Restrictions on the natural frequencies of an RC-RL network☆

Abstract The following problem is investigated: given any polynomial N ( s ) of the form N(s)= ∏ l n (s+p i ) ∏ l m (s+s i ) what are the necessary and sufficient conditions that N ( s ) be formed in the following manner: N(s)=A 0 ∏(s+a i ) + B i ∏(s+b i ) A 0 , B 0 > O where the − a i and − b i alternate at most in pairs along the negative real axis. Such formation is necessary for the − p i and − s i to be the natural frequencies of a network consisting of RC and RL subnetworks joined at two terminals. It is shown that N ( s ) may be so formed if and only if (a) for n=0, Σ l m arg s i ≥ π 2 (b) for n > 0, p i > 0 and Σ l m arg s i π 2 Also, a simple method is presented for finding suitable a i and b i whenever formation is possible. The result is expected to have application in the synthesis of both passive and active networks.