ELECTRIC WAVE-FILTERS 297 



positive branch. When Vk = 0, as when no dissipation is assumed, 

 the formulas become functions of Uk only but contain a certain in- 

 determinateness regarding the signs attributable to the phase constants 

 and image impedance reactances of the two branches. This difficulty 

 vanishes when dissipation is present to give W a value different from 

 zero, as in a physical wave-filter. 



With dissipation such as to preserve the "constant k'' relation it is 

 readily shown that Vu is negative in a negative branch and positive in 

 a positive one; that is, Vk has the sign of xu- This follows directly 

 from the formula 



since ru- must be a positive resistance in a passive network. On the 

 basis of this result it follows from formulas (6) that ^ in a negative 

 branch 



Xik, Vk, Bk and Xik are negative; 

 X2k and X^k are positive. 



In a positive branch these signs are reversed. 



The characteristics of two such representative branches are shown 

 in Fig. 3, joined as they would be to form an internal transmitting 

 band. The scale of abscissas is Uk rather than frequency in order to 

 be general, and Uk varies in going from left to right from — oo to for 

 the negative branch and from to — qo for the positive branch. In 

 this way a movement along the abscissa-axis from left to right always 

 corresponds to an increase in frequency. A translation from the Uk 

 to the frequency-scale can be obtained in any particular case through 

 the known relationship between Uk and frequency. Such a translation 

 would be equivalent to a variable expansion or contraction of the above 

 characteristics parallel to the abscissa-axis. The effects of dissipation 

 on the different characteristics are indicated by broken lines and show 

 a rounding-off of abrupt changes. Here, for convenience, it was 

 assumed that Vk = .01 Z7a- in a negative branch and Vk = — .Olt/i in 

 a positive branch. If each pair of characteristics Is considered as 

 separated by an imaginary line perpendicular to the Z/t-axis at 

 Uk = 0, then a comparison will yield the statement that corresponding 

 pairs of Ak, Rik and R^k are images of each other with respect to such 

 lines, while pairs of Bk, X^k and X^k are images but also opposite in 

 sign. 



^ See also page 577 of paper in footnote 2. 



