?/0^P which we used in Chapter II. 



Let us now imagine that there is an impressed current J which flows in 

 the z direction and has the form 



/ = J(x, y)g~J (6.31) 



According to Maxwell's equations we must have 



dx dy 



Now, we will assume that the fields are given by some overall stream func- 

 tion TT which varies with x and y and with z as exp(— F^). 



In terms of this function tt, Hx , Hy and Ex , Ey will be given by relations 

 (6.7), (6.8), (6.10), (6.11). However, the relation used in obtaining Ez is 

 not valid in the presence of the convection current. Instead of (6.16) we 

 have 



dHy dHx • z. , r 



dx dy 



(6.33) 





