Specifics of determination sensitivity of linear parametric circuits in the frequency domain

Yu.I. Shapovalov

Lviv Polytechnic National University

In software which are dedicated for the design of electronic circuits with constant parameters is widely used function of sensitivity, which allows you to focused  solve the tasks of multivariate  analysis and optimization of such circuits. The function of sensitivity of linear parametric circuits not used widely because there was no reliable methods of symbolic computation of parametric transfer functions. We proposed such method (frequency symbol method [3]), so, filling the existing gap in article discusses the possibility of forming functions of sensitivity of linear parametric circuits and is given account the peculiarities of their determination.
For a basis of definition the concept of sensitivity of linear parametric circuit was taken definition of the concept of sensitivity of linear circuits with constant parameters. This feature of sensitivity of linear parametric circuit which consists in the fact that this sensitivity: a) is a function of two variables -  complex variable and time; b) by the frequency symbolic method is determined based on approximated parametric transfer functions.
It is shown that, typically, the sensitivity of parametric linear circuit varies periodically in time with period T, which is chosen in presentation the transfer function by trigonometric polynomial of Fourier. It was emphasized that the sensitivity of some parameters of parametric circuit in time can grow infinitely. This applies to the parameters that define the period T.
We considered the example of elementary parametric linear circle, consisting of a single parametric capacity. The example illustrates the change the different functions of sensitivity of such circuit in time.  
It is shown that at sufficient accuracy of approximation of parametric transfer function by trigonometric polynomial  of Fourier, derivatives on parameters of elements of circuit obtained from approximated parametric transfer functions enough to accurately reproduce derivatives on parameters of elements of circuit the original  parametric transfer functions and, therefore, can be used to form the functions of sensitivity, as well as in solution of tasks multivariate analysis and optimization of  linear parametric circuits. This conclusion shows a perspective application of frequency symbolic method to the mentioned problems.
For illustration of material relating to the correctness of the definition of derivatives of parametric transfer functions on parameters of elements of parametric circuit is selected single-circuit parametric amplifier, in oscillatory contour, which the capacity varies with a period T. It is shown when the number of harmonic components in the approximated parametric transfer function k 4 then results in the calculation of the derivative of this function on the parameter does not change. Therefore, the value of derivatives that were obtained when k=4 were taken as a basis in these calculations. Obviously, the calculation for the selected k<4 will be inaccurate, and if k>4 it takes unjustified big time. It is shown the dependence the sensitivity of module of transfer coefficient of the input current to the output voltage from the depth of modulation   of parametric capacity and time.