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

Abstract

Abstract. 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.Key words: periodically time-varying networks, operational calculus, stability, synthesis, optimization. L 1 -impulses and periodic signalsThe L 1 -impulse is an absolutely summable signaland its periodic extension is a T -period functioñwhere p ∈ Z (integers). Series (1) always converges. It results from the fact that every L 1 -impulse is majorised by an exponential signal:where a, b, c, d -positive numbers, 1(t) -step function. Applying the formula (1) to the (2) we getfor t ∈ [0, T ).The inner product of the L 1 -impulses is defined as followsand the linear operator Hwhere L ∞ -space of bounded signals, R -real numbers. The special case of the (5) operator is the convolution operatorIt results that the sequence of two convolution operators act as L ∞ into L ∞ mapping, which means at the same time thatThus the convolution of the L 1 impulses produce also the L

L1-impulses method as an alternative method of harmonic components in the power theory of discrete time systems

“…In the article the distribution directly in the discrete time domain to the active and reactive currents using the original method proposed by one of the authors -the L 1 -impulse method [1,2] is presented. This method uses the isomorphism between the rational function set of complex variable and the exponential L 1 impulses set of positive and negative time domain and their N -periodic extensions set.…”

“…For the convolution operator (3) the adjoint ratio for its characterizing function (the impulse function) has the form [1] …”

Energy-optimal current distribution in a complex linear electrical network with pulse or periodic voltage and current signals. Optimal control

“…Optimal distribution itself is not as reachable as the distribution of a current divider, but in order to trigger it off, it is necessary to use optimal control carried out by the control operator X st (s) (13), generating an appropriately distributed signal of the voltage source e st (12): It also appears that distributions: energy-optimal (7) and the current divider (8) The optimal control operator X st (s) is skew-Hermitian, i.e. X st (−s) = −X st (s) which makes that skew-Hermitian is also an operator:…”

“…(7) and (8) have the same structure, but the system (7) is Hermitian type, part of, the system of Eq. (8).…”

Reactive compensator synthesis in time-domain

“…Czarnecki [1] where the author considers if it is possible to make a current decomposition into active, reactive and unbalanced current in a time domain and basing on it to build reactance compensators. The current decomposition in a time domain was presented in the previous articles [2,3] and in this article the reactive compensators design in a time-domain is presented.…”