24 BELL SYSTEM TECHNICAL JOURNAL 



like the original fields and velocity of one line, and the fields and velocity of 

 the other mode will be almost like the original fields and velocity of the other 

 line. However, if the coupling is strong enough compared with the original 

 separation of phase velocities, both lines will participate almost equally in 

 each mode. One mode will be a "longitudinal mode" for which the excitations 

 on the two lines are substantially equal, and the other mode will be a "trans- 

 verse" mode for which the excitations are substantially equal and opposite. 

 The ratios of the voltages on the lines for the two modes are given by 

 (3.75). Here it is assumed that the series reactances A' and shunt susceptances 

 B of the lines are almost equal, differing only enough to make a difference 

 AFo in the propagation constants. Bn and X12 are the mutual susceptance 

 and reactance. We see that to make the voltages on the two lines nearly 

 equal or equal and opposite, B12 and Xn should have the same sign, so that 

 capacitive and inductive couplings add. 



Fig. 3.1 — A helically conducting sheet of radius a. The sheet is conducting along hehcal 

 paths making an angle xp with a plane normal to the axis. 



Increasing the coupling increases the velocity separation between the two 

 modes, and this is desirable. When there is a substantial difference in ve- 

 locity, operation in the desired mode can be secured by making the electron 

 velocity equal to the phase velocity of the desired mode. 



To make the capacitive and inductive coupHngs add in the case of con- 

 centric helices (Fig. 3.17), the helices should be wound in opposite directions. 



3.1 The Helically Conducting Sheet 



In computing the properties of a helix, the actual helix is usually replaced 

 by a helically conducting cylindrical sheet of the same mean radius. Such a 

 sheet is illustrated in Fig. 3.1. This sheet is perfectly conducting in a helical 

 direction making an angle ^, the pitch angle, with a plane normal to the 

 axis (the direction of propagation), and is non-conducting in a helical direction 

 normal to this \p direction, the direction of conduction. Appropriate solutions 

 of Maxwell's equations are chosen inside and outside of the cylindrical sheet. 

 At the sheet, the components of the electric field in the \}/ direction are made 

 zero, and those normal to the \p direction are made equal inside and outside. 

 Since there can be no current in the sheet normal to the ^ direction, the 



