Arrhenius average temperature - IEEE Xplore

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IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGIES, VOL. 27, NO. 3, SEPTEMBER 2004

Arrhenius Average Temperature: The Effective Temperature for Non-Fatigue Wearout and Long Term Reliability in Variable Thermal Conditions and Climates Michal Tencer, John Seaborn Moss, and Trevor Zapach

Abstract—This paper presents a method of assessing the effective temperature essential for predicting the temperature acceleration of the wearout mechanisms (other than thermal fatigue) of electronic equipment. This is particularly important for equipment experiencing variable thermal conditions. The approach, based on weighting of thermal acceleration factors, leads to the Arrhenius average temperature e given by (4). e is related to wearout processes and allows one to compare predictions from the thermal design to results of accelerated testing. It has no relation to the maximum component temperature which influences functionality. The method is applicable to outside plant telecommunication equipment as well as in the automotive and aerospace industries. The effect of climate and design constraints on e are discussed. Index Terms—Arrhenius average temperature, thermal acceleration factors, thermal fatigue.

I. INTRODUCTION

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HE CURRENT trend is to place much of the electronic functionality of telecommunication equipment in outside plant (outdoor) conditions and in customer premises in variety of climates. This has created a situation where the temperatures experienced by the critical components and surfaces (e.g., junction temperature ) undergo changes due to: a) seasonal temperature variations; b) diurnal cycles; c) traffic related duty cycles. In this situation the attention of the reliability engineer is mostly paid, and deservedly so, to thermal fatigue and related failures. However, there are many other, thermally activated, failure mechanisms which are not related to fatigue but may decide about a component life span. As their rates are temperature dependent, the equipment designer requires a clear definition of what should be understood as the “long term” effective , relevant to wearout processes and long term temperature, reliability is needed (i.e., for the junction temperature, some ). Such a definition would kind of average temperature enable one to calculate a temperature value characteristic of a Manuscript received June 7, 2001; revised April 29, 2004. This work was recommended for publication by Associate Editor S. H. Bhavani upon evaluation of the reviewers’ comments. M. Tencer is with the Nortel Networks, Ottawa, ON K1Y 4H7, Canada (e-mail: [email protected]). J. S. Moss is with Dependable Technology Planning, Ottawa, ON Canada. T. Zapach is with Nortel Networks, Calgary, AB T3J 3R2, Canada. Digital Object Identifier 10.1109/TCAPT.2004.831834

given design and environment for a given failure mechanism which could be compared with long term reliability data derived from accelerated testing for this mechanism. With the central office equipment and most other indoor electronics the operating temperature is quite constant and is the effective temperature for acceleration assuming that air conditioning or cooling fan failures are rare and short-lived. With outdoor equipment, the maximum temperature and the effective temperature for acceleration of wearout may differ greatly, especially in continental climates. This problem has always existed in the automotive and aerospace electronics. Even though, due to continuous developments in reliability physics, thermally activated wearout processes play a smaller role than they used to, especially in integrated circuits [1], [7], they are still important with many parts (e.g., with electrolytic capacitors, fans, batteries, and disk drives). Several approaches in design practice and they include, have been used to assess without much theoretical or practical justification, the arithmetic and logarithmic (or geometric) average. This paper presents the approach to calculating the average based on weighting of thermal acceleration factemperature tors which is being used in Nortel Networks for thermal design of reliable electronics expected to operate in highly variable thermal conditions, especially wireless and broadband equipment. The properties of this average with respect to different climate conditions are also discussed in the paper. It is stressed here once more that the fatigue wearout processes are not addressed here. Also, the concept of the maximum operating tem, which is related to functionality of compoperature, e.g., nents remains unchanged. II. DEFINITION OF EFFECTIVE TEMPERATURE For the purpose of further considerations we will define the as follows. temperature of a component The effective or average temperature experiencing variable thermal conditions is such temperature which would lead to the same degree of wearout if it was the component’s constant temperature over the same time period. Thus, the component in the field conditions will experience the same time to failure as the component at constant tempera. ture This definition applies to wearout processes that involve thermally activated mechanisms such as a chemical reaction, diffu-

1521-3331/04$20.00 © 2004 IEEE

TENCER et al.: ARRHENIUS AVERAGE TEMPERATURE

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TABLE I HYPOTHETICAL CASE WITH TWO TEMPERATURES: T

= 20 C (293 15 K), = 40 C (313 15 K), 1 = 50 C :

sion, creep etc. Thermally driven fatigue wearout, on the other hand, is not included. It does not apply to the component’s functionality or random failures. Its applicability to defect related early failures is limited to those ones which are thermally activated, e.g., in the case of tantalum capacitors.

T

:

T

If the component’s temperature through its lifetime was con, it would be stant and such that its failure rate coefficient was , according the definition of Secit’s effective temperature tion II (3)

III. ANALYSIS We assume that the long term (wearout) component failure depends on decomposition of a substance “A” whose amount’ measure is . After is depleted to a critically low number ,the failure occurs. can be, e.g., metallization cross section or dielectric thickness in semiconductors or the amount of electrolyte in a capacitor. The rate of such process can be zero , first order , order, i.e., independent of , etc. [2]. In the temperature depensecond order dent proportionality coefficient or rate coefficient, , are hidden material properties and externally imposed conditions like geometry, voltage or current density. Let us divide the life of the component is into time segments (run at different temperatures) in which the rate coefficient is . As shown in the Appendix A, the effective rate coefficient of the process is, independently of its order, given as

Comparing (1)–(3) and solving for for the effective temperature as

we get the expression

(4a) If, as it is often the case, we have the temperature data for equal discrete intervals we have upon small modification of the previous equation (4b) and if we have a theoretical climate model with a continuous temperature change the sum can be replaced with an integral (4c)

(1) In other words, the effective rate constant is the time-weighted average of all the segmental rate constants. Equation (1) applies to a single failure mechanism. We assume here that the failure is an activated process and the temperature dependence of each of the rate constants is given by the Arrhenius equation (2) where is the activation energy, is the Boltzmann constant, and is the absolute temperature (K) during the given time segment . In more advanced treatments of reaction kinetics the Arrhenius equation is replaced by a similar Eyring equation where, additionally, a weak temperature dependence of the preexponential factor A is introduced. However, in the area of reliability the field or accelerated data never have enough resolution to distinguish between these two models and thus we settle here for the Arrhenius dependence which, although mathematically simpler, can still be traced back to statistical mechanics and the Boltzmann energy distribution. It is important to stress here that the Arrhenius equation applies only to thermally activated processes where the process (chemical reaction, diffusion, etc.) has to overcome an energy barrier and does not apply to nonthermal mechanisms.

The effective temperature thus derived will be called here the Arrhenius average temperature . As we can see, unlike with the arithmetic and logarithmic average, is strongly dependent on the activation energy of the process. Thus, different components and even different failure mechanisms of the same component may have different effective temperatures. Thus, it important to bear in mind that there is no universal effective temperature, even for one component, as there is no single activation energy which can be assigned to a component. Usually, however, only one of the failure mechanisms (the fastest one) may be of practical importance and has to be dealt with. IV. ARRHENIUS AVERAGE TEMPERATURE AND CLIMATE The following examples will illustrate the properties of . Let’s assume that a piece of equipment is experiencing two different temperatures of equal intervals: nighttime and daytime . If there is no energy dissipation we get for different activation energies from (cli(4) Arrhenius averages which are given in Table I as increases strongly with the activamate). We can see that tion energy and is considerably higher that either the arithmetic

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TABLE II CLIMATE ARRHENIUS AVERAGE TEMPERATURES FROM HOURLY DATA FOR REPRESENTATIVE US LOCATIONS (AVERAGE FOR YEARS 1961–1990)

or geometric average, especially for high . Now, if additionof 50 due to ally a component experiences a constant thermal energy dissipation, the Arrhenius average temperature calculated from (4) for this situation will be appropriately higher (comand its values for different are given in Table I as are also ponent). For comparison, values of given in the table. value We can see that by calculating (component) we make a very small rather than the actual error on the side of caution because the latter is close in value but greater than the former. It can be easily shown, that these increases. two values are even closer when in deThus, a simple and effective approach to estimate (climate) for different geographical sign practice is to have locations tabulated and to add the characteristic of a given design and component. A few examples of such climactic data for several locations are given in Table II. The data was obtained using measured hourly ambient temperature data obtained from the Samson Data Set [3] for 30 years (1961 to 1990).1 When comparing Phoenix versus Key West in Table II we can see that a location with a lower arithmetic average can . This is because in Key West temperature is have a higher fairly constant while in Phoenix it is more variable and, in (4) hot periods have a higher weight than colder ones. The notion of the climate Arrhenius temperature serves two purposes: 1) to show an approximate, quick, way of deriving TArrh for a given mechanism; 2) to compare the severity of different climates vs. each other and versus indoor conditions. For instance, it can be seen that for failure mechanisms with activation energy above 0.75 eV the climate in Arizona is more thermally severe than that in Florida even though the latter has a higher yearly average temperature. Another important observation with most locations, even s do not exceed those expected in an fairly hot ones, is that air conditioned central office as specified by Bellcore [4]. Thus, in most cases the Arrhenius average temperature is not the most 1Interestingly, when compared year by year, a steady increase of T is observed over the period of last 30 years. Although this could be attributed to global warming, in most cases it is likely due to the cities growing and getting closer to the airports where temperatures were measured. This points out the necessity of taking microclimate into considerations when designing equipment.

critical reliability concern on going to outdoors conditions, influencing Rather, more important factors are usually functionality and diurnal temperature swings inducing fatigue. Hour-by-hour data are not available for all locations. What is usually available is daily minimum (night) and maximum (day). Using these values rather than the hourly ones will overemphasize the high temperatures, as the temperature tends to be constant at night and peaks during the day. This may overvalued by up to 2–3 for locations with high lead to thermal variations. Fortunately, when using the minimum/maximum data is our only choice, it errs on the side of caution. World based on this type of data used by Nortel designers maps of were created by Rolt [5]. It is also possible to calculate from (4c) assuming a certain continuous time function for . As proposed by Maxwell [6], assuming sinusoidal diurnal and annual variations and taking into account solar loading this function of time would look as

where is time in hours, is the mean temperature , is the component temperature rise, is the annual variation, is the diurnal variation, is the temperature rise due to solar loading, is the diurnal frequency, and is the annual frequency. Using such form with the help of Mathematica, makes it easy to calculate Mathcad, or similar software packages. Some designs use heaters to ensure that, for functionality reasons, temperature never falls below a certain value, typically 0 . Other designs use air conditioning (AC) systems to limit the air temperature from the top, typically ensuring that it never increases above 30 . Details of such design modifications for discussed above locations are given in Appendix B. Here, we will only discuss general conclusions. Inclusion of heaters does , either in hot (Phoenix, Key West) or in modnot influence erate locations (Seattle), increasing it only in cold places like Fairbanks (from 7.2 to 8.7 ) where it is low anyway and is not a reliability concern. Limiting top temperatures with air condionly in hot continental locations like Phoenix tioners lowers (from 26.7 to 23.7 ) where extreme temperatures occur often but neither in Key West where it only rarely strays from the average (25.3 ) nor in moderate or cold climates.


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V. CONCLUSION Although for most electronics due to the progress in the physics of failure thermally driven wearout (Arrhenius mechanisms) was replaced by fatigue as the primary cause of failure, there are components where Arrhenius wearout does limit the for calcufailure rate. The proper effective temperature lating the nonfatigue wearout life in conditions of variable temperature, especially outdoors, is the Arrhenius average temincreases strongly with perature calculated from (4a)–(4c). and for typically encountered values the activation energy of is higher that either the arithmetic or geometric average. Different failure mechanisms may have different Arrhenius is to add average temperatures. A good approximation of characteristic of a given design to the temperature rise (climate) which can be tabulated for different locations. Proper averaging shows that there is minimal contribution to Arrhenius degradation from winter and night time in all continental and even many maritime climates. Typically, from the point of view of thermally activated failure mechanisms outside plant applications are little if any more severe than equivalent central office applications. Including heaters in the design only influin very cold climates where it is low anyway while ences only in hot continental climates. air conditioners influences APPENDIX A EFFECTIVE RATE CONSTANT We assume that the long term (wearout) component failure depends on decomposition of a substance “A” whose initial amount is . After A is depleted to a critically low number , the failure occurs. The kinetics of this process is either 0th order in which the rate of the process is independent on the amount of A, first order where the rate of the process is proportional to the amount of A at a given time, or the second order where it is proportional to . These three cases can be expressed, respectively, as order

(6a)

first order

(6b)

second order

(6c)

where is the rate coefficient of the process. The initial confor , and we have, dition for all three cases is respectively [2] order

(7a)

first order

(7b)

second order

(7c)

Lets now subject A consecutively to an array of conditions where rate constants are: over the period , over the peover the period , etc. The final amount of A riod left unchanged after the th segment is, respectively order

(8a)

first order

(8b)

second order

(8c)

We can visualize an equivalent process with such a rate conover the total elapsed time that would lead to stant value. This rate constant will be called here the same final the effective (or average) rate constant order first order

(9a) (9b)

second order

(9c)

Comparing (8) and (9) in series we arrive for each case at (10) It can be easily shown that the same result would be also obtained if any other rate order (third, etc., or a fractional one, e.g., 1/2) was operative. It may be worth stressing here that the fact that there is an exponential in the applied kinetic equations does not mean or imply that we have to have exponential distribution of failures. The kinetic equation refers to rates of depletion of a structural material in the component leading to failure, not to the distribution of failures in time. In fact, wearout failures characteristically have narrow distribution and in the same conditions happen almost simultaneously while exponential distribution is characteristic of random failures. APPENDIX B EFFECT OF HEATING AND AIR CONDITIONING FOR SELECTED LOCATIONS The analysis presented here was conducted with measured hourly ambient temperature data collected from the Samson Data Set [3] for the locations of Table II. The values quoted are arithmetic averages of 30 yearly Arrhenius averages over the years 1961 to 1990 for the stated location. All averages were calculated using (4) and a range of activation energies (0.25–1.50 eV). The effect of heaters, air conditioning (AC), temperature offset and climate type were investigated. Heaters “on” mean . that minimum ambient temperature was constrained to 0 Air conditioning “on” means that maximum ambient tem. If an offset is indicated perature was constrained to 30 was added to the ambient temperature then a constant after it was modified by the heater or air conditioning and . Combinations of then subtracted from the thus calculated heater, AC, offset, and climate was simulated. The nine test cases are given in Table III. The result of simulations are given in Table IV. The following conclusions were drawn from the simulation.

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TABLE III TEST CASES FOR NUMERICAL STUDY OF ARRHENIUS AVERAGE TEMPERATURE

TABLE IV ARRHENIUS AVERAGE TEMPERATURES IN DEGREES CELSIUS. SIMULATION RESULTS FOR TEST CASES IN TABLE III

Effect of activation energy: Higher activation energy results in higher Arrhenius average temperatures. When a climate has large temperature variations, Arrhenius average temperatures increase more dramatically as activation energy in increased. Effect of offset: Increasing temperature offset reduces Ar), rhenius average temperatures. In the limit (very large the Arrhenius average temperature approaches the arithmetic average temperature. This is counterintuitive and results from the reduced relative difference between absolute temperatures.

Effect of heaters: As expected, in cold climates heaters increase the Arrhenius average temperature. This does not effect reliability of equipment because the climatic Arrhenius average is very lower in cold climates anyway. Effect of AC: As expected, in hot continental climates AC reduces the Arrhenius average temperature. Effect of climate: Climate has the most significant influence on Arrhenius average temperature. High Arrhenius average temperatures occur in tropical and desert climates. The worst case Arrhenius average temperature for North America occurs in the South West desert and ranges from 24 to


30 . Air conditioning reduces this average by no more than . Adding heaters to equipment in cold climates increases 5 the Arrhenius average temperature but not to a significant value. Arrhenius averages produced in moderate and humid climates are not effected by heaters or AC. Adding an offset temperature to a climatic Arrhenius average will result in a conservative (high) component Arrhenius average temperature. REFERENCES

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John Seaborn Moss was born in London, ON, Canada in 1939. He received the B.Sc. degree in physics and mathematics from the University of Western Ontario, London, ON, Canada and the Ph.D. degree in solid state physics from McMaster University, Hamilton, ON. From 1973 to 1999, he was with Nortel Networks on the design, materials, dependability, manufacturing, and environmental aspects of a wide range of different systems. Since 1999, he has been President of Dependable Technology Planning, Ottawa, ON.

[1] F. Jensen, Electronic Component Reliability. New York: Wiley, 1995. [2] C. Capellos and B. H. J. Bielski, Kinetic Systems. New York: WileyInterscience, 1972. [3] Solar and Meteorological Surface Observation Network 1961–1990, U.S. Department of Commerce, National Climatic Data Center, Sept. 1993. [4] Bellcore, “NEBS, TR-NWT-000 063,” Bellcore Inc., Sept. 1993. [5] S. Rolt, “Personal Communication,” Nortel Networks Europe, Harlow, U.K., 2003. [6] D. Maxwell, “Personal Communication,” Nortel Networks Europe, Paignton, U.K.. [7] M. Pecht, “Physics-of-failure approach to design and reliability assessment of microelectronic packages,” in Proc. 1st Int. Symp. Microelectronic Package and PCB Technology, Beijing, China, Sept. 19–23, 1994, pp. 175–180.

Michal Tencer received the M.Sc. degree in organic chemistry from Warsaw University of Technology, Warsaw, Poland, in 1970 and the Ph.D. degree in physical organic chemistry from the Institute of Organic Chemistry, Polish Academy of Sciences, Warsaw, in 1975. From 1975 to 1981, he was with the Institute of Nuclear Research, Swierk, Poland. From 1981 to 1988, he held several academic positions in physical organic chemistry, polymer chemistry, and photophysics and materials for microelectronics at the Swiss Federal Institute of Technology (ETH), Zurich, the University of Toronto, Toronto, ON, Canada, and Lehigh University, Bethlehem, PA, respectively. From 1988 to 1991, he was Technical Director ofAcrylium-Microbex, Inc., Unionville, ON. Since 1991, he has been with Nortel Networks (formerly Bell-Northern Research), Ottawa, ON, where he is currently Senior Member of Scientific Staff working on materials, reliability, packaging, and weatherproofing for wireless telecommunication and optoelectronics.

Trevor Zapach received the B.Sc. degree in mechanical engineering from the University of Alberta, Edmonton, AB, Canada, in 1991 and the M.A.Sc. degree in computational fluid mechanics and heat transfer from the University of Victoria, Victoria, BC, Canada, in 1994. He has been with Nortel Networks, Calgary, AB, Canada, since 1994 where he advises on the thermal and mechanical design aspects of cell phone base stations.