j Ii{ya)Ki{ya) — Io{ya)Ko{ya)) , 



(The /'s and A''s are modified Bessel functions). The admittance inside the 

 beam is 



V =^' =^— h{yb) , . 



' E,i y h{yh)- ^ ^ 



Boundary conditions require that E^o = Ez^ = Ez and Hzq — Hzi = z-^ . 

 Combining the boundary conditions, we see that 



'''-'''-Li' (5) 



where the ratio of ^r is given by (2). Thus the field method gives two equa- 



tions which are equivalent to the circuit and electronic equations of the 

 normal mode method. 



A6,l Normal Mode Par.-^meters for TraN Beam 



The constants appearing in eq. (1) can be evaluated by equating the cir- 

 cuit equation (1) to the circuit equation (5). Thus if Yc = Vo — Vi , 



The constants can be obtained by expanding each side of eq. (6) in terms of 

 the zero and pole occurring in the vicinity of To . Thus if 70 and 7^ are the 

 zero and pole of Yc , respectively, 



Y.^-(y,-yJp) (r^^'), (7) 



and the two sides of eq. (6) will be equivalent if 



To = -70 - I3l , (8) 



-1/2 



f = (-f"-^^' 



7p — 7o 



