Full Model and Characterization of Noise In

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: REGULAR PAPERS, VOL. 56, NO. 1, JANUARY 2009 97

Full Model and Characterization of Noise inOperational Amplifier

Gino Giusi, Felice Crupi, Calogero Pace, and Paolo Magnone

AbstractIn 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 inputcurrent noise(EICN) and equivalent input voltage noisegeneratorsbut also of their cross-correlation coefficients, which are routinelydiscarded in noise analysis of op-amps. The method is applied toextract all noise parameters of the low-noise bipolar-input op-ampOP27 and is validatedthrough noise measurementsin a test circuit.A key finding is that neglecting the cross-correlation coefficient be-tween the two EICN generators can lead to severe errors in noiseanalysis.

Index TermsCross correlation, noise measurements, noise

model, operational amplifiers (op-amps).

I. INTRODUCTION

ACCURATE 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) genera-

tors and one equivalent input voltage noise (EIVN) generator.

Thethreenoisesources areusually assumed uncorrelatedto each

other.Moreover, thetwo EICNsare usually assumed equaldue to

the symmetry of the input differential amplifier. Based on these

assumptions, the op-amp noisemodelingrequires the knowledgeof only two noise quantities, the EIVN and the EICN, which are

usually reported in theop-amp data sheets. This popular model is

an incomplete representation of the op-amp noise, andit can lead

to severe errors in noise analysis. A complete noise model re-

quires also the knowledge of the correlationcoefficients 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 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

Manuscript received February 27, 2008; revised April 24, 2008. First pub-lished June 6, 2008; current version published February 4, 2009. This work wassupported by the Ministero degli Affari Esteri under the RHESSA Project. Thispaper was recommended by H. Schmid.

The authors are with the Dipartimento di Elettronica, Informatica eSistemistica, University of Calabria, 87036 Arcavacata di Rende, Italy(e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TCSI.2008.927011

Fig. 1. E 0 I model for a linear two-port network. E is a voltage noisegenerator, 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 rolein 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 model (Fig. 1), the noise

coming from a general linear two-port network is modeled by

two noise generators and located at the input port.

is a voltage noise generator, while is a current noise gener-

ator which are generally correlated through a correlation coeffi-

cient. 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 diffusedop-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 in-

verting input port.

Generally, there should exist a corresponding correlation co-

efficient between each of these four quantities. Noise generators

at the two input ports are usually assumed equal to one another

so that , due to the high symmetry of

the input differential amplifier. The other op-amp noise model

[see Fig. 2(b)] is based on three noise generators [16]: and

are the current noise generators between the noninverting

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98 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: 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 and of the previous model

are in series through the differential op-amp input impedance,

we have . Moreover, under the hypothesis that

and are 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

(1)

where and are the cross-correlation coefficients be-

tween , and , , respectively; is the correlation

coefficient between , and , ; are the PSDs of

, , and . is the cross spectrum between and

. Note that, because the cross spectra have real and imaginary

components, , , and are complex functions of the

frequency. Which is the relationship between and ? Be-

cause of the high symmetry of the op-amp input, it is licit to

assume that and so

that and . As discussed in theintroduction, noise analysis typically assumes that all the corre-

lation coefficients equal to zero. To our knowledge, only and

have been experimentally investigated. In the next section,

we will describe a method to extract also , which cannot be

negligible, as it will be shown in Section V.

III. DESCRIPTION OF THE METHOD

As discussedin the previous section, a complete op-amp noise

characterization requires the evaluation of six noise quantities,

the three spectra , , and and the three cross spectra

, , and , which allow us to calculate

the correlation coefficients according to (1). Fig. 3 shows aschematic of the system proposed to evaluate these six noise

parameters. The system has four outputs , which corre-

spond to the outputs of voltage amplifiers . The op-amp

under test (OA4) works in a transimpedance amplifier config-

uration with gain . Voltage amplifiers and are

connected to the output of while and are connected

to its noninverting input. Voltage amplifier gains must be equalone to the other. Moreover, the particular implementation of am-

plifiers , , and is not important. They are modeled with

the classical two-port noise model. Differently from

the previous voltage amplifiers, is specifically an op-amp

(OA3)-based voltage amplifier. Outputs are the inputs

of a spectrum analyzer which performs cross correlationsamong

the four channels. We will refer the output values with respect to

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.

In the first measurement step, we use the circuit configuration

shown in Fig. 3. The input-referred outputs are

(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

the cross spectrum between and . If the current noise

and are negligible, we have

(4)



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GIUSI et al.: FULL MODEL AND CHARACTERIZATION OF NOISE IN OPERATIONAL AMPLIFIER 99

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 sameratio. C , which reduces the measurement bandwidth, is due to the op-ampcommon-mode input capacitances.

By assuming that is a simple resistor , after step 1, we

obtain three of the six noise quantities

(5)

where and are t he P SD o f and , respectively. I n

the second measurement step, op-amps OA4 and OA3 exchange

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)

(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 , which is the target of the suc-

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 and are increased by the factor in order to

maintain the same gain. Now, (9) can be written as

(10)

By combining (9) and (10), we can obtain and

. 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 , , and are negligible with respect to the

PSD of the EICN of the op-amp under test , . Conse-

quently, the method works well if we characterize the noise in

bipolar-input op-amps by using MOS input op-amps in the mea-

suring 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

the cross correlation between and (configuration of step1). Moreover, in this case, amplifiers and are not neces-

sary, 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

op-amp OP27. Data sheets report pA Hz at 1 kHz

with a corner frequency Hz, and nV Hz at

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 externalinterferences. 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 are op-amp based, but as dis-

cussed in the previous section, they are not necessary. 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 fA Hz, so as to

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 impedanceof amplifier is a resistance with in parallel a capacitor



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Fig. 4. Electrical implementation of the schematic of Fig. 3. All the voltageamplifiers are op-amp based. The op-amp under test is the OP27, while op-ampTLC070 is used in voltage amplifiers because of its low EICN.

Fig. 5. OP27 voltage noise extracted with the proposed method. The flat valueis 3 nV =

p

Hz and the corner frequency is about 2.25 Hz (Table I). These valueswell agree with data reported on the OP27 data sheet.

used for the stability compensation of . The cross-corre-

lation 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

to obtain a good tradeoff, we chose k . Figs. 5

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

of correlation coefficients , , and . Spectra were fitted

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

Fig. 6. OP27 current noise extracted with the proposed method. It is apparentthat current noise generators of the two inputs are equal. The flat value is 0.6pA=

p

Hz and the corner frequency is about 63 Hz (Table I). These values wellagree with data reported on the OP27 data sheet.

Fig. 7. Real components of cross-correlation coefficients. C is about 0.5 and

it has a flat spectrum.C

andC

are about 0.02 at higher frequencies butbecame higher (about 0.05 at 1 Hz) toward lower frequencies.

Fig. 8. Imaginarycomponents of cross-correlation coefficients. All of them arenegligible.

real parts of and are about 0.02 at higher frequencies

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 arecollected in a stray capacitance .



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GIUSI et al.: FULL MODEL AND CHARACTERIZATION OF NOISE IN OPERATIONAL AMPLIFIER 101

TABLE IEXTRACTED NOISE PARAMETERS BY FITTING THE SPECTRA OF FIGS. 58

WITH THE LAW A = f + N . VOLTAGE NOISE AND CURRENT NOISEIN 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-

ment 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 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 parameter remarkably im-

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 ,to a voltage amplifier , or to a differential amplifier in

which case is the parallel between and . The output

voltage referred at the op-amp input is

(11)

where is t he p arallel b etween and and and

are the thermal noise coming from and , respectively.

The PSD is

(12)

Fig. 10. Measured and expected PSDs at the output of the test circuit as shownin Fig. 9. Expected PSD well agree with the measured data. In addition, it isshown the PSD neglecting the cross-correlation coefficients. An error of about40% 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), it is evident that cross-correlation contribution de-

pends on t he and values. In this example, k ,

M , and is equal to their parallel. Notice that this

is just the case of a differential amplifier configuration. In par-

ticular, increases the overall noise while lowers it. In this

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

(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 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 experi-

mental 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. Themethod allows the extraction not only of the spectra

of the EICN and EIVN generators but also of their cross-corre-

lation coefficients, which are routinely neglected in noise anal-

ysis 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-cor-

relation coefficients is necessary to perfectly predict the noisebe-

havior of op-amp-basedcircuits. In particular,we reported a case

in which neglecting the cross-correlation coefficient betweenthe two EICN generators leads to an error of about 40%.



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[7] C. Ciofi, F. Crupi,C. Pace,and G.Scandurra, How to enlarge theband-width without increasing the noise in OP-AMP-based transimpedanceamplifier, IEEE Trans. Instrum. Meas., vol. 55, no. 3, pp. 814819,Jun. 2006.

[8] F. Crupi, G. Giusi, and C. Pace, Two-channel amplifier for high-sen-sitivity voltage noise measurements, in Proc. IEEE Instrum. Meas.Technol. Conf., May 13, 2007, pp. 14.

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[12] T. Robe, Taming noise in IC OP AMPS, Electron. Design, vol. 15,pp. 6470, Jul. 19, 1974.

[13] D. F. Stout, Handbook of OperationalAmplifier Circuit Design. NewYork: McGraw-Hill, 1976, pp. 4551.

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[15] G. B. Clayton and B. W. G. Newby, B. H. Newnes, Ed. , OperationalAmplifiers, 1992, ch. 2-3.

[16] C. D. Motchenbacher and J. A. Connelly, Low Noise Electronic SystemDesign. New York: Wiley, 1993.

Gino Giusi received the M.Sc. and Ph.D. degreesin electronic engineering from the Universityof Messina, Messina, Italy, in 2002 and 2005,respectively.

In 2005, he was a Visiting Scientist at the Interuni-versity Microelectronics Center, Leuven, Belgium.In 2006, he was with the National Research Center,Catania, Italy. He is currently a Researcher withthe Dipartimento di Elettronica, Informatica eSistemistica, University of Calabria, Arcavacata diRende, Italy. His main research interests include

the design of ultralow-noise instrumentation, the characterization of devices

through noise measurements, and the electrical characterization of modernCMOS devices and memories.

Felice Crupi received the M.Sc. degree in elec-tronic engineering from the University of Messina,Messina, Italy, in 1997 and the Ph.D. degree in elec-tronic engineering from the University of Firenze,Firenze, Italy, in 2001.

Since1998, he has beena Visiting Scientistrepeat-edly at the Interuniversity Microelectronics Center,Leuven, Belgium. In 2000, he was a Visiting Scien-

tist at IBM Thomas J. WatsonResearch Center, York-town Heights, NY. Since 2002, he has been with theUniversity of Calabria, Arcavacata di Rende, Italy,

where he is currently an Associate Professor of electronics in the Dipartimentodi Elettronica, Informatica e Sistemistica. In 2006, he was a Visiting Scientist atthe Universitat Autonoma de Barcelona, Barcelona, Spain. His mainresearch in-terests include reliabilityof verylarge scaleintegrated CMOSdevices, electricalcharacterization techniques for solid state electronic devices, and the design ofultralow-noise electronic instrumentation. He has authored or coauthored morethan 80 publications in international scientific journals and in international con-ference proceedings.

Calogero Pace was born in Palermo, Italy, in 1965.He received the Laurea degree and the Ph.D. degreein electronic engineering from the University ofPalermo, Palermo, in 1990 and 1994, respectively.

In 1996, he was an Assistant Professor withthe University of Messina, Messina, Italy. Since2002, he has been with the University of Calabria,Arcavacata di Rende, Italy, where he is currentlyan Associate Professor of electronics in the Dipar-timento 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 electronicinstrumentation, and on the design and characterization of optoelectronic gassensors.

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, Arcava-cata di Rende, Italy, in 2003 and 2005, respectively,where he is currently working toward the Ph.D. de-gree in the Dipartimento di Elettronica, Informaticae Sistemistica.

From 2006 to 2007 and 2007 to 2008, he was withthe Interuniversity Microelectronics Center, Leuven,Belgium, within the APROTHIN project (MarieCurie Actions), where he worked on parameter

extraction and matching analysis of FinFET devices. His research interestsinclude the electrical characterization of semiconductor devices with particularemphasis on the study of low-frequency noise.

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Full Model and Characterization of Noise In