320 BELL SYSTEM TECHNICAL JOURNAL 



Equation (2.2-5) is more general than the expression (2.1-5) for )u(s) in that 

 it contains waves going in both directions, but is more special in that To is a 

 diagonal matrix. 



In the curved guide we take /x(s) to be given by (2.1-5) , thus 



n{z) = e-'^f, 2 > (2.2-6) 



where F is a square matrix whose elements are assumed to be known and/+ is 

 a column matrix whose elements are to be determined along with those of/"". 

 The conditions that the transverse components of the electric and mag- 

 netic intensities be continuous at the junction of the two guides are assumed 

 to lead to the requirements 



|ju(z) in straight portion] = [iu(s) in curved portion] 



Yd 1 r ^ 1 (2-2-7) 



— /i(z) in straight portion = V — ii{z) in curved portion 



where the quantities within the brackets are evaluated at the junction and V 

 is a square matrix whose elements are constants. When the curvature of the 

 curved portion becomes small V approaches the unit matrix. For the 

 problem at hand (2.2-7) may be written as 



W^)].=-^ = [m(2)].=^o (2.2-8) 



r^M«i =Fr;^M«] (2.2-9) 



\_az Jz=-o \_az Jz=+o 



in which the subscripts z = — 0, 2 = +0 refer to the straight and curved 

 portions, respectively, of the guide at 2 = 0. 



The requirements (2.2-7) have been established for the rectangular guide 

 in Section 1.4. Their form is also suggested by the conditions that the 

 voltage and current be continuous at the junction of two transmission lines. 

 Thus if we let ju(z) play the role of the voltage, the current in the first line is 

 — Z^^ d\i.{z)ldz and the current in the second is — Z^^ dij.{z)/dz where Zi and 

 Z2 denote the distributed series impedances of the two Hues. It is seen that 

 this leads to scalar equations which look like (2.2-7), but now V denotes the 

 scalar Z1/Z2 instead of a square matrix. 



Setting the expressions (2.2-5) and (2.2-6) for ^{z) in the conditions 

 (2.2-8) and (2.2-9) gives two equations which may be solved simultaneously 

 to obtain/" and/+ in terms of /?, To, F and V: 



h+f-=f-^ 



(2.2-10) 

 -Toh + Tof-= -FF/+ 



/- = (Fo + FD-KFo - VT)h (2.2-11) 



/+ = (Fo 4- VT)-'2Toh (2.2-12) 



