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Open access

M. Siwczyński and M. Jaraczewski

The L1-impulse method as an alternative to the Fourier series in the power theory of continuous time systems

The Fourier series method is frequently applied to analyze periodical phenomena in electric circuits. Besides its virtues it has many drawbacks. Fourier series usually have slow convergence and fail for fast changing signals, especially for discontinues ones. Therefore they are suitable to describe only quasiharmonic phenomena.

For strongly nonsinusoidal signal analysis we propose the L 1-impulse method.

The L 1-impulse method consists in an equivalent notation of a function belonging to L 1 as a sum of exponential functions. Such exponential functions have rational counterparts with poles in both sides of imaginary axis. With the L 1-impulse functions we can describe periodical signals, thus we get the homomorfizm between periodical signals and a rational functions sets. This approach is especially adapted to strongly deformed signals (even discontinues ones) in linear power systems, and thanks to that we can easily calculate optimal signals of such systems using the loss operator of the circuit. The loss operator is exactly the rational function with central symmetry of poles [1].

In this paper the relation between the L 1-impulse and the Fourier series method was presented.

It was also proved that in the case of strong signal deformation the L 1-impulse method gains advantage.

Open access

M. Siwczyński and M. Jaraczewski

Application of L 1-impulse method to the optimization problems in power theory

In optimization power theory we can distinguish the three approaches:

• the theory of instant power values

• the theory of average power values (integral power)

• the theory of instant-average power value.

The theory of instant power uses the instant power and signals values i.e. p(t) = u(t)i(t) whereas the theory of average power uses the energy or average power terms i.e. P = (u(t), i(t)) (the dot the product of signals). The main problem in the average power theory comes from the Schwartz inequality:

This inequality causes numerous optimization problems, among which the norm of the current minimization is the most important one:

Whereas the theory of instant-average power values joins both aforementioned methods and uses so-called ‘instant active power’:

The mathematic methods used in these theories derive from the theorems of signals and instant power modulation. This article deals only with the average power theory which uses the L 1 impulses as an alternative to the Fourier series method. This technique is efficient when the energy is transmitted with highly distorted periodic signals.

Open access

M. Siwczyński and M. Jaraczewski

Reactive compensator synthesis in time-domain

The source reactive-current compensation is crucial in the energy transmission efficiency. The compensator design in a frequency-domain has already been widely discussed and examined. This paper presents results of a study on how to design reactive compensators in a time-domain. It is the first time the reactive compensator has been designed in a time domain. The example of a compensator is presented.

Open access

M. Siwczyński and K. Hawron

Abstract

The paper presents the new optimal real-time control algorithm of the power source. The minimum of the square-instantaneous current was assumed as an optimal criterion, with an additional constraint on source instantaneous power. The mathematical model of a multiphase source was applied as a voltage-current convolution in the discrete time domain. The resulting control algorithm was the recursive digital filter with infinite recursion.

Open access

M. Siwczyński, A. Drwal and S. Żaba

L 1-impulses method as an alternative method of harmonic components in the power theory of discrete time systems

The article presents the basic mathematical theory of the operational calculus of the L 1-impulses in the discrete time domain. It presents the isomorphism between the rational function set of complex variable and the exponential L 1 impulses set of positive and negative time domain. The paper shows how for any factorization of the rational function consisting of casual and noncasual parts can be directly obtained the N - periodic version of the original signal using for the individual components of the L 1 impulses N - copy formula. It is done by the distribution of the convolution - the type admitance operator Y of electrical circuit to the two commutative convolution operators and on this basis is obtained the distribution of electrical circuit current to two components: the active current and the reactive current in the discrete time domain using the cyclic convolutions. The distribution of current in the time domain for signals significantly different from the sinusoidal is much more favorable than the distribution in the frequency domain.

Open access

M. Siwczyński and M. Jaraczewski

Abstract

This paper describes a new method of determining the reactive power factor. The reactive power factor herein is calculated on the basis of time samples and not] with the Fourier transform of signals, like it was done previously. The new reactive power factor calculation results from the receiver admittance-operator decomposition into the product of self-adjoint and unitary operators. This is an alternative decomposition to another one, namely into a sum of the Hermitian and skew-Hemiitian operators.

Open access

M. Siwczyński, S. Żaba and A. Drwal

Abstract

The article presents that in the circuits of electrical signals belonging to the L1-impulses space or periodic signals space, real distribution of electrical currents occurs which does not meet the principle of minimum energy losses. The paper presents a solution of this problem by using the control system in the form of current-dependent voltage sources entering it into a meshes set of a complex RLC network. It has been shown that the control is energy-neutral.

Open access

M. Siwczyński, A. Drwal and S. Żaba

Abstract

The simple digital filters are not sufficient for digital modeling of systems with distributed parameters. It is necessary to apply more complex digital filters. In this work, a set of filters, called the digital function filters, is proposed. It consists of digital filters, which are obtained from causal and stable filters through some function transformation. In this paper, for several basic functions: exponential, logarithm, square root and the real power of input filter, the recursive algorithms of the digital function filters have been determined The digital function filters of exponential type can be obtained from direct recursive formulas. Whereas, the other function filters, such as the logarithm, the square root and the real power, require using the implicit recursive formulas. Some applications of the digital function filters for the analysis and synthesis of systems with lumped and distributed parameters (a long line, phase shifters, infinite ladder circuits) are given as well.