854
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 3, JUNE 2004
Experimental Characterization of Operational
Amplifiers: A System Identification Approach—
Part I: Theory and Simulations
Rik Pintelon, Fellow, IEEE, Gerd Vandersteen, Member, IEEE, Ludwig De Locht, Yves Rolain, Senior Member, IEEE,
and Johan Schoukens, Fellow, IEEE
Abstract—Using specially designed broadband periodic random
excitation signals, the open loop, the common mode, and the power
supply gains of operational amplifiers are measured and modeled.
The proposed modeling technique 1) takes into account the measurement uncertainty and the nonlinear distortions, 2) gives information about possible unmodeled dynamics, 3) detects, quantifies,
and classifies the nonlinear distortions, and 4) provides opamp parameters (time constants, gain-bandwidth product, etc.) with confidence bounds. The approach is suitable for the experimental characterization of operational amplifiers (see [23]) as well as the fast
evaluation of new operational amplifiers designs using network
simulators (see Part I). Part I describes the modeling approach and
illustrates the theory on simulations.
Index Terms—Common mode rejection, linear characteristics,
nonlinear distortions, open loop gain, operational amplifier, power
supply rejection, system identification.
I. INTRODUCTION
A
BUNDANT literature exists on the measurement and/or
simulation of the frequency-dependent operational amplifiers (opamp) characteristics such as the open loop gain
([1]–[6]), the common mode gain
or common mode
([1], [7]–[9]), and
rejection ratio
the power supply gains
,
or power supply re,
jection ratios
([10], [11]). In all cases, single-sine measurements and/or simulations are performed to obtain the operational amplifier characteristics. Except for in [1], the calculation
of the opamp parameters (poles, gain-bandwidth product, etc.)
is based on a fixed first- or second-order model. The proposed
estimation procedures 1) do not take into account the measurement uncertainty, 2) give no information about the possible unmodeled dynamics [a first- or second-order model may be in,
,
or
] and
sufficient to explain
Manuscript received June 24, 2003; revised January 6, 2004. This work was
supported in part by the Fund for Scientific Research (FWO-Vlaanderen), the
Flemish Government (GOA-IMMI), and the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office,
Science Policy programming (IUAP 5).
R. Pintelon, L. De Locht, Y. Rolain, and J. Schoukens are with the Electrical
Measurement Department (ELEC), Vrije Universiteit Brussel, 1050 Brussels,
Belgium (e-mail: Rik.Pintelon@vub.ac.be).
G. Vandersteen and L. De Locht are with the IMEC, dev. DESICS, 3001 Heverlee, Belgium.
Digital Object Identifier 10.1109/TIM.2004.827094
the nonlinear distortions (an operational amplifier behaves nonlinearly for “sufficiently large” inputs), and 3) provide no confidence bounds on the estimated opamp parameters.
In this series of two papers, we present a system identification approach to the experimental characterization of operational amplifiers. The proposed method uses the matched resistor circuit shown in Fig. 1 in combination with specially designed broadband periodic random excitations signals (multi, the common
sines [12]) for measuring the open loop gain
, and the power supply gains
and
mode gain
. Following the measurement procedure of [13], [14],
the periodic random excitation signal allows us to gather, in
one single measurement, 1) the frequency response functions
(FRFs) in a broad frequency band, 2) the noise levels, and 3)
the class (even and/or odd degree) and the level of the nonlinear
distortions in the output spectrum. Averaging of the FRF measurements over different realizations of the periodic random excitation gives the variance information needed in the parameter
estimation procedure. Next, a parametric model is identified (estimation of the model parameters and model order selection),
resulting in a best linear approximation of the characteristic together with its uncertainty (due to measurement noise and/or
nonlinear distortions). Note that no prior first- or second-order
model is imposed in the procedure.
Before going to silicon, the performance of new operational
amplifier designs is verified via simulations. The number of
transistors in such circuits is usually so large that the calculation time of network simulators is a non-negligible part of the
design loop. Hence, there is a need for a fast evaluation of the
linear and the nonlinear behavior of operational amplifiers. In
Part I, a method is proposed that allows us to evaluate simultaneously the linear opamp characteristics, the level of the nonlinear distortions, and the type of nonlinearity (even and/or odd
degree distortions).
Since the measurement of the common mode rejection ratio
is (very) sensitive to the resistor mismatch
in Fig. 1, a simple calibration procedure is proposed in [23] to
eliminate this error. It consists of making a bridge measurement
with the four resistances when the opamp is removed. Note that
the calibration procedure also accounts for the nonzero output
impedance of the voltage buffer in the feedback loop, and for
the possible frequency-dependent behavior of the resistances
(megahertz range).
0018-9456/04$20.00 © 2004 IEEE
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PINTELON et al.: SYSTEM IDENTIFICATION APPROACH—PART I: THEORY AND SIMULATIONS
855
in the feedback loop is an ideal voltage buffer (infinite input
and that
impedance, zero output impedance) with gain
. Proceeding
the resistors are not matched
in this way,
also accounts for the nonzero output impedance
of the buffer.
A. Open Loop Gain
,
The switch in Fig. 1 is in position 1,
V, and
,
are measured. Using
, it follows from (1) that
assuming that
Fig. 1. Basic scheme for measuring/simulating the characteristics of an
R and R R ).
operational amplifier (R
=
=
The contributions of this series of two papers are as follows:
1) (simultaneous) measurement of the opamp characteristics, the noise levels, and the levels of the nonlinear distortions using specially designed multisines;
2) classification of the nonlinear distortions of the opamp in
odd and even degree contributions;
3) parametric identification of the opamp characteristics,
taking into account the disturbing noise and the nonlinear
distortions;
4) calculation of opamp parameters (time constants, gainbandwidth product, etc.) with uncertainty bounds;
5) a simple calibration procedure to eliminate the resistor
mismatch in the common mode rejection ratio measurement;
6) fast evaluation of the performance of a new opamp design
using a network simulator in combination with specially
designed multisines.
Part I elaborates the measurement/identification procedure
and illustrates the theory on a network simulation of an LM741
operational amplifier. [23] discusses the calibration of the experimental setup and illustrates the whole identification approach
on real measurements of two UA741s.
V,
, and
(2)
B. Common Mode Rejection Ratio
,
The switch in Fig. 1 is in position 2,
V, and
,
are measured. Using
with
V,
(3)
it follows from (1) that
with
and
(4)
(see [1]). Hence, the common mode rejection ratio (CMRR)
equals
II. BASIC MEASUREMENT SETUP
In this section, we study the basic matched resistor circuit
, the common mode
used for measuring the open loop
, and the power supply
,
gains (see
Fig. 1). The circuit consists of four resistances, a voltage buffer,
and the device under test (DUT). The voltage buffer in series
(ideally, infinite input impedance, zero
with the resistor
output impedance) suppresses the effect of the nonzero output
impedance of the operational amplifier on the measurements.
The analysis of the circuit starts from the basic open loop
equation of the unloaded DUT
(5)
C. Power Supply Rejection Ratio
The switch in Fig. 1 is in position 2,
,
, either
V
and
V, or
V and
V
, and
and
or
are measured.
and
, and assuming that
Using
, it follows from (1) that
with
(1)
and
the voltages of the “ ” and “ ” inputs of
with
and
the varying signals at the positive
the DUT, and
and negative
power supply terminals. The calculations made in the sequel of this section assume that the buffer
(6)
where superscript means that the signal source is connected to
or negative
power supply terminal.
either the positive
Hence, the power supply rejection ratios (PSRR) equal
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(7)
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 3, JUNE 2004
III. MEASUREMENT PROCEDURE
A. Introduction
The operational amplifier characteristics (2)–(7) are measured using random phase multisines. These are periodic
harmonically related sine
signals consisting of the sum of
and random phases
waves with user defined amplitudes
(8)
where
,
, or
(see [13], [14]). The phases
are
randomly chosen such that
, with
the expected
is uniformly distributed in
. In
value; for example,
[14]–[16] it has been shown that the FRF of a wide class of
nonlinear systems, obtained using a random phase multisine (8)
with sufficiently large, can be written as
(9)
the FRF of the true underlying linear system,
the
with
measurement noise, and
.
is the bias or
deterministic nonlinear contribution which depends on the odd
degree nonlinear distortions and the power spectrum of the input
is the zero mean stochastic nonlinear contribution
only, and
for
(10)
where the overline denotes the complex conjugate, and where
the expected values are taken w.r.t. the different random phase
realizations of the excitation (8). Due to (10), the stochastic nonact as circular complex noise for suflinear contributions
ficiently large. Hence, over different realizations of the random
cannot be distinguished from the measurephase multisine,
. The sum
ment noise
(11)
is the best linear approximation to the nonlinear system for the
class of Gaussian excitation signals (normally distributed noise,
periodic Gaussian noise, and random phase multisines) with a
given power spectrum (see [14]–[16] for the details). It can be
approximated arbitrarily well by a rational form in the Laplace
variable .
Since the even degree nonlinear distortions do not affect the
while they increase the variance of
, the varibias term
ability of the FRF measurement (9) can be reduced by using
random phase multisines which excite the odd harmonics only
. These so-called odd random phase
[12], [13]: (8) with
multisines allow us to detect the presence and the level of even
degree nonlinear distortions by looking at the even harmonics in
the output spectrum. To detect the presence and the level of the
odd degree nonlinear distortions, one should leave out some of
Gf
M
Fig. 2. FRF ( ) measurement: applying
different random phase
realizations of the excitation and measuring each time periods of length
after a waiting time
.
T
P
T
the odd harmonics in the odd random phase multisine. The optimal strategy consists of splitting the odd harmonics in groups
of equal numbers of consecutive lines and eliminating randomly
one line out of each group. This can be done for a linear, as well
as a logarithmic, frequency distribution (see [12] for the linear
case). The resulting excitation, an odd random phase multisine
with random harmonic grid (linear or logarithmic frequency distribution), is used throughout the paper.
Two measurement strategies are proposed in the sequel of this
section (see Fig. 2): the first uses one-phase realization of the
odd random phase multisine with random harmonic grid and is
only, while the second uses multiple phase
suitable for
realizations (each time with the same random harmonic grid)
,
, and
.
and is suitable for
B. First Measurement Strategy
The first strategy for measuring the open loop gain
(2)
and its uncertainty consists of the following steps.
1) Choose the amplitude spectrum
and the frequency resolution
of the odd
.
random phase multisine (8)
2) Split the excited odd harmonics (linear or logarithmic frequency distribution) in groups of equal number of consecutive lines, and randomly eliminate one odd harmonic out
of each group (for example, 100 excited odd harmonics
are split in 25 groups of four consecutive excited odd harmonics, and one out of the four odd harmonics is randomly eliminated in each group).
of the nonzero
3) Make a random choice of the phases
harmonics of the random phase multisine (8), and calcu.
late the corresponding time signal
4) Apply the excitation
to the circuit (see Fig. 1) and
consecutive periods 1 of the steady-state
measure
and output
(see Fig. 2, one horizontal
input
line).
1At least six periods are needed to preserve the properties of the maximum
likelihood estimator used in the parametric modeling step [14].
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PINTELON et al.: SYSTEM IDENTIFICATION APPROACH—PART I: THEORY AND SIMULATIONS
From the
noisy input/output spectra
,
,
, one can calculate the average open loop
gain and its sample variance
857
excited odd harmonics is obtained by linear or cubic interpolation of the level at the nonexcited odd harmonics [12]. The
at the excited odd harmonics in
is
bias contribution
bounded by
(15)
with
(12)
where is a heuristic factor depending on the power spectrum
of the excitation and the system (see [12], [13] for the linear
frequency distribution). Typical values for lie between two
and ten. Finally, calculating
and
(16)
is calculated over consecutive
Since the sample variance
periods of one particular realization of the random phase
multisine, it is clear that it only contains the contribution of
the measurement noise to the open loop gain measurement
.
The presence and the level of the odd and even degree nonlinear distortions is revealed by analyzing the nonexcited frequencies [ missing harmonics in
] in the output spectrum
. However, straightforward interpretation of the output
spectrum is impossible. Indeed, due to the feedback loop in the
(see Fig. 1, feedback resistor
),
setup for measuring
is also contaminated by the nonthe input spectrum
linear distortions of the device under test (DUT), and, hence, the
are partially due to the linear
nonexcited frequencies in
. Since the DUT is
feed through of the distorted input
is obtained by
dominantly linear, a first-order correction
subtracting the linear contribution of the DUT from the output
at the nonexcited harmonics
The first three steps of the second strategy for measuring the
(2),
(4) and
operational amplifier characteristics
(6) are identical to the first strategy for measuring
(see Section III-B). In addition to steps 1)–4) of Section III-B
(
,
we have the following. 4) Apply the excitation
, or
) to the circuit in Fig. 1 and measure FRFs
(
,
, or
) from
consecutive periods of the
times3
steady-state response.2 5) Repeat steps 3) and 4)
(see Fig. 2, the
horizontal lines).
noisy FRFs
,
and
From the
, one can calculate for each experiment the avand its sample variance
erage FRF
excited harmonic in
nonexcited harmonic in
(13)
(17)
where, according to the frequency resolution,
at the
nonexcited frequencies is obtained through linear or cubic interpolation of the values at the excited frequencies. Further, the
sample mean and sample variance of the corrected output spectrum
at the excited odd harmonics gives the level of the stochastic
and the bias
contributions on the open loop gain
measurement
.
C. Second Measurement Strategy
An additional averaging over
whole measurement procedure
gives the final FRF
of the
(18)
(14)
, the
are calculated. Within the measurement uncertainty
presence and the level of the even and nonexcited odd harmonics
reveals the presence and the level of respectively the even
in
and odd degree nonlinear distortions at the nonexcited frequencies [12], [13]. What about the nonlinear distortions at the ex? For odd random phase multisines
cited odd harmonics in
with random harmonic grid (linear or logarithmic distribution),
at the
the level of the stochastic nonlinear contributions
together with its sample variance
.
From (9), (17), and (18), it follows that
and
(19)
2At
least two periods are needed to calculate a sample variance.
least six periods are needed to preserve the properties of the maximum
likelihood estimator used in the parametric modeling step [14].
3At
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 3, JUNE 2004
Hence, if the system is linear
, then
should be
divided by
approximately equal to the mean value of
. The model paramwith
eters are obtained by minimizing the sample maximum likelihood cost function
(20)
(18) is larger than
(20), then this is an indication that
If
, and
the systems behaves nonlinearly
(21)
(24)
w.r.t. , where
and where the summation index
runs over the excited harmonics [14], [19]. The values of and
are obtained by minimizing the minimum description length
(MDL) model selection criterion
otherwise
is an estimate of
linear bias contribution
(22)
where
(25)
. Using (15), (16), and (21), the nonin (9) can be bounded by
is defined in (15) (see [12] and [13] for the details).
D. Discussion
The advantages of the first measurement strategy are 1) the
reduced measurement time (all information is gathered in one
single experiment and 2) the classification in odd and even degree nonlinear distortions. Its disadvantages are 1) the variance
of the FRF measurement only accounts for the measurement
noise and 2) an approximation (extrapolation) is needed to characterize the nonlinear stochastic contributions on the FRF measurement.
The advantages of the second measurement strategy are 1) the
contribution of the stochastic nonlinear distortions to the FRF
measurement are obtained without any approximation (extrapolation), and 2) the variance of the FRF measurement accounts
for the measurement noise as well as the stochastic nonlinear
distortions. Its disadvantages are 1) the increased measurement
time (several experiments are needed and 2), except for the open
loop gain, no classification in odd and even degree nonlinear distortions can be made.
If the input signal-to-noise ratio is smaller than 6 dB, then
the relative bias on the FRF measurement can no longer be neglected [14], [17]. It can be reduced by appropriate averaging
noisy input/output spectra before calculating the
of the
FRF. At the cost of knowing exactly a reference signal (typically the signal stored in the arbitrary waveform generator),
the second measurement strategy can be generalized to handle
input/output spectra (see [18] for the details).
IV. MODELING PROCEDURE
The measured FRF
by a rational form in the Laplace variable
is modeled
(23)
with the minimizer of (24),
the number of free parameters in (23), and the number of frequencies in (24) (see [20]).
The result is an estimate together with its covariance matrix
. Finally, from and
the poles, the zeros, the
gain, and their uncertainties are calculated (see [14] for the theoretical background and [21] for the software implementation).
Finally, the value of cost function at its minimum
is compared to the 95% uncertainty interval of the cost function
constructed under the assumption that no modeling errors are
present
(26)
where
and
(see [14]). This hypothesis test allows us to verify whether or
not residual model errors are present.
V. SIMULATION RESULTS
A. Introduction
The whole measurement/modeling procedure is illustrated on
a network simulation (Spectre RF circuit simulator version 4.4.6
of Cadence Design Systems, Inc.) of an LM741 operational amplifier (see [22], p. 424 for the circuit). For the open loop and
with
common mode gain simulations, the voltage source
in Fig. 1 can be implemented in two ways in the
impedance
circuit simulator: either as one voltage source where the random
phase multisine signal is defined by an array of numbers, or as
the series connection of ( number of excited harmonics in
the multisine) ideal sinewave voltage sources with appropriate
amplitude and phase in series with the impedance . The first
solution has the disadvantage that the time derivatives of the
simulated random phase multisine signal are not exact. The time
derivatives of the second solution are exact; however, it has the
disadvantage that the number of nodes of the circuit increases
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PINTELON et al.: SYSTEM IDENTIFICATION APPROACH—PART I: THEORY AND SIMULATIONS
859
with the number of excited harmonics . Replacing the seby parries connected ideal voltage sources in series with
removes
allel connected ideal current sources in parallel with
the drawback of the second solution. The solution with current
sources is used in all simulations.
For the power supply gain simulations, an ideal voltage
in series with an ideal dc source must be imsource
V
and
V or
plemented [
V and
V
]. The ideal voltage
is realized by parallel connected ideal sinewave
source
current sources and one ideal current controlled voltage source.
The goal of the simulation is the validation of the first measurement strategy. Therefore, the open loop gain is measured
using the first and second measurement strategy. These results
are reported in Section V-B.
Although no disturbing noise is added in the simulations,
the two measurement strategies of Section III are applied with
. It allows us to verify the level of the arithmetic noise
of the simulator ( variability of the steady-state response from
one period to the other), but gives no information about the systematic error of the integration method used (here the trapezium
rule). For linear circuits, the systematic error of the trapezium
rule is eliminated by a bilinear warping of the frequency axis
(27)
where is the sample period of the simulator. For nonlinear circuits with a dominantly linear behavior, the frequency warping
is a first-order correction. It is applied to the simulation data for
the parametric modeling of the operational amplifier characteristics.
B. Simulation of the Operational Amplifier Characteristics
The setup of Fig. 1 is used with
,
,
k ,
V, ideal
voltage buffers, and odd random phase multisine excitations
with random harmonic grid (see Section III). Of each signal,
periods and
points per period are calkHz. The frequencies of
culated at the sampling rate
the odd random phase multisines are logarithmically distributed
Hz and
between
kHz. Of each group of three consecutive odd harmonics,
one odd harmonic is randomly eliminated. The resulting odd
random phase multisines with random harmonic grid contain
odd excited harmonics. All FRF calculations are performed
25 times with different phase realizations of the
odd random phase multisines.
is
For the open loop gain simulation, the source level
mVrms. Figs. 3 and 4 show the results.
such that
From Fig. 3, it can be seen that
is contaminated by the
(“o”
nonlinear distortions: the nonexcited harmonics of
and ) are well above the arithmetic noise level (solid line) in
. After correction, as in (13), it follows from the output
spectrum
that the odd degree nonlinear distortions are
dominant (the level of the even harmonics is 20 dB below the
level of the nonexcited odd harmonics “o”), and that the nonlinear distortions are quite large in the band [1 Hz, 30 Hz] and
Fig. 3. Spectral content of the signals of one experiment (realization
excitation) of the open loop gain simulation. Top row: minus terminal DUT
v (f ). Middle row: output DUT v (f ). Bottom row: corrected output DUT
v (f ). +: excited odd harmonics. o: odd nonexcited harmonics. Gray :
even nonexcited harmonics. : standard deviation excited harmonics. …:
standard deviation nonexcited harmonics.
0
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3
860
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 3, JUNE 2004
Fig. 4. Open loop gain simulation. Top row: first measurement strategy. Bottom row: second measurement strategy. Bold black line: FRF. Solid line: standard
deviation measurement noise. Bold gray lines: total sample variance. Gray : predicted level of the stochastic nonlinear distortions.
+
decrease with increasing frequency. The latter is not visible in
the original (uncorrected) output spectrum because of the linin Fig. 1).
earizing effect of the feedback loop (resistor
Fig. 4 compares the open loop gain obtained from one realization of the multisine excitation (first measurement strategy, see
25 multisine realizaSection III-B) to that obtained from
tions (second measurement strategy, see Section III-C). It can be
seen that the second measurement strategy leads to a less noisy
FRF (see the phase characteristic), and a lower level of the stochastic nonlinear distortions. Both effects are explained by the
25 multisine realizations
averaging of the FRF over the
leading to a reduction of the stochastic nonlinear distortions
in (9) by a factor
dB. Since the prediction of the
(16) in the first
level of the stochastic nonlinear distortions
measurement strategy (gray ) is based on a single experiment,
(18) obtained by the second meait is much noisier than
surement strategy (bold gray line). From Fig. 5, it follows that
, averaged over the
25 multisine experiments, equals
((21) with
) within a few decibels.
The second measurement strategy has also been applied to the
common mode and power supply gain measurements. It turns
out that the level of the nonlinear distortions on the common
Fig. 5. Comparison of the predicted levels of the stochastic nonlinear
distortions A on the open loop gain obtained by (gray ) the first and (solid
line) second measurement strategy.
+
mode and power supply gains is much smaller than that on the
open loop gain.
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PINTELON et al.: SYSTEM IDENTIFICATION APPROACH—PART I: THEORY AND SIMULATIONS
TABLE I
SOME OPEN LOOP GAIN MODEL PARAMETERS AND THEIR UNCERTAINTY
C. Modeling of the Operational Amplifier Characteristics
To suppress the systematic error of the trapezium integration
rule, the frequency axis is warped according to (27). It transforms the original digital frequency band [0.95 Hz, 9.5 kHz]
to the analog frequency band [0.95 Hz, 10.3 kHz]. The open
loop gain measurements (18) obtained by the second measurement strategy are modeled by a rational form (23). Using the
modeling procedure of Section IV, it follows that a model of
,
is necessary to explain the data: the minorder
and lies
imum of the cost function equals
in the 95% confidence region [269.4, 344.2] constructed under
the hypothesis that no model errors are present (26). A correlation analysis of the residuals shows that the residuals are white,
which confirms that all dynamics are captured by the model.
As could be expected for a nonlinear RC network, all poles and
zeros of the identified model lie on the real axis. This is not the
case when no frequency warping (27) is applied to the simulation data.
and the
Table I gives the estimated dc open loop gain
cut-off frequency
. Although the influence of the stochastic
nonlinear distortions on the estimate tends to zero as either
or
or
,
still depends on the nonlinear distortions through the bias contribution
in (9). Hence, the values given in Table I depend on the particular power spectrum of the excitation. However, they can be
used to predict the response to Gaussian noise, periodic noise,
and random phase multisines excitations [16].
VI. CONCLUSION
A system identification approach for modeling the linear
operational amplifier characteristics has been presented. It
includes the choice of the excitation, the detection, qualification, and quantification of the nonlinear distortions, and the
parametric modeling. Two measurement procedures have been
proposed that allow us to measure (simultaneously) the linear
operational amplifier characteristics, the noise level, and the
level of the odd and even nonlinear distortions. The first measurement approach is especially useful for the fast evaluation
of (large scale) nonlinear circuits using network simulators. In
Part I of this series of two papers, the theory has been validated
via simulations on an LM741 operational amplifier.
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Meas., vol. 53, June 2004.
Rik Pintelon (M’90–SM’96–F’98) was born in
Gent, Belgium, on December 4, 1959. He received
the degree of Electrical Engineer (burgerlijk ingenieur) in July 1982, the degree of Doctor in applied
sciences in January 1988, and the qualification to
teach at university level (geaggregeerde voor het
hoger onderwijs) in April 1994, from the Vrije
Universiteit Brussel (VUB), Brussels, Belgium.
From October 1982 until September 2000, he was
a Researcher of the Fund for Scientific Research,
Flanders, VUB. Since October 2000, he has been a
Professor in the Electrical Measurement Department, VUB. His main research
interests are in the fields of parameter estimation/system identification and
signal processing.
Authorized licensed use limited to: Rik Pintelon. Downloaded on December 2, 2008 at 08:33 from IEEE Xplore. Restrictions apply.
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 53, NO. 3, JUNE 2004
Gerd Vandersteen (M’01) was born in Belgium
in 1968. He received the degree in electrical
engineering and the Ph.D. degree from the Vrije
Universiteit Brussel (VUB), Brussels, Belgium, in
1991 and 1997, respectively.
He is presently a Principal Scientist in the MixedSignal Design Group, IMEC/DESICS. His main interests are in the field of modeling, measurement, and
simulation of nonlinear microwave devices.
Ludwig De Locht was born in Leuven, Belgium, on
December 31, 1979. He received the degree of Electrical Engineer from the Vrije Universiteit Brussel
(VUB), Brussels, Belgium, in 2002.
In August 2002, he joined the Electrical Measurement Department, VUB, and the DESICS group of
IMEC, Leuven, Belgium, as a Research Associate. In
December 2002, he recieved an IWT fellowship. The
goal of his research is to develop tools for analog designers giving insight into the nonlinear behavior of
power amplifiers for telecommunication systems.
Yves Rolain (SM’96) is with the Electrical Measurement Department, Vrije Universiteit Brussel (VUB),
Brussels, Belgium.
His main research interests are nonlinear microwave measurement techniques, applied digital
signal processing, parameter estimation/system
identification, and biological agriculture.
Johan Schoukens (M’90–SM’92–F’97) was born in
Belgium in 1957. He received the Engineer degree
in 1980 and the Doctor degree in applied sciences
in 1985, both from the Vrije Universiteit Brussel
(VUB), Brussels, Belgium.
He is presently a Professor at the VUB. The prime
factors of his interest are in the field of system
identification for linear and nonlinear systems, and
growing tomatoes in his green house.
Authorized licensed use limited to: Rik Pintelon. Downloaded on December 2, 2008 at 08:33 from IEEE Xplore. Restrictions apply.