32 BELL SYSTEM TECHNICAL JOURNAL 



K 2 i(ni = .6), as in Fig. 10. Dissipation in the inductances is included 



by assuming effective coil resistance = coil reactance; it has the 



effect of eliminating abrupt changes in the attenuation and impedance 

 characteristics. Computations made on this basis give the sum of the 

 three attenuation constants and the impedance X 2 i(.6) as shown in 

 the figure. 



The attenuation over a range of 2500 cycles about the center of 

 the transmitting band is less than .19 attenuation units and for fre- 

 quencies in the attenuating band is high, remaining after the first 

 maximum on either side of the transmitting band above a value 

 7.30 in the lower frequency attenuating band and a value 7.10 in the 

 upper. The characteristic impedance K 2 i(.6) over the 2500 cycle 

 range is everywhere within 3% of the desired resistance value, 600 

 ohms, and has here a negligible reactance. Its resistance component 

 has maxima at the critical frequencies and decreases rapidly to small 

 values in both attenuating bands. The reactance component is nega- 

 tive like a capacity reactance at very low frequencies, has a positive 

 maximum at the lower critical frequency and negative minimum at 

 the upper critical frequency, and is positive like an inductive react- 

 ance at very high frequencies. This demonstrates the possibilities 

 of the composite structure method. 



APPENDIX I 



Derivation of Fundamental Formulae 



Although formulae for the propagation constant and characteristic 

 impedances of the 'adder type of recurrent network are well known 

 and follow readily from a consideration of the current and voltage 

 relations shown in Fig. 1, it is perhaps of interest to derive them as 

 a special case of general formulae which involve admittances 3 and 

 which are directly applicable to any type of recurrent passive 

 structure including loaded lines. 



Let the periodic section of the recurrent structure be defined by the 

 one-point and two-point admittances A aa , A bb , and A ab , where the 

 subscripts a and b , respectively, refer to its two pairs of terminals. 

 Then the current at the junction, q, in terms of the voltages at the 

 junctions q- 1, q, and g+1, is 



Iq = AabV t -i-AbbVg = A aa V q -AabV q+1 , (1) 



3 The solution by Difference Equations in terms of the admittances was suggested 

 by J. R. Carson and is a convenient form for expressing the general results. 



