IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 1, JANUARY 2009
97
Full Model and Characterization of Noise in
Operational Amplifier
Gino Giusi, Felice Crupi, Calogero Pace, and Paolo Magnone
Abstract—In this paper, we propose a method to fully characterize noise in operational amplifiers (op-amps). The method allows the extraction not only of the spectra of the equivalent input
current noise (EICN) and equivalent input voltage noise generators
but also of their cross-correlation coefficients, which are routinely
discarded in noise analysis of op-amps. The method is applied to
extract all noise parameters of the low-noise bipolar-input op-amp
OP27 and is validated through noise measurements in a test circuit.
A key finding is that neglecting the cross-correlation coefficient between the two EICN generators can lead to severe errors in noise
analysis.
Index Terms—Cross correlation, noise measurements, noise
model, operational amplifiers (op-amps).
I. INTRODUCTION
A
CCURATE modeling of operational amplifier (op-amp)
noise is fundamental, since op-amps are vastly used as
building blocks to implement low-noise amplifiers in discrete
and integrated circuits [1]–[8]. Noise in op-amps is routinely
modeled by two equivalent input current noise (EICN) generators and one equivalent input voltage noise (EIVN) generator.
The three noise sources are usually assumed uncorrelated to each
other. Moreover, the two EICNs are usually assumed equal due to
the symmetry of the input differential amplifier. Based on these
assumptions, the op-amp noise modeling requires the knowledge
of only two noise quantities, the EIVN and the EICN, which are
usually reported in the op-amp data sheets. This popular model is
an incomplete representation of the op-amp noise, and it can lead
to severe errors in noise analysis. A complete noise model requires also the knowledge of the correlation coefficients between
each couple of noise sources. The noise sources are, in general,
correlated simply because they may include the contribution of
the same noise physical mechanism. In the past, a method [9]
was proposed to evaluate the correlation coefficient between the
EIVN and the EICN along with the three noise sources. This
method has two main drawbacks: 1) It neglects the correlation
coefficient between the two EICNs and 2) the proposed procedure is very complicated, requiring seven measurement steps.
In this paper, we propose a cross-correlation-based method to
evaluate the three noise sources and the correlation coefficients
between each couple of noise sources. The full op-amp noise
Manuscript received February 27, 2008; revised April 24, 2008. First published June 6, 2008; current version published February 4, 2009. This work was
supported by the Ministero degli Affari Esteri under the RHESSA Project. This
paper was recommended by H. Schmid.
The authors are with the Dipartimento di Elettronica, Informatica e
Sistemistica, University of Calabria, 87036 Arcavacata di Rende, Italy
(e-mail: ggiusi@deis.unical.it; crupi@unical.it; cpace@unical.it; magnonep@
deis.unical.it).
Digital Object Identifier 10.1109/TCSI.2008.927011
0
Fig. 1. E I model for a linear two-port network. E is a voltage noise
generator, while I is a current noise generator. Generally, they are correlated.
characterization is obtained with a three-step procedure. Our
key finding is that the usually neglected and seldom measured
correlation coefficient between the two EICNs can play a role
in noise behavior of op-amp-based circuits.
The remainder of this work is organized as follows. In
Section II, the basic theoretical background of the op-amp
noise model is discussed. In Section III, we illustrate the
proposed procedure for the complete op-amp noise characterization. In Section IV, we report the experimental results
obtained by applying the proposed method to the low-noise
bipolar-input op-amp OP27. Experimental results obtained on
a test circuit validating the proposed method are reported in
Section V. Finally, in Section VI, we present our conclusions.
II. OP-AMP NOISE MODEL
First studies on noise modeling of a general linear two-port
network were reported by Rothe and Dahlike and Haus in [10]
model (Fig. 1), the noise
and [11], respectively. In their
coming from a general linear two-port network is modeled by
and
located at the input port.
two noise generators
is a voltage noise generator, while
is a current noise generator which are generally correlated through a correlation coefficient. Modeling of a more general -port linear network requires
at least noise generators. In this case, it is necessary also to
take into account correlation coefficients between each couple
of noise generators. Since op-amps are three-port network, at
least three noise generators and three correlation coefficients are
required to model their noise behavior. The two most diffused
op-amp noise models are shown in Fig. 2. As shown in Fig. 2(a),
the first model is based on four noise generators [12]–[15]:
and
are the noise generators related to the noninverting input
port, while
and
are the noise generators related to the inverting input port.
Generally, there should exist a corresponding correlation coefficient between each of these four quantities. Noise generators
at the two input ports are usually assumed equal to one another
,
due to the high symmetry of
so that
the input differential amplifier. The other op-amp noise model
and
[see Fig. 2(b)] is based on three noise generators [16]:
are the current noise generators between the noninverting
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 1, JANUARY 2009
Fig. 2. Two popular op-amp noise models with (a) four and (b) three equivalent input noise sources. Generally, noise generators are correlated one to the other.
and ground and between the inverting input and ground and
is the voltage noise source in series with one or the other input
and
of the previous model
terminal. By assuming that
are in series through the differential op-amp input impedance,
. Moreover, under the hypothesis that
we have
and
are uncorrelated, the power spectral density (PSD)
is
. Note that, in the particular case in
of
which the noninverting input terminal is connected to ground,
the op-amp is reduced to a single input port device, and the
model of Fig. 1 applies. In the noise model of
simple
Fig. 2(b), we have three noise generators, and hence, we can
compute three different cross-correlation coefficients
, which correparameters. The system has four outputs
. The op-amp
spond to the outputs of voltage amplifiers
under test (OA4) works in a transimpedance amplifier configwith gain
. Voltage amplifiers
and
are
uration
while
and
are connected
connected to the output of
to its noninverting input. Voltage amplifier gains must be equal
one to the other. Moreover, the particular implementation of amis not important. They are modeled with
plifiers , , and
noise model. Differently from
the classical two-port
is specifically an op-amp
the previous voltage amplifiers,
are the inputs
(OA3)-based voltage amplifier. Outputs
of a spectrum analyzer which performs cross correlations among
the four channels. We will refer the output values with respect to
the input of the voltage amplifiers in order to render the discussion independent on the particular choice of their gains. The
proposed method consists of three measurement steps.
In the first measurement step, we use the circuit configuration
are
shown in Fig. 3. The input-referred outputs
(1)
where
and
are the cross-correlation coefficients be,
and
, , respectively;
is the correlation
tween
and
,
;
are the PSDs of
coefficient between ,
,
, and
.
is the cross spectrum between
and
. Note that, because the cross spectra have real and imaginary
,
, and
are complex functions of the
components,
and
? Befrequency. Which is the relationship between
cause of the high symmetry of the op-amp input, it is licit to
and
so
assume that
and
. As discussed in the
that
introduction, noise analysis typically assumes that all the correand
lation coefficients equal to zero. To our knowledge, only
have been experimentally investigated. In the next section,
, which cannot be
we will describe a method to extract also
negligible, as it will be shown in Section V.
III. DESCRIPTION OF THE METHOD
As discussed in the previous section, a complete op-amp noise
characterization requires the evaluation of six noise quantities,
,
, and
and the three cross spectra
the three spectra
,
, and
, which allow us to calculate
the correlation coefficients according to (1). Fig. 3 shows a
schematic of the system proposed to evaluate these six noise
(2)
where
is parallel between
and
and
is the total
noise coming from these resistors. By taking the cross spectra,
we obtain
(3)
where
is the cross spectrum between
and
in step 1,
is the cross spectrum between
and
, and
is
and
. If the current noise
the cross spectrum between
and
are negligible, we have
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(4)
GIUSI et al.: FULL MODEL AND CHARACTERIZATION OF NOISE IN OPERATIONAL AMPLIFIER
99
Therefore, after step 2, we obtain a fourth noise parameter
(8)
and a relationship between the remaining two noise quantities
(9)
The problem now boils down to determine another equation
and
, which is the target of the sucrelating
cessive step. In the third measurement step, op-amps OA4 and
OA3 maintain the same configuration as in step 2 but the values
and
are increased by the factor in order to
of resistors
maintain the same gain. Now, (9) can be written as
(10)
Fig. 3. Schematic of the system used to evaluate the op-amp noise parameters.
OA4 is the op-amp under test. In step 2, OA4 and OA3 exchange their position.
In step 3, resistances R and R change their values, maintaining the same
, which reduces the measurement bandwidth, is due to the op-amp
ratio. C
common-mode input capacitances.
By assuming that
is a simple resistor
obtain three of the six noise quantities
, after step 1, we
(5)
where
and
are the PSD of
and
, respectively. In
the second measurement step, op-amps OA4 and OA3 exchange
their position. Therefore, to obtain the new equations, it is sufficient to exchange the subscripts three and four in the right-hand
side of (3)
(6)
Neglecting
and
, we obtain
(7)
and
By combining (9) and (10), we can obtain
. Therefore, the proposed three-measurement-step
procedure allows us to evaluate all the six noise quantities.
It is worth noting that the validity of our method is limited
by the approximations done in (4) and (7). These assumptions
are usually verified if the PSDs of the EICNs of the measuring
,
, and
are negligible with respect to the
amplifiers
,
. ConsePSD of the EICN of the op-amp under test
quently, the method works well if we characterize the noise in
bipolar-input op-amps by using MOS input op-amps in the measuring system, as it will be shown in the next section. Really, for
op-amps with a MOS input stage, current noise generators have
a very low value, so their contribution is negligible in most of
the practical cases. The only significant noise parameter is
which can be obtained in a single measurement step by taking
and
(configuration of step
the cross correlation between
1). Moreover, in this case, amplifiers
and
are not necessary, so the whole system reduces to only two outputs.
IV. APPLICATION OF THE METHOD
The proposed method has been applied to perform the full
op-amp noise characterization of the low-noise bipolar-input
pA Hz at 1 kHz
op-amp OP27. Data sheets report
Hz, and
nV Hz at
with a corner frequency
1 kHz with a corner frequency
Hz. Fig. 4 shows the
electrical implementation of the proposed system. The electrical
circuit is enclosed in a metal box for shielding against external
interferences. The acquisition system is a PC-based spectrum
analyzer composed of a PC equipped with an eight-channelinput DSA board (PXI 4472) manufactured by National Instruare op-amp based, but as disments. Voltage amplifiers
cussed in the previous section, they are not necessary. Unless the
op-amp is under test, all the other op-amps are TLC070 which
has a MOS input stage. Op-amp TLC070 has been chosen befA Hz, so as to
cause of its very low current noise
make valid the approximations of (4) and (7). Voltage amplifier
gain is equal to 101 in order to have a sufficient signal-to-noise
ratio at the input of the PC-based spectrum analyzer. In step 2,
,
k , while in step 3
k and
k so that
in (10). Feedback impedance
of amplifier
is a resistance
with in parallel a capacitor
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Fig. 6. OP27 current noise extracted with the proposed method. It is apparent
that current noise generators of the two inputs are equal. The flat value is 0.6
pA= Hz and the corner frequency is about 63 Hz (Table I). These values well
agree with data reported on the OP27 data sheet.
p
Fig. 4. Electrical implementation of the schematic of Fig. 3. All the voltage
amplifiers are op-amp based. The op-amp under test is the OP27, while op-amp
TLC070 is used in voltage amplifiers because of its low EICN.
Fig. 7. Real components of cross-correlation coefficients. C is about 0.5 and
it has a flat spectrum. C and C are about 0.02 at higher frequencies but
became higher (about 0.05 at 1 Hz) toward lower frequencies.
p
Fig. 5. OP27 voltage noise extracted with the proposed method. The flat value
is 3 nV= Hz and the corner frequency is about 2.25 Hz (Table I). These values
well agree with data reported on the OP27 data sheet.
used for the stability compensation of . The cross-correlation contributions in (3) depend on the
value. The higher
is, the higher is the sensitivity of the method to extract the
correlation coefficients but the lower the bandwidth. In order
k . Figs. 5
to obtain a good tradeoff, we chose
and 6 show the extracted
,
, and
spectra which well
agree with data reported in data sheets. It is apparent that
and
are identical as tacitly assumed in op-amp data sheets.
Figs. 7 and 8 show the extracted real and imaginary components
,
, and
. Spectra were fitted
of correlation coefficients
, and the results are shown in Table I.
with the law
Imaginary components are null while real components are not
negligible in the low-frequency range near 1 Hz. In particular,
is about 0.5, and it has a flat spectrum, while
real part of
Fig. 8. Imaginary components of cross-correlation coefficients. All of them are
negligible.
real parts of
and
are about 0.02 at higher frequencies
but become higher (about 0.05 at 1 Hz) toward lower frequencies. A factor that was not taken into account is the effect of the
common-mode capacitances of the op-amp input terminals toward ground. In Fig. 3, the common-mode capacitances of the
inverting terminal of OA4, the common-mode capacitance of
are
noninverting input of OA3 and the input capacitance of
collected in a stray capacitance
.
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GIUSI et al.: FULL MODEL AND CHARACTERIZATION OF NOISE IN OPERATIONAL AMPLIFIER
101
TABLE I
EXTRACTED NOISE PARAMETERS BY FITTING THE SPECTRA OF FIGS. 5–8
N . VOLTAGE NOISE AND CURRENT NOISE
WITH THE LAW A=f
IN THE FLAT PART OF THE SPECTRUM AGREE WELL WITH DATA
REPORTED ON THE OP27 DATASHEET
+
Fig. 10. Measured and expected PSDs at the output of the test circuit as shown
in Fig. 9. Expected PSD well agree with the measured data. In addition, it is
shown the PSD neglecting the cross-correlation coefficients. An error of about
40% is calculated in the whole frequency range.
To make the analysis simpler, we can use the well-verified apand
to obtain
proximations
(13)
Fig. 9. Test circuit used for the validation of the proposed method.
The effect of this capacitance is that of reducing the measurement bandwidth to a few kilohertz. For this reason, the sampling
frequency has been chosen equal to 2 kHz, and the spectra are
shown only until 1 kHz. Large variance in correlation coefficients (Figs. 7 and 8) is due to the cross-correlation operation
which is intrinsically very slow in obtaining convergence. Measurement time depends on the desired variance in the spectra.
Useful information can be obtained after some hours of measurement for each step.
V. VALIDATION OF THE METHOD
In order to validate the proposed method, we compared noise
measurements obtained in the test circuit shown in Fig. 9 with
the results expected by using the noise parameters extracted in
the previous section on the same physical op-amp. To highlight
the usefulness of the proposed procedure, we considered a case
parameter remarkably imin which the always neglected
pacts the noise behavior of the circuit. The test circuit consists of
the general topology for op-amp-based amplifiers. Indeed, it can
,
be reduced to a transimpedance amplifier
, or to a differential amplifier in
to a voltage amplifier
which case
is the parallel between
and
. The output
voltage referred at the op-amp input is
(11)
is the parallel between
and
and
and
where
are the thermal noise coming from
and
, respectively.
The PSD is
(12)
From (13), it is evident that cross-correlation contribution deand
values. In this example,
k ,
pends on the
M , and
is equal to their parallel. Notice that this
is just the case of a differential amplifier configuration. In parlowers it. In this
ticular, increases the overall noise while
contribution is very low due the very low
example, the
value. In addition, the voltage-noise contribution is negligible,
so that (13) can be written as
(14)
Fig. 10 shows the measured output PSD and the expected
PSD according to (13). Noise parameters in (13) are the same
as calculated in the measurements reported in the previous sections. Measured and extracted PSD perfectly coincide. In addition, shown in Fig. 10 is the PSD when one neglects the
contribution. It can be easily shown from (14) that the maximum error in neglecting
corresponds to the case in which
which was just our particular choice. The measured
error in the whole frequency range is about 40%. This experimental result clearly indicates that it is not always licit to discard
in noise analysis of op-amp-based circuits.
VI. CONCLUSION
We proposed a novel approach to fully characterize noise in
op-amp. The method allows the extraction not only of the spectra
of the EICN and EIVN generators but also of their cross-correlation coefficients, which are routinely neglected in noise analysis of op-amps. As an example of the application of the method,
we extracted all noise parameters of the low-noise bipolar-input
op-amp OP27. We showed how the knowledge of the cross-correlation coefficients is necessary to perfectly predict the noise behavior of op-amp-based circuits. In particular, we reported a case
in which neglecting the cross-correlation coefficient between
the two EICN generators leads to an error of about 40%.
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[13] D. F. Stout, Handbook of Operational Amplifier Circuit Design. New
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[14] J. R. Hufault, Operational Amplifiers Network Design. Hoboken, NJ:
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[15] G. B. Clayton and B. W. G. Newby, B. H. Newnes, Ed., Operational
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Gino Giusi received the M.Sc. and Ph.D. degrees
in electronic engineering from the University
of Messina, Messina, Italy, in 2002 and 2005,
respectively.
In 2005, he was a Visiting Scientist at the Interuniversity Microelectronics Center, Leuven, Belgium.
In 2006, he was with the National Research Center,
Catania, Italy. He is currently a Researcher with
the Dipartimento di Elettronica, Informatica e
Sistemistica, University of Calabria, Arcavacata di
Rende, Italy. His main research interests include
the design of ultralow-noise instrumentation, the characterization of devices
through noise measurements, and the electrical characterization of modern
CMOS devices and memories.
Felice Crupi received the M.Sc. degree in electronic engineering from the University of Messina,
Messina, Italy, in 1997 and the Ph.D. degree in electronic engineering from the University of Firenze,
Firenze, Italy, in 2001.
Since 1998, he has been a Visiting Scientist repeatedly at the Interuniversity Microelectronics Center,
Leuven, Belgium. In 2000, he was a Visiting Scientist at IBM Thomas J. Watson Research Center, Yorktown Heights, NY. Since 2002, he has been with the
University of Calabria, Arcavacata di Rende, Italy,
where he is currently an Associate Professor of electronics in the Dipartimento
di Elettronica, Informatica e Sistemistica. In 2006, he was a Visiting Scientist at
the Universitat Autonoma de Barcelona, Barcelona, Spain. His main research interests include reliability of very large scale integrated CMOS devices, electrical
characterization techniques for solid state electronic devices, and the design of
ultralow-noise electronic instrumentation. He has authored or coauthored more
than 80 publications in international scientific journals and in international conference proceedings.
Calogero Pace was born in Palermo, Italy, in 1965.
He received the Laurea degree and the Ph.D. degree
in electronic engineering from the University of
Palermo, Palermo, in 1990 and 1994, respectively.
In 1996, he was an Assistant Professor with
the University of Messina, Messina, Italy. Since
2002, he has been with the University of Calabria,
Arcavacata di Rende, Italy, where he is currently
an Associate Professor of electronics in the Dipartimento di Elettronica, Informatica e Sistemistica.
He is currently involved in research projects on the
study of nanocrystal memory devices, on the design of low-noise electronic
instrumentation, and on the design and characterization of optoelectronic gas
sensors.
Paolo Magnone was born in Italy on June 22, 1981.
He received the B.S. and M.S. degrees in electronic
engineering from the University of Calabria, Arcavacata di Rende, Italy, in 2003 and 2005, respectively,
where he is currently working toward the Ph.D. degree in the Dipartimento di Elettronica, Informatica
e Sistemistica.
From 2006 to 2007 and 2007 to 2008, he was with
the Interuniversity Microelectronics Center, Leuven,
Belgium, within the APROTHIN project (Marie
Curie Actions), where he worked on parameter
extraction and matching analysis of FinFET devices. His research interests
include the electrical characterization of semiconductor devices with particular
emphasis on the study of low-frequency noise.
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