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Showing papers on "Harmonics published in 2003"



Journal ArticleDOI
TL;DR: In this paper, a robust control strategy for regulating the grid current entering a distribution network from a three-phase VSI system connected via a LCL filter is proposed, which integrates an outer loop grid current regulator with inner capacitor current regulation to stabilize the system.
Abstract: Many grid connected power electronic systems, such as STATCOMs, UPFCs, and distributed generation system interfaces, use a voltage source inverter (VSI) connected to the supply network through a filter. This filter, typically a series inductance, acts to reduce the switching harmonics entering the distribution network. An alternative filter is a LCL network, which can achieve reduced levels of harmonic distortion at lower switching frequencies and with less inductance, and therefore has potential benefits for higher power applications. However, systems incorporating LCL filters require more complex control strategies and are not commonly presented in literature. This paper proposes a robust strategy for regulating the grid current entering a distribution network from a three-phase VSI system connected via a LCL filter. The strategy integrates an outer loop grid current regulator with inner capacitor current regulation to stabilize the system. A synchronous frame PI current regulation strategy is used for the outer grid current control loop. Linear analysis, simulation, and experimental results are used to verify the stability of the control algorithm across a range of operating conditions. Finally, expressions for ""harmonic impedance" of the system are derived to study the effects of supply voltage distortion on the harmonic performance of the system.

869 citations


Journal ArticleDOI
28 Nov 2003-Science
TL;DR: It is shown that the synchronization could be improved considerably by controlling the underlying ultrafast electron dynamics, to provide pulses of 130 attoseconds in duration, which would allow us to track fast electron processes in matter.
Abstract: Subfemtosecond light pulses can be obtained by superposing several high harmonics of an intense laser pulse. Provided that the harmonics are emitted simultaneously, increasing their number should result in shorter pulses. However, we found that the high harmonics were not synchronized on an attosecond time scale, thus setting a lower limit to the achievable x-ray pulse duration. We showed that the synchronization could be improved considerably by controlling the underlying ultrafast electron dynamics, to provide pulses of 130 attoseconds in duration. We discuss the possibility of achieving even shorter pulses, which would allow us to track fast electron processes in matter.

694 citations


Journal ArticleDOI
TL;DR: A complete analysis is given for a seven-level converter (three dc sources), where it is shown that for a range of the modulation index m/sub I/, the switching angles can be chosen to produce the desired fundamental V/sub 1/=m/ sub I/(s4V/sub dc///spl pi/) while making the fifth and seventh harmonics identically zero.
Abstract: In this work, a method is given to compute the switching angles in a multilevel converter to produce the required fundamental voltage while at the same time cancel out specified higher order harmonics. Specifically, a complete analysis is given for a seven-level converter (three dc sources), where it is shown that for a range of the modulation index m/sub I/, the switching angles can be chosen to produce the desired fundamental V/sub 1/=m/sub I/(s4V/sub dc///spl pi/) while making the fifth and seventh harmonics identically zero.

324 citations


Journal ArticleDOI
TL;DR: In this article, a Z-source inverter system and control for adjustable speed drive (ASD) control is presented, which employs a unique LC network to couple the inverter main circuit to the diode front end.
Abstract: This paper presents a Z-source inverter system and control for adjustable speed drives (ASD). The Z-source inverter employs a unique LC network to couple the inverter main circuit to the diode front end. By controlling the shoot-through duty cycle, the Z-source can produce any desired output AC voltage, even greater than the line voltage. As results, the new Z-source inverter system provides ride-through capability under voltage sags, reduces line harmonics, and extends output voltage range. Simulation results are presented to demonstrate the new features.

319 citations


Proceedings ArticleDOI
19 Feb 2003
TL;DR: In this article, all possible solutions to the problem of eliminating harmonics in a switching converter were found by converting the transcendental equations that specify the harmonic elimination problem into an equivalent set of polynomial equations.
Abstract: The problem of eliminating harmonics in a switching converter is considered That is, given a desired fundamental output voltage, the problem is to find the switching times (angles) that produce the fundamental while not generating specifically chosen harmonics In contrast to the well known work of Patel and Hoft (1973, 1974) and others, here all possible solutions to the problem are found This is done by first converting the transcendental equations that specify the harmonic elimination problem into an equivalent set of polynomial equations Then, using the mathematical theory of resultants, all solutions to this equivalent problem can be found In particular, it is shown that there are new solutions that have not been previously reported in the literature The complete solutions for both unipolar and bipolar switching patterns to eliminate the 5th and 7th harmonics are given Finally, the unipolar case is again considered where the 5th 7th, 11th, and 13th harmonics are eliminated along with corroborative experimental results

264 citations


Journal ArticleDOI
12 Oct 2003
TL;DR: In this article, a selective harmonic feedback control strategy was proposed for medium-voltage DVR systems to provide voltage harmonic compensation capabilities with minimal effect on the sag compensation performance of the basic DVR, which was experimentally verified on a medium voltage (10 kV) three-phase DVR prototype under a range of conditions, including distorted supply voltages, nonlinear loads, and operation during distorted voltage sags.
Abstract: Dynamic voltage restorers (DVRs) are now becoming more established in industry to reduce the impact of voltage sags to sensitive loads. However, DVRs spend most of their time in standby mode, since voltage sags occur very infrequently, and hence their utilization is low. In principle, it would be advantageous if the series-connected inverter of a DVR could also be used to compensate for any steady-state load voltage harmonics, since this would increase the power quality "value-added" benefits to the grid system. However, before this can be done, consideration must be given to the control of steady-state power through the DVR, the increased losses, and the low modulation depths at which the scheme must operate to achieve acceptable harmonic compensation performance. This paper presents a selective harmonic feedback control strategy that can be easily added to medium-voltage DVR systems to provide voltage harmonic compensation capabilities with minimal effect on the sag compensation performance of the basic DVR. The proposed controller has been experimentally verified on a medium-voltage (10 kV) three-phase DVR prototype under a range of conditions, including distorted supply voltages, nonlinear loads, and operation during distorted voltage sags.

248 citations


Journal ArticleDOI
12 Oct 2003
TL;DR: In this paper, the authors presented a method for estimating the machine parameters of a synchronous motor, both the interior and surface-mount types, and compared the results with the finite element and the experimental results to demonstrate the effectiveness of the presented method.
Abstract: This paper presents a method for estimating the machine parameters of a synchronous motor. The presented method is equally applicable for a wound field synchronous motor, synchronous reluctance motor, or permanent magnet (PM) synchronous motor, both the interior and surface-mount types. The method works particularly well for machines having a significant amount of space harmonics, such as synchronous reluctance and interior PM (IPM) machines, where the harmonics are predominantly the slot harmonics. It is also well suited for surface PM machines operating under saturation, where the harmonics are saturation-induced. The presence of these harmonics makes the parameter identification difficult. Most of the methods presented so far in the literature have failed to properly identify machine parameters in the presence of space harmonics. In this paper, the machine parameters, identified by using a proposed algorithm, are compared with the finite element and the experimental results to demonstrate the effectiveness of the presented method. Both interior PM and surface PM machines are considered.

224 citations


Journal ArticleDOI
TL;DR: Results under diotic and dichotic conditions indicate that the auditory system, in performing f0 discrimination, is unable to utilize the additional peripherally resolved harmonics in the dichotic case, and suggest that harmonic number, regardless of peripheral resolvability, governs the transition between two different pitch percepts.
Abstract: Three experiments investigated the relationship between harmonic number, harmonic resolvability, and the perception of harmonic complexes. Complexes with successive equal-amplitude sine- or random-phase harmonic components of a 100- or 200-Hz fundamental frequency (f0) were presented dichotically, with even and odd components to opposite ears, or diotically, with all harmonics presented to both ears. Experiment 1 measured performance in discriminating a 3.5%-5% frequency difference between a component of a harmonic complex and a pure tone in isolation. Listeners achieved at least 75% correct for approximately the first 10 and 20 individual harmonics in the diotic and dichotic conditions, respectively, verifying that only processes before the binaural combination of information limit frequency selectivity. Experiment 2 measured fundamental frequency difference limens (f0 DLs) as a function of the average lowest harmonic number. Similar results at both f0's provide further evidence that harmonic number, not absolute frequency, underlies the order-of-magnitude increase observed in f0 DLs when only harmonics above about the 10th are presented. Similar results under diotic and dichotic conditions indicate that the auditory system, in performing f0 discrimination, is unable to utilize the additional peripherally resolved harmonics in the dichotic case. In experiment 3, dichotic complexes containing harmonics below the 12th, or only above the 15th, elicited pitches of the f0 and twice the f0, respectively. Together, experiments 2 and 3 suggest that harmonic number, regardless of peripheral resolvability, governs the transition between two different pitch percepts, one based on the frequencies of individual resolved harmonics and the other based on the periodicity of the temporal envelope.

193 citations


Journal ArticleDOI
TL;DR: In this paper, an effective technique for analyzing the generation of second harmonics of Lamb modes in elastic plates is presented, where the second-order perturbation of the linear elastic response is treated as a second order perturbations of the elastic response.
Abstract: An effective technique for analyzing the generation of second harmonics of Lamb modes in elastic plates is presented. The nonlinearity of the wave equation governing the wave propagation ensures that there is second-harmonic generation accompanying primary Lamb mode propagation. This nonlinearity may be treated as a second-order perturbation of the linear elastic response. Using a second-order perturbation approximation and a modal analysis approach, the complicated problems of the generation of second harmonics of Lamb modes have been investigated. The fields of the second harmonics of Lamb modes in elastic plates are considered as superpositions of the fields of a series of double-frequency Lamb modes. The solutions provide physical insight into the generation of second harmonics with a cumulative growth effect, and the corresponding second-harmonic solutions. Although Lamb modes are dispersive, the cumulative growth effect of the second harmonics does exist under some conditions. The influence of Lamb mode dispersion on second-harmonic generation is considered.

193 citations


Journal ArticleDOI
TL;DR: In this paper, a correlation has been found between the fundamental and low frequency harmonic components of leakage current and different forms of aging, where erosion could be associated with an increase in the level of both the low frequency and fundamental components.
Abstract: The paper presents the results of using the fundamental and the low frequency harmonic components of leakage current to study aging of silicone rubber in salt-fog. Experiments have been conducted on RTV and HTV coated rods at different fields (0.25-0.6 kV/cm) and conductivities (1000 to 2500 /spl mu/S/cm). The onset of dry-band arcing on samples could be determined by measuring the low frequency harmonic components. A correlation has been found between the fundamental and low harmonic components of leakage current and different forms of aging. Where erosion could be associated with an increase in the level of both the fundamental and low frequency harmonic components of leakage current. For example, surface damage for HTV rods occurred when the fundamental component of leakage current was greater than 2 mA. On the other hand, when the samples approached failure, the fundamental component of leakage current reached relatively high values ( > 6 mA for HTV rods and > 2 mA for RTV rods) and the low frequency harmonic components of the leakage current tended to decrease. The results suggest that both the fundamental and low frequency harmonics of leakage current can be used as a tool to determine both the beginning of aging and end of life of silicone rubber in salt-fog.

Journal ArticleDOI
TL;DR: A novel approach to harmonic and interharmonic analysis, based on the "subspace" methods, is proposed, and Min-norm harmonic retrieval method is an example of high-resolution eigenstructure-based methods.
Abstract: Modern frequency power converters generate a wide spectrum of harmonic components. Large converters systems can also generate noncharacteristic harmonics and interharmonics. Standard tools of harmonic analysis based on the Fourier transform assume that only harmonics are present and the periodicity intervals are fixed, while periodicity intervals in the presence of interharmonics are variable and very long. A novel approach to harmonic and interharmonic analysis, based on the "subspace" methods, is proposed. Min-norm harmonic retrieval method is an example of high-resolution eigenstructure-based methods. The Prony method as applied for signal analysis was also tested for this purpose. Both high-resolution methods do not show the disadvantages of the traditional tools and allow exact estimation of the interharmonics frequencies. To investigate the methods several experiments were performed using simulated signals, current waveforms at the output of a simulated frequency converter, and current waveforms at the output of an industrial frequency converter. For comparison, similar experiments were repeated using the fast Fourier transform (FFT). The comparison proved the superiority of the new methods. However, their computation is much more complex than FFT.

Journal ArticleDOI
TL;DR: By using triangular carrier wave and proper modulating waves for each inverter, the open-end winding induction motor can be operated in the entire modulation range, eliminating all the 6n /spl plusmn/ 1 harmonics coupled with 11th and 13th harmonic suppression.
Abstract: In this paper, a harmonic elimination and suppression scheme for a dual-inverter-fed open-end winding induction motor drive is presented. Two isolated dc-link sources with voltage ratio of approximately 1 : 0.366 are required for the present drive. These two isolated do links feeding two inverters to drive the open-end winding induction motor eliminate the triplen harmonic currents from the motor phase. The pulsewidth-modulation scheme proposed enables the cancellation of all the 5th- and 7th-order (6n /spl plusmn/ 1, where n = 1, 3, 5, 7, etc.) harmonic voltages and suppresses the 11th- and 13th-order harmonic voltage amplitudes in the motor phase voltage, in all modulation ranges. The next higher order harmonics present in the motor phase voltages are 23rd, 25th, 35th, 37th etc. (6n /spl plusmn/ 1, n = 4, 6, etc.). By using triangular carrier wave and proper modulating waves for each inverter, the open-end winding induction motor can be operated in the entire modulation range, eliminating all the 6n /spl plusmn/ 1 harmonics (n = 1, 3, 5, 7, etc.) coupled with 11th and 13th harmonic suppression. The proposed scheme also gives a smooth transition to the overmodulation region.

Journal ArticleDOI
TL;DR: In this paper, a completely digitally controlled high-performance low-harmonic rectifier with self-tuning digital comb filter is described. But the dynamics of the outer voltage loop can be significantly improved using a self tunable digital comb filtering.
Abstract: This paper describes a completely digitally controlled high-performance low-harmonic rectifier. It is shown that the dynamics of the outer voltage loop can be significantly improved using a self-tuning digital comb filter. Low input current harmonics and fast voltage transient responses are experimentally verified on a 200 W universal-input boost rectifier operating at the switching frequency of 200 kHz.

Journal ArticleDOI
TL;DR: In this paper, a lock-in amplifier is used to detect higher harmonics in the dynamic atomic-force microscopy signal, which is useful to monitor the imaging conditions in tapping mode and can be applied for nanoscale imaging with a material contrast.
Abstract: In tapping-mode atomic-force microscopy usually amplitude and phase of the cantilever motion are acquired. These signals are related to the fundamental oscillation frequency neglecting information at higher frequencies. However, the nonlinear contact between tip and sample induces higher frequency vibrations that are harmonics of the fundamental. In order to recover the available information the full tip motion has to be analyzed. The higher harmonics can be employed for image formation. A setup that consists of two independently operated lock-in amplifiers is used to detect higher harmonics in the dynamic atomic-force microscopy signal. Higher harmonic imaging proves to be useful to monitor the imaging conditions in tapping mode and can be applied for nanoscale imaging with a material contrast.

Proceedings ArticleDOI
21 Nov 2003
TL;DR: In this article, a control strategy of the single-phase active filter using a novel d-q transformation is proposed, which is implemented by using Hilbert transform, by which the instantaneous single phase voltage and current are converted into complex vectors (analytic signals) on the instantaneous basis.
Abstract: In this paper, a control strategy of the single-phase active filter using a novel d-q transformation is proposed This d-q transformation is implemented by using Hilbert transform, by which the instantaneous single-phase voltage and current are converted into complex vectors (analytic signals) on the instantaneous basis Since the fundamental components of voltages and currents are converted into DC components on the d-q coordinate both in the steady and transient states, the harmonics of voltages and currents can be obtained precisely through the low-pass filer installed on the d-q coordinate This results in the economical active filter with a small capacitor suppressing the DC bus voltage fluctuation because the fundamental power can be fully removed from the active filter input

Journal ArticleDOI
08 Apr 2003
TL;DR: In this article, a method for controlling an active power filter using neural networks is presented, where adaptive networks estimate the reference compensation currents and a multilayer perceptron feedforward network is used as a hysteresis band comparator.
Abstract: A method for controlling an active power filter using neural networks is presented. Currently, there is an increase of voltage and current harmonics in power systems, caused by nonlinear loads. The active power filters (APFs) are used to compensate the generated harmonics and to correct the load power factor. The proposed control design is a pulse width modulation control (PWM) with two blocks that include neural networks. Adaptive networks estimate the reference compensation currents. On the other hand, a multilayer perceptron feedforward network (trained by a backpropagation algorithm) that works as a hysteresis band comparator is used. Two practical cases with Matlab-Simulink are presented to check the proposed control performance.

Journal ArticleDOI
TL;DR: In this paper, an extended harmonic domain (EHD) method is presented for transient and steady-state analysis of electric networks containing nonlinearities and switching plants components, which is a natural approach for conducting dynamic and steady state studies of the evolution of harmonics in power circuits.
Abstract: A novel methodology is presented in this paper for conducting transient and steady-state analysis of electric networks containing nonlinearities and switching plants components. The method is termed the extended harmonic domain (EHD) method as may be seen as an extension of the harmonic domain (HD) method used in steady-state analysis. It is shown in the paper that EHD is a natural approach for conducting dynamic and steady-state studies of the evolution of harmonics in power circuits containing nonlinear components and flexible AC transmission systems (FACTS) devices. It is also shown in the paper that EHD provides a suitable basis for extending the well-known steady-state power quality indices to the transient range. To illustrate the use of the theory, a three-phase linear circuit and a three-phase PWM-STATCOM are modeled using EHD. The results are compared against those provided by industry standard tools as PSCAD/EMTDC and Matlab.

Journal ArticleDOI
TL;DR: In this article, a microstrip patch antenna using a defected ground structure (DGS) to suppress higher order harmonics is presented, where an H-shaped defect on the ground plane with only one or more unit lattices has been utilized and yielded bandstop characteristics.
Abstract: A microstrip patch antenna using a defected ground structure (DGS) to suppress higher order harmonics is presented. An H-shaped defect on the ground plane with only one or more unit lattices has been utilized and yielded bandstop characteristics. Compared with a conventional microstrip patch antenna without the DGS unit cell, the radiated power of the DGS patch antenna at harmonic frequencies has been drastically decreased.

Journal ArticleDOI
Er-Wei Bai1
TL;DR: In this paper, a frequency domain approach was proposed to identify the Hammerstein model in the frequency domain using sampled input-output data, and its convergence was shown for both the linear and nonlinear subsystems in the presence of noise.
Abstract: Discusses Hammerstein model identification in the frequency domain using sampled input-output data. By exploring the fundamental frequency and harmonics generated by the unknown nonlinearity, we propose a frequency domain approach and show its convergence for both the linear and nonlinear subsystems in the presence of noise. No a priori knowledge of the structure of the nonlinearity is required and the linear part can be nonparametric.


Journal ArticleDOI
TL;DR: In this article, a 3-phase shunt active power filter is proposed to compensate harmonics and the reactive power requirement of nonlinear loads, based on sensing line currents only.
Abstract: This paper presents complete design, simulation, and experimental investigations on a 3-phase shunt active power filter to compensate harmonics and the reactive power requirement of nonlinear loads. The paper describes the complete design aspects of power circuit elements and control circuit parameters. The compensation process is based on sensing line currents only, an approach different from conventional methods that require the harmonics and reactive volt-ampere requirement of the load. Various simulation results are presented to study the performance during steady-state and transient conditions to validate the design. A laboratory prototype has been developed to verify the simulation results. The control scheme is realized on a dedicated micro-controller-based system. PWM pattern generation is based on carrierless hysteresis-based current control to obtain the switching signals. Based on simulation and experimental results it can be concluded that the compensation process is simple and easy to impleme...

Journal ArticleDOI
TL;DR: A detailed experimental and theoretical study of the third harmonic generation in high-delta microstructured fibers including phase-matching wavelengths, far-field intensity distributions and polarization dependence is presented.
Abstract: Strong guiding provided by the high-delta microstructured fibers allows for efficient intermodally phase-matched harmonic generation with femtosecond pumping at telecom wavelengths. Visible harmonics are generated in a number of distinct transverse modes of the structure. We present a detailed experimental and theoretical study of the third harmonic generation in such fibers including phase-matching wavelengths, far-field intensity distributions and polarization dependence. Good agreement between the theory and experiment is achieved.

Journal ArticleDOI
TL;DR: In this paper, an overview of power-electronic based devices for mitigation of power quality phenomena is given, where the concept of custom power is highlighted and the authors focus on medium-voltage applications.

Journal ArticleDOI
TL;DR: In this paper, the authors used an 800-nm fundamental pulse whose polarization evolves with time to generate high-order harmonics by controlling the ellipticity modulation of the fundamental field, which can continuously confine the harmonic emission from an estimated minimum value of 7 fs $(1\mathrm{fs}^{\ensuremath{-}15}s})$ up to more than the 35 fs input pulse duration.
Abstract: We generate high-order harmonics by using an 800-nm fundamental pulse whose polarization evolves with time. Controlling the ellipticity modulation of the fundamental field allows us to continuously confine the harmonic emission from an estimated minimum value of 7 fs $(1\mathrm{fs}{=10}^{\ensuremath{-}15}\mathrm{s})$ up to more than the 35 fs input pulse duration. Depending on the observed harmonic, the harmonic spectrum can show either a narrowing or a broadening when the ellipticity is changed in good agreement with an effective confinement of the high-order harmonic generation.

Proceedings ArticleDOI
09 Dec 2003
TL;DR: In this paper, the switching angles in a multilevel converter were computed to produce the required fundamental voltage while at the same time not generating higher order harmonics, and it was shown that the theory of symmetric polynomials can be exploited to reduce the degree of the polynomial equations that must be solved.
Abstract: A method is presented to compute the switching angles in a multilevel converter so as to produce the required fundamental voltage while at the same time not generate higher order harmonics. Previous work has shown that the transcendental equations characterizing the harmonic content can be converted to polynomial equations which are then solved using the method of resultants from elimination theory. A difficulty with this approach is that when there are several DC sources, the degrees of the polynomials are quite large making the computational burden of their resultant polynomials (as required by elimination theory) quite high. In this paper, it is shown that the theory of symmetric polynomials can be exploited to reduce the degree of the polynomial equations that must be solved which in turn greatly reduces the computational burden. In contrast to results reported in the literature that use iterative numerical techniques to solve these equations, the approach here produces all possible solutions.

Journal ArticleDOI
Er-Wei Bai1
TL;DR: A frequency domain algorithm for Wiener model identifications based on exploring the fundamental frequency and harmonics generated by the unknown nonlinearity is proposed.

Journal ArticleDOI
12 Oct 2003
TL;DR: In this paper, the authors investigated the effect of slot permeance on the production of principal slot harmonics (RSH) in three-phase and single-phase induction motors.
Abstract: Three-phase as well as single-phase induction machines form the backbone of a substantial portion of the adjustable-speed drives' applications. Among the current estimation techniques available for speed-sensorless induction motor drives, speed measurement based on rotor or principal slot harmonics (RSH or PSH) detection in machine line current happens to be a prominent one. In practice, two predominant RSH can be seen. The present paper attempts to investigate the effect of slot permeance on the production of RSH. Simulation studies on a small motor but with a popular stator, rotor slot, and pole combination suggest that their effect is less significant compared to rotor space harmonics effect. Also, out of the two aforementioned RSH, the cause of the less significant slot harmonic is shown to be the negative-sequence-current-induced reverse rotating field. Voltage unbalances and constructional imperfections can cause negative-sequence current even in a healthy machine. Thus, although the less significant RSH is absent in the simulated line current of an idealized machine running from a balanced supply, it can be detected in an actual machine.

Journal ArticleDOI
TL;DR: In this paper, a three-dimensional space vector algorithm of multilevel converters for compensating harmonics and zero sequence in three-phase four-wire systems with neutral is presented.
Abstract: A three-dimensional (3-D) space vector algorithm of multilevel converters for compensating harmonics and zero sequence in three-phase four-wire systems with neutral is presented. The low computational cost of the proposed method is always the same and it is independent of the number of levels of the converter. The conventional two-dimensional (2-D) space vector algorithms are particular cases of the proposed generalized modulation algorithm. In general, the presented algorithm is useful in systems with or without neutral, unbalanced load, triple harmonics and for generating 3-D control vectors.

Proceedings ArticleDOI
09 Jun 2003
TL;DR: In this paper, it was shown that the instantaneous power waveform presents symmetries if 1/6, 1/3 or 1 cycle of the power system fundamental frequency, depending on the system being balanced or not, and having or not even harmonics.
Abstract: The "generalized theory of the instantaneous reactive power in three-phase circuits", proposed by Akagi et al., and also known as the p-q theory, is an interesting tool to apply to the control of active power filters, or even to analyze three-phase power system in order to detect problems related to harmonics, reactive power and unbalance. In this paper it will be shown that in three phase electrical systems the instantaneous power waveform presents symmetries if 1/6, 1/3, 1/2 or 1 cycle of the power system fundamental frequency, depending on the system being balanced or not, and having or not even harmonics (interharmonics and subharmonics are not considered in this analysis). These symmetries can be exploited to accelerate the calculations for active filters controllers based on the p-q theory. In the case of the conventional reactive power or zero-sequence compensation, it is shown that the theoretical control system dynamic response delay is zero.