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S.B. Karavashkin, O.N. Karavashkina

In this paper we will modify the approach to the ladder filters calculation, using the original Dynamical ElectroMechanical Analogy (DEMA) whose foundations have been stated in [6]. The conventional electromechanical analogy has the following known basis: “to each dependent variable and all derivatives of one system (i.e., x, dx/dt, d²x/dt²), the variable having similar derivatives (q, dq/dt, d²q/dt²) corresponds, and the related functions are interconnected so that if some function of one system is known, the corresponding function of the second system can be found” [7, pp. 88 – 89]. In distinction from this, we will use the known property of the modelling system of equations for ladder filters, which admits to separate the time and spatial variables. And we will compare the mechanical and electrical models as the whole, grounding on the correspondence of modelling systems of algebraic equations obtained from those differential after the time dependence eliminating. It will enable us to be not fixed so rigidly to the correspondence of the circuit particular elements and to be not limited in the range of analogues by differentiating or integrating the modelling differential equations. So, with the new relationship of models, the circuits vibration pattern can basically differ, retaining the correspondence between the models, since the base principles will be more generalised when choosing the analogues. These principles will be connected with the boundary conditions and with the features of mechanical system of an elastic line as a whole, but the main, with the correspondence of modelling systems of algebraic equations. Namely the systems of algebraic equations, not one differential equation of the electrical circuit on whose base the analogy relationship was established up to now (see, e.g., [7, p. 89], [8, pp. 32 – 34]), referring to the two-port method. The possibility to introduce such an analogy was indicated by Atkinson in [9, p. 32 – 33]. However the matrix technique to obtain the solutions for mechanical systems enabled Atkinson neither to develop this direction nor to exceed the frames of simplest LC and CL ladder filters nor to write in general form the relationship between compared systems nor to obtain the solutions in the analytical form.

To avoid the complication appearing when using the matrix methods, we will use, in addition to DEMA, the exact analytical solutions for mechanical lines presented in [10] and [11]. Their main distinction is that when finding the solutions using the non-matrix method, we have no need to study the eigenvalues and natural frequencies of a mechanical line as the base to find these solutions. The exact solution is found directly for a specific modelling system of differential equations which accounts all the features defined by the initial and boundary conditions, and the solutions are presented not in the matrix form or as the recurrent relationship but in the analytical form.

Combining these two methods, we have no need to consider a ladder filter as a simple assemblage of specific two-ports. As we will show in this paper, it enables us to extend essentially the calculation scope and to study the exact patterns of processes for the much broader range of models than the conventional two-port method enables.