As an important parameter in power electronic converters, the leakage inductance of high-frequency transformers is of great significance in improving the operating mode and power transmission characteristics of isolated DC-DC converters. Compared with the solid round wire, the Litz wire can reduce eddy current losses in high-frequency magnetic components. However, the complicated structure of the Litz-wire windings poses a serious challenge to predicting leakage inductance in high-frequency transformers. On the one hand, it is difficult to precisely extract the magnetic field energy in various regions of the core window. On the other hand, it is hard to accurately characterize the multi-stranded and twisting structures of the Litz wires. Therefore, this paper presents a fast calculation method of leakage inductance in the high-frequency transformer with Litz-wire winding.

Firstly, a homogenized equivalent process for Litz wire is proposed to enhance the flexibility of modeling and the efficiency of computation. The magnetic field energy variation with frequency inside the Litz-wire conductors are analyzed. Then, the 2-D magnetic field energy inside the core window is extracted based on the method of images to eliminate the impact of the edge effect. The internal and external magnetic fields at different locations in the winding are accurately characterized by introducing the meshing into the method of images, and a coordinate transformation method is proposed to consider the twisting structure of the Litz wires. Finally, two high-frequency transformer prototypes with different structures are designed and fabricated. Compared with the measurement results and two existing methods, the accuracy and efficiency of the proposed approach are verified.

The following conclusions can be drawn. (1) A homogenized equivalent model of the Litz-wire twisting structure is developed by introducing the relative complex permeability, which simplifies the model building and reduces the computational cost. The variation of the magnetic field energy in the Litz wires with frequency is analyzed, and the magnetic field energy stored in the Litz wires gradually decreases with the frequency increase. (2) The meshing process is introduced into the method of images, and the coordinate transformation method is proposed to characterize the twisting structure of the Litz wire. It can counteract a part of the external magnetic field and reduce the magnetic field energy in the conductors. (3) Considering the twisting characteristics of the Litz wire and the high-frequency effect, a leakage inductance prediction model is developed based on the magnetic field energy variation with frequency. (4) The accuracy and efficiency of the proposed method are verified compared with the measurement and the current two analytical methods. The maximum error does not exceed 4% throughout the measurement frequency range, and the calculation time is about 20 seconds. Moreover, the proposed method can be effectively applied to fast iterative calculations in the optimal design of high-frequency transformers.

Permanent magnet linear synchronous motors with section power supply are affected by disturbances like load force, detent force, and friction force. In the field of electromagnetic drive, the stator track is long. Linear motors usually adopt a segmented structure to save inverter capacity. However, it is difficult to ensure that the air gap of each segmented stator is equal during installation. Therefore, the mover is affected by the normal force. In addition, the load is usually accelerated to the target speed in a short time, so electromagnetic drive devices are usually operated under high current and high acceleration conditions, where the motor parameters are prone to change. Due to the lack of intermediate transmission devices in PMLSMs, these disturbances will directly affect the motor drive system and are included in the output of the controller. An SMSC with a novel convergence law is designed to ensure the fast convergence of speed and suppress the chattering. The designed TSMDO observes the disturbance output to ensure the disturbance suppression performance. Then, the acceleration fluctuations are reduced, and the thrust fluctuations are suppressed in the motor output.

Firstly, a mechanical motion model of PMLSM is established based on Newton's second law. It takes the disturbances as the lumped disturbance d(t) caused by load force, detent force, friction force, normal force, and parameter variation. Secondly, the shortcomings of conventional sliding mode control are analyzed, and a new adaptive sliding mode approaching law is proposed. The designed sliding mode approaching law ensures that variables can approach the sliding mode surface at a fast speed. As the state variables of the system gradually approach the sliding mode surface, the designed sliding mode approaching law can reduce the speed to weaken chattering. Then, due to the presence of d(t) in the output of the sliding mode speed controller, the TSMDO is designed to compensate for it. This observer is equivalent to a first-order low-pass filter. Finally, the proposed SMSC strategy is compared with PI control and conventional SMSC.

The experiments show that the PI controller can improve the dynamic tracking performance of speed by increasing h. However, this increases the acceleration fluctuation. At the same time, there is a noticeable overshoot when the rotor enters the constant speed range. The conventional SMSC has better dynamic tracking performance than PI control, resulting in small acceleration fluctuations. However, due to the fixed sliding mode gain, the increasing sliding mode gain increases the overshoot. The proposed SMSC has a faster tracking speed and better dynamic response than the conventional approaching law because of the new sliding mode approaching law. The proposed SMSC can adaptively change its gain when the speed state changes, ensuring good speed-tracking performance and reducing overshoot. The designed TSMDO effectively reduces the impact of thrust disturbances, acceleration fluctuations, and thrust fluctuations.

The main conclusions are as follows. (1) The proposed new sliding mode approaching law has a fast convergence speed due to the adaptive gain function f(x₁, s) for adaptively adjusting the sliding mode gain. It can weaken chattering, ensuring speed dynamic tracking performance and convergence speed. (2) The designed TSMDO has a small chattering phenomenon. The introduction of TSMDO effectively suppresses the total disturbance d(t) in the SMSC output, reduces acceleration fluctuations, and ensures stable thrust output of the motor.

In wireless power transfer (WPT) systems, achieving accurate voltage regulation and efficient operation are critical. Current research achieves constant voltage output and zero voltage switch (ZVS) with additional DC-DC converters and variable resonant networks. However, these approaches increase system losses and costs. Therefore, this paper proposes a two-sided LCL phase-shifting control strategy. The internal phase shift angle of the inverter and active rectifier (AR) is used to achieve constant voltage output and maximum efficiency tracking, and the external phase shift angle between the two converters achieves ZVS of all switching tubes. By analyzing the power loss, the constraint condition between the internal phase shift angle is obtained. The minimum external phase shift angle δ_opt of ZVS is further determined, and the system’s high efficiency is realized. In addition, the power angle θ_power is introduced as the intermediate variable, and the frequency synchronization of the primary and secondary sides is realized using the voltage-controlled oscillator (VCO).

Firstly, utilizing the fundamental wave equivalent model, the constant voltage characteristics of the system and the constraint conditions of the inverter output voltage and AR input voltage pulse-width ratio D₁ and D₂ are analyzed. The results show that transmission efficiency peaks when the AC voltage ratio α =1. With load variations, achieving constant voltage output and maximum efficiency tracking is feasible by adjusting D₁ and D₂. Secondly, based on the time-domain harmonics model, the derivation and simplification of the time-domain expression of inductance current are conducted. The simplified model is then analyzed to determine the external phase shift angle δ. By comparing δ of the inverter and AR, the δ_opt is obtained. Thirdly, the overall control strategy is introduced. The constant voltage output is achieved by adjusting D₁ and D₂. The introduction of θ_power as an intermediate variable establishes the relationship between δ and θ_power, enabling indirect control of δ through the regulation of θ_power. Subsequently, the frequency synchronization of the primary and secondary sides is realized using a VCO, effectively solving the synchronization challenge associated with an active rectifier.

Finally, system simulations and experiments were conducted. The experimental results show that the system can achieve ZVS for all MOSFETs and maintain a constant voltage output regardless of load variations. Moreover, the proposed synchronous control strategy can effectively track the switching frequency of the inverter and precisely adjust the required δ_opt. As D₁ and D₂ consistently adhere to the maximum efficiency constraints during system adjustments, the system also achieves maximum efficiency tracking. When the coupling coefficient k is 0.31, the transmission efficiency of the system is the highest, and the maximum efficiency is 93.8%.

The single-stage Totem pole dual active bridge (DAB) AC-DC converter has the advantages of low component count, high power density, and low cost, which has a broad application prospect in the field of on-board chargers (OBC). However, in the available research, the traditional single phase shift (SPS) and extended phase shift (EPS) modulation strategies are unable to optimize the quality of grid-connected current and efficiency of the Totem pole DAB AC-DC converter at the same time due to the problem of insufficient modulation degrees of freedom, limiting the further application in on-board chargers.

This paper introduces the asymmetric modulation based on the extended phase shift modulation strategy when the duty cycle of the secondary side switching tubes is no longer 50%. An asymmetric extended phase shift (AEPS) modulation strategy with three degrees of freedom is proposed. Accordingly, a multi-objective optimal modulation strategy is solved by considering the simultaneous optimization of the grid-connected current quality and efficiency of the Totem pole DAB AC-DC converter.

Firstly, the steady-state analytical model of AEPS modulation is established by using the time domain analysis method. The initial value decoupling constraint of the inductor current is considered to optimize the quality of grid-connected current, and the peak-to-peak inductor current is taken as the optimization objective. According to the Lagrange algorithm and Karush Kuhn Tucker conditions, the above multi-objective optimization problem is transformed into mathematical equations to solve the optimization solution of modulation variables. Matlab simulations show that the inductor current’s initial value decoupling and peak-to-peak value optimization are realized under the AEPS optimization modulation strategy. Compared with the SPS and EPS modulation, the proposed AEPS optimization modulation strategy reduces the peak-to-peak and RMS levels of the inductor current in the full power band, which reduces the conduction loss of the converter. Moreover, the optimized solutions in different operating modes under APES modulation are continuous, making seamless switching between different operating modes available.

An experimental prototype of a totem pole DAB AC-DC converter with a rated power of 800 W is constructed. Experimental results show that the converter achieves a peak efficiency of 93.5% under the proposed AEPS optimized modulation strategy, 5% and 14.7% higher than the SPS strategy at full load and light load, respectively; 1.5% and 14.7% higher than the EPS modulation strategy at full load and light load, respectively. The converter's grid-connected current THD is significantly reduced in the full power range, improving its grid-connected current quality. Simulation and experimental results verify the effectiveness of the proposed AEPS-optimized modulation strategy.

In servo systems, the bandwidth of the current loop is increased by raising the switching frequency. However, the dead-time nonlinearity of the voltage source inverter (VSI) intensifies with the increase of switching frequency, causing the deviation between the actual and the theoretical output voltage, resulting in serious distortion of the inverter output current waveform.

The requirement of computational power constrains the implementation of high switching frequency control. Consequently, the dead-time nonlinearity compensation strategy should be more straightforward to decrease computational time, especially for high switching frequency applications. The amplitude increase is relatively modest at a low switching frequency but significantly surges at a high one, bringing on a severe degradation of the linear modulation region. It is an imperfect solution to compensate after the occurrence of dead-time, which unavoidably introduces compensation errors. Furthermore, the accurate solution of inverter nonlinear voltage error (INVE) under different currents represents a crucial aspect of achieving exact INVE compensation. Nevertheless, the existing methods are complex.

This paper proposes a novel strategy that combines no-dead-time double modulation wave pulse-width modulation (PWM) and inverter nonlinearity compensation for analyzing the dead-time nonlinearity and the nonideal characteristics of the inverter on the output voltage error. Firstly, according to the continuous current characteristics of the anti-parallel diode, the drive vacancy area is added between the complementary drive pulses to avoid the introduction of dead time. Compared with the ideal space vector pulse width modulation (SVPWM), an auxiliary modulating wave is added. Depending on the current polarity, its amplitude is adjusted up or down from the original modulating waveform. The underlap periods are generated between the complementary drive pulse by contrasting the double-modulating and the triangular carrier waves. It is practical to avoid both the bridge arm shoot-through and the introduction of dead time. Most importantly, the actual output voltage of this method is identical to the optimal voltage, which directly eliminates the dead-time nonlinearity and removes the limitation of dead time on the output duty cycle at a high switching frequency. Moreover, the control signals of each switching device are obtained based on the comparison between the double-modulating and the carrier wave. No additional control loop calculations are required, while the generated PWM signals are symmetric about the carrier midpoint.

Secondly, the inverter nonlinearity is equated to the INVE, which varies with the current. When the motor is at a standstill of ${{\theta }_{\text{e}}}={{0}^{\circ }}$, by injecting the ramp current signal into the direct axis and applying Kirchhoff's voltage law, the sum of INVE containing the nonlinear factor of the two-phase VSI is obtained. Finally, the relationship between the INVE and the current amplitude is calculated using the linear iterative interpolation approach, and the online compensation of the INVE is achieved.

The results show that the proposed strategy can increase the linear modulation region of the output voltage, eliminate the output duty cycle limitation derived from the dead-time, and effectively suppress the current harmonic distortion phenomenon caused by the dead-time nonlinearity of high switching frequency inverters. In addition, the strategy is easy to implement without additional control loop calculations, which can be applied to servo drive control systems requiring high switching frequencies.

Fractional-order elements (FOEs) serve as fundamental components in fractional-order circuits, forming the cornerstone of research into fractional-order circuit systems. Unlike single-component counterparts, the multi-component method offers greater flexibility in selecting constituent elements, enabling the adjustment of order and impedance coefficients for enhanced practicality. This paper provides a comprehensive overview of prevailing construction methods to facilitate the selection of appropriate multi-component FOEs for specific application needs. These methods are classified into three categories based on the type of constituent devices: passive devices, operational amplifiers, and power electronic converters.

In the construction method based on passive devices, two approaches involving Foster RLC ladder circuits are discussed. Passive RLC networks are used to create circuits that match the impedance characteristics of the transfer function. Reducing phase error requires increasing the order and using numerous components, leading to complex structures and cumbersome calculations. When standard off-the-shelf components cannot be used, replacing analytical parameters with standard ones increases the phase angle deviation. Adjusting the FOE order or impedance coefficients requires replacing all circuit components. This method is most suitable for fixed FOE order and impedance coefficients, which are effective in medium to low-frequency scenarios.

Next, the paper introduces the construction method based on operational amplifiers. The method of constructing FOEs based on generalized impedance converter (GIC) circuits can realize FOEs with orders varying from -2 to 2. However, the limited open-loop gain and gain instability of operational amplifiers often result in significant deviations of the obtained FOE performance from its ideal characteristics. Therefore, exploring how to use other active devices, such as operational transconductance amplifiers (OTA) and current feedback operational amplifiers (CFOA), to construct FOEs is an exploration direction. GIC circuits offer great integration and functionality, making them suitable for applications where precise impedance and phase characteristics are crucial.

The construction method based on power electronic converters is also discussed in detail. Multi-component FOEs based on power electronic converters have wide applications because their power level depends on the inverter, and their order and impedance can be adjusted by changing the control parameters. However, different structures of filters affect the operating performance of fractional-order elements. Therefore, exploring the application of different filter structures on multi-component FOEs and optimizing the parameters of the filters become the direction of development for power electronic converter-based FOEs. Power electronic converters provide the advantage of handling higher power levels and dynamic adaptability. The ability to digitally control the fractional order and impedance in real-time makes these elements highly versatile.

Lastly, this paper proposes a three-phase fractional-order electrical spring (TPFES). TPFES controls the order of the equivalent fractional-order capacitance of each phase and the pseudo-capacitance. The effect of stabilizing the load voltage is realized, the power factor is improved, and the power is balanced. The application in grid power compensation demonstrates that multi-component FOEs can effectively enhance the performance of practical circuit systems. This work provides a reference for future applications of fractional order components in electrical engineering.

The equivalent circuit is valuable for designing an induction machine and its driving system, and its parameters are essential for analyzing the electromagnetic performance and establishing the driving model, especially the magnetizing inductance that characterizes the main flux distribution in the machine. The finite element analysis can precisely determine the magnetizing inductance. However, it is unsuitable for the initial design stage when the design parameters need frequent adjustment. Traditional analytical calculations like the flux linkage method neglect the influence factors, such as core saturation, tooth-slot effect, and rotor movement in the actual operation, causing accuracy issues. The improved analytical calculation method can significantly enhance the accuracy. However, the magnetic circuit in each core segment still needs to be more accurate, and the local saturation points are easily ignored. Besides, the solving process contains nonlinear iterations of multi-segment of the magnetic circuit, which has repeating calculations under different slips.

This paper proposes an elementary layer method based on the main and leakage magnetic circuits to calculate the magnetizing inductance. Firstly, the magnetic voltage drop of each pole of the main magnetic circuit under different air-gap flux densities is calculated, and the magnetic voltage drop and the air-gap flux density are converted into the electromotive force and the magnetizing current to obtain the objective function. Secondly, the distribution of the leakage flux in the stator slot and the current induced in the rotor bar is analyzed to calculate the slot leakage inductance of the stator and rotor, together with the rotor AC resistance. The stator terminal voltage expression is constructed as the constraint condition. Finally, each set of the electromotive force and the magnetizing current on the objective function are substituted into the constraint condition to make the results equal to the rated phase voltage of the stator. The ratio of the set is the value of the magnetizing reactance, and therefore, the magnetizing inductance is obtained. In the main and leakage flux circuits, the irregular and nonlinear magnetic and electric circuits are regularized and linearized using the thin layer elements to substitute theoretical integral. Hence, the tooth-slot structure and the nonlinear material properties can be considered more accurately when calculating the magnetic voltage drop and leakage inductance.

A wet submersible induction machine is an example of the analytical calculation of the magnetizing inductance using the proposed elementary layer method and the traditional flux leakage method. Besides, the steady-state outputs of the equivalent circuit are obtained. The prototype test and the finite element simulation under the magnetic saturation of the stator teeth are conducted. The results show that the elementary layer method considers the distribution of the magnetic voltage drop in the core segment, which is more effective than the flux linkage method when the tooth magnetic circuit is saturated. Therefore, the calculation of the magnetizing inductance is more accurate, and in the steady-state outputs of the equivalent circuit, the stator current, input power, and power factor curves are consistent with the finite element analysis. The error is less than 0.5% when the finite element results are used as the reference. The proposed method links the design parameters and the magnetizing inductance, providing convenience for the initial design and optimization of the submersible induction machine and other types of machines.