518 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 57, NO. 2, FEBRUARY 2010 IMD of Closed-Loop Filterless Class D Amplifiers Wei Shu and Joseph S. Chang, Member, IEEE Abstract—Audio amplifiers, including the contemporary Class D amplifiers (CDAs), are typically qualified by their several nonlin- earities, including total harmonic distortion and intermodulation distortion (IMD). In the case of filterless CDAs, IMD remains largely unexplored, and the mechanism thereof is largely un- known. This paper presents an analytical modeling of IMD for the prevalent first- and second-order filterless CDAs and shows that the dominant mechanisms of the former is phase error while of the latter is duty-cycle error. By means of multidimensional Fourier series analysis, analytical expressions for the IMD components and thereafter, the IMD expression for these CDAs, are derived. The derived expressions depict that the IMDs of these CDAs are significant, and the IMD of the first-order filterless CDA, in spite of its lower noise-suppression attribute, is somewhat unexpectedly superior to the second-order filterless CDA. Furthermore, the derived expressions delineate the parameters that affect IMD and are insightful to designers to optimize/vary pertinent parameters to reduce IMD. The derived IMD expressions are verified against HSPICE simulations and on the basis of measurements on a prototype CDA IC and other CDAs realized discretely. Index Terms—Class D amplifier (CDA), duty-cycle error, inter- modulation distortion (IMD), phase error. I. INTRODUCTION R ECENT ANALOG and digital [1], [2] Class D ampli- fiers (CDAs) embodying advanced semiconductor tech- nology (such as DMOS and high-voltage CMOS fabrication processes [3]) and innovative circuit design techniques [4] have yielded designs that are routinely accepted for mainstream con- sumer electronics, including cellular phones, LCD televisions, etc. The primary worthy attribute of CDAs over their classical linear counterparts, such as Class A and Class AB amplifiers, is their substantially higher power efficiency, and this is largely due to the switch-mode operation in the output stage of CDAs [5]. In a conventional CDA [6], an filter is typically em- ployed between the CDA output stage and the load to atten- uate the high-frequency carrier, thereby recovering the audio signal. A more recent CDA architecture, the “filterless” CDA [7], has gained some acceptance in applications where cost and form factor are important considerations. That is because filter- less CDAs do not require the aforementioned filter which is costly and bulky, and the savings are typically 30% of the cost and 75% of the printed-circuit-board area [8]. Furthermore, because of the absence of the filter, the nonlinearities thereof arising from the iron/ferrite core inductor and capacitor are Manuscript received March 17, 2009. First published July 14, 2009; current version published February 10, 2010. This paper was recommended by Asso- ciate Editor P. K. T. Mok. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: shuw0001@ntu.edu.sg). Digital Object Identifier 10.1109/TCSI.2009.2027801 hence nonexistent, unlike those in conventional CDAs. In short, the filterless CDA lends itself to lower cost and, potentially, to higher fidelity audio applications than conventional CDAs. The parameters that qualify the fidelity of an amplifier are its nonlinearities, including power-supply noise (and power-supply-induced intermodulation distortion for CDAs [9]), total harmonic distortion (THD), and intermodulation dis- tortion (IMD). In the case of conventional closed-loop CDAs, the mechanisms of power-supply noise [9], THD [10], [11], and IMD [8], and their pertinent parameters (and associated trade- offs) have been reported in literature. In the case of closed-loop filterless CDAs, the power-supply noise [9] and THD [10], [11] have also been reported—but IMD remains unreported. IMD is a distortion measure of an input signal comprising two or more frequency components, defined by the Society of Motion Picture and Television Engineers (SMPTE) [12] as (1) where and the output voltages are at input frequencies and . Compared with THD, IMD is sometimes inappropriately considered a secondary measure of distortion in CDAs. Unlike linear amplifiers where IMD is typically specified, IMD is often, somewhat unexpectedly, neglected in commercial CDAs. This contrasts with what is generally practiced within the high-fidelity audio community, where IMD is often consid- ered an equally pertinent parameter to THD. This is because audiophiles often assert that IMD has a better correlation with the audible quality than THD, in part due to the absence of a harmonic relationship between the IMD components and the input signals (as opposed to THD). In other words, IMD tends to be more audible and aurally unpleasant. The IMD in commercial CDAs based on the conventional CDA architecture is typically poor, for example, 1% [8], and as delineated earlier, this is largely attributed to the nonlinear- ities of the filter; for completeness, note that the IMD in- vestigated in this paper is also applicable to conventional CDAs (although the IMD due to the nonlinear low-pass filter will dominate the overall IMD). Although it is generally thought that, due to the absence of the filter, the IMD of the filter- less CDA may be negligible, we will later show that it remains a significant nonlinearity and comparable with THD—for ex- ample, the IMD of a typical closed-loop filterless CDA is usu- ally 0.1% (compared with 0.01% IMD in classical linear amplifiers). Arguably, the poor IMD performance is one of the obstacles of the filterless CDAs against their general acceptance in high-fidelity audio applications. Simply put, in view of the 1549-8328/$26.00 © 2010 IEEE