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 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.

These can be calculated using analytical method (suitable for systems of low orders); numerical method:

where should be determined either analytically or numerically or experimentally in very small time instant; discrete method using Z-transform

and can be determined as above; experimental method by measuring of state-variable at the time instant.

Describing discrete determination method using Z-transform-by iterative process.

As mentioned, recursive formula

with where a are discrete impulse responses of state-va- riables gained by any of computation (above) or identification method [14] , is calculation step, works with discretized time

Calculation step should be short enough e.g. or step of the sequence. Usually, equal 1 - 2 el. Decomposing the state Equation (16) into two scalar equations yields

where under understanding electrical L-C//R circuitry with parameters Figure 5:

:

Time discretization using Euler explicit method:

where is calculation (integration) step.

Then, taking as above one gets for

and

.

Taking equal to 0.0001 sec the coefficients and are, respectively

So, in matrix form

Regarding to:

Replacing n in Equation (23) by

one gets (30)

where “fix” is notation for rounding of numbers to zero [15] .

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 will be possible using, see Figure 6(a) and Figure 6(b).

After choosing, k will be the in the range of 0 - 30, thus

and

Then

Finally the values are

Since

Thus

Calculated sequences and of, state variables are given in Table 1. The values of state-variables, in the frame of one half period are presented in detail in Table 2.

The sequences and of, state variables are also depicted in Figure 7, interconnected by polynomial of the 1st order because of continuous quantities.

Let’s note that values of state variables and calculated with step can be presented as sequences (a) or time waveforms, with bonding points by linear interpolation (b); verificated by LT Spice emulator (c).

The same result can be obtained by numerical solution using explicit or implicit Euler

method for the second order system with integration step and taking in account the same time instants:

So, sequences are similarly the same as calculated using Equation (39) q.e.d.

The sequences and can also be worked-out using Z-transform of Equation (22)

where

By adapting

Or, by decomposition of into two scalar equations

where and

And applying Z-transform

where is the same as above.

So, and can be derived and separated:

Since it flows from Equation (47), (48)

and

Executing an inverse Z-transform of Equations (32), (33) or (29) one obtains

where n is a number of roots of the polynomial of denominator of, i.e. of, and are roots of the of equation

Similarly

with the same roots as of above. Those lead to sequences and worked-out and given in Table 1 or Figure 6, respectively.

But, it can be seen, that this method using residua theorem is rather arduous because of need of evaluation of denominator of.

System behaviour during transient for longer time-practically up to the steady state can be describe using Equation (18), (10) and theory given in [15] with computation step:

For

, , ,

be valid

By graduated calculation and using mathematical induction the general relation can be derived

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

where

Behaviour of the system under load switched-on during 8 periods, i.e. 2880 of k is shown in Figure 9.

Let’s note that values of state variables and are drawn with computation step connected by linear interpolation, too.

Confirmation of transient behavior using the fundamental harmonic method:

Analytical calculation of Fourier coefficient [2] , [11] :

Taking in account symmetry of impulse waveform the magnitude of fundamental harmonic will be

This is the same value as can be obtained using Equation (34), [17]

where

―is order of harmonics;

2N―number of pulses in period;

―relative pulse width 0 - 1;

―supply voltage of the 3-phase inverter.

For:

what indicates equality of both calculations.

Now, one can use the harmonic voltage with magnitude as exciting function applied to system (19).

Behaviour of the system under load switched-on during 8 periods, i.e. is shown in Figure 10.

Let’s note that values of state variables, and also are drawn with computation step under method of fundamental harmonics while impulse waveform of supply voltage was substituted by its fundamental harmonic.

Verification of transient behavior using circuit emulator LT Spice:

Verification of transient behavior was done using circuit LT Spice emulator. The scheme of electronic circuitry is shown in Figure 11. Schematics of R-L-C load is being shown in Figure 5.

The result is shown in Figure 12.

Let’s note that values of state variables, and also have been obtained from circuit emulator with the same sampling as computation step used above.

By comparing Figures 8-10 and Figure 12 one can conclude that behaviour of the system-step switching-on of impulse discontinuous exciting function-calculated by different methods is practically the same. Transient waveforms show that the over-shoot during the first period is around multiple 2, and settling time of the transient is about 10 periods.