MODELLING OF POWER CONVERTERS USING Z-TRANSFORM MODELLING OF POWER CONVERTERS USING Z-TRANSFORM

There are many methods of analyses of power converters. Classical analytic methods, Laplace transform or/and Fourier analysis are suitable mainly for steady state operation [1], [2]. Transient analysis uses dynamic state-space modeling and/or Z-transform method. One of the fastest methods is that, which uses impulse switching functions (ISF). This method is used in signal theory, lesser in electrical engineering. Obviously, caused by power converters nature, those ISFs are strongly non-harmonic; sometime piece wise constant with zero spaces between pulses [3]. Then, it deals with power series of time pulses. From those series the impulse switching functions can be derived which are again orthogonal ones. Derived relations for voltages sequences can be used for current sequences calculations in electrical engineering system using impulse transfer function of the used plant and its time discretization. Similarly, one can derive relation for continuous time functions of voltages and currents. They are often grouped in two orthogonal α and β axes [4], [5].


Introduction -Impulse switching functions
There are many methods of analyses of power converters. Classical analytic methods, Laplace transform or/and Fourier analysis are suitable mainly for steady state operation [1], [2]. Transient analysis uses dynamic state-space modeling and/or Z-transform method. One of the fastest methods is that, which uses impulse switching functions (ISF). This method is used in signal theory, lesser in electrical engineering. Obviously, caused by power converters nature, those ISFs are strongly non-harmonic; sometime piece wise constant with zero spaces between pulses [3]. Then, it deals with power series of time pulses. From those series the impulse switching functions can be derived which are again orthogonal ones. Derived relations for voltages sequences can be used for current sequences calculations in electrical engineering system using impulse transfer function of the used plant and its time discretization. Sim-ilarly, one can derive relation for continuous time functions of voltages and currents. They are often grouped in two orthogonal α and β axes [4], [5].
Examples of impulse switching functions belonging to output voltage of single-and three-phase inverters are shown in Fig. 1a and 1b, respectively. -for single/two-phase α, β-system

MODELLING OF POWER CONVERTERS
where U α (z), U β (z) are voltages in orthogonal axes and roots of polynomial of denominator are , placed on boundary of stability in unit circle [6], [7] see placed again on boundary of stability in unit circle [6], [7], Fig. 2.
Applying inverse Z-transform for converter output phase voltages in Z-domain we can create orthogonal impulse switching functions.
Taking discrete state-space model for three-phase converter output current as state-variable considering the 1 st order load (resistive-inductive or resistive-capacitive) where F T ⁄ 6 , G T ⁄ 6 are fundamental and transition matrices (in general) of system parameters.
Applying Z-transform and considering the 1 st order load where X(z) is image of founded state-variable (inductor current or capacitor voltage), and Y ∞ (z) is maximum of steady state value (U ⁄ R or U, respectively).
And after adaption and simplification (11) where Applying inverse Z-transform Finally we get discrete form of state variable (converter output state variable -inductor currents or capacitor voltages) where n ϭ 0, 1, 2, …

Calculation of ISF function values using series and sequences
The ISF functions presented above are numerical series (sequences) or trigonometric ones, respectively.
At first it is necessary to determine the F T ⁄ 6 , G T ⁄ 6 functions values which are the state-variable values in 1⁄ 6-instant of time period (i.e. they are state-and transition responses). These can be obtained e.g. by using recursive relation for one-pulse solution: thus recursive relation where F ∆ a G ∆ are discrete impulse responses of state-variables gained by some of identification methods. The second fraction term is z-image of the partial sum of voltage impulses (1 Ϭ 60) [6], since r an и z Ϫn ; a Ͻ 0 is geometrical series, see Fig. 3.
After choosing ∆ ϭ T/360, k will be the in the range of 0 Ϭ 59, thus G T ⁄ 6 ϭ y(60) and F T ⁄ 6 ϭ 1 Ϫ G T ⁄ 6 . Supposing time constant of the load equal to T/2 and F ∆ ϭ 0.994 4598; G ∆ ϭ 0.005 5401 those values of F T ⁄ 6 , G T ⁄ 6 will be Now, one can calculate state-variable x for any n, Fig. 4.
It is also possible to change the step of series (sequences) e.g. for step equal T⁄ 2, by determining of F T ⁄ 2 and G T ⁄ 2 : , and regarding to G T ⁄ 6 : . So, then one can calculate where .

Steady state determination
The steady state value of 'periodical' sequence can be determined, based on condition or vice versa,i.e. .
Setting this value into {x n } or x(n ϩ 1) we get: ; .
Based on zero order hold function [6] and total mathematical induction one finally yields: For three-phase Clarke transformed system . ,