Magnitude Of Harmonics Research Articles

An artificial neural network based approach for harmonic component prediction in a distribution line

With the increasing use of nonlinear devices in both generation and consumption of power, it is essential that we develop accurate and quick control for active filters to suppress harmonics. Time delays between input and output are catastrophic for such filters which rely on real-time operation. Artificial Neural Networks (ANNs) are capable of modeling complex nonlinear systems through adjustments in their learned parameters. Once properly trained, they can produce highly accurate predictions at an instantaneous time frame. Leveraging these qualities, various complex control systems may be replaced or aided by neural networks to provide quick and precise responses. This paper proposes an ANN-based approach for the prediction of individual harmonic components using minimal inputs. By extracting and analyzing the nature of harmonic component magnitudes obtained from the survey of a particular area through real-time measurements, a sequential pattern in their occurrence is observed. Various neural network architectures are trained using the collected data and their performances are evaluated. The best-performing model, whose losses are minimal, is then used to observe the harmonic cancellation for multiple unseen cases through a simplified simulation in hardware-in-the-loop. These neural network structures, which produce instantaneous and accurate outputs, are effective in harmonic filtering.

Comparison of Approaches for Assessing Linearity of Impedance Measurements

There are two experimental methods available to assess the linearity of electrochemical impedance measurements. Observation of current as a function of potential, known as Lissajous plots, provides an instantaneous visual representation of the data.1 Deviation from an elliptical shape suggests a perturbation amplitude that is too large, resulting in nonlinear impedance data. A second approach is to measure, along with the fundamental, the higher harmonics of the measurement signal. Under potentiostatic control, the harmonics of current that exceed the noise level suggest nonlinear behavior. These harmonics may be expressed in terms of a frequency-dependent total harmonic distortion, THD, written as the ratio between the sum of the root mean square of the harmonic magnitude and fundamental frequency magnitude for electrochemical applications.The impedance was measured for a model system consisting of a Pt disk immersed in an electrolyte containing 0.01 M potassium ferricyanide, 0.01 M potassium ferrocyanide, and 1 M KCl. Measurements were performed using a Gamry Reference 3000 with THD measurement enabled and disabled. Low-frequency (10 mHz) Lissajous plots were recorded to facilitate comparison of the two methods for assessing linearity. The impedance measurements with THD enabled or disabled were in good agreement, including the frequency-dependent stochastic error structure, the shapes of Lissajous curves, and the values of the impedance. Both the low-frequency THD and the low-frequency Lissajous plots gave similar sensitivity to nonlinear behavior. Perturbation amplitudes of 1, 5, and 10 mV rms yielded elliptical Lissajous plots with THD values on the order of the background noise. Interestingly, the THD for the 10 mV perturbation was slightly above the background noise with a value of 0.01 (or 1 % of the fundamental). Perturbation amplitudes of 20, 60, and 100 mV rms yielded both non-elliptical Lissajous plots and low-frequency THD values much larger than the background noise.The regression of the measurement model 2 to the imaginary part of the impedance was applied to assess the consistency of the measured data to the Kramers-Kronig relations.3 Data measured at amplitudes of 1, 5, 10, and 20 mV rms were found to be consistent with the Kramers-Kronig relations, and data collected with amplitudes of 60 or 100 mV rms were found to be inconsistent with the Kramers-Kronig relations. As the measurement with a 20 mV rms amplitude was found to be nonlinear by observation of non-elliptical Lissajous plots and a THD of 4.2% at a frequency of 10 mHz, the post-measurement analysis of consistency with the Kramers-Kronig relations was found to be less sensitive than experimental observation of Lissajous plots and calculation of THD. Comparison of Lissajous plots to THD values suggests that experiments with THD less than 1% of the fundamental may be considered to be linear.

MEDIUM-POWER DRIVE INSTALLATIONS BASED ON TRIPLE VOLTAGE SOURCE INVERTERS ADJUSTED BY ALGORITHMS OF SYNCHRONOUS MULTI-ZONE PWM

This paper presents short overview of the development and dissemination of specialized schemes and algorithms of space-vector-based synchronous multi-zone pulsewidth modulation (MZ PWM) for control of triple inverters of the medium-power variable speed drives characterized by relatively low switching frequency of inverters. It insures providing synchronization and symmetry of the basic voltage waveforms in triple-inverter-based configurations of drive installations on the base of standard voltage source inverters (VSIs). It assures also minimization of magnitudes of even-order harmonics and undesirable subharmonics in spectra of the basic voltages of drive installations, leading to reducing of losses in the corresponding apparatuses and to increasing of its efficiency. Examples of application of the basic techniques of multi-zone PWM for regulation of three typical structures of triple-VSI-based medium-power motor drives are presented. Modeling and simulations give a behavior of drive installations based on triple inverters adjusted by algorithms of synchronous MZ PWM. References 12, figures 11, table 1.

Magnetic nanoparticles in square-wave fields for breakthrough performance in hyperthermia and magnetic particle imaging

Driving immobilized, single-domain magnetic nanoparticles at high frequency by square wave fields instead of sinusoidal waveforms leads to qualitative and quantitative improvements in their performance both as point-like heat sources for magnetic hyperthermia and as sensing elements in frequency-resolved techniques such as magnetic particle imaging and magnetic particle spectroscopy. The time evolution and the frequency spectrum of the cyclic magnetization of magnetite nanoparticles with random easy axes are obtained by means of a rate-equation method able to describe time-dependent effects for the particle sizes and frequencies of interest in most applications to biomedicine. In the presence of a high-frequency square-wave field, the rate equations are shown to admit an analytical solution and the periodic magnetization can be therefore described with accuracy, allowing one to single out effects which take place on different timescales. Magnetic hysteresis effects arising from the specific features of the square-wave driving field results in a breakthrough improvement of both the magnetic power released as heat to an environment in magnetic hyperthermia treatments and the magnitude of the third harmonic of the frequency spectrum of the magnetization, which plays a central role in magnetic particle imaging.

Attentional decoder networks for chest X-ray image recognition on high-resolution features

Background and objective:This paper introduces an encoder–decoder-based attentional decoder network to recognize small-size lesions in chest X-ray images. In the encoder-only network, small-size lesions disappear during the down-sampling steps or are indistinguishable in the low-resolution feature maps. To address these issues, the proposed network processes images in the encoder–decoder architecture similar to U-Net families and classifies lesions by globally pooling high-resolution feature maps. However, two challenging obstacles prohibit U-Net families from being extended to classification: (1) the up-sampling procedure consumes considerable resources, and (2) there needs to be an effective pooling approach for the high-resolution feature maps. Methods:Therefore, the proposed network employs a lightweight attentional decoder and harmonic magnitude transform. The attentional decoder up-samples the given features with the low-resolution features as the key and value while the high-resolution features as the query. Since multi-scaled features interact, up-sampled features embody global context at a high resolution, maintaining pathological locality. In addition, harmonic magnitude transform is devised for pooling high-resolution feature maps in the frequency domain. We borrow the shift theorem of the Fourier transform to preserve the translation invariant property and further reduce the parameters of the pooling layer by an efficient embedding strategy. Results:The proposed network achieves state-of-the-art classification performance on the three public chest X-ray datasets, such as NIH, CheXpert, and MIMIC-CXR. Conclusions:In conclusion, the proposed efficient encoder–decoder network recognizes small-size lesions well in chest X-ray images by efficiently up-sampling feature maps through an attentional decoder and processing high-resolution feature maps with harmonic magnitude transform. We open-source our implementation at https://github.com/Lab-LVM/ADNet.

A Novel Approach for Harmonic Assessment of Power Systems With Large Penetration of IBRs – A U.K. Case Study

This work provides a high-level assessment of the harmonic performance of the future UK power system with high penetration of Renewable Energy Sources (RESs). A few variations to the standard harmonic analysis are proposed to allow for a more accurate representation of Inverter-Based Resources (IBRs) in frequency-domain studies. In the first part of the paper, two reduced network models (for 2020 and 2050, respectively) are developed, including both transmission and distribution system representation. A frequency-dependent impedance model for individual and aggregated IBRs is adopted, and a novel distribution load model with harmonic injection is proposed. Additionally, the variability of harmonic phase angles is modelled taking into account the physical behavior of the IBRs. Probabilistic Harmonic Load Flows (PHLFs) are carried out on the reduced networks by varying the harmonic injection magnitudes and phase angles of each harmonic source. This work shows that, while overall harmonic levels are dependent on the changes in demand, the inverter impedance has significant impact on local resonances, and accurate models for IBRs are required.

Phase modulation of carriers for cascaded multilevel three-phase inverters

Cascaded multilevel three-phase inverters are commonly used in industrial applications and the quality of these inverters is highly dependent on the modulation techniques. The common mode voltage is one of the factors affecting the quality of the three-phase inverter output voltage. This paper proposes a carrier phase modulation method for cascade multilevel inverters to reduce the magnitude of common-mode voltage magnitude and harmonics. In the proposed method, the carrier phase modulator is considered as an adaptive controller and the phase of carrier is completely controlled by the magnitude of control signal. The phase modulation has a high immunity to the noises compared with the modulations of frequency and amplitude because it is difficult for noises to intrude into the phase of carrier. In addition, the algorithm of carrier phase modulation is simple and requires little memory. Moreover, the number of switching commutations of the proposed method also reduces significantly. Simulation and experiment results based on a cascaded five-level three-phase inverter system have confirmed the effectiveness of the proposed method compared with that of the popular methods using unmodulated carriers such as phase disposition (PD) and phase opposite disposition (POD).

Reset control performance improvement using fractional derivatives

When a linear controller is replaced by a reset controller, it is possible to keep the gain behaviour essentially the same, while improving the phase behaviour. However, because reset control is nonlinear, higher order harmonics appear, which may deteriorate the results. In this paper, reset controllers are combined with fractional derivatives, decreasing the magnitude of higher order harmonics. In this way, better results are found in simulations of a Clegg integrator, of a FORE (first order reset element), and of a CgLp (constant in gain lead in phase) controller. The reason why better results are found is explained.

STUDY OF INFLUENCE OF HIGHER HARMONICS ON RELIABILITY CABLE NETWORKS

This study is devoted to the analysis of the cable network of industrial enterprises with a significant number of cable lines with a voltage of 10 kV. A feature of this electric power supply unit is a powerful direct current valve electric drive for rolling production. The schematic diagram of the cable network is shown in the figure, the total length of the cables is more than 20 km. The results of previous experimental studies made it possible to determine the magnitude and spectrum of harmonics in this node, namely: in 5 harmonics do not exceed 1.5%, but abnormal n = 22—40 determine the value of the non-sinusoidal voltage coefficient within the limits of 5.5—8.9%, which exceeds the 5% norm. The plans of the conducted research also included the measurement of single-phase ground fault currents, which gave the following results for workshop substations: PS-4 — 24.2 A; PS-5 — 17.3 A; PS-5 — 14.8 A. Analyzing the problem of the reliability of cable networks, it is necessary to take into account one more indicator — the value of the leakage currents. A significant number of high-voltage cables are insulated with cable paper impregnated with oil-rosin mastic. Such insulation contains gas inclusions that, under the influence of high-frequency electric shocks, are ionized with subsequent neutralization, which leads to its local damage. Oscillograms of the supply voltage show a sharp shape of the curve as a result of the presence of higher harmonics, which causes the aging of the insulation due to the acceleration of physicochemical processes in the dielectric. Thus, the results of theoretical and experimental studies made it possible to substantiate the main factors that determine the operational reliability of cables, namely: the dynamics of leakage currents, the value of the coefficient of non-sinusoidal voltage caused by abnormal harmonics. Analysis of the specified characteristics allows you to assess the current state of the cable network and predict the reliability of work in the future.

An Online Condition Monitoring Method for Series-Connected IGBTs to Avoid Cascading Failure

The series-connected insulated gate bipolar transistors (IGBTs) have numerous applications in industry and research fields. One of the risks of the resultant series high-voltage switch (RSHVS) is the IGBTs short circuit failure. The number of the IGBTs in a RSHVS is usually designed with an appropriate voltage de-rating factor to make it resilient against a limited number of the IGBTs short circuit failures. However, by the increased number of failures, the withstand voltage of the healthy IGBTs finally violates the IGBTs' tolerable voltage in the off-state. In such conditions, a cascading failure occurs and breaks all remaining healthy IGBTs. This paper proposes a condition monitoring (CM) method to avoid cascading failure in RSHVSs. The proposed method works based on the change of the rising edge of the load current in the RSHVS turning-on time. It will be shown when some of the IGBTs in the series structure fail short, the turning-on time of the RSHVS increases. Thus, the rising edge of the current also increases which leads to the decrement of high-order harmonics magnitude. The proposed CM analyzes such harmonics and provides a constant voltage as the CM output. The performance of the proposed CM is evaluated using simulations and experiments for a RSHVS consisting of eight IGBTs.

Nonlinear interactions between the fundamental and its harmonics of self-excited azimuthal thermoacoustic instabilities

Thermoacoustic instabilities in combustors are inherently nonlinear processes, manifested by multiple peaks in the spectrum. Analyzing these signals is crucial to understand the mechanism of nonlinear limit cycle oscillations. The goal of this study is to investigate the acoustic modal dynamic relation between the fundamental of azimuthal instabilities and its first harmonic. The paper first demonstrates that the nonlinear interaction of the fundamental azimuthal mode produces not only higher order azimuthal modes but also an axial mode in the harmonic. This is confirmed by the experimental data showing a correlation between the axial mode in the harmonic and two counter-rotating modes in the fundamental. Furthermore, the study shows that neglecting this axial mode may result in inaccurate estimation of the maximum pressure in the chamber, potentially leading to safety concern. The combustor is operated under various conditions to examine the correlation between the fundamental and its harmonic in terms of dominant spinning direction, orientation of the acoustic structure, and the maximum pressure magnitude. It is found that the azimuthal components of the fundamental and its harmonic rotate in the same direction under a wide range of operating conditions. Additionally, the maximum pressure magnitudes are positively correlated with each other despite the difference in the order of magnitude between them. Specifically, the harmonic's magnitude shows quadratic dependence on the fundamental's, even when the mode transitions from a standing wave to a spinning wave. However, the anti-node locations of the fundamental and its harmonic exhibit no correlation, and their difference in location varies with operating conditions.

Torque-Ripple Reduction of Permanent Magnet Synchronous Machine Drives Based on Novel Speed Harmonic Control at Low-Speed Operation

Torque ripple reduction methods based on harmonic current injection for PMSM drives have been widely discussed, while the performance of torque-ripple-model (TRM) based methods are limited due to model accuracy as well as the rotor position errors, and the speed harmonic control based methods still cannot get rid of the impact of the phase information of speed harmonics, which results in remaining torque ripples and the difficulty in designing speed harmonic controller. In this paper, the unique relationships between the quadrature magnitudes of speed harmonics and harmonic currents are derived considering the minimal conduction copper loss by the harmonic currents. Based on that, a novel torque ripple reduction method based on speed harmonic control is proposed, wherein a novel speed harmonic controller is presented to regulate the quadrature magnitudes of speed harmonics, so that the phase of speed harmonics is avoided in the speed harmonic controller. Also, the proposed speed harmonic controller aims to generate harmonic current references. The proposed methodology is evaluated by experiments and is verified to reduce torque ripples of PMSM drives effectively.

An Adaptive Virtual Impedance Method for Grid-Connected Current Quality Improvement of a Single-Phase Virtual Synchronous Generator under Distorted Grid Voltage.

The proportion of distributed generation systems in power grids is increasing, leading to the gradual emergence of weak grid characteristics. Moreover, using voltage-sourced grid-connected inverters can enhance the stability of a weak grid. However, due to the presence of background harmonics in weak grids, the grid voltage can cause significant distortions in the grid-connected current, which adversely affects the quality of the grid-connected current. This paper begins by briefly introducing the principle of the virtual synchronous generator (VSG). Then, the output current of the voltage source inverter is analyzed to elucidate the mechanism of harmonic current generation. Considering the distortion in the grid-connected current of the voltage source grid-connected inverter caused by background harmonics in the grid voltage, a harmonic current suppression strategy based on an adaptive virtual harmonic resistor is proposed. The proposed strategy employs a signal separation module based on multiple second-order generalized integrators connected through a cross-feedback network. This module effectively separates the fundamental and harmonic currents from the grid-connected current, extracts the amplitudes of the fundamental and harmonic currents through coordinate transformation, and adaptively adjusts the virtual harmonic resistance magnitude through the negative feedback control of the harmonic content (the ratio of the harmonic current amplitude to the fundamental current amplitude). These measures are used to enhance the quality of the grid-connected current. Additionally, the stability of the system is analyzed using the root locus of the open-loop transfer function. Finally, the effectiveness of the proposed method is validated through a combination of MATLAB/Simulink simulations and experimental results.

Aeroacoustic radiation of low Reynolds number rotors in interaction with beams.

The radiation characteristics of rotor-beam interaction noise are studied experimentally for low Reynolds number small-scale rotors in interaction with beams of different shapes, sizes, and downstream positions. The number of blades ranges from two to four. For the two-bladed rotor, the presence of the beam has no effect on the mean aerodynamic performance. Moreover, the blade passing frequency (BPF) and the high frequency broadband noise (BBN) appear not to be affected by the presence of the beam. On the contrary, the magnitude of the 2×BPF-25×BPF harmonics increases up to 30 dB compared to the case without beam, with an envelope consisting of two humps: one centered around 5×BPF and another around 20×BPF-25×BPF. For the first hump, a dipole-like pattern with minimal amplitude aligned with the beam can be observed, whereas another dipole-like pattern is observed for the higher frequency hump, but with a minimal amplitude over all the rotor disk plane. Compared to the two-bladed rotor, the presence of the beam has an effect on the mean aerodynamic performance of the three- and four-bladed rotors, increasing both the torque and the thrust at iso-rotational speed. This change leads to a change in the directivity of the BPF tone that decreases at a latitude angle of θ=0° and increases at a latitude angle of θ=40°. Moreover, the same two competing humps are observed on the BPF harmonics envelope. Interestingly, the frequency range over which an amplification of the harmonic magnitude is observed seems not to be influenced by the number of blades. Finally, the magnitude of the low frequency hump increases with the beam diameter, the rotational speed, and the number of blades but decreases with the rotor-beam distance. That of the high frequency hump increases also with the rotational speed and the number of blades, but not anymore with the beam diameter, and reaches a maximum value when the rotor-beam distance is at an intermediate distance of L = 25 mm. This hump is also influenced, to a lesser extent, by the shape of the beam. The two different evolutions permit us to conclude that the noise generation mechanisms leading to the two humps must be different. Scaling laws of the acoustical energy are derived for all those parameters. As already done for previous experiments without beam, all of the results are made available as an open database, at https://dataverse.isae-supaero.fr/.

Harmonic Distortion and Hosting Capacity in Electrical Distribution Systems with High Photovoltaic Penetration: The Impact of Electric Vehicles

Electric vehicles and the charging stations and their operation require a thorough examination to evaluate the effects on the electrical network. This becomes particularly challenging in the case of high photovoltaic penetration, due to the variability of the solar resource and vehicle connection patterns, which cater to individual user preferences. The current study investigates the impact of harmonics generated by charging stations and electric vehicles on different photovoltaic penetration scenarios within an electrical distribution system. DC and AC charging stations are analyzed. The findings reveal a third harmonic magnitude increase exceeding 300% compared to other cases. Furthermore, this study demonstrates the effects of current and voltage variations on end-users and substation transformers. The impact of harmonics on the hosting capacity of the network is also analyzed, resulting in a 37.5% reduction in the number of vehicles.

Harmonics of Pulsatile Pressure At Different Ages and Its Effect On Other Pulsatile Parameters and Waveform-based Clinical Indices

Abstract Pulsatile pressure at an artery is a collection of harmonics of the heartbeat. This study examines harmonics of pulsatile pressure at different ages and its effect on other pulsatile parameters and waveform-based clinical indices. Based on a vibrating-string model of the arterial tree, wave velocity and characteristic impedance are related to arterial stiffness and radius. Blood velocity, wall shear stress (WSS), and driving force on the left ventricle (LV) are related to pulsatile pressure. Reflection magnitude and return time are related to input impedance. These relations are applied to pulsatile pressure and blood velocity at the ascending aorta (AA) and the carotid artery (CA) at different ages in a database to calculate harmonics of all the pulsatile parameters and reflection magnitude and return time at each harmonic. Harmonics of pulsatile pressure varies with aging and between the two arteries. Reflection magnitude and return time vary between harmonics. While wave reflection manifests the arterial tree (i.e., arterial stiffness and radius) and termination, harmonics of pulsatile pressure is a combination of the LV, the arterial tree, and termination. Harmonics of pulsatile pressure dictates harmonics of WSS and affects endothelial function. Harmonics of pulsatile pressure needs to serve as an independent clinical index indicative of the LV function and endothelial function. Reflection magnitude and return time of the 1st harmonic of pulsatile pressure serve as clinical indices indicative of arterial stiffness and radius.

Detecting Mild Phonotrauma in Daily Life.

The aim of this study was to gain quantitative insights into the role of daily voice use associated with mild phonotrauma via the Daily Phonotrauma Index (DPI), a measure derived from neck-surface acceleration magnitude (NSAM) and difference between the first two harmonic magnitudes (H1 - H2). An ambulatory voice monitor recorded weeklong voice use for 151 female patients with phonotraumatic vocal hyperfunction (PVH) and 181 female vocally healthy controls. Three laryngologists rated phonotrauma severity from each patient's laryngoscopy. Mixed generalized linear models evaluated the accuracy, sensitivity, and specificity of the original DPI trained on all patients versus a mild DPI version trained on only patients rated with mild phonotrauma. Individual contribution of NSAM and H1 - H2 to each DPI model was also evaluated. Reliability across the laryngologists' phonotrauma ratings was moderate (Fleiss κ = 0.41). There were 70, 69, and 12 patients with mild, moderate, and severe phonotrauma, respectively. The mild DPI, compared to the original DPI, correctly classified more patients with mild phonotrauma (Cohen's d = 0.9) and less controls (d = -0.9) and did not change in overall accuracy. H1 - H2 contributed less to mild phonotrauma classification than NSAM for mild DPI. Compared with the original DPI, the mild DPI exhibited higher sensitivity to mild phonotrauma and lower specificity to controls, but the same overall classification accuracy. These results support the mild DPI as a promising detector of early phonotrauma and that NSAM may be associated with early phonotrauma, and H1 - H2 may be a biomarker associated with vocal fold vibration in the presence of lesions. Level 4, case-control study Laryngoscope, 133:3094-3099, 2023.

A Novel Error-Injected Solution for Compensation of Current Measurement Errors in PMSM Drive

Current measurement errors including offset error and scaling error can generate asymmetric phase currents and 1st- and 2nd-order harmonics in motor steady-state torque/speed. In this paper, a novel method using artificial error injection is proposed to calculate and compensate for the offset error and scaling error in permanent magnet synchronous motor (PMSM) drive. The artificial errors are injected to introduce additional equations for the calculation of current measurement errors because of the problem of rank deficiency existing in the equation about the offset/scaling error and the 1st/2nd-order harmonic magnitude. In order to guarantee the calculation accuracy of the offset error and scaling error, the cascade multiple second-order generalized integrators (Cascade-Multi-SOGIs) are employed to extract the magnitudes of the 1st- and 2nd-order harmonics in steady-state speed error under different error injections. The method does not require the values of motor parameters, and it can be used either as part of an initial debugging or start-up program, or periodically during the regular operation of the motor. The experiments can verify its validity and practicability.

Runge–Kutta optimization‐based selective harmonic elimination in an H‐bridge multilevel inverter

AbstractThe application of iterative techniques for solving the Selective Harmonic Elimination Pulse Width Modulation (SHE‐PWM) problem, such as the Newton–Raphson (NR) method, can tend to get stuck at local optima in the solution space. Additionally, these methods may be sensitive to the initial value estimation of the solution. In contrast, metaheuristic approaches demonstrate resilience in seeking out the optimal solution. As such, this study utilizes the Runge–Kutta (RUN) metaheuristic optimization algorithm to demonstrate the SHE‐PWM technique in multilevel inverters (MLIs), including 5‐ and 7‐level modified H‐bridge (MHB) topology and a 9‐level asymmetric cascaded H‐bridge (CHB) inverter topology. The switching angles are obtained by varying the modulation index from 0.01 to 1.0 in steps of 0.01 for 5‐, 7‐, and 9‐level MLIs. The performance of the RUN algorithm in minimizing the total harmonic distortion (THD) value is verified through simulations in MATLAB/Simulink software. The superiority of RUN is established by comparing the results with state‐of‐the‐art metaheuristic algorithms, such as the Differential Evolution (DE), Genetic Algorithm (GA), and Grey Wolf Optimizer (GWO). Additionally, the switching angles obtained through RUN are validated by hardware experiments. According to the simulation and experimental results, the proposed RUN method exhibits better performance in terms of objective function values, algorithm robustness, fundamental harmonic magnitude, and THD values. The findings confirm the elimination of fifth harmonic, fifth, and seventh harmonics, and fifth, seventh, and ninth harmonics in 5‐, 7‐, and 9‐level MLIs, respectively.

Dynamic Motions of Beam-type Structural Element Subjected to a Harmonic Magnitude Partially Distributed Load at Uniform Velocity

Dynamic Motions of Beam-type Structural Element Subjected to a Harmonic Magnitude Partially Distributed Load at Uniform Velocity