Full Model and Characterization of Noise in Operational Amplifier

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.

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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 charac-
terize noise in operational amplifiers (op-amps). The method al-
lows the extraction not only of the spectra of the equivalent input
currentnoise(EICN)andequivalentinputvoltagenoisegenerators
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
OP27andisvalidatedthroughnoisemeasurementsinatestcircuit.
A key finding is that neglecting the cross-correlation coefficient be-
tween 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
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) genera-
tors and one equivalent input voltage noise (EIVN) generator.
Thethreenoisesourcesareusuallyassumeduncorrelatedtoeach
other.Moreover,thetwoEICNsareusuallyassumedequaldueto
the symmetry of the input differential amplifier. Based on these
assumptions,theop-ampnoisemodelingrequirestheknowledge
of only two noise quantities, the EIVN and the EICN, which are
usuallyreportedintheop-ampdatasheets.Thispopularmodelis
anincompleterepresentationoftheop-ampnoise,anditcanlead
to severe errors in noise analysis. A complete noise model re-
quiresalsotheknowledgeofthecorrelationcoefficientsbetween
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 thecorrelation 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 proce-
dure 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
CCURATE modeling of operational amplifier (op-amp)
noise is fundamental, since op-amps are vastly used as
Manuscript received February 27, 2008; revised April 24, 2008. First pub-
lished 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
Fig. 1. ? ??
generator, while ? is a current noise generator. Generally, they are correlated.
model for a linear two-port network. ?
is a voltage noise
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 charac-
terization. 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]
and [11], respectively. In their
coming from a general linear two-port network is modeled by
two noise generators
and
is a voltage noise generator, while
ator which are generally correlated through a correlation coeffi-
cient.Modelingofamoregeneral -portlinearnetworkrequires
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
leastthreenoisegeneratorsandthreecorrelationcoefficientsare
required to model their noise behavior. The two most diffused
op-ampnoisemodelsareshowninFig.2.AsshowninFig.2(a),
the first model is based on four noise generators [12]–[15]:
andare the noise generators relatedto thenoninverting input
port, while
andare the noise generators relatedto the in-
verting input port.
Generally, there should exist a corresponding correlation co-
efficientbetweeneachof thesefour quantities.Noise generators
at the two input ports are usually assumed equal to one another
so that
,
the input differential amplifier. The other op-amp noise model
[see Fig. 2(b)] is based on three noise generators [16]:
are the current noise generators between the noninverting
model (Fig. 1), the noise
located at the input port.
is a current noise gener-
due to the high symmetry of
and
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98IEEE 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
terminal. By assuming that
are in series through the differential op-amp input impedance,
we have
. Moreover, under the hypothesis that
andare uncorrelated, the power spectral density (PSD)
of
is . Note that, in the particular case in
which the noninverting input terminal is connected to ground,
the op-amp is reduced to a single input port device, and the
simple
model of Fig. 1 applies. In the noise model of
Fig. 2(b), we have three noise generators, and hence, we can
compute three different cross-correlation coefficients
andof the previous model
(1)
where
tween
coefficient between
,, and
. Note that, because the cross spectra have real and imaginary
components,
,, and
frequency. Which is the relationship between
cause of the high symmetry of the op-amp input, it is licit to
assume that
that
and
introduction, noise analysis typically assumes that all the corre-
lationcoefficientsequaltozero.Toourknowledge,only
have been experimentally investigated. In the next section,
we will describe a method to extract also
negligible, as it will be shown in Section V.
and are the cross-correlation coefficients be-
,, respectively;
,and,;
is the cross spectrum between
, and is the correlation
are the PSDs of
.and
are complex functions of the
and? Be-
andso
. As discussed in the
and
, which cannot be
III. DESCRIPTION OF THE METHOD
Asdiscussedintheprevioussection,acompleteop-ampnoise
characterization requires the evaluation of six noise quantities,
the three spectra
,, and
,, and
the correlation coefficients according to (1). Fig. 3 shows a
schematic of the system proposed to evaluate these six noise
and the three cross spectra
, which allow us to calculate
parameters. The system has four outputs
spond to the outputs of voltage amplifiers
under test (OA4) works in a transimpedance amplifier config-
uration
with gain. Voltage amplifiers
connected to the output of
while
to its noninverting input. Voltage amplifier gains must be equal
onetotheother.Moreover,theparticularimplementationofam-
plifiers
,, andis not important. They are modeled with
the classical two-port
the previous voltage amplifiers,
(OA3)-based voltage amplifier. Outputs
ofaspectrumanalyzerwhichperformscrosscorrelationsamong
thefourchannels.Wewillrefertheoutputvalueswithrespectto
the input of the voltage amplifiers in order to render the discus-
sion independent on the particular choice of their gains. The
proposed method consists of three measurement steps.
Inthefirstmeasurement step,weusethecircuit configuration
shown in Fig. 3. The input-referred outputs
, which corre-
. The op-amp
andare
andare connected
noise model. Differently from
is specifically an op-amp
are the inputs
are
(2)
where
noise coming from these resistors. By taking the cross spectra,
we obtain
is parallel betweenandandis the total
(3)
where
is the cross spectrum between
is the cross spectrum between
the cross spectrum between
andare negligible, we have
and in step 1,
, and and is
and. If the current noise
(4)
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GIUSI et al.: FULL MODEL AND CHARACTERIZATION OF NOISE IN OPERATIONAL AMPLIFIER99
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 ? and ? change their values, maintaining the same
ratio. ?
, which reduces the measurement bandwidth, is due to the op-amp
common-mode input capacitances.
By assuming that
obtain three of the six noise quantities
is a simple resistor, after step 1, we
(5)
where
thesecondmeasurementstep,op-ampsOA4andOA3exchange
their position. Therefore, to obtain the new equations, it is suffi-
cient to exchange the subscripts three and four in the right-hand
side of (3)
andarethePSDofand ,respectively.In
(6)
Neglecting
and, we obtain
(7)
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
relating
and
cessive step. In the third measurement step, op-amps OA4 and
OA3 maintain the same configuration as in step 2 but the values
of resistors
andare increased by the factor
maintain the same gain. Now, (9) can be written as
, which is the target of the suc-
in order to
(10)
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
amplifiers
, , andare negligible with respect to the
PSD of the EICN of the op-amp under test
quently, the method works well if we characterize the noise in
bipolar-inputop-ampsbyusingMOSinputop-ampsinthemea-
suring system, as it will be shown in thenext 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
the cross correlation between
1). Moreover, in this case, amplifiers
sary, so the whole system reduces to only two outputs.
and
,. Conse-
and(configuration of step
andare not neces-
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
op-amp OP27. Data sheets report
with a corner frequency
1 kHz with a corner frequency
electricalimplementationoftheproposedsystem.Theelectrical
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-channel-
input DSA board (PXI 4472) manufactured by National Instru-
ments. Voltage amplifiers
cussedintheprevioussection,theyarenotnecessary.Unlessthe
op-amp is under test, all the other op-amps are TLC070 which
has a MOS input stage. Op-amp TLC070 has been chosen be-
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
kso thatin (10). Feedback impedance
of amplifieris a resistance
pAHz at 1 kHz
nV Hz, andHz at
Hz. Fig. 4 shows the
are op-amp based, but as dis-
fA Hz, so as to
kand
with in parallel a capacitor
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100IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 56, NO. 1, JANUARY 2009
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. 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
lation contributions in (3) depend on the
is, the higher is the sensitivity of the method to extract the
correlation coefficients but the lower the bandwidth. In order
to obtain a good tradeoff, we chose
and 6 show the extracted
,
agree with data reported in data sheets. It is apparent that
and are identical as tacitly assumed in op-amp data sheets.
Figs. 7and 8show theextractedrealand imaginarycomponents
of correlation coefficients
,
with the law
, and the results are shown in Table I.
Imaginary components are null while real components are not
negligible in the low-frequency range near 1 Hz. In particular,
real part of
is about 0.5, and it has a flat spectrum, while
. The cross-corre-
value. The higher
k . Figs. 5
, and spectra which well
, and. Spectra were fitted
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.
Fig. 7. Real components of cross-correlation coefficients. ?
it has a flat spectrum. ?
and ?
became higher (about 0.05 at 1 Hz) toward lower frequencies.
is about 0.5 and
are about 0.02 at higher frequencies but
Fig.8. Imaginarycomponentsofcross-correlationcoefficients.Allofthemare
negligible.
real parts of
but become higher (about 0.05 at 1 Hz) toward lower frequen-
cies. A factor that was not taken into account is the effect of the
common-mode capacitances of the op-amp input terminals to-
ward ground. In Fig. 3, the common-mode capacitances of the
inverting terminal of OA4, the common-mode capacitance of
noninverting input of OA3 and the input capacitance of
collected in a stray capacitance
andare about 0.02 at higher frequencies
are
.
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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
WITH THE LAW ??? ? ? . VOLTAGE NOISE AND CURRENT NOISE
IN THE FLAT PART OF THE SPECTRUM AGREE WELL WITH DATA
REPORTED ON THE OP27 DATASHEET
Fig. 9. Test circuit used for the validation of the proposed method.
The effect of this capacitance is that of reducing the measure-
mentbandwidthtoafewkilohertz.Forthisreason,thesampling
frequency has been chosen equal to 2 kHz, and the spectra are
shown only until 1 kHz. Large variance in correlation coeffi-
cients (Figs. 7 and 8) is due to the cross-correlation operation
which is intrinsically very slow in obtaining convergence. Mea-
surement time depends on the desired variance in the spectra.
Useful information can be obtained after some hours of mea-
surement 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
in which the always neglected
pactsthenoisebehaviorofthecircuit.Thetestcircuitconsistsof
thegeneraltopologyforop-amp-basedamplifiers.Indeed,itcan
be reduced to a transimpedance amplifier
to a voltage amplifier
, or to a differential amplifier in
which case
is the parallel between
voltage referred at the op-amp input is
parameter remarkably im-
,
and. The output
(11)
where
are the thermal noise coming from
The PSD is
istheparallelbetween andandand
and, respectively.
(12)
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 ap-
proximations
and to obtain
(13)
From(13), itis evidentthatcross-correlationcontribution de-
pendsonthe
andvalues. Inthisexample,
M , andis equal to their parallel. Notice that this
is just the case of a differential amplifier configuration. In par-
ticular,
increases the overall noise while
example, the
contribution is very low due the very low
value. In addition, the voltage-noise contribution is negligible,
so that (13) can be written as
k ,
lowers it. In this
(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 sec-
tions. Measured and extracted PSD perfectly coincide. In addi-
tion, shown in Fig. 10 is the PSD when one neglects the
contribution. It can be easily shown from (14) that the max-
imum error in neglectingcorresponds to the case in which
which was just our particular choice. The measured
error in the whole frequency range is about 40%. This experi-
mentalresultclearlyindicatesthatitisnotalwayslicittodiscard
in noise analysis of op-amp-based circuits.
VI. CONCLUSION
We proposed a novel approach to fully characterize noise in
op-amp.Themethodallowstheextractionnotonlyofthespectra
of the EICN and EIVN generators but also of their cross-corre-
lation coefficients, which are routinely neglected in noise anal-
ysisofop-amps.Asanexampleoftheapplicationofthemethod,
we extracted all noise parameters of the low-noise bipolar-input
op-amp OP27. We showed how the knowledge of the cross-cor-
relationcoefficientsisnecessarytoperfectlypredictthenoisebe-
haviorofop-amp-basedcircuits.Inparticular,wereportedacase
in which neglecting the cross-correlation coefficient between
the two EICN generators leads to an error of about 40%.
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