202 BELL SYSTEM TECHNICAL JOURNAL 



elements we shall most often encounter are shunt resonant near the fre- 

 quencies considered; their susceptance is near zero and changing slowly but 

 their reactance is near infinity. 



If these ladders are continued endlessly to the right (or terminated in a 

 reflectionless manner) and if a signal is impressed on the left-hand end, the 

 voltages, currents and fields at corresponding points in successive sections 

 will be in the ratio exp(-r) so that we can write the voltages, 



Vn = Fo r"'' (4.35) 



If the admittances Yx and Y^ are pure susceptances (lossless reactors), V 

 is either purely real (an exponential decay with distance) or purely imaginary 

 (a pass band). In this case F is usually replaced by 7/3. In order to avoid 

 confusion of notation, we will use jd instead, and write for the lossless case 

 in the pass band 



Vn = Fo «"'■"' (4.35a) 



Thus, d is the phase lag in radians in going from one section to the next. 

 In terms of the susceptances,* 



cos = 1 + ^2/251 (4.36) 



We will henceforward assume that all elements are lossless. 



Two characteristic impedances are associated with such iterated networks. 

 If the network starts with a shunt susceptance 5i/2, as in a of Fig. 4.12, then 

 we see the mid-shunt characteristic impedance K.^ 



K, = 2{-B,{B2 + 45i))-i/2 (4.37) 



If the network starts with a series susceptance 2Bx we see the mid-series 

 characteristic impedance Kt 



Kr = ±(l/2^i)(-^2 + 450/52)1/2 (4.38) 



Here the sign is chosen to make the impedance positive in the pass band. 

 When such networks are used as circuits for a traveling-wave tube, the 

 voltage acting on the electron stream may be the voltage across B^ or the 

 voltage across Bi or the voltage across some capacitive element of B^ or 

 Bi . We will wish to relate this peak voltage F to the power flow P. If the 

 voltage across B2 acts on the electron stream 



FyP = 2K, (4.39) 



If the voltage across Yi acts on the electron stream 



F = I/jBx 



* The reader can work sucli relations out or look them up in a variety of books or hand- 

 books. They are in Schelkunoli's Electromagnetic Waves. 



