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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 10, OCTOBER 2008 991 Miller Theorem for Weakly Nonlinear Feedback Circuits and Application to CE Amplifier Gaetano Palumbo, Fellow, IEEE, Melita Pennisi, Member, IEEE, and Salvatore Pennisi, Senior Member, IEEE Abstract—The paper presents the derivation of Miller formulas for weakly nonlinear feedback networks. The expressions found are simple and compact and constitute a generalization of the wellknown linear case. As an application example, the formulas are applied to a common-emitter amplifier to straightforwardly derive the closed-form expressions of second- and third-harmonic distortion factors. The results found, validated by Spectre simulations with a VBIC bipolar model, allow to understand in depth the contribution of each nonlinear element. Index Terms—Amplifiers, analog circuits, harmonic distortion, Miller theorem, nonlinear network analysis. Fig. 1. (a) Generic and (b) equivalent network. II. MILLER FORMULAE DERIVATION I. INTRODUCTION HE theoretical evaluation of harmonic distortion in the frequency domain is an important topic recently emerged and taken into consideration by several researchers and designers within the analog circuits and systems community [1]–[13]. To this purpose, the approach traditionally followed is based on the Volterra series [7]–[10] or on its extension to the circuit domain [1]–[3]. However, in the authors’ opinion, these tools often lead to analytical relationships too difficult to be exploited during the design phase. To overcome this problem, other approaches have been proposed such as that in [4]–[6] or the authors’ one, which exploits the phasor notation in the frequency domain [11]–[13]. Although the approach based on phasors demonstrated its strengths in contests such as feedback amplifiers [11] and particularly in two- or three-stage amplifiers [12], [13], a great complication may still arise when nonlinearity is generated also by feedback blocks. As is well known, the Miller theorem and its derivations provide simple and very powerful tools for the analysis of feedback networks [14]–[16]. However, the theorem is demonstrated for linear circuits only, where the superposition criterion holds. In this paper, applying the authors’ methodology based on the phasor notation, the Miller theorem is extended also to weakly nonlinear networks. Simple formulas are analytically derived to account for the nonlinear contribution of nonlinear feedback blocks during the harmonic distortion evaluation. To demonstrate the simplicity of the harmonic distortion analysis by using the Miller theorem extension, we applied it to analyze the harmonic distortion of a bipolar transistor in common-emitter (CE) configuration. T A. General Formulation Let us consider the generic weakly nonlinear network in Fig. 1(a), having a weakly nonlinear feedback element connected between nodes and . In the following analysis, for the nonlinear feedback element, we will consider a nonlinear admittance described by only three Taylor coefficients , , and . The case of a nonlinear feedback impedance will be discussed at the end of the same subsection. The phasor of the current through the element is (1) where the symbol “ ” was defined in [11] and [13] and means that the function at the right must be evaluated at the frequency of the incoming signal on the left. In order to derive an equivalent network with the topology depicted in Fig. 1(b), we have to equate the current (1) to those currents flowing through and as follows: (2) (3) Assuming that voltages on node and the following nonlinear relationship: are related through (4) and using relationships from (1) to (3), we get the three Taylor coefficients of and given as Manuscript received November 27, 2007; revised April 10, 2008. Current version published October 15, 2008. This paper was recommended by Associate Editor P. P. Sotiriadis. The authors are with the Dipartimento di Ingegneria Elettrica, Elettronica, e dei Sistemi (DIEES), University of Catania, 6, I-95125 Catania, Italy (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCSII.2008.2001976 (5a) (5b) (5c) 1549-7747/$25.00 © 2008 IEEE Authorized licensed use limited to: CALIF STATE UNIV NORTHRIDGE. Downloaded on February 17,2010 at 13:52:17 EST from IEEE Xplore. Restrictions apply. 992 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 10, OCTOBER 2008 (6a) (6b) (6c) It is worth noting that, if the nonlinear feedback element is represented by a nonlinear impedance, we can convert it to a nonlinear admittance by using the following relationships derived in Appendix A: (7a) (7b) (7c) which are an extension in the frequency domain of those given in [1]. B. Simplified Relationships in High Voltage-Gain Cases Often, we apply the Miller theorem to circuit elements between two nodes characterized by a high (linear) voltage gain, such as common-emitter or common-source amplifiers. For these typical cases, the effect of nonlinear terms , is much smaller than that of the linear term . Hence, relationships (5) and (6) simplify to (8) (9) for , 2, and 3. It is apparent that (8) and (9) are very simple and are a close extension of the original Miller theorem. III. HARMONIC DISTORTION ANALYSIS OF THE COMMON-EMITTER STAGE To show a valuable application and concurrently validate the analysis carried out in the previous section, we consider the CE configuration [shown in Fig. 2(a)] and derive the analytic expression of second- and third-order harmonic distortion factors of the output voltage. The CE small-signal equivalent nonlinear circuit is shown in Fig. 2(b), where and are the load resistance and the base current, respectively. The model in Fig. 2(b) accounts for both static and frequency-dependent nonlinearities, and, neglecting the term , has the dominant pole given by (10) Fig. 2. Common-emitter schematic diagram. (a) Nonlinear small-signal model and (b) simplified model. The main static nonlinear contribution is due to the nonlinear collector current, modeled by the nonlinear transconductance with the factors , , and , whereas the nonlinear contribution due to the base current (i.e., terms and ) can be neglected. The frequency-dependent contribution to nonlinearity is mainly due to both the exceeding base charges, modeled by a base-emitter nonlinear capacitance, with factors , , and , and the base-collector nonlinear junction effects, with factors , , and . Indeed, even if sometimes high-frequency nonlinearity evaluation was carried out neglecting the base-collector capacitance nonlinearity contribution [7], it has been observed that this simplification is not generally acceptable [8], [9], [17]–[19]. On the other hand, assuming the base-emitter diode to be forward biased, we can neglect the base-emitter junction nonlinearities with respect to the diffusion capacitances contribution. A. Simplified Equivalent Nonlinear Model In order to simplify the nonlinear model in Fig. 2(b) into an equivalent one without the nonlinear feedback element, we can profitably apply the generalized Miller formulae previously derived to the nonlinear base-collector capacitance . Moreover, since the common emitter is a high-gain stage, to simplify the analysis, we can apply the simplified relationships (8) and (9), together with the usually adopted approximation of considering . However, in the obtained equivalent model of Fig. 2(c), we only use relationships (8), which represent the dominant terms, avoiding considering the contribution of to the output node, i.e., disregarding relationships (9). As shown in Fig. 2(c), the equivalent base-collector nonlinear capacitance, , whose nonlinear factors equals , with , 2, 3, and where , the dc amplifier gain, is in parallel with the base-emitter nonlinear capacitance. Thus, their effect on the harmonic distortion can be analyzed combining together the nonlinear factors of the same order. It is worth noting that, although the nonlinear capacitive factors of base-collector capacitance are in general much lower Authorized licensed use limited to: CALIF STATE UNIV NORTHRIDGE. Downloaded on February 17,2010 at 13:52:17 EST from IEEE Xplore. Restrictions apply. PALUMBO et al.: MILLER THEOREM FOR WEAKLY NONLINEAR FEEDBACK CIRCUITS AND APPLICATION TO CE AMPLIFIER 993 than the base-emitter counterparts, since their contribution is , they can dominate nonmagnified by the dc amplifier gain linearity effects, especially in high-gain amplifiers. B. Harmonic Distortion Evaluation Under the weakly distortion condition we can consider only the first three output signal harmonics and we can separately evaluate each nonlinear element contribution. In particular, with angular frequency , given a sinusoidal base current the three main output voltage components are Fig. 3. Magnitude of (15) plus (17) with k equal to 2 and 3. Thus, the output harmonics which depend on the dc gain are (15) (11) where in the third harmonic the terms containing the mutual products between the second nonlinear coefficients of the elements considered have been neglected. This procedure allows to analyze in detail the impact of each nonlinear element on the harmonic distortion terms, given by and their maximum values which are in the two ranges and are (16) Let us consider now the nonlinearity due to only the transconductance . The th-order harmonic at the output is given by (17) (12) where parameter can be 2 or 3 and identifies the second- or the third-order harmonic distortion factor, respectively. If a more accurate estimation of harmonic distortion is needed, we can include second-order effects by following the approach discussed in detail in [11]–[13]. C. Nonlinear Contribution Let us analyze the nonlinearity due to only the base-emitter and base-collector capacitances, inside the dashed box in Fig. 2(c). By inspection of the figure, the input nonlinear admittance is simply given by the sum of the admittance components of the same order where is equal to 2 or 3 to get the second and third harmonic, respectively, and the expression is reported in (14a). It is apparent that nonlinear transconductance for both and gives a contribution that is constant up to frequency and then it starts to decrease. In order to understand the impact of each nonlinear element on harmonic distortion of the common-emitter stage, we have to combine together relationships (15) and (17). In particular, if the dc value of (17) is lower than the maximum value given by (16), which holds when (13a) (18) for equal to 2 or 3, as shown in Fig. 3, the contribution due to the input nonlinearity becomes the dominant one starting from frequencies and , respectively, up to the amplifier dominant pole, being (13b) (19) (13c) It is worth noting that, when the dc gain is low, the effect of nonlinear elements of the input node may be approximately neglected. However, since the input signal is a current and we are interested in evaluating the base-emitter voltage, we need to represent the input element as nonlinear impedance. Hence, by using the relationships in Appendix B and the index equal to 2 or 3 to specify the second and third terms, respectively, we get (14a) (14b) D. Simulations and Discussion To validate the results carried out in the previous subsections, we compared harmonic distortion factors expressed by (12) with simulation results performed on a VBIC bipolar transistor provided by Austria Micro Systems. Small-signal linear and nonlinear parameter extraction has been derived assuming a base current, load resistance, and power supply equal to 10 A, 10 k and 5 V, respectively, and using the main parameter summarized in Table I. Moreover, to Authorized licensed use limited to: CALIF STATE UNIV NORTHRIDGE. Downloaded on February 17,2010 at 13:52:17 EST from IEEE Xplore. Restrictions apply. 994 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 10, OCTOBER 2008 TABLE I KEY SPECTRE PARAMETER VALUES FOR THE VBIC BJT MODEL TABLE II NONLINEAR COMPONENTS Fig. 4. Expected and simulated data for HD and HD . the theoretical curves are in very good agreement with simulated results and has a maximum error of about 1 dB up to approximately 1 GHz. It is worth noting that, even if inaccuracy at high frequency arises at values lower than , they are due to the inconsistent simulator’s results for high-linear high-frequency signals. IV. CONCLUSION evaluate both linear and nonlinear small-signal parameters of the amplifier in Fig. 2(a), we performed a preliminary dc simulation. As is well known, GM, CPI, and CMU are the linear parts of , , and , while the nonlinear components are given by the relationships reported in Table II and developed in [1]. The dc open-loop gain of the amplifier, , is (i.e., 42 dB), and the nonlinear factors are summarized in Table II. Moreover, the pole and the zero of the amplifier are located at frequency 4.6 MHz and 46.5 GHz, respectively, whereas the transition frequency is equal to 4.1 GHz. In Fig. 4, we plotted expected and simulated data of and . Moreover, in order to verify the validity of the analysis, we also applied the modified Miller theorem to a commonemitter driven by a sinusoidal voltage. In this case, the second and third harmonic of the output voltage can be derived by using (13) for the nonlinear components of the input admittance, and they are given by (20) and (21), shown at the bottom of the page, where and are the amplitude and the series resistance of the voltage generator, respectively. Considering mV and , the expected and simulated results are reported in Fig. 4. It is apparent that, for both the cases considered, In this paper, a generalization of the Miller formulas useful for analyzing the nonlinear behavior in the frequency domain of feedback circuits was derived. An approximation of these expressions was used to analyze the harmonic distortion of a common-emitter amplifier. The analysis is straightforward and provides accurate results while allowing an in-depth understanding of the nonlinear contribution due to each nonlinear element. APPENDIX A Consider a nonlinear frequency-dependent impedance whose nonlinear voltage–current relationship is given by (A1) are known, and and are expressed where coefficients up to the third order (due to the weakly nonlinear assumption), respectively, in (A2) (A3) (20) (21) Authorized licensed use limited to: CALIF STATE UNIV NORTHRIDGE. Downloaded on February 17,2010 at 13:52:17 EST from IEEE Xplore. Restrictions apply. PALUMBO et al.: MILLER THEOREM FOR WEAKLY NONLINEAR FEEDBACK CIRCUITS AND APPLICATION TO CE AMPLIFIER Substituting (A3) into (A1) and neglecting higher harmonics, , for , 2, 3 we find the components (A4a) (A4b) 995 It can be approximated to 1 since poles and zeros are close to may be further approximated, resulting each other. Besides, in (14b) with , given in the main text, since the zero inside the last brackets in typical applications falls over the cutoff frequency. (A4c) Impedance may be regarded also as an equivalent nonlinear admittance , whose nonlinear current–voltage relationship is given by (A5) Substituting (A2) into (A5) leads to (A6a) (A6b) (A6c) Replacing of (A6) with expressions found in (A4) yields a nonlinear system for , whose solution is given in (7) in the main text. APPENDIX B In order to derive the nonlinear components of , we exploit relationships (7) for the inverse problem of finding an equivalent nonlinear impedance given a nonlinear admittance. In this case, we obtain (B1a) (B1b) (B1c) The expressions in (14a) and (14b) for are straightforwardly found by substituting (13a) and (13b) into (B1a) and (B1b), respectively. Substituting (13c) into (B1c), we get (B2) where the function is given by (B3) REFERENCES [1] P. Wambacq and W. 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