CIRCUIT DESCRIBED IN TERMS OF NORMAL MODES 241 



Applying Maxwell's equations again, we have 



dEz dEy 



3 ^ = ico/x£?x 



ay dz 



j d fd T d''7t \ _|_ ir ^x _ . dit 



coe dy ydx^ dy^ / coe dy dy 



(6.15) 



This is certainly true if 



■^2 



iSo = cov^ = co/c (6.17) 



We find that this satisfies the other curl E relations as well. 

 From (6.16) and (6.14) we see that 



E. = (-i/coe)(r2 + ^l^^^^ 3,),-r^ (^ Ig) 



For a given physical circuit, it will be found that there are certain real 

 functions 7r„(x, 3') which are zero over the conducting boundaries of the 

 circuit, assuring zero tangential field at the surface of the conductor, and 

 which satisfy (6.16) with some particular value of F, which we will call r„ . 

 Thus, as a particular example, for a square waveguide of width W some 

 (but not all) of these functions are 



T^n(x, y) = cos (mry/W) cos (utx/W) (6.19) 



where n is an integer. We see from (6.10), (6.11) and (6.18) that this makes 

 Ex , Ey and Ez zero at the conducting walls x = ±:W/2, y = ±W/2. 



Each possible real function Ttn{x, y) is associated with two values of 

 r„ , one the negative of the other. The r„'s are the natural propagation 

 constants of the normal modes, and the tt^'s are the functions giving their 

 field distribution in the x^ y plane. The 7r„'s can be shown to be orthogonal, 

 at least in typical cases. That is, integrating over the region in the x, y 

 plane in which there is field 



/ / Ttn{x, y) 7r„(x, y) dx dy ^ 



(6.20) 

 n 9^ m 



For a lossless circuit the various field distributions fall into two classes: 

 those for which r„ is imaginary, called active modes, which represent 

 waves which propagate without attenuation; and those for which r„ is 

 real, which change exponentially with amplitude in the z direction but do 

 not change in phase. The latter can be used to represent the disturbance 

 in a waveguide below cutoff frequency, for instance. 



