IEEE TRANSACTIONS ON CIRCUITS ANDSYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 5, MAY 2006 961 Distortion Analysis of Miller-Compensated Three-Stage Ampliﬁers Salvatore Omar Cannizzaro, Gaetano Palumbo, Senior Member, IEEE, and Salvatore Pennisi, Senior Member, IEEE Abstract—This paper proposes a theoretical approach for evalu- ating distortion in the frequency domain of three-stage ampliﬁers adopting two commonly used compensation techniques, namely the nested Miller (NM) and the reversed NM. The analysis is based on appropriate ampliﬁer modeling and on the assumption that the nonlinearity generated by each stage is static. Calculations are thus greatly simpliﬁed avoiding complex methods based on the Volterra series. Only dominant contributions need to be taken into account, thereby highlighting those mechanisms generating distortion and their features in the frequency domain. Moreover, the adopted ap- proach provides useful design guidelines and explains why the NM compensation technique allows generally better linearity perfor- mance at low frequency and why the reversed NM is best suited to high frequencies. Simulation results with Spectre on two tran- sistor-level CMOS circuits are also provided and found to be in very good agreement with expected results. Index Terms—Feedback circuits, harmonic distortion, Miller compensation, multistage ampliﬁers. I. INTRODUCTION C ONSTANT reduction of supply voltages has meant that cascoding techniques are no longer suitable for guaran- teeing the gain and voltage swing performance of operational ampliﬁers. Therefore, they have been progressively replaced by alternative design techniques exploiting the cascade of simpler gain stages. Moreover, there are applications where the gain of two-stage ampliﬁers is insufﬁcient and using multistage ar- chitectures becomes mandatory [1]–[5]. Unfortunately, multi- stage ampliﬁers exhibit multiple low-frequency poles in their frequency response, which complicates frequency compensa- tion. This explains why the main research focus has been dedi- cated to developing optimized compensation techniques to max- imize frequency performance [6]–[11]. The best known of these are the nested Miller (NM) and reversed NM (RNM) techniques. The former is used when only the last ampliﬁer’s stage is voltage inverting, whereas the latter is used when the second stage is the only inverting one and provides larger bandwidths. 1 [7] These techniques have been exploited in several low-voltage high-gain ampliﬁers implemented both in bipolar and CMOS technology. Of course, the main reason underlying the use of such high-gain Manuscript received February 14, 2005; revised June 23, 2005. This paper was recommended by Associate Editor I. F. Filanovsky. The authors are with the Dipartimento di Ingegneria Elettrica Elettronica e dei Sistemi (DIEES), University of Catania, I-95125 Catania, Italy(e-mail: scanni@diees.unict.it; gpalumbo@diees.unict.it; spennisi@diees.unict.it). Digital Object Identiﬁer 10.1109/TCSI.2005.862287 1 In the NM approach both compensation capacitors have one terminal con- nected to the output and, in turn, to the load capacitor, whereas in the RNM approach one compensation capacitor has both terminals connected to (low ca- pacitive) internal nodes. architectures is stringent requirements in terms of closed-loop accuracy and linearity. For this reason, a systematic theoretical procedure capable of predicting the latest closed-loop linearity performance with reasonable precision is a key issue, especially if the equations obtained are simple enough for use during syn- thesis even in a preliminary pencil-and-paper design. Distortion in the frequency domain due to weakly nonlinear sources is usually analyzed with methods based on the Volterra series [12]–[19]. These are general in the sense that they can be applied to circuits whose nonlinear elements are either static (re- sistors and controlled sources) or frequency dependent (capac- itors and inductors). Unfortunately, despite attempts to better interrelate these approaches at the circuit and/or system level [17]–[19], such methods still remain undoubtedly complex and difﬁcult to apply, particularly when the circuits have a large number of nonlinear elements as in the case of multistage am- pliﬁers. In the authors’ view, high-frequency distortion of feedback ampliﬁers is one of the signiﬁcant cases that can be studied without referring to the Volterra series. In its place, a simpler method, similar to one valid at low frequencies and based on power series analysis, can be proﬁtably adopted to obtain sufﬁciently accurate results. In fact, to guarantee closed-loop stability, feedback ampliﬁers must be frequency compensated through a suitable network that produces a dominant-pole frequency behavior [20], [21]. Up to the ﬁrst order, we can ne- glect the effect of the (nonlinear) parasitic capacitors included in each gain stage and formulate the fundamental hypothesis that nonlinearity is generated by the static (or memoryless) subsection of each stage, exclusively modeled by resistors and/or controlled sources. We can then evaluate the effects on the closed-loop performance of the compensation capacitors after treating them as if they were perfectly linear. For this purpose, it is well known that compensation capacitors made up customarily of two polysilicon layers are more linear than junction capacitors. This is important if closed-loop linearity is needed, because it is also well known that in a feedback system we cannot obtain a closed-loop linearity performance which is better than that of the feedback network itself [21], [22]. 2 The above considerations lay the basis for an approach al- ready adopted by the authors to analytically characterize distor- tion in the frequency domain of both single and two-stage am- pliﬁers [21], [23], and in an approximated way, of three-stage ampliﬁers adopting NM compensation [24]. 2 In this case, we are not referring to the overall feedback but to the local feedback provided by the compensation capacitors, as will become clear in the following. 1057-7122/$20.00 © 2006 IEEE