e distortion reveal considerable information
on its frequency and bias dependence. Points on the
^ versus
collector current curve yielding optimum distortion performance
are identified and interpreted in terms of current cancellation.
Both second- and third-order distortion are considered, and the
results are validated by both simulation and experiment.
Index
TermsCurrent
cancellation,
harmonic
distortion,
heterojunction
and
homojunction
bipolar
transistors,
high-frequency distortion, intermodulation distortion, linearity,
nonlinear distortion, unitycurrentgain frequency, Volterra
series.
I. I
NTRODUCTION
O
VER THE years, the problem of high-frequency distor-
tion in bipolar transistors has been extensively studied.
Among the earlier works, Narayanan [1], [2] was the first to
present a detailed examination of distortion using Volterra
series; Chisholm and Nagel [3] and Kuo [4] focused on com-
puter algorithms for calculating distortion in transistor circuits;
Poon and Narayanan [5][7] combined Volterra series with a
charge-control approach; Abraham and Meyer [8] employed
a simplified transistor model to suggest design guidelines for
low distortion; and many others contributed to the literature, as
cited by these authors, and in a comprehensive review paper
by Lotsch [9]. More recently, Maas et al. [10] attributed the
surprisingly good linearity of heterojunction bipolar transis-
tors (HBTs) working at high frequencies to a cancellation
of nonlinear currents arising from the dynamic resistance
and capacitance of the emitterbase junction. This analysis
was followed by a number of studies that were mostly of an
empirical nature, leading to various observations on the factors
Manuscript received June 26, 2002. This work was supported in part by the
Natural Sciences and Engineering Research Council of Canada, by IBM under
a University Partnership Program grant, by Agilent Technologies, by the Center
for Wireless Communications at the University of California at San Diego, and
by the Army Research Office under the Multiuniversity Research Initiative Pro-
gram Digital Communications Based on Nonlinear Dynamics and Chaos.
M. Vaidyanathan, M. Iwamoto, L. E. Larson, and P. M. Asbeck are with the
Center for Wireless Communications, Department of Electrical and Computer
Engineering, University of California at San Diego, La Jolla, CA 92093-0407
USA.
P. S. Gudem was with the West Coast Design Center of Excellence, IBM T. J.
Watson Research Center, Encinitas, CA 92024 USA. He is now with Qualcomm
Inc., San Diego, CA 92121 USA.
Digital Object Identifier 10.1109/TMTT.2002.807821
affecting high-frequency distortion; for example, comments
were made on the role of current cancellation [11][14], the
feedback effect of the parasitic base and emitter resistances
[15], [16], the impact of the nonlinear, collectorbase depletion
capacitance [14][19], the importance of basecollector transit
time [16], [19], and the choice of bias voltage and current [14],
[16][20]. Studies have also been undertaken to examine the
distortion behavior of other important microwave devices; for
example, Pedro et al. [21][23] recently examined distortion
in metalsemiconductor field-effect transistors (MESFETs).
Despite all of these investigations, a general description of
high-frequency distortion, which offers good physical insight
and can be applied to a wide variety of devices, is still lacking.
This shortcoming stems mainly from the fact that nonlinear
distortion is an involved problem, which is not amenable to
easy solution; even when expressions for transistor distortion
can be found, they are typically very complex, and involve
numerous terms that offer little intuition.
In this study, we develop a basic theory of high-frequency
distortion in bipolar transistors by employing the charge-control
approach suggested by Poon and Narayanan [5][7]. Use of the
charge-control approach alleviates much of the usual difficulty
in analyzing distortion, and leads to powerful expressions that
relate the distortion generated by the transistor to its transcon-
ductance and unitycurrentgain frequency, and to the deriva-
tives of these quantities with respect to baseemitter voltage and
collector current, respectively. In particular, the connections be-
tween the distortion and unitycurrentgain frequency provide
substantial information on the frequency and bias dependence of
the distortion characteristics, and offer new insight into the can-
cellation phenomenon described by Maas et al. [10]. The result,
which we validate using both simulation and experiment, is a
useful step toward a general theory of distortion.
In Section II, the transistor model used for the analysis is pre-
sented, and the equations needed to combine the charge-control
approach with Volterra series are formulated. In Section III,
expressions are found for the second-order distortion charac-
teristics, and their predictions are compared with simulation
and experiment. In Section IV, expressions are derived for
the third-order intermodulation distortion and then applied to
practical devices. Section V summarizes the conclusions.
II. A
NALYTIC
A
PPROACH
A. Model
Fig. 1 shows the transistor model used in the analysis. The
elements and their assumed functional dependencies on the total
0018-9480/03$17.00 © 2003 IEEE VAIDYANATHAN et al.: THEORY OF HIGH-FREQUENCY DISTORTION IN BIPOLAR TRANSISTORS
449
Fig. 1.
Large-signal transistor model used in the analysis. The definitions of
the elements are given in the text. Nonlinear elements are marked in the usual
fashion.
(static plus small-signal) values of the internal baseemitter and
basecollector voltages,
and
, are as follows.
1)
is the constant supply voltage.
2)
is the total source voltage.
3)
is the sum of the external source and device base
resistances.
4)
is the sum of the external load and device collector
resistances.
5)
is the sum of any external emitter and device emitter
resistances.
6)
is the charge associated with the emitterbase
depletion capacitance.
7)
is the emitter portion of the stored free charge.
8)
is the total collector charge, which in-
cludes both the collector portion of the stored free
charge and the charge associated with the collectorbase
depletion capacitance.
9)
is the quasi-static collector current.
10)
is the quasi-static base current.
The circuit in Fig. 1 represents the simplest adequate model
with which to examine the problem. Important features of this
model are as follows.
Effects arising from the falloff of the transistors
unitycurrentgain frequency at high currents are au-
tomatically included, since the charges
and
can have an arbitrary
(or
) and
dependence.
The terminations
,
, and
are assumed to be
purely resistive. This assumption not only simplifies the
analysis, but also leads to useful results that connect
(through charge-control relations) the distortion gener-
ated by the transistor to its unitycurrentgain frequency,
the latter being defined under conditions in which
,
, and
take on resistive values determined solely
by the device parasitics. In addition, characterizing
distortion with resistive terminations offers advantages
with respect to evaluating the capabilities of a technology
under standardized conditions [20, p. 1530]. From these
perspectives, the distortion expressions derived in this
study can be viewed as figures-of-merit for device lin-
earity. We have also found these expressions to be useful
in the design of real broad-band power amplifiers [24].
However, in specific circuit applications, it should be
noted that the distortion performance will depend on the
exact nature of the terminations [25], [26].
A number of other simplifications in the model are nec-
essary to keep the analysis manageable, including the
neglect of the Early effect, the neglect of avalanche break-
down, the neglect of self-heating [27], the assumption
of a linear base resistance (which is lumped into
),
and the neglect of collectorsubstrate capacitance, which
is present in Si-based devices. However, each of these
should have only secondary impacts, as illustrated, for
example, by the work in [20], where numerical results
from a model [20, Fig. 1(b)] that is very similar to the
one used here were shown to be in good agreement [20,
Figs. 47] with those from a much more involved model
[20, Fig. 1(a)] as well as experiment.
B. Formulation
Using Kirchoffs laws, it is possible to write the following
circuit equations:
(1)
(2)
where
and
are effective
load and source resistances, respectively, and where the terminal
currents are
(3)
and
(4)
with
being the total charge in the transistor, given by
(5)
If the current gain is high and the operating frequency is re-
stricted to a few times below the unitycurrentgain frequency,
then combining (1)(4) and retaining only the most important
terms, it is easy to obtain the following simplified set of circuit
equations:
(6)
(7)
Since
in Fig. 1 is a known function of
, the voltage
can be eliminated between (5) and (6), and the small-signal
baseemitter voltage
, and the small-signal collector current
, can each be expanded as a Taylor series in the small-signal
charge
:
(8)
(9)
As shown in Appendix I, the series coefficients in (8) and (9)
can be expressed in terms of the transconductance
and its 450
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 51, NO. 2, FEBRUARY 2003
derivatives (
and
) with respect to the baseemitter voltage
, and in terms of the loaded unitycurrentgain frequency,
denoted
, and its derivatives (
and
) with respect to
.
The definition of
, given by (56) in Appendix I, differs
from the usual un