A Complete Operational Amplifier Noise Model: Analysis and Measurement of Correlation Coefficient

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATION, VOL. 47, NO. 3, MARCH for ractical alication, oening the ath for widesread adotion of the clock-gating technique in low-ower design of custom IC s. REFERENCES [] M. Pedram, Power minimization in IC design: Princiles and alications, ACM Trans. Design Automation, vol., no.,. 3 56, Jan [] G. Friedman, Clock distribution design in VLSI circuits: An overview, in Proc. IEEE ISCAS, San Jose, CA, May 994, [3] E. Tellez, A. Farrah, and M. Sarrafzadeh, Activity-driven clock design for low ower circuits, in Proc. IEEE ICCAD, San Jose, CA, Nov. 995, [4] M. Alidina and J. Monteiro et al., Precomutation-based sequential logic otimization for low ower, IEEE Trans. VLSI Syst., vol., , Dec [5] L. Benini and G. De Micheli, Symbolic techniques of clock-gating logic for ower otimization of control-oriented synchronous networks, in Proc. Euroean Design Test Conf., Paris, France, 997, A Comlete Oerational Amlifier Noise Model: Analysis and Measurement of Correlation Coefficient Jiansheng Xu, Yisong Dai, and Derek Abbott Abstract In contrast to the general oerational amlifier (o am) noise model widely used, we roose a more comlete and alicable noise model, which considers the correlation between equivalent inut voltage noise source and current noise source. Based on the suer-osition theorem and equivalent circuit noise theory, our formulae for the equivalent inut noise sectrum density of an o am noise are alied to both the inverting and noninverting inut terminals. By measurement, we demonstrate that the new exressions are significantly more accurate. In addition, details of the measurement method for our noise model arameters are given. A commercial oerational amlifier (Burr Brown OPA37A) is measured by means of a low-frequency noise ower sectrum measuring system and the measured results of its noise model arameters, including the sectral correlation coefficient (SCC), are finally given. Index Terms Noise models, oerational amlifiers, sectral correlation coefficient. I. INTRODUCTION Recently, integrated oerational amlifiers (o ams) have been used in more and more ractical alications. With the continual imrovement of their noise characteristics, they have been commonly found in the design of reamlifier circuits. For this reason, the calculation of the circuit noise of an o am and its low-noise design are aid more attention than ever. At resent, the noise models [] [3] of the overwhelming majority of o ams are illustrated as in Fig. (a) and (b). The commonly acceted two-ort noise model is in Fig. (a). The o am is considered noiseless and the equivalent voltage noise source e n Manuscrit received June, 998; revised May 0, 999. This work was suorted in art by the China Natural Science Foundation under Contract This aer was recommended by Associate Editor K. Halonen. J. Xu and Y. Dai are with the School of Information Science and Engineering, Jilin University of Technology, Changchun, China D. Abbott is with the Centre for Biomedical Engineering (CBME), Electrical and Electronic Engineering Deartment, the University of Adelaide, Adelaide, SA5005, Australia. Publisher Item Identifier S 057-7(00) and current noise source i n are referred back to the inut terminals. Fig. (b) is commonly adoted when the ositive terminal is grounded. To simlify calculation, in some models only e n is adoted and i n is neglected [4], [5]. The advantage of these equivalent circuits is simlicity and convenience. However, in the area of small-signal detection, the requirements of noise secifications in the course of calculation and design of a low-noise circuit become higher. The shortcoming of Fig. (a) and (b) is obvious: the correlation between voltage noise source e n and current noise source i n is not considered, giving rise to inaccuracy. At resent, methods for measuring e n and i n [6], [7] use a small value of source resistance to measure an equivalent inut voltage noise e n and use a very large source resistance to measure an equivalent inut current noise i n. Because the correlation is not considered in this method, the measuring method is only an aroximate solution. In fact, it can be calculated that the neglect of the correlation item can lead to, at most, a 40% measurement error [7]. Thus, it is commonly believed that the method can give only an aroximate solution, and cannot give an accurate solution. To solve this roblem, a more comlete o am noise model is resented in this aer, based on Fig. (c), which considers the correlation between e n and i n for each inut terminal and then the formula of equivalent inut noise ower sectrum density for the inverting and noninverting inut terminals can be derived. With different source resistors, the noise model arameters of an o am have been measured by means of a low-frequency noise measuring system and the noise model arameters, including the sectral correlation coefficient, are resented. II. A COMPLETE NOISE MODEL AND ITS EQUIVALENT INPUT NOISE POWER SPECTRUM In order to imrove recision of the noise model, based on Fig. (a) and (b), we use one equivalent voltage noise source and one equivalent current noise source at each o am inut terminal in our model. Second, it should be ointed out that the correlation between e n and i n at each inut terminal should be considered for comleteness. Let = + j be the sectral correlation coefficient (SCC), given by = S ei (f )= S e (f )S i (f), in which S e (f ), S i (f) are the ower sectral densities of the voltage noise e n and current noise i n, resectively, and S ei (f) is the cross-sectral density [8] between e n and i n. Also let 0 = 0 + j 0 be the SCC between e n and i n, in which i n and i n are current noises at two inut terminals of an o am. Thus, it can be concluded that there is no correlation between them. Fig. (c) is a comlete o am noise model including eight arameters, i.e., e n, i n, = + j, e n, i n, and 0 = 0 + j 0, each of which varies with frequency. It is obvious that all these arameters cannot be calculated by use of internal noise sources of an o am, for noise sources in an o am are so many that it is very difficult to calculate them searately and accurately. However, they can be calculated by measuring equivalent inut noise ower sectrum with different source resistors. Now the relation between the eight arameters and equivalent inut noise ower sectrum can be derived as follows. Let Z = R + jx, Z = R + jx and Z f = R f + jx f, e =4T R, e =4T R, i =4T =R f f, where e and e are the thermal noise sectrum of resistance R and R, i f is the current noise sectrum of resistance R f. According to Fig. (a), its equivalent noise circuit can be drawn as in Fig. (b). According to the suerosition theorem, the gain of each noise source can be calculated first and then the total outut noise can be obtained by addition of each noise source ower. Multilication by the square of the noise bandwidth finally gives the outut noise ower 057 7/00$ IEEE Authorized licensed use limited to: Adelaide University. Downloaded on October 3, 008 at 0:58 from IEEE Xlore. Restrictions aly.

2 4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATION, VOL. 47, NO. 3, MARCH 000 (a) (b) (c) Fig.. Equivalent noise models for an o am. (a) (b) Fig.. (a) Inverting inut o am circuit. (b) The equivalent noise circuit of the inverting inut o am. sectrum. Therefore, from Fig. (b), the contribution of inverting inut terminal noise sources to the outut noise can be exressed as S 0 0(f )=e Z + fe n +(i n + i f ) jz == j +eninre[(z == ) 3 ]g + Z f : () Z The contribution of noninverting inut terminal noise sources to the outut noise could be exressed as S+(f 0 )= fe n + i n jz j + e +eninre[ 0 Z 3 ]g + Z f : () Z Therefore, the total outut noises contributed by the two inut terminals are S 0(f) =S 0 0(f) +S 0 +(f ): (3) The exressions of total outut noise referred to inverting inut terminal and noninverting inut terminal can be exressed as follows. The equivalent noise ower sectrum of inverting inut terminal is S 0 (f) = S 0(f) = e + fe n +(i n + i f )jz == j Z +eninre[(z == ) 3 ]g + Z + fe n + i n jzj + e +eninre[ 0 Z 3 ]g + Z : (4) The equivalent noise ower sectrum of noninverting inut terminal is S +(f )= S 0(f) + Z f Z = e + Z + fe n +(i n + i f ) jz == j +eninre[(z == ) 3 ]g + fe n + i n jzj + e +eninre[ 0 Z 3 ]g: (5) Now let us discuss (4) and (5). If the correlation between en and in, en, and in is neglected, then (4) can be simlified as S 0(f) =e + fe n + e n + e ) + Z +(i n + i f ) jz j + i n jz j + Z : (6) When Z = R, Z = R, =, (6) is equal to [7,. 59, (3-7)], which means that (4) is more general. Under the same conditions, (5) can be simlified as S +(f )= e + Z + fe n + e n + e + i n jzj +(i n + i f ) jz == j g: (7) According to the same conditions, (7) is equal to [7,. 60, (3-8)], which means that (5) is also more general. In addition, from (4) and (5) it can be theoretically concluded that the equivalent voltage noise sources en and en cannot be calculated Authorized licensed use limited to: Adelaide University. Downloaded on October 3, 008 at 0:58 from IEEE Xlore. Restrictions aly.

3 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATION, VOL. 47, NO. 3, MARCH Fig. 3. The measurement system block diagram. searately. Therefore, for calculation convenience, it is suosed that e n is equal in magnitude to e n. Hence, let e n = e n + e n. Then (4) and (5) can be changed to TABLE I AND THE SPECTRAL DENSITY THE VALUES OF R, R S0(f )=e + fe n +(i n + i f ) jz ==Z f j + e ni nre[(z ==Z f ) 3 ] + i n jz j + e + e ni nre[ 0 Z 3 ]g + Z Z f (8) TABLE II THE VALUES OF R, R, R, AND THE SPECTRAL DENSITY S + (f )= e + e n +(i n + i f ) jz ==Z f j + Z Z + e n i n Re[(Z ==Z f ) 3 ] + i n jz j + e + e ni nre[ 0 Z 3 ]: (9) It should be noted that S + (f ) and S 0 (f) are different because the two inut terminal voltage gains are different. III. MEASUREMENT SYSTEM AND MEASUREMENT METHOD Fig. 3 is the measurement system block diagram. In order to refer the measured outut noise to the inut terminals, it is necessary to have measured the frequency resonse A(f ) of the o am and measuring system. Therefore switch S is at A first. Then the switch S is at B to measure the outut noise ower sectrum of an o am. We measure the outut noise ower sectrum S o (f) by means of an FFT sectrum analyzer (model: CF-90), then the equivalent inut noise ower sectrum is given by S i (f) = S o(f) A (f ) where A(f ) is the gain of measurement system, including the gain A (f ) of amlifier measured and the measuring system gain A (f ), namely, A(f )=A (f )A (f). In this system the cross-sectrum estimation method is used to reduce reamlifier noise contribution because the noise of the two reamlifiers are uncorrelated (owered by different batteries) and their cross-sectrum value is very small and, as a consequence, a small noise value (nv = Hz) can be measured. The cross-sectrum estimation is measured in the frequency range of Hz 00 khz. The measuring rocess and data rocessing are automated by comuter. The measurement software is chiefly made u of two arts: ) the IEEE-488 interface to control the CF-90 and ) the data rocessing arameter calculation from the measured results and dislay rograms. In order to accurately measure equivalent inut noise sectrum, A(f ) is obtained by measuring the swet sine wave resonse. Simultaneously, in this measurement system 5 sectral averages are erformed to maintain the high recision of sectrum estimation. The measured results have shown that the accuracy of measuring system is suerior to 4% [9]. The measuring method is as follows. According to (4) and (5), it is seen that the equivalent noise ower sectrum of inverting or noninverting inut terminals can be measured by varying source imedances and then the noise model arameters can be calculated accurately. The calculation formulae are derived as follows. A. Inverting Inut Terminal ) Let X = X = X f =0, R =0, then (8) changes into S 0 (f) =e + e n + R R f +(i n + i f )R + e n i n R + R : (0) R f The values of three different resistors R and R f are shown in Table I. The equivalent inut noise ower sectra of three source resistances are measured, resectively, and the results are as follows (where K = R f =R ): i n = R R 0 R R 3 S (f)r 0 S (f)r 0 S (f)r 3 0 S 3 (f)r R 0 R R 0 R 3 () e n = i nr R 0 S (f)r 0 S (f)r R 0 R + K () Authorized licensed use limited to: Adelaide University. Downloaded on October 3, 008 at 0:58 from IEEE Xlore. Restrictions aly.

4 43 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATION, VOL. 47, NO. 3, MARCH 000 Fig. 4. The measured results of an o am. (a) Equivalent inut voltage noise e versus frequency. (b) Equivalent inut current noise i and i versus frequency. (c) Real art of SCC and versus frequency. (d) Imaginary art of SCC and versus frequency. TABLE III COMPARISON BETWEEN OUR MEASURED VALUES AND TYPICAL DATASHEET VALUES AT 3SPOT FREQUENCIES (nv= Hz) = eninr + K S (f ) 0 4T R 0 e n + K 0 (in + i f )R : (3) ) Let X = Xf = R =0, =k, C =4F, R = 00, the corresonding equivalent inut noise ower can be measured and the exression is obtained as S0(f) =4T R + [X +(R + ) ] R f en +(in + i f )(R + X ) [R (R + )+X ] + enin + enin X : (4) Then, can be calculated as = S0(f) 0 4T R 0 [X +(R + ) ] R f e n 0 (i n + i f )(R + X ) 0 [R (R + )+X ] enin eninx : (5) 3) For the case of X = X = Xf =0, R = 00, =k, then it leads to S0(f) =e + e n + R +(in + i f )R + eninr + R + i nr + e + eninr 0 + R : (6) Let S(f) =S 0 (f) 0 e 0 en + R 0 (in + i f )R 0 eninr + R : (7) When source resistance R varies with Table II, the equivalent inut noise ower sectra can be measured, resectively, and then we can obtain i n = S 4(f)R 5 0 S 5(f)R 4 + R 4R 5(R 4 0 R 5) K 0 = eninr 4 S 4 (f) + K 4) For the case of X = Xf =0, then we have (8) 0 i nr 4 0 4T R 4 : (9) S 0 (f) =e + en + R +(in + i f ) R + eninr + R +[i n(r + X )+4T R + enin (R 0 + X] 0 + R (0) Authorized licensed use limited to: Adelaide University. Downloaded on October 3, 008 at 0:58 from IEEE Xlore. Restrictions aly.

5 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATION, VOL. 47, NO. 3, MARCH TABLE IV COMPARISONS BETWEEN OUR MEASURED VALUE AND TYPICAL DATASHEET VALUES AT THREE SPOT FREQUENCIES (A= Hz) than that of high-frequency region. If the correlation is neglected, then the errors in the low-frequency region become bigger. In addition, the results also show that the noise model and measurement method roosed in this aer can imrove measurement and calculation recision in low-noise circuit design. where R = 00 ; = k; R = 00 ; C = 4F; R = R +! R C!RC X = +! R. C According to the noise ower sectrum of the measured equivalent inut S0(f ), then we can have 0 = eninx S 0(f) 0 e 0 e n + R R 0 eninr + R 0 i n(r + X )+4T R + 0 (in + i f ) + R eninr 0 : () In this way, all the noise model arameters (en, in, = + j, in, and 0 = 0 + j) 0 of an o am can be obtained by means of measuring the equivalent inut noise ower sectrum with varying source resistance. B. Noninverting Inut Terminal In the same way, for the case of the noninverting inut terminal, all the noise model arameters can be also calculated with varying source imedance. For brevity, the formulae are not given here. IV. THE MEASURED RESULTS By use of the method and the measuring system above, a commercial o am (OPA37A) has been measured and the results are shown in Fig. 4. From Fig. 4(a) and (b) it can be seen that when the correlations between en and in, en, and in are neglected, then the measured results of both en and in are larger than the values when the correlation are considered. Thus, an overestimate results when the widely used method to measure en and in is carried out, where correlation is neglected and only noise contributions of a small source resistance and a large source resistance are considered in the course of noise model arameter calculation. Our measured results also show that the correlation between en and in, en, and in do exist, esecially in low frequency. When =f noise dominates, the correlation coefficients between en and in, en, and in become bigger. From Table III and IV (tyical values can be found in the Burr Brown datasheet) it can be seen that in low-frequency region the errors are bigger than those in the high-frequency region, which in another way demonstrates that the correlation does exist in the low frequency region and is stronger V. CONCULSION According to the analysis and measured results, the following conclusions can be reached. ) In contrast to the noise model commonly used at resent, a comlete noise model and its measurement method for an o am are roosed in the aer. Because contributions of all noise sources to outut noise and the correlation between en and in, en, and in are considered adequately, the noise model arameters can be obtained accurately through this way. ) The formulae of comlete equivalent noise ower sectra for inverting inut and noninverting inut terminals are derived, in which the contributions of all noise sources to outut terminal and correlative coefficients between them are included. 3) The main advantage of this method is that all internal arameters of an o am do not need to be known in advance for noise model arameter calculations. And the exerimental results are in good agreement with theoretical analysis. ACKNOWLEDGMENT The authors wish to thank the associated editor and reviewers for a number of suggestions. REFERENCES [] D. F. Stout, Handbook of Oerational Amlifier Circuit Design. New York: McGraw-Hill, 976, [] J. R. Hufault, Oerational Amlifiers Network Design. New York: Wiley, 986, [3] G. B. Clayton and B. W. G. Newby, Oerational Amlifiers, B. H. Newnes, Ed., 99, ch. 3. [4] G. Esinosa et al., Noise erformance of OTA-C circuits, in Proc. IEEE/ISCAS, San Jose, CA, 988, [5] P. Bowron and K. A. Mezher, Noise analysis of second-order analog active filters, Proc. Inst. Elect. Eng., vol. 4, no. 5, , Oct [6] J. G. Graeme et al., Oerational Amlifiers: Design and Alications. New York: McGraw-Hill, 97, [7] C. D. Motchenbacher and J. A. Connely, Low Noise Electronic System Design. New York: Wiley, 993. [8] J. Xu, Y. Dai, and Y. Li, The study of the relation between R G noise model and E I noise model of an amlifier, IEEE Trans. Circuits Syst. I, vol. 45, , Feb [9] Y. Dai, Performance analysis of cross-sectral density estimation and its alication, Int. J. Electron., vol. 7, no., , 99. Authorized licensed use limited to: Adelaide University. Downloaded on October 3, 008 at 0:58 from IEEE Xlore. Restrictions aly.

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