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Analysis of a hysteresis-controlled self-oscillating class-D amplifier

Cox, Stephen M.; Yu, Jun; Goh, Wang Ling; Tan, Meng Tong


Stephen M. Cox

Jun Yu

Wang Ling Goh

Meng Tong Tan


This paper gives the first systematic perturbation analysis of the audio distortion and mean switching period for a self-oscillating class-D amplifier. Explicit expressions are given for all the principal components of audio distortion, for a general audio input signal; the specific example of a sinusoidal input is also discussed in some detail, yielding an explicit closed-form expression for the total harmonic distortion (THD). A class-D amplifier works by converting a low-frequency audio input signal to a high-frequency train of rectangular pulses, whose widths are slowly modulated according to the audio signal. The audiofrequency components of the pulse-train are designed to agree with those of the audio signal. In many varieties of class-D amplifier, the pulse-train is generated using a carrier wave of fixed frequency, well above the audio range. In other varieties, as here, there is no such fixed-frequency clock, and the local frequency of the pulse-train varies in response to the audio input. Such self-oscillating designs pose a particular challenge for comprehensive mathematical modelling; we show that in order to properly account for the local frequency variations, a warped-time transformation is necessary. The systematic nature of our calculation means it can potentially be applied to a range of other self-oscillating topologies. Our results for a general input allow ready calculation of distortion diagnostics such as the intermodulation distortion (IMD), which prior analyses, based on sinusoidal input, cannot provide.


Cox, S. M., Yu, J., Goh, W. L., & Tan, M. T. (in press). Analysis of a hysteresis-controlled self-oscillating class-D amplifier. IMA Journal of Applied Mathematics, 82(2),

Journal Article Type Article
Acceptance Date Oct 17, 2016
Online Publication Date Dec 26, 2016
Deposit Date Nov 18, 2016
Publicly Available Date Dec 26, 2016
Journal IMA Journal of Applied Mathematics
Print ISSN 0272-4960
Electronic ISSN 1464-3634
Publisher Oxford University Press
Peer Reviewed Peer Reviewed
Volume 82
Issue 2
Keywords Class-D amplifier, Mathematical model, Astable integrating modulator, Hysteresis-controlled
Self-oscillating modulator, Hysteresis comparator
Public URL
Publisher URL
Additional Information This is a pre-copyedited, author-produced version of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The version of record Stephen M. Cox, Jun Yu, Wang Ling Goh, and Meng Tong Tan
Analysis of a hysteresis-controlled self-oscillating class-D amplifier IMA J Appl Math first published online December 26, 2016 doi:10.1093/imamat/hxw053 is available online at:


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