Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

5Passive Immittancesand Positive-RealFunctionsWai-Kai ChenUniversity of Illinois at Chicago References.............................................................................................. 5-8In this chapter on passive filters, we deal with the design of one-port networks composed exclusively ofpassive elements such as resistors R, inductors L, capacitors C, and coupled inductors M. The one-portsare specified by their driving-point immittances, impedances, oradmittances. Our basic problem is thatgiven an immittance function, is it possible to find a one-port composed only of R, L, C, and M elementscalled the RLCM one-port network that realizes the given immittance function? This is known as therealizability problem, and its complete solution was first given by Brune [1].Consider a linear RLCM one-port network of Figure 5.1 excited by a voltage source V 1 (s). For ourpurposes, we assume that there are b branches and the branch corresponding to the voltage source V 1 (s)is numbered branch 1 and all other branches are numbered from 2 to b. The Laplace transformedKirchhoff current law equation can be written aswhereA is the basis incidence matrixI(s) is the branch-current vector of the networkAIðÞ¼0 s(5:1)If V n (s) is the nodal voltage vector, then the branch-voltage vector V(s) can be expressed in terms ofV n (s) byVðÞ¼A s0 V n ðÞ s(5:2)where the prime denotes the matrix transpose. Taking the complex conjugate of Equation 5.1 inconjunction with Equation 5.2 givesV 0 ðÞI s ðÞ¼V s0 n ðÞAI s ðÞ¼V0 s nðÞ0 s ¼ 0 (5:3)5-1