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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
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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
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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
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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)
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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)
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