966,326. Two-terminal impedance networks. ASSOCIATED ELECTRICAL INDUSTRIES Ltd. July 9, 1962 [July 17, 1961], No. 25806/61. Heading H3U. A two terminal impedance comprises passive RC impedance networks and a negative impedance converter (NIC). Specifically, the networks (Figs. 1, 2) are of the type described in the paper " Synthesis of Driving-Point Impedances with active RC Networks," published in the Bell System Technical Journal, Vol. 39 (July, 1960), pp. 947-962. Both forms of the network comprise passive RC networks Zv, Zw arranged in parallel arms as shown, with a NIC of ratio - Zx/Zy arranged with input and output terminals respectively IC, OC as shown. According to the invention, a prescribed impedance function Z (p) is realized for networks of this type by putting ZvkSP1/SP/pZa, Zw=kSP1/SPZa, and - Zx/Zy = - Za/Zb, where kSP1/SP= kfor the first form (Fig. 1) and kSP1/SP= 1/k for the second form (Fig. 2), k being a positive or negative real constant. The expression k(Za - Zb)/(1 - pZaZb) gives the impedance Z AB for the first form of the network and for the second form the impedance expression is equal to 1/PZAB (p. is the normalized complex frequency variable) It can be shown that RC impedance functions Za, Zb can be found for any positive or negative k which satisfies Z (p) ZAB or Z (p) = 1/pZAB, where Z (p) is any real rational impedance function, including functions which are non- positive on the negative real p axis. The Specification gives a detailed procedure for determining Za and Zb. They are expressed as where q=#p and the subscripts o and e denote the odd and even parts of polynomials in q defined by Qa=II(q - qa) and Qb=II(q - qb). a b qa and qb are the values of q in the q plane at which qZ/k is equal to + 1. These values are found for the required function Z (p) and the polynomials formed. After separation into odd and even parts, the functional forms of Za and Zb may be determined. k may be chosen to realize any desired advantage, e.g. simplification of calculation, reduction of components &c. For example, choosing ZAB p and k=1, values Za=1, Zb=1/(1+p) are obtained, giving Zv= 1/p, Zw=1. The resultant network is shown in Fig. 4 where a NIC of ratio -1 is used. In order to realize the conversion ratio - Zx/Zy = - Za/Zb, the NIC is associated in the manner shown with impedances Zx=1, Zy= 1/(1 +p)- A network corresponding to Fig. 2 is also described. In general, the required conversion ratio may be obtained by taking Zx and Zy equal to GZa and GZb or to G/pZb and G/pZa respectively, G being an arbitrary constant or a suitable function of p.
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