456 BELL SYSTEM TECHNICAL JOURNAL 



For an impedance Zu which is made up of lumped elements of 

 resistance, inductance, and capacity we may express the impedance 

 ratio Zul2R as the ratio of two frequency-polynomials in {if), where 

 i = V— 1 and /is frequency. Thus, 



Zu ._ gp + aSf) + a^iifY + • • • _ ■ • /.gx 



2R bo + bM) + bMr- + • • • ^' 



The impedance coefficients ao, &o, etc., of which one is unity and some 

 may be zero, are positive quantities and are algebraic combinations 

 of the network elements. Their number is equal to, or greater than, 

 the number of independent elements. For any given type of network 

 the coefficients are fixed by the elements, and vice versa. 



Putting this expression in any of the formulae (12) to (15), there 

 results for the propagation constant a form 



ho + h,(if) + h^iify + • • • ^" ^ 



in which go, ho, etc., are algebraic functions of ao, bo, etc., also of c 

 if the network is a bridged-T type. From this the attenuation con- 

 stant and phase constant can also be derived and expressed separately 

 as functions of frequency. 



For the attenuation constant, a form is obtained 



which is the ratio of two frequency-polynomials both in even powers 

 of frequency. One of the attenuation coefficients is unity. 

 For the phase constant, a form 



in which one of the phase coefficients is unity, is the ratio of two 

 frequency-polynomials, odd powers of frequency in the numerator 

 and even powers in the denominator. (It is sometimes convenient to 

 use tan (3/2).) In (21) and (22) the attenuation coefficients Po, Qo, 

 etc., and the phase coefficients Mi, No, etc., are expressible in terms 

 of the impedance coefficients ao, bo, etc. 



It should be mentioned here that in deriving the above expressions 

 certain assumptions have been made; namely, invariable elements 

 and non-dissipative inductances and capacities. These restrictions 

 are well justified from the fact that such departures are usually small 



