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# Mathematical model of a converters and its analysis.

The basis circuit of a single-phase voltage controller used for the analysis, is showed in fig. 1, where the load is the parallel connection of an active resistance RÍ and inductance LÍ. The input LC-filter is not taken into account yet.

The method of an algebraization of the differential equations (ADE) is used for the analysis of the circuit. It is possible to write down the following system of the differential equations to the circuit in fig.1:

. (1)

and  are switching functions of the keys Ê1 and Ê2 in a system (1). It is possible to receive one equation from a set of equations (1) by means of substitution

. (2)

Further, the standard procedures of a method of the ADE is applied to the equation (2). The expression for an effective value of the first harmonics of output voltage of a controller with reference effective value an input voltage Å: turns out

. (3)

In the ratio (3)  and  are the average values ??of switching functions  and  correspondently. The expression (3) does not give the phase of output voltage. To determine the phase characteristics of the all currents and voltages circuit it is convenient to use the method of variables states with decomposition of an obtained variables on active and reactive components.

The system of differential equations of the first order for a variables states in the matrix shape looks like

, (4)

where: x =  - Vector of a variables;

, ,  . (5)

With the purpose of deriving the algebraic equations for the first harmonics of variables (for effective values ??active (à) and reactive (r) of variables states) we multiply the equation (4) serially on  , Then on  and make it averaged for a period [6]. Finally we have:

. (6)

Or in the matrix shape

. (7)

The solution looks like

. (8)

The same we can define the effective values ??of all variables states [6, 7].

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