Continuants, signal flow graphs, and ladder networks

Transfer function of multi-stage active filters: a solution based on Pascal's triangle and a general expression

“…Operational amplifiers are also used in active filters. The ratio of x may change for these types of filters [10], but the form of the expression for the transfer function for the unit of a ladder remains the same as in equation (3). Therefore, we shall use the ratio x (= Z 1 /Z 2 ) as it is more convenient for the analysis described further.…”

“…The appearance of Fibonacci numbers in electrical ladder networks generated considerable interest, which goes back to about 50 years ago [1,2] during a period when results for the transfer functions were required in a more calculable form [3]. In general, a polynomial characterizes the electrical network over a frequency range.…”

“…New methods were developed to find the coefficients of these characteristic polynomials using recurrence relations [4], which generate the correct coefficients for a ladder network of any given number of stages. These methods are less accessible to third or fourth year undergraduate students as they involve the matrix formalism or knowledge of theorems, such as Cagney-Hamilton [3] theorems applied to ladder networks, from which the modified numerical triangle is being generated. In this paper, a recurrence relation is obtained by applying the fundamental Kirchhoff's current law [5,6] at a given node of the ladder network.…”