Vakulenko Konstantin Evgeney
THE PERFECTIONMENT OF PARTIEL REACTIONS METHOD WITH THE HELP OF DEVELOPMENT OF PARTIEL PART PARAMETERS TABLES
MASTER`S THESIS SHORTLY
At the estimation of influences of hindrance of x(t) on electro-receivers it is necessary to take into account the reaction at(t) an electro-receiver on the hindrance. Therefore in the mathematical models of electromagnetic compatibility (EMS) of electro-receivers there is a linear filter which designs a reaction. The filter is described the transmission function of W (p) as a relation of polynomials of f(p) and z(p) degrees of m and n at m < n. For example, filter of flicermetra. M has an international electrical engineering commission = 2 and n = 4.
At stationary casual hindrances a task consists of determination of mean value whisker and dispersions of Dy in the stationary state by means of value of xc and spectral closeness of S(w) or correlation function to (t) (CF) hindrances. Spectral amplitudes And(w) and the phase-frequency j(w) functions (ACHF and FCHF) of filter are determined on the transmission function the known methods.
Calculation of dispersion on the general formula:
(1)
with the usage of theorem about deductions does not meet of principle difficulties, but it is attended with the bulky laying out. In addition, for every system, having other transmission function, it is necessary to execute all of lying out. In this connection there is a necessity for development of compatible method of decision of tasks of linear filtration, that is the purpose of the article.
The offered method of «partial reactions» is based on decomposition of transmission function of W(p) on the polynomials of the first order. Two types of task are also possible: at first, to find a decision for the different kinds of hindrances and, secondly, consider that in all of cases an entrance process is white noise of x(t).
We will consider the first type of task. Let ð1, ð2, ..., ðn are simple roots of characteristic equalization of z(p) = 0. It is known that the relation of polynomials can be presented as a sum of shots:
(2)
with coefficients
If 
- a multiplier at 
in a denominator
We will designate through
(3)
As a result we will get expression
which it ensues from, that the filter is presented as the parallel included inertia (aperiodic) links of the first order with permanent time And by the coefficients of transmission to. Parameters of links can be complex sizes. Thus, task of calculation of descriptions «partial» reaction is taken to determination of descriptions of reactions of inertia links.
There is the mean value in the stationary state:
there fore
(4)
For the inertia link of ACHF
(5)
in this connection, dispersion of reaction of first link will make
(6)
The principle difference of casual «partial» reactions from periodic is their mutual dependence, conditioned, that the same process of x(t) acts on the entrances of inertia links. This circumstance must be taken into account at determination of dispersion of reactions sum. For this purpose besides dispersions (6) will find mutual correlation moments between the reactions of first and r links taken in pairs. Taking into account that FCHF of inertia link is given expression:
general formula
will present the theory of casual processes in the kind
(7)
Then the sought after dispersion will make
(8)
In spite of increase of elements amount calculations on this formula are simpler, because one time enough to calculate Dyi and kir for the set spectral closeness or CF.
We will pass to consideration of the second type task. Complete standardization can attain, based on fundamental position that the stationary casual processes looked after in practice are the result of passing of white noise of x(t) through the hypothetical linear system. The transmission function of G(p) of such system is determined the spectral closeness (CF) of hindrance (pic. 2):
Picture 2 - the Flow diagram of filter with the receipt of entrance process by transformation of white noise
The use of white noise simplifies calculations considerably, as its spectral closeness is permanent:
(9)
and CF looks like
(10)
where ñ - a permanent size; d(t) - delta-function.
We will consider the task of estimation of measuring error of casual process with exponential CF by a measuring device, having a transmission function:
(11)
where Ò1, Ò2 - permanent time of device.
The dynamic error of measuring is determined dispersion of Dy of the measured process. In the task of the first type (character ~) an entrance process remains unchanging. As in this case n = 2, the device is designed by two inertia links. In the task of the second type an inertia link is added with permanent time of Nd. Substantially, that permanent time of other links the same as well as in the previous case, but coefficients of transmission of links to will be other.
Realization of method of "partial reactions" on PCM is taken to the calculations on eventual analytical expressions. It allows avoid an error, arising up at numeral integration with an endless top limit.
Conclusion. Presentation of the linear systems as the parallel included inertia links allows unify and simplify the decision of tasks of linear filtration of electro energy casual processes.