ELECTRIC CIRCUIT THEORY 85 



If we invert the structure, that is, make the series impedance Zi 

 a capacity C and the shunt impedance Z2 an inductance L, so that 



1 . ^ 1 



21=7-7=;, 22=^C0i., P=— . 2 7-/"' 



we get, corresponding to (226) and (227), 



™=''^ = l-2;;SC' (228) 



This type of artificial Hne transmits without attenuation currents 

 of all frequencies for which the right hand side of (228-a) lies between 

 + 1 and —1; that is, all frequencies from infinity to a lower limiting 

 frequency l/47r\/LC, while it attenuates all frequencies below this 

 range. It is known, on this account, as the high-pass filter. 



It is possible by using more complicated impedances to design 

 filters which transmit a series of bands of frequencies. We cannot, 

 however, go into the complicated theory of wave filters here, which 

 has been covered in a series of important papers. One point should 

 be noted, however: transmission without attenuation implies that 

 the impedance elements are non-dissipative. Actually, of course, 

 all the elements introduce some loss, so that in practice the filter 

 attenuates all frequencies. Careful design, however, keeps the 

 attenuation very low in the transmission bands. 



We shall now derive the indicial admittance formulas for some 

 representative types of artificial lines and wave filters from the oper- 

 ational formula 



An = -y=L==Wl+p + .yj]-^\ (229) 



\/(l+p)2l22 ^ ^ 



This equation follows directly from (225) on putting Fo = l. 



We start with the so-called low-pass filter on account of its sim- 

 plicity and also its great importance in technical applications. This 

 type of filter consists of series inductance L and shunt capacity C. 

 The general case which includes series resistance R and shunt leakage 

 G has been worked out (see Transient Oscillations, Trans. A. I. E. E., 

 1919). The solution is, however, extremely complicated and will not 

 be dealt with here. We shall, instead, consider the important and 

 illuminating case where the series and shunt losses are so related 



