THE ELECTRIC WAVE-FILTER 15 



Information as to the location of the bands is often obtained most 

 readily by plotting both Zi and —AZ^, as illustrated in Fig. 7, and 

 determining the critical frequencies by noting where the curves cross 

 each other and the abscissa axis, as well as where they become in- 

 finite. Any particular band is then a pass band, a stop ( + ) band 

 or a stop ( — ) band, according as Zi, the abscissa axis, or — 4Z2 lies 

 between the other two of the three lines. In Fig. 7 the pass bands 

 are Pi, P2, Pz, Pi\ the stop ( + ) bands are 52, 54, 56; and the stop 

 ( — ) bands are 5i, S^, St, 5?, and they illustrate quite a variety of 

 sequences. By altering the curves the bands may be shifted, may 

 be made to coalesce, or may be made to vanish. 



Wave-Filter Curves 



The pass band and stop band characteristics of wave-filters are 

 concretely illustrated for a few typical cases by the curves of Figs. 

 8-13, which show the attenuation constant A, the phase constant B, 

 and both the resistance R and reactance X components of the itera- 

 tive impedance for a range of frequencies which include all of the 

 critical frequencies, except infinity. The heavy curves apply to the 

 ideal resistanceless case, while the dotted curves assume a power 

 factor equal to l/(207r) for each inductance which is a value readily 

 obtained in practice. This value is, however, not sufficiently large to 

 make these small scale curves entirely clear, since considerable por- 

 tions of the dotted curves appear to be coincident with the heavy line 

 curves; but this, as far as it goes, proves the value of the present dis- 

 cussion which rests upon a close approximation of actual wave-filters 

 to the ideal resistanceless case. 



The low pass resistanceless wave-filter, as shown by Fig. 8, pre- 

 sents no attenuation below 1,000 cycles; above this frequency the 

 attenuation constant increases rapidly, in fact, the full line attenuation 

 curve increases at the start with maximum rapidity, since it is there 

 at right angles to the axis. The dotted attenuation curve, which in- 

 cludes the effective resistance in the inductance coils, follows the 

 ideal attenuation curve closely, except in the neighborhood of 1,000 

 cycles, where resistance rounds ofi the abrupt corner which is present 

 in the ideal A curve. The phase constant B is, at the start, propor- 

 tional to the frequency, as for an ordinary uniform transmission line; 

 its slope becomes steeper as the critical frequency 1,000 is approached 

 where the curve reaches the ordinate tt, at which value it remains 

 constant for all higher frequencies. As shown by the dotted B 

 curve, resistance rounds off the corner at the critical frequency, but 



