Extended Describing Function Research Articles

The Effect of Cross-Coupling on the Dynamic Performance of a Dual Phase Shift Controlled BiWPT System

In a bidirectional wireless power transfer (BiWPT) system, dual phase-shift (DPS) control is used to maintain the optimal efficiency at constant output power over varying misalignment. The presence of multiple control and output variables in DPS control creates cross-coupling between transmitter and receiver side, making BiWPT a multiple-input multiple-output (MIMO) system. Due to cross coupling, any change in control variable in one side of the BiWPT system will affect the output of the same side as well as output and control input in the other side. This results in an uncertain dynamic behaviour of the output response. The cross-coupling effect has not been adequately researched in the literature. Hence, this paper presents an accurate dynamic model using a generalized state-space averaging approach (GSSA) and an extended describing function (EDF) method for a multivariable control of the S-S compensated BiWPT system. This model is used to derive the system’s relative gain array (RGA), which is used to study the effect of cross-coupling terms on the output variables. A 1 kW S-S compensated BiWPT experimental set-up is used to validate the effect of cross-coupling on the dynamic behavior of the system by comparing DPS and single phase-shift (SPS) controlled experimental waveforms. The steady-state and transient behavior of the BiWPT system has also been investigated with simulation and experimental results.

Dynamic improvement for phase-shift LLC resonant converter based on disturbance observer

LLC resonant converters have been widely used in DC/DC conversion because of their high efficiency. However, the converters are highly nonlinear, making it hard to use typical linear control methods. This paper proposes a disturbance-observer-based control approach, which achieves better performance compared with the conventional PI controller. First, according to the extended describing function (EDF) method, the small-signal model of the phase-shift LLC resonant converter is established, which is further simplified to a third-order system. Then, the disturbance-observer-based control is presented, which incorporates a disturbance observer (Dob) to estimate the uncertain effects resulting from external disturbances, circuit parameters variation and model mismatch, and compensate for them in a feedforward way. The main advantage of this control structure is that it can improve the stability and dynamic response simultaneously, where the Dob optimizes the dynamic responses and the PI controller is designed to satisfy the stability requirement. Finally, a 600 W simulation model is built in MATLAB/Simulink to verify the proposed control method, whose superiority over the classic PI controller is proved by the simulation results.

Decoupling Control of an LLC-Quad-Active-Bridge Cascaded Power Electronic Transformer Based on Accurate Small-Signal Modeling

In modular ac–dc power electronic transformers (PETs) based on cascaded H-bridges, usually, a bulky module capacitor is required to store the low-frequency ripple powers introduced from ac phases, which degrades the power density of the system. A considerable amount of power devices arises a lot of switching loss, thus compromising the system efficiency. In this article, an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LLC</i> -type quad-active-bridge-based power channel PET ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LLC</i> -QAB-based PC-PET) topology is first presented. It is intended for improving both the system power density and efficiency. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LLC</i> -QAB is featured by the ripple power counteracting channels as well as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LLC</i> resonant tanks. By using the frequency control combined with ripple power decoupling control, this topology can perfectly prevent ripple power spread and realize soft switching for power devices as well. In order to enhance the dynamic performance and make the control parameter tuning easy, a detailed small-signal model is derived based on extended describing function (EDF). Simulation and experimental results have verified the proposed topology, linearized model, and control strategy. Their effectiveness on soft-switching capability, system efficiency promotion, as well as power density improvement are also analyzed.

Small-Signal Modeling for Phase-Shift Controlled Resonant Converters

The extended describing function (EDF) method is widely used in modeling the resonant converters. This article finds that the model of the phase-shift controlled resonant converter deduced by this method might be inaccurate especially when the duty-cycle is close to 1. The reason behind it is investigated, showing that derivation of the inverter bridge model based on its fundamental harmonic will cause error. A novel model of the inverter bridge by Fourier analysis is proposed. Incorporating the novel model into the EDF-based resonant model, the accuracy of the phase-shift controlled resonant converter is greatly improved. A series-series compensated inductive power transfer system is fabricated in the lab to verify the validity of the proposed model.

Small Signal Modeling and Control Loop Design of Critical Conduction Mode Active Clamp Flyback Converter

Critical conduction mode (CrCM) active clamp flyback (ACF) converter is regarded as a good candidate to be applied in high-frequency small power supplies. This article focuses on the modeling, analysis, and controller design of CrCM-ACF converter. The small signal model of the converter is derived based on a zero-input resonant equivalent circuit and extended describing function (EDF) method for the first time, then the obtained full-order model is simplified to a low-order model. This procedure establishes an insightful understanding about the effects of the converter parameters and working conditions. Based on the simplified model, a control loop compensator is designed in a single-loop voltage-feedback control scheme for high dynamic performance. An experimental prototype of digital controlled CrCM-ACF converter is built, the accuracy of the derived model is verified by comparing with experimental and simulation results. The performance of the compensator is experimentally evaluated by applying abrupt load current and reference voltage changes in the operating conditions of the converter. From the experimental results, it is demonstrated that the derived analytical model and designed compensator are effective in regulating the output voltage in CrCM-ACF converter.

Reduced-Order Modeling and Design of Single-Stage LCL Compensated IPT System for Low Voltage Vehicle Charging Applications

In this paper, single-stage LCL inductive power transfer (IPT) system is proposed in low voltage vehicle charging applications to minimize the system complexity, and its reduced-order modeling and design approach is presented. To deal with the high-order resonant tank, the reduced-order methodology is applied throughout the paper, with which the behavior of resonant tank is investigated under multiple scales. Using the reduced-order time-domain model, the resonant waveform within switching period is accurately described with only one cycle's numerical computation. A novel parameter design principle for realizing ZVS condition and high averaged efficiency within full load range is then proposed and serves as parametric basis. Next, the low-order linear frequency-domain model is deduced using Extended Describing Function (EDF). It is found that six-order LCL-type system behaves as second-order underdamped system within frequency range of interest. Compared to existing models, the proposed reduced-order model features much more simplicity as it provides intuitive understanding into both steady and dynamic characteristic of the converter while facilitating the parameter and controller design. Through above work, the overall design consideration towards a single-stage LCL IPT system is elaborated. Finally, experimental results are given to verify accuracy of the models as well as the effectiveness of designed parameters and controller.

Analysis and Design of Soft-Switching Active-Clamping Half-Bridge Boost Inverter for Inductive Wireless Charging Applications

This paper presents an accurate analysis and a design methodology for a fixed-frequency-controlled active-clamping half-bridge boost inverter (HBBI)-based series–series compensated inductive power transfer (SS-IPT) charging system. First, the operation principles of the active-clamping HBBI in the charging system are analyzed. Consequently, both the steady-state model and the small-signal model are correctly derived by using the extended describing function (EDF) method and are used to design the system. The derived steady-state model is employed to develop a new design approach to achieve zero-voltage switching (ZVS) for the inverter. The dynamic behavior of the system is investigated, and a digital controller for charging current regulation is designed based on the derived small-signal model. The proposed methodology enables not only reducing switching losses but also avoiding bifurcation. Finally, a 1-kW laboratory prototype is implemented to verify the accuracy of the theoretical analyses. Simulation and experimental results demonstrate that the control method can effectively regulate the charging current with a fast response and no steady-state errors and can enable the inverter to achieve ZVS over a wide variation of the charging current and the battery voltage. The hardware prototype achieves a peak dc-to-dc efficiency of 93.4% at a 170-mm air gap, which is comparable to other IPT systems at similar power levels in the literature.

Circuit Topology and Small Signal Modeling of Variable Duty Cycle Controlled Three-Level LLC Converter

With a view to regulate the output voltage with fixed frequency, without using any additional component or complex modulation technique, a topology of a variable duty controlled three-level LLC converter is proposed and an equivalent small signal model for Electric Vehicle (EV) charger applications is deduced in this paper. The steady state equations of each operating region are derived in time domain. Based on an Extended Describing Function (EDF) approach, a small signal equivalent circuit is modeled which includes both frequency and duty controlled terms. The equivalent circuit is further simplified to derive a transfer function of duty control to output voltage. The transfer function is verified through simulation software. Analyzing the transfer function, a voltage controller is designed and implemented with a PI compensator. The simulation results of the proposed control schemes are illustrated and discussed. The topology is compared to a conventional frequency control topology and the merits of the proposed topology are presented.

Linearization and Control of Series-Series Compensated Inductive Power Transfer System Based on Extended Describing Function Concept

The extended describing function (EDF) is a well-known method for modelling resonant converters due to its high accuracy. However, it requires complex mathematical formulation effort. This paper presents a simplified non-linear mathematical model of series-series (SS) compensated inductive power transfer (IPT) system, considering zero-voltage switching in the inverter. This simplified mathematical model permits the user to derive the small-signal model using the EDF method, with less computational effort, while maintaining the accuracy of an actual physical model. The derived model has been verified using a frequency sweep method in PLECS. The small-signal model has been used to design the voltage loop controller for a SS compensated IPT system. The designed controller was implemented on a 3.6 kW experimental setup, to test its robustness.