Ripple compensation for a class-D amplifier

This paper presents the first detailed mathematical analysis of the ripple compensation technique for reducing audio distortion in a class-D amplifier with negative feedback. The amplifier converts a relatively low-frequency audio signal to a high-frequency train of rectangular pulses whose widths are slowly modulated according to the audio signal (pulse-width modulation, PWM). Distortion manifests itself through unwanted audio-frequency harmonics that arise in the output due to nonlinearities inherent in the design. In this paper, we first develop a small-signal model, which describes the fate of small-amplitude perturbations to a constant input, and demonstrate how this traditional engineering tool may be extended to allow one to infer the most significant contributions to the full output in response to a general audio input. We then compute the audio output of the amplifier through a perturbation expansion based on the ratio between audio and switching frequencies. Our results explicitly demonstrate how the ripple compensation technique significantly linearizes the output, thereby reducing the distortion

One-dimensional spectral analysis techniques for multilevel PWM strategies

This paper presents a novel spectral analysis technique for multilevel modulation. Conventionally, such analyses use a double Fourier series technique, but this approach can become intractable when complex reference waveforms (e.g., multilevel space vector offsets) and regular sampling processes are considered. In contrast, the strategy proposed in this paper separates the multilevel pulse width modulation (PWM) waveform into a spectral image of the reference, and sideband basis functions which are then expanded using a one-dimensional Fourier series. The coefficients of this Fourier series are defined by a one-dimensional Fourier integral that is simpler in form compared to the corresponding double integral associated with the double Fourier series. This analysis technique naturally incorporates regular sampling, and a discrete formulation is developed that enables complex PWM reference waveforms, including centered space vector offsets, to be solved. Results of this analysis are validated against previously published multilevel inverter double Fourier series results and matching switched simulations

Integration of inverter constraints in geometrical quantification of the optimal solution to an MPC controller

Published Conference ProceedingsThis paper considers a model predictive controller with reference tracking that manipulates the integer switch positions of a power converter. It can be shown that the optimal switch position can be computed without solving an optimization problem. Specifically, in a new coordinate system, the optimization problem can be solved offline, leading to a polyhedral partition of the solution space. The optimal switch position can then be found using a binary search tree. This concept is exemplified for a three-level single-phase converter with an RL load

Multistep Model Predictive Control for Cascaded H-Bridge Inverters: Formulation and Analysis

© 1986-2012 IEEE. In this paper, a suitable long prediction horizon (multistep) model predictive control (MPC) formulation for cascaded H-bridge inverters is proposed. The MPC is formulated to include the full steady-state system information in terms of output current and output voltage references. Generally, basic single-step predictive controllers only track the current references. As a distinctive feature, the proposed MPC also tracks the control input references, which in this case is designed to minimize the common-mode voltage (CMV). This allows the controller to address both output current and CMV targets in a single optimization. To reduce the computational effort introduced by a long prediction horizon implementation, the proposed MPC formulation is transformed into an equivalent optimization problem that can be solved by a fast sphere decoding algorithm. Moreover, the benefits of including the control input references in the proposed formulation are analyzed based on this equivalent optimization problem. This analysis is key to understand how the proposed MPC formulation can handle both control targets. Experimental results show that the proposal provides an improved steady-state performance in terms of current distortion, inverter voltages symmetry, and CMV

A third-order class-D amplifier with and without ripple compensation

We analyse the nonlinear behaviour of a third-order class-D amplifier, and demonstrate the remarkable effectiveness of the recently introduced ripple compensation (RC) technique in reducing the audio distortion of the device. The amplifier converts an input audio signal to a high-frequency train of rectangular pulses, whose widths are modulated according to the input signal (pulse-width modulation) and employs negative feedback. After determining the steady-state operating point for constant input and calculating its stability, we derive a small-signal model (SSM), which yields in closed form the transfer function relating (infinitesimal) input and output disturbances. This SSM shows how the RC technique is able to linearise the small-signal response of the device. We extend this SSM through a fully nonlinear perturbation calculation of the dynamics of the amplifier, based on the disparity in time scales between the pulse train and the audio signal. We obtain the nonlinear response of the amplifier to a general audio signal, avoiding the linearisation inherent in the SSM; we thereby more precisely quantify the reduction in distortion achieved through RC. Finally, simulations corroborate our theoretical predictions and illustrate the dramatic deterioration in performance that occurs when the amplifier is operated in an unstable regime. The perturbation calculation is rather general, and may be adapted to quantify the way in which other nonlinear negative-feedback pulse-modulated devices track a time-varying input signal that slowly modulates the system parameters

Stability analysis of multicell converters using Gerschgorin circles

The multicell converter topology possesses a natural voltage balancing property. This paper uses models obtained previously to perform a stability analysis of multicell converters. It uses a generic model in steady-state- and time constant analyses to determine the stability of multicell converters. In this paper stability refers to balanced cell capacitor voltages, whereas instability refers to unbalanced cell capacitor voltages. The stability analysis is performed for both sinusoidal modulation as well as fixed duty-cycle modulation. The model used in the stability analyses in this paper is valid for any modulation method. The conditions that lead to stability of the cell capacitor voltages as well as that leading to instability are presented. Theoretical results are included to verify the presented stability analyses and properties

ITERATIVE SLICING AS SOLUTION TO THE MODEL PREDIC- TIVE CONTROL PROBLEM OF A THREE-LEVEL INVERTER

Conference ProceedingsThis paper considers a model predictive controller with reference tracking that manipulates the integer switch positions of a power converter. It can be shown that the optimal switch position can be computed in a new coordinate system by solving the closest vector problem in a lattice by iterative slicing. A list of Voronoi relevant vectors defining the basic Voronoi cell of a lattice is used to find the Voronoi cell containing the unconstrained optimum in an iterative manner. This concept is exemplified for a three-level single-phase converter with an RL load

Stability analysis of multicell converters

The multicell converter topology possesses a natural voltage balancing property. This paper uses models obtained previously to perform a stability analysis of multicell converters for the general case of p-cells. It uses a genetic model in steady-state- and time constant analyses to determine the stability of multicell converters. In this paper stability refers to balanced cell capacitor voltages, whereas instability refers to unbalanced cell capacitor voltages. The stability analysis Is performed for both sinusoidal modulation as well as fixed duty-cycle modulation. The model used in the stability analyses in this paper is valid for any modulation method. The conditions that lead to stability of the cell capacitor voltages as well as that leading to instability are presented. Theoretical results are Included to verify the presented stability analyses and properties

Natural balance of multicell converters: The two-cell case

The multicell converter topology is said to possess a natural voltage balancing property. This paper is the first of a two-part series in which multicell converters are modelled for the general case of p-cells. This paper focuses on the development of the natural balancing theory for the two-cell case. An understanding of the two-cell case is fundamental to understanding the general balancing theory. The switching functions used in switching these converters are mathematically analyzed. Equivalent circuits are derived and presented. The switching and balancing properties of these converters are mathematically analyzed. The main conclusion of the analysis is that the natural balancing of these converters are influenced by three factors namely, the harmonic content of the reference waveform, the switching frequency and the load impedance. Mathematical tools are presented that can help designers to predict if balancing problems would occur for a particular set of operating conditions. As a result of the detailed understanding of the balancing mechanism that is gained through this theory it is shown that by adding a balance booster, the load impedance can be manipulated to improve the natural balancing of the converter. Simulation results are included to verify the presented balance theory and properties

Natural balance of multicell converters

The multicell converter topology is said to possess a natural voltage balancing property. This paper models multicell converters for the general case of p-cells. It describes the relationship between the models for different numbers of cells in a generic model for a p-cell multicell converter. The switching functions used in switching these converters are mathematically analyzed. Equivalent circuits are derived and presented. The switching and balancing properties of these converters are mathematically analyzed. Multicell converters were modeled previously for both sinusoidal and fixed duty-cycle references. The model discussed is valid for any modulation method. The conditions for balanced cell capacitor voltages are presented from the aforementioned mathematical synthesis. The balancing property of these converters is also proved mathematically from the presented properties and conditions for natural balancing are discussed. The balancing properties and conditions are formulated and presented as a "natural balance theory for multicell converters". Simulation and experimental results are included to verify the presented balance theory and properties. Conclusions presented include conditions for the theory to be valid as well as possible extensions to the theory and practical implications on converter design