Analysis of Higher Order System with Impulse Exciting Functions in Z-Domain

This paper deals with mathematical modelling of impulse waveforms and impulse switching functions used in electrical engineering. Impulse switching functions are later investigated using direct and inverse z-transformation. The results make possible to present those functions as infinite series expressed in pure numerical, exponential or trigonometric forms. The main advantage of used approach is the possibility to calculate investigated variables directly in any instant of time; dynamic state can be solved with the step of sequences (T/6, T/12) that means very fast. Theoretically derived waveforms are compared with simulation worked-out results as well as results of circuit emulator LT spice which are given in the paper.


Introduction
It is known that periodical non-harmonic discontinuous function is possible to portray in compact closed form using Fourier infinite series [1] [2].One of the lesser known methods is using of Fischer-Turbar definition of It is also possible to express the rectangular waveform using Laplace or Laplace-Carson transform but inverse transform is not easy calculation, particularly for higher order systems.Classical solution leads to results in Fourier series form, otherwise the Heaviside calculus is to be used [2], [6].
Assuming finite switch-on and switch-off times of real-time waveforms the normalized derivative impulse function for given waveforms can be created [7], Figure 1.
Further, based on zero order hold function and unipolar modulation [8]- [10], the switch-off impulses will be substituted by zero points, and result waveforms can be presented as follow from, Figure 2.
The impulse switching functions as in Figure 2 can be easily described in Z-domain using basic definitions and rules of Z-transformation.

Description of Impulse Switching Functions in Z-Domain
Using basic definition of Z-transform-taking into account z-images of constant and alternating series and based on the rules of the Z-transform it can be written [10].The sum of that geometric series with quotient where root of the denominator is one can use different methods [11]: Cauchy integral residua theorem [12] { } ( ) ( ) Applying inverse Z-transform for converter output phase voltages in Z-domain one can create impulse switching functions.Residua theorem described above can be used for inverse Z-transform Let's consider following different discontinuous type of waveforms:

Impulse Functions of Rectangular Half Width Waveform
Using theorem for displacement in the Z-transformation [10] [11] ( ) the Z-image of the 1/2-pulse length rectangular waveform will be: ( ) where roots of the denominator 1,2 z j = ± are placed on boundary of stability in unit circle [1], [10], Figure 3(a).
Applying inverse Z-transform one can write This result can be expressed in different forms: purely numerical-, exponential-, and trigonometric ones The all poles of denominator polynomials are placed on boundary of stability of unit circle and can be used for further analytic solution.

Three-Pulse Modulated Waveform
Above given approach can also be used for rectangular waveform with half-width of the pulse.Graphical interpretation of this switching function is shown in the Figure 4(a).

Z-transform image ( )
F z of that function will be: Formula for voltage impulse sequence { } n f can also be worked-out by inverse z-transform using the lema for residua.

Three-Phase Impulse Waveform
The Z-image for three-phase system with discontinuous waveform, Figure where roots of the denominator are Applying inverse Z-transform for this three-phase system After adapting ( ) Formula ( 17) can be expressed in exponential form ( ) e e e e , ph j jn j jn f n and also in trigonometric one ( ) Proof within the frame of one time period: So, { } 1; 2;1; 1; 2; 1, Presented in figure worked-out sequences express impulse nature and represent the impulse switching functions which can be easily described in Z-domain using basic definitions and rules of Z-transformation.From the Figure 4(c) and pole displacement of three-phase impulse system 3_ ph F , Figure 3(c) implies that it will feature by 2N-multiple symmetry and therefore analysis can be done within one T/6-th of time period [13].

Modelling and Simulation of 2nd Order System with Non-Harmonic Periodical Exciting Functions Based on ISF
Dynamical state model of the systems include exciting functions ( ) u t as an input vector.The models can be expressed in a continuous form: or discrete form, respectively { } where k is order of computation step (not the step of sequence).
Discrete form of state space model of the investigated system with the step of impulse switching function can be obtained directly from the impulse switching functions generated above: where the step is equal to the step or period, respectively to the impulse sequences p T of switching functions.So, when step is equal e.g.π/6 i.e.T/12 (see Equation ( 17)) then { }

Calculation of T F 12 , T G 12 Matrix Coefficients
These can be calculated using analytical method (suitable for systems of low orders); numerical method: , , where , ∆ ∆ F G should be determined either analytically or numerically or experimen- tally in very small time instant ∆ ; discrete method using Z-transform , F G can be determined as above; experi- mental method by measuring of state-variable at the time instant 12 T .
Describing discrete determination method using Z-transform-by iterative process.
As mentioned, recursive formula where under understanding electrical L-C//R circuitry with parameters Figure 5: { } where ∆ is calculation (integration) step.
Based on total mathematical induction it can be derived with the help from [16], { } derivation of this formula see below.Then .
Using Equation (28) the determination of ; T T F G will be possible using ; ∆ ∆ F G , see Figure 6(a) and Figure 6(b).
After choosing 360 T ∆ = , k will be the in the range of 0 -30, thus .

Calculation of State Variable Values
Since ( ) ( ) ( ) Calculated sequences { } x in the frame of one half period are pre- sented in detail in Table 2.
The sequences { } x state variables are also depicted in

Alternative Way of T F 12 , T G 12 Matrix Coefficients Calculation and State Variable Values Calculation
The same result can be obtained by numerical solution using explicit or implicit Euler where ( ) ( ) By graduated calculation and using mathematical induction the general relation can be derived where { } ( ) ( ) Behaviour of the system under load switched-on during 8 periods, i.e. 96 of T/12 is shown in Figure 8.
Another way using computation step Δ leads to and using above approach This is the same value as can be obtained using Equation (34), [17]

Conclusion
The method given in the paper demonstrated how is possible to write impulse switching functions which can be describable by z-transformation by application of unipolar modulation and zero order function.Results presented in paper demonstrated exceptionality of the formulated method-calculation of variable quantities of investigated linear dynamical system at any time, without knowing the values of foregoing time(s).This is not possible in case of pure numerical computing.Moreover, dynamical state can be solved very fast using step of calculation equal step of sequences (T/6, T/12).Comparing results worked-out by four different methods one can see that they reached waveform practically the same.Presented techniques are suitable for analysis of both transient and steady-state behaviour of investigated system mainly in electrical engineering.
on a standardization of trigonometric function modulo π [3]-[5].So, increasing saw-tooth function with angular frequency ω can be expressed in closed form

Figure 2 .
Figure 2. Impulse switching functions with unipolar control of: rectangular waveform with half width.

Figure 3 .
Figure 3. Pole placements of denominator polynomials of (a)

Figure 7 ,calculated with step 12 T
Figure 7, interconnected by polynomial of the 1st order because of continuous quantities.Let's note that values of state variables ( ) L i n and

58 )Figure 8 .
Figure 8. Transient of the 2nd order system under impulse exciting function with the step of T/12.
order of harmonics; 2N-number of pulses in period; γ -relative pulse width 0 -1; dc U -supply voltage of the 3-phase inverter.

Figure 9 .
Figure 9. Transient of the 2nd order system under impulse exciting function with the step of T/360.

Figure 11 .
Figure 11.Schematics of generating modulated impulse voltage in LT spice environment.

Figure 12 .
Figure 12.Transient of the 2nd order system under impulse exciting function verificated by LT spice.

Table 1 .
State variable values during the first period after switching the load on.

Table 2 .
Proof within the frame of one half period.