NETWORKS FOR I.MI'KDANCK FrXCTlOXS 383 



P(p) — P(0), as given by (3), the following coiiditions are ohtaiiied: 

 P(p) will be a positive real funetion if, and oiilij ij\ 



aa - bi3 > 0; i.e. 7^(0) > (4) 



and 



aa + bid > 

 P{p) — P{0) will be a positive real funetion if, (uid onltj if, 



aa - bis < 0; i.e. P(0) < (5) 



and 



aa' - 3abi3 - SaajS' + biS' < 0. 



If all terms of the series satisfy one or the other of these conditions, 

 network branches can be devised to represent all the terms and all the 

 R, L, G, C elements of the branches will be positive. 



In case all the terms are of the type where P„(0) is positive, so that 

 the network branches are made to represent Pnip), the left-over constant 

 terms can be collected and added to the first term, F{0), of the series. 

 This collection of terms then must be represented by a final branch of 

 pure resistance, or conductance, of value, 



FiO) - Z PM 



n=o 



If the sum of the variable terms approaches zero for p —^ ±zoo, the 

 final constant term supplies the high frecjuency resistance of the func- 

 tion F(p) and since this must be positive, if condition (3) is satisfied, the 

 final resistive element will be positive. If the series converges non-uni- 

 formly, the sum of the variable terms can have a value other than zero 

 as p ^ ±/oo in spite of the fact that every term approaches zero indi- 

 vidually. In that case (see example 1) all or part of the high frequency 

 resistance may be supplied by the sum of the variable terms. 



In case all the terms are of the type where Pn(0) is negative, so that 

 the network branches are made to represent the sum, P„(p) — Pn(0), 

 of the variable and constant terms and the series is uniformly conver- 

 gent, all the high frecjuency resistance is provided by the branches rep- 

 resenting these terms. The first term, F(0) then supplies the dc re- 

 sistance, which is positive by condition (3). Non-uniform convergence 

 can modify this division of high- and low-freciuency resistance, however. 



Cases can arise in which the series contains terms of both types. In 

 such a case the dc resistance, or high frecjuency resistance, or both, of 



