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Convention Paper 7997 Presented at the 128Th Convention 2010 May 22–25 London, UK

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Convention Paper 7997 Presented at the 128Th Convention 2010 May 22–25 London, UK Audio Engineering Society Convention Paper 7997 Presented at the 128th Convention 2010 May 22–25 London, UK The papers at this Convention have been selected on the basis of a submitted abstract and extended precis that have been peer reviewed by at least two qualified anonymous reviewers. This convention paper has been reproduced from the author’s advance manuscript, without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending request and remittance to Audio nd Engineering Society, 60 East 42 Street, New York, New York 10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society. Modeling Distortion Effects in Class-D 8094 Amplifier Filter Inductors Arnold Knott1, Tore Stegenborg-Andersen 1, Ole C. Thomsen 1, Dominik Bortis 2, Johann W. Kolar 2 Gerhard Pfaffinger3 and Michael A.E. Andersen1 1Technical University of Denmark, 2800 Kgs. Lyngby, Denmark 2ETH Zurich / Power Electronics System Laboratory, CH-8092 Zurich, Switzerland 3Harman/Becker Automotive Systems GmbH, 94315 Straubing, Germany Correspondence should be addressed to Arnold Knott ([email protected]) ABSTRACT Distortion is generally accepted as a quantifier to judge the quality of audio power amplifiers. In switch- mode power amplifiers various mechanisms influence this performance measure. After giving an overview of those, this paper focuses on the particular effect of the nonlinearity of the output filter components on the audio performance. While the physical reasons for both, the capacitor and the inductor induced distortion are given, the practical in depth demonstration is done for the inductor only. This includes measuring the inductors performance, modeling through fitting and resulting into simulation models. The fitted models achieve distortion values between 0.03 % and 0.2 % as a basis to enable the design of a 200 W amplifier. 1. INTRODUCTION it makes sense to break the origin of the distortion Even other measures, like intermodulation distor- mechanisms down into the various parts of an am- tion are more in depth measures, a distortion figure plifier. For linear audio power amplifiers this was is the fundamental starting point to distinguish am- done in [2]. For the more efficient switch-mode au- plifiers. It reveals the noise level of the amplifier dio power amplifiers a number of publications cov- and provides the basics for more advanced tests as ered these mechanisms. The different stages of these described in [1]. As the desired figures necessitate amplifiers can be broken down according to the block the precision of signal levels to be in the µV range, diagram in figure 1. Knott et al. Modeling Distortion Effects in Class-D Amplifier Filter Inductors The blocks and their distortion sources are: Fig. 2 Tradeoffs in output filter design. audio Fig. 1 Block diagram of a switch-mode audio power performance amplifier A B C D E F electromagnetic energy A. Input and Control The distortion is dom- compatibility losses inated in the input stage and regulator by either the used operational amplifier for Fig. 3 Circuit of an output filter for an audio power time-continuous inputs and specified in their amplifier. datasheets or, for time-discrete inputs, by the L sampling process [3, 4, 5] or clock induced noise level [6]. Vps C R B. Modulator The modulator induced distortion is mainly based on linearity of the carrier [7]. C. Level Shifter The impact of the level shifter on the audio performance has not been researched up to now and leaves room for further research. power stage and Vout the output voltage of the am- plifier, which is applied across the speaker terminals. D. Power Stage The power stages influence on the audio performance has been described in [8]. Equation 2.1 Transfer function of filter in figure 3. E. Output Filter This paper is dealing with the V 1 H (s) = out = influence of the output filters properties on lin- L 2 V ps 1 + s R + s LC earity of the amplifier. F. Loudspeaker The transducers influence on au- dio performance has been described in [9] and This paper is specifically investigating the nature broken down into single mechanisms in [10] and impact of the output filter on the audio per- formance. Therefore both filter components, the ca- pacitors and the inductors physical properties are 2. OUTPUT FILTER investigated in this section. The output filter is required to suppress the energy, 2.1. Capacitor which is used to operate the output stage in an effi- Capacitance C is defined as stored charge q per cient mode. The frequency of this energy is beyond voltage V 2.2. the audible frequency range [11] and generally caus- ing trouble in electromagnetic compatibility. The Equation 2.2 Definition of capacitance. insertion of the filter is solving those, however gen- q erating audible effects and losses, which leads to the C = V tradeoffs visualized in figure 2. A simplified circuit diagram of the output filter is shown in figure 3 and its transfer function is given in 2.1 where Vps denotes the output voltage of the Applying Gauss Law to the charge allows itemiza- AES 128th Convention, London, UK, 2010 May 22–25 Page 2 of 9 Knott et al. Modeling Distortion Effects in Class-D Amplifier Filter Inductors tion into electrical field E~ with vacuum permittivity This reveals the Pockels effect to be responsible for ~ ε0 and displacement vector P via charge density ̺ second order nonlinearities and the Kerr Effect to be and electric displacement D~ 2.3. the reason for third order effects. For ferroelectric materials, the description of non- Equation 2.3 Gauss Law. linearity is getting somewhat more complicated, as the displacement vector has a hysteretic dependency ~ ~ ~ on the electrical field. Theses hysteretic curves have q = ̺δVol = D δ~s = ε0E + P δ~s ZZZ I I been shown quantitativelty in [13]. A A 2.2. Inductor The equivalent physical derivation of the reasons for The denominator of 2.2 can be expressed by Far- the nonlinearity of the inductor start with the defi- radys law 2.4. nition of inductance 2.7 in dependency on magnetic flux Φ and electrical current I. Equation 2.4 Farradays Law. Equation 2.7 Definition of inductance. V = E~ δ~s I φ s L = I Through both of those physical principles, the defi- nition of capacitance can be rewritten as in 2.5 Through Gauss Law of Magnetism Φ is expressed Equation 2.5 Definition of capacitance taking in 2.8 as a function of magnetic field H~ and the Gauss and Farradays Law into account. magnetization M~ with the aid of the permeability ~ ~ in vacuum via the magnetic flux density B~ . ε0E δ~s P δ~s C = AH + AH ~ ~ E δ~s E δ~s Equation 2.8 Gauss Law of Magnetism. Hs Hs ~ ~ ~ ~ ~ linear part polarization Φ = B δA = µ0 H + M δA | {z } | {z } I I dependent S S The nonlinearity of the capacitor is therefore origi- nated in the polarization defined through the electric Through Amperes Circuit Law the current is ex- susceptibility χ 2.6 for anisotropic dielectric materi- pressed as a function of the magnetic field H~ as in als [12]. 2.9. Equation 2.6 Linear and nonlinear parts of dis- placement vector. Equation 2.9 Amperes Circuit Law. I = H~ δ~l ~ P (1) (2) (3) I = χ E~ + χ E~ E~ + χ E~ E~ E~ C ǫ ij j ijk j k ijkl j k l 0 Xj Xjk Xjkl linear Pockels Kerr Effect |suscepti-{z } | Effect{z } | {z } bility Taking both of those two laws into account, the def- inition of the inductance is rewritten in 2.10. AES 128th Convention, London, UK, 2010 May 22–25 Page 3 of 9 Knott et al. Modeling Distortion Effects in Class-D Amplifier Filter Inductors Equation 2.10 Definition of inductance taking 3. MODELING Gauss and Amperes Law into account. Section 2 showed the duality between capacitor and ~ ~ ~ ~ inductor in theory. Consequently the rest of the pa- µ0H δA µ0M δA per is dealing with one of them only, without loosing L = SH + SH H~ δ~l H~ δ~l generality for the other one. It is the inductor, which CH CH is generally dominating size constraints, electromag- netic compatibility challenges and showing the most linear part polarization interesting saturation effects. Therefore the induc- | {z } dependent| {z } tors linearity is pursued furtheron in this paper. Through the desired power rating of an amplifier and neglect of the ripple current, the current rat- The polarization dependent part M~ is not following ing of the filter inductor is given. For the following the BH-curve, which has been numerically fitted analysis of inductors, a power rating of 200 W into in [14], but rather the Rayleigh Loop [15] which a 4 Ω transducer is arbitrarily chosen. This leads has been extended to symmetry of the loop as to a peak current of 10 A through the filter induc- only limitation, by [16] as described in [17]. While tor. Only few technical documentations of suitable the current I is linear dependent on the magnetic components give the dependency of inductance on field H~ , its relation to the magnetic polarization the current flowing through the windings like in [18]. M~ contains higher order terms. This nonlinear Therefore a measurement setup is described follow- dependency of magnetization on the magnetic field ing, allowing to derive this curve. is covered by the high order terms in 2.11 by the named references.
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