V.2 No 1

37

Mismatched ladder filters

Before we get down the main purpose of our investigation, let us generalise the relationships of the electromechanical analogy presented in [6]. Realising the approach stated in the introduction to set up the electromechanical analogy between the systems, obtain the relationship of our interest by direct comparison of the modelling systems of the algebraic equations of related models. It will be sufficient to use any two models of the studied kind, for example, a semi-finite mechanical line and the electrical filter corresponding to it. Their general form is shown in Fig. 1.

Fig. 1. The schematic diagram of a semi-finite mechanical elastic line (a) and of corresponding ladder filter (b)

The modelling system of differential equations for a semi-finite mechanical elastic line, Fig. 1a, has the following form:

where F (t) = F₀ is the external acting force; _n is the momentary displacement of the nth mass from the state of rest; s is the line stiffness coefficient; m is the line elements mass; ₀ is the initial phase of the external excitation; n = 1, 2, 3, … is the line element number.

To establish the electromechanical analogy, we have to transform (1) to the form convenient for it. Present _n in the form

where _n is the displacement amplitude of the nth element. We yield

We will determine the modelling system of differential equations for the ladder filter conventionally, using the first Kirchhoff law for the node indicated in Fig. 1b. Direct the currents in the filter longitudinal elements with the external current, and in the parallel elements – to the selected nodes of the filter. With it the modelling system takes the following form: