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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 solve d 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.

It is known that periodical non-harmonic discontinuous function is possible to portray in compact closed form using Fourier infinite series [

increasing saw-tooth function with angular frequency

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 [

Assuming finite switch-on and switch-off times of real-time waveforms the normalized derivative impulse function for given waveforms can be created [

Further, based on zero order hold function and unipolar modulation [

The impulse switching functions as in

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 [

The sum of that geometric series with quotient

where root of the denominator is

For inverse Z-transform

Cauchy integral residua theorem [

where

Taking example

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:

Using theorem for displacement in the Z-transformation [

the Z-image of the 1/2-pulse length rectangular waveform will be:

where roots of the denominator

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.

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

Z-transform image

Formula for voltage impulse sequence

where roots of the polynomial

see

Proof within the frame of one half period:

So,

The Z-image for three-phase system with discontinuous waveform,

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

and also in trigonometric one

Proof within the frame of one time period:

So,

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

Dynamical state model of the systems include exciting functions

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

where

Determining

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

where

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

As mentioned, recursive formula

with

Calculation step

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

Time discretization using Euler explicit method:

where

Then, taking

and

Taking

So, in matrix form

Regarding to

Replacing n in Equation (23) by

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

Based on total mathematical induction it can be derived with the help from [

derivation of this formula see below. Then

Using Equation (28) the determination of

After choosing

and

Then

Finally the values are

Since

Thus

Calculated sequences

The sequences

Let’s note that values of state variables

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

n | u | ||
---|---|---|---|

0 | 1 | 0.0000 | 0.0000 |

1 | 0 | 0.0299 | 0.0077 |

2 | 2 | 0.0291 | 0.0176 |

3 | 0 | 0.0878 | 0.0380 |

4 | 1 | 0.0858 | 0.0601 |

5 | 0 | 0.1133 | 0.0787 |

6 | −1 | 0.1103 | 0.0934 |

7 | 0 | 0.0775 | 0.0922 |

8 | −2 | 0.0750 | 0.0844 |

9 | 0 | 0.0132 | 0.0637 |

10 | −1 | 0.0122 | 0.0401 |

11 | 0 | -0.0183 | 0.0193 |

12 | 1 | -0.0022 | 0.0182 |

k | u_{k} | x_{1,k} | x_{2}_{,k} |
---|---|---|---|

0 | 1 | 0.0000 | 0.0000 |

30 | 0 | 0.0299 | 0.0076 |

60 | 2 | 0.0295 | 0.0176 |

90 | 0 | 0.0886 | 0.0381 |

120 | 1 | 0.0881 | 0.0605 |

150 | 0 | 0.1150 | 0.0795 |

180 | -1 | 0.1113 | 0.0946 |

method for the second order system with integration step

So, sequences

The sequences

where

By adapting

Or, by decomposition of

where

And applying Z-transform

where

So,

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

Similarly

with the same roots

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 [

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

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

Let’s note that values of state variables

Confirmation of transient behavior using the fundamental harmonic method:

Analytical calculation of Fourier coefficient

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

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

where

2N―number of pulses in period;

For

what indicates equality of both calculations.

Now, one can use the harmonic voltage with magnitude

Behaviour of the system under load switched-on during 8 periods, i.e.

Let’s note that values of state variables

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

The result is shown in

Let’s note that values of state variables

By comparing Figures 8-10 and

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.

The paper was supported from R&D operational program Centre of excellence of power electronics systems and materials for their component No OPVaV-2008/2.1/01- SORO ITMS 26220120003, and also from Slovak Grant Agency VEGA by the grant No 1/0928/15.

Dobrucký, B., Šte- fanec, P., Beňová, M., Chernoyarov, O.V. and Pokorný, M. (2016) Analysis of Higher Order System with Impulse Exciting Functions in Z-Domain. Circuits and Systems, 7, 3951-3970. http://dx.doi.org/10.4236/cs.2016.711328