Monte Carlo Reliability Model for Microwave Monolithic Integrated Circuits

Transcription

1 Monte Carlo Relablty Model for Mcrowave Monolthc Integrated Crcuts Ars Chrstou Materals Scence and Engneerng, Relablty Engneerng Unversty of Maryland College Park, MD Abstract A Monte Carlo smulaton s reported for analog ntegrated crcuts and s based on the modfcaton of the falure rate of each component due to nteracton effects of the faled components. The Monte Carlo technque s the methodology used to treat such crcuts, snce they are ndependent of the number of components and the degree of system complexty. The relablty model s applcable over a wde temperature and bas range and may be used to predct relablty of mcrowave systems. The model s compared to accelerated test results of two analog mcrowave crcuts. Excellent agreement has been obtaned for a low nose amplfer as well as for a transmpedence amplfer. INTRODUCTION Assessng the hgh temperature behavor of MMICs (Monolthc Mcrowave Integrated Crcuts) from ndvdual (Feld Effect Transstor) relablty s an mportant practcal problem. The relabltes are often assessed by lfe tests conducted under controlled test envronments - accelerated lfe testng. Testng an entre MMIC, or even ts components, under the actual operatonal envronments s rarely feasble. In assessng the MMIC relablty, prevous nvestgatons were based on MIL-HDBK-27 [] and smply assumed that the actve and passve components are statstcally ndependent of each other. Ths s often napproprate, and therefore correlaton coeffcents must be used. In the case of a complex MMIC crcut, t s not plausble to attan the analytcal relablty by the Markov approach [2] for constant falure rate, whch perhaps s the best and most straghtforward analytcal approach to computatons n systems wth dependence. The equatons become numerous and out of control for a large MMIC system, and the Markov method may break down when falure rates become nonconstant. The Monte Carlo technque s an approprate methodology used to treat such crcuts, snce they are ndependent of the number of components and the degree of system complexty [3]. The present report ams at establshng a relablty model to predct the relablty of MMICs by usng Monte Carlo technques. The relablty model wll be applcable over a wde temperature range and hence may be used for mcrowave systems. I. THE METHODOLOGY TO ESTIMATE MMIC HIGH TEMPERATURE PERFORMANCE The Jont Probablty Method va Monte Carlo Smulaton Theoretcally, a component-dependent MMIC system can be represented by a seres of jont probablty densty functons for the remanng tme to falure of the survvng components. For a set of n correlated (dependent) components wth random tmes-to-falure t, t 2,, t n, the jont probablty densty and cumulatve dstrbuton functons can be expressed as f F ( t,t,,t ) f ( t ) f ( t t ) f ( t t,t ) f ( t t,,t, ) = (), 2,, n 2 n , 2,, n n n 2 t,,,! ( t,t,,t ) F ( t ) F ( t t ) F ( t t,t ) F ( t t,,t, ) = (.2), 2,, n 2 n , 2,, n n n 2 t,,,!

2 where f (t ) and F (t ) are probablty densty and cumulatve dstrbuton functons of component (the frst faled component), and f,2,, (t t -,, t 2, t ) and F, 2,, (t t -,, t 2, t ) are the condtonal probablty densty and cumulatve dstrbuton functons of component gven that components, 2,, - have faled. Snce the random tmes-to-falure are dependent, a set of unformly dstrbuted numbers can not be used to generate the tmes-to-falure correspondng to components, 2,, n. An alternatve method s to let (x, x 2,, x n ) denote a set of unformly dstrbuted random numbers whch are between 0 and. Then the random tme to falure t correspondng to the frst faled component can be determned from! t = F (3) ( ) x Wth the value of t known, the condtonal dstrbuton functon F, 2 (t 2 t ) becomes a functon only of t 2, and t can be nverted to fnd t 2 as ( x )! 2 F2 2 t Ths recursve procedure s contnued untl the last tme to falure t n s generated as: t n t! n = (4) ( x t,,t,t ) = F (5) We can repeat the above procedure untl a desred samplng sze N s obtaned. The relablty and mean tme to falure (MTTF) of the system can then be estmated as n n! 2 Ns R = (6) N N! = TTF MTTF = (7) N where N s s the number of successes (.e., random tme to falure s greater than desgnated lfetme) and TTF s the random tme to falure for samplng. The above technque s applcable for cases where the jont cumulatve dstrbuton functons are known. In the case of a complcated MMIC, however, the jont cumulatve dstrbuton functons are not easly obtaned. Therefore, an alternatve method has been proposed and appled to estmate the relablty of MMIC by ntroducng a weghng factor w(n f, t) whch wll be dscussed later. The Non-Markovan Method va Monte Carlo Smulaton Most IC system relablty studes assume that the components' falure rates λ are constant [4]. Ths s a very common assumpton for most applcatons. However, f the assumpton of constant falure rate s not vald such as n MMIC crcuts, then the system becomes non-markovan [5] and addtonal technques are requred for handlng ths process (MMIC crcuts are non-lnear). The way of generatng the hstores for a non-markovan system s the same as that for a Markovan system [6]. Any one of the generated hstores s composed of many tme-segments and each tme-segment represents a state change. The total falure rate of the system s gven as: # n! ( t) = " ( t) = (8)

3 where λ (t) s the falure rate of component at tme t. The probablty densty f(t) that the state change wll occur at tme T + t, f the prevous state change occurred at T, would be f ' % % & t ( t) = (( t) exp ) *(( x) dx, T < t <! Therefore, the cumulatve probablty F(t) that there s a state change before t, f the last state change at T, s gven by T $ " " # t ' y $ % F dy (0)! %! dx " "" & T # ( t) = (( y) exp ) (( x) In a Monte Carlo smulaton, a random number s generated as x, unformly dstrbuted between 0 and, to stand for the cumulatve probablty functon F(t),.e., T T +" ( t % & # x = F( ") =! )( t) exp! ( ) &* ) x dx # dt () T ' T $ The tme to falure t for ths partcular tme-segment wll then be calculated by the followng equaton. " = F! x (2) Ths nverson of F(t) can be carred out ether analytcally or numercally. The above procedure s repeated untl the desred samplng sze s obtaned. The relablty and MTTF are determned by Equatons (8) and (9). The MMIC Monte Carlo Technque For the MMIC Monte Carlo smulaton, t s convenent to defne the nter-component dependence by modfyng the falure rate for each survvng component due to the nteracton effects of the faled components. The falure of a component would then nvolve choosng the proper combnaton of components and the correspondng falure rates to generate the remanng tmes-to-falure. The modfcaton of the falure rates of dependent-components may not have any dentfable pattern, and may nvolve changng the type or parameters of probablty densty functon. For a gven component, the falure rate changes are expected to depend on the faled components. The modfed falure rate can be expressed as: where ( ) ( t) =! ( t) +! ( t) +! ( t) = W( n, n )" ( t)!' (3) j cc f! λ s the falure rate due to component tself, λ j s the falure rate due to nteractons between components and j, λ cc s the falure rate due to common cause, and W(n, n f ) s a weghtng functon of component n and faled components n f. The functon W s always equal to or greater than. If t s, then there s no nteracton between components. If t s very large, then there s strong nteracton between components and these components can be put n seres n the relablty block dagram. (9)

4 II. MMIC CIRCUIT RELIABILITY MODEL The Gven Condtons for MMIC Relablty Model In general, several condtons must be gven n order to establsh a practcal MMIC crcut relablty model, and these are summarzed as follows: ) The MMIC system s composed of m statstcally-dependent subsystems (or stages, Fgure ), whle the th ( =,, m) subsystem (or stage) conssts of n statstcally-dependent and non-reparable components. Therefore, the MMIC system conssts of n components where: n = m! = n (4) and where each component s n ether a faled or operatng state. 2) Dependent falures can be due to a common cause (the falure of multple components due to a sngle mechansm such as catastrophc or envronmental falure), to nteractons wthn a subsystem, and to nteractons between subsystems. Due to the component falure nteracton, the falure rate of the component (or subsystem) would ncrease once the neghborng components (or subsystems) have faled.

5 Out PC: : Passve Component Feld Effect Transstor, Actve Component PC Subsystem PC2 Subsystem 2 PC3 PC4 PC5 Subsystem 3 PC6 PC7 PC8 PC9 PC0 PC PC2 In Fgure A Typcal Relablty Block Dagram of a Mult-Stage MMIC 3) A sngle falure mechansm can affect several components, and a gven component can be affected by several mechansms and these mechansms are statstcally-ndependent. 4) Falure rate λ of a component would be the sum of λ (due to falure mechansm for component tself), λ j (due to nteracton by component j) and λ cc (due to a common cause such as catastrophc falures whch result a system falure as a whole). The nteractons between passve components wll be neglected. 5) Fgure 2 shows the relablty schematc of a TIA MMIC system. The effects of nteractons among seres connectons may be neglected, snce the path assocated wth the faled component has also faled. For example, f component 5 n Fgure 2 has faled, the path (5-6) through t has also faled (open crcut). Component 6 s assumed to be non-operatng. The nteracton caused by components 5 and 6 therefore can be gnored.

6 In Out Fgure 2 The Relablty Block Dagram of TIA 6) The effects due to nteractons are the same for the survvng components for the same subsystem. However, the effects of nteractons among parallel actve redundant s wll be taken nto account. As known, the stress of an actve redundant component wll ncrease once the neghborng components have faled. The stress wll also ncrease wth respect to the number of faled components, and ths causes the survvors to have a hgher falure rate. Referrng to Fgure.2, where! 2 >! >! 0 λ 2 s the falure rate of component 9 (or 0) for gven faled paths -2 and 7-8, λ s the falure rate of component 9 (or 0) for gven faled path ether -2 or 7-8, and λ 0 s the falure rate of component 9 (or 0) for no faled path. (7) The effects due to nteractons are the same for the survvng components for the same subsystem. For example, f component 7 (or 8) has faled, ts effects on components, 2, 9 and 0 are the same. (8) Interactons among components and subsystems are estmated through correlatons determned expermentally f t s possble, or may be estmated by SPICE crcut analyss. (9) The falure dstrbuton functon s gven for each ndependent component. It can be a mxture of several known falure dstrbuton functons,.e., f = a f + a 2 f 2 +! + a n f n (5) where a s the fracton of the effects due to falure dstrbuton functon f and a + a a n =. The weghtng factors however must be modfed after each component falure. Procedures to Model MMIC Relablty Two cases have been nvestgated, and the results as well as the procedures used are summarzed as follows: Case : If the nteractons between components can be estmated by the correlaton matrx obtaned through SPICE crcut analyss or by experment, then the steps to model the MMIC system relablty are:

7 ) Determne the nteractons between components through SPICE crcut analyss so that the falure weghtng factor W(n, n f ) can be determned. 2) Identfy the falure dstrbuton functon for each ndependent component. Based on the falure dstrbuton functon, select a random number for each component and through the nverse transformaton method calculate a tme to falure for each component. The tme to falure t of a component (.e., ) related to a random number x s obtaned by the proper selecton of the dstrbuton functon. 3) If the predcted tme to falure of a component s greater than a pre-specfed lfe, then the component s operatonal, otherwse t s a falure. Identfy the frst faled component, and set the component tme to falure to be T. 4) Modfy the remanng tme to falure of the survvng components by W(n, n f ). The new tme to falure T ' ( = 2, 3, 4,, n, and n s the number of components consstng the MMIC crcut) wll be " T T ' = T! + (6) W n, n ( )! or ( n, n! ) ( n, n )! W T ' = T + " T (7) W! where ΔT s the dfference between tme to falure T of the survvng component and tme to falure T - of the faled component -. (5) Step (4) s repeated untl the modfed T's of all components are obtaned, determne the system's tme to falure as the modfed T' of the fnal faled component, compare t to the system's msson lfe, and record t as a success or a falure. (6) Step (2) s repeated untl a statstcally adequate samplng sze s obtaned. (7) Calculate the relablty and MTTF by Equatons (8) and (9), and error by the functon, R error = 200 (8) N (! R) % The flow chart for the methodology s shown n Fgure.3. Two types of MMICs, whch are the TIA (Transmpedence Amplfer) and LNAs (Low Nose Amplfer), have been analyzed by applyng ths method. Equaton (8) as s noted, has been derved from the defnton of falure rate λ(t) whch s ( t) ( t +! t) " N f ( t) N( t)! t N f # = (9) The falure rate λ(t) s an approxmately nverse proporton to the survvng tme to falure Δt for a fxed number of falures at tme from t to t + Δt. If λ(t) s ncreased by a weghtng factor W(n, n f ), then Δt wll be reduced by a factor of W(n, n f ). The new survvng tme to falure, therefore, s modfed by Δt/W(n, n f ), and the modfed tme to falure wll be determned by Equaton (8). The relatonshp between the correlaton coeffcent and the weghtng factor s obtaned by assumng that the dfference of tme to falure between the survvng components and the faled component s proportonal to the assocated dfference of current drft [7],.e.,! TTF "!I d (20)

8 Select the desred number N of smulatons to be conduced, and start smulaton and set I = Identfy the falure dstrbutons functon for each ndependent component. Generate a unformly dstrbuted random number [ 0 - ] for each component. Transform the generated random number to the correspondng tme to falure based on ts dstrbuton. Identfy the frst faled component, and set the component tme to falure to be T. Modfy the remanng tmes to falure of the survvng components by W(n, n f ) tll the fnal faled component has been determned. Determne the falure weghtng factor W(n, n f ) by SPICE analyss Determne the system s tme to falure as the modfed T of the fnal faled component Compare the modfed T wth the desred performance to evaluate the MMIC s whether the smulaton s a success or a falure. Is = N? Usng the randomly generated set of parameters to evaluate the random performance of the system and compare t wth the desred performance to determne f t s a success or falure. Fgure 3 The flow chart for the relablty estmaton of TIA and LNA holds. We also note that equaton (8) s applcable for both the TIA and LNA. Based on the lnear regresson method, f two varables (s) have the same (current drft) dsperson,.e., S = S j then the correlaton coeffcent s dentcal to the regresson coeffcent b j and b j,.e., r j = b j = b j, and the followng relaton! T = r! T (2) j j! T = r! T (22) j j where ΔT (or ΔT j ) s the dfference of tme to falure between component (or j) and the faled component j (or ) (Fgure.4).

9 Equaton (23) can be generalzed as: j j j ( T T ) T " = T! r! (23) j ( T T ) " = T! r,!!! T (24) Comparng Equaton (9) wth Equaton (23), the relatonshp between correlaton coeffcent and the weghtng factor s determned by the followng equaton: ( n, n! ) ( n, n )! W! r,! = (25) W W ( n, n ) j! =! r Steps (4) and (5) can be explaned as n Fgure 4 n whch random tmes to falure generated can be arranged so that T < T 2 < < T n and W(, 2,, n - ) s the weghtng factor due to falures of component, 2,, n -. j (26) T ) T 2 T 2 T 2 ' = ) T /W() + T ) T 2 3 T 3 T 3 ' = ) T 2 /W(, 2) + ) T /W() + T ) T 2 /W(, 2) + T 2 ' 4 T 4 ) T 3 T 4 ' = ) T 3 /W(, 2, 3) + ) T 2 /W(, 2) + ) T /W() + T = ) T 3 /W(, 2, 3) + T 3 ' n T n Tn' = ) T n- /W(, 2, 3,..., n-) +... ) T 2 /W(, 2) + ) T /W() + T = ) T N- /W(, 2, 3,..., N-) + T n- ' Fgure 4 The Methodology to Determne the Tme To Falure Case 2: If the correlaton between components can not be estmated by SPICE crcut analyss or any other method, then the steps to model the MMIC system relablty by Monte Carlo technques can be stated as follows: ) Determne from the MMIC crcut the specfc groups of s-dependent components and groups of s-ndependent components, for example, through 4 n Fgure 5 are n an s-dependent group. The falure rate of a component n the s-dependent groups wll be affected by the state (faled or operatonal) of any other components whch are n the same group. 2) Identfy the falure dstrbuton functon for each ndependent component. Based on the falure dstrbuton functon, select a random number for each component and through the nverse transformaton method calculate a tme to falure for each component. 3) If the predcted tme to falure of a component s greater than a pre-specfed lfe, then the component s operatonal, otherwse t s a falure. Determne n whch s-dependent subsystem the faled components belong to f a falure has occurred, and then set the component falure tme to be T.

10 4) If the faled components belong to an s-dependent group, then modfy the remanng lfe of the survvng components n the same by W(n, n f ). W(n, n f ) s determned by assumng that the total stress upon the s-dependent subsystem (stage) s fxed and also the stress upon the component s proporton to the falure rate λ of the component. The new falure rate of the survvng component s obtaned as: n! = n " n (27) ( ) ' o f! & # $ ( o (' = n! (28) % n ' n f " where n s the total number of components n the MMIC system, n f s the number of faled components, λ o s the orgnal falure rate, and λ' s the new falure rate. The weghtng factor W(n, n f ) s estmated by n W( n, n j) = (29) n! n The new tme to falure can stll be determned by Equaton (8). 5) Step (4) s repeated untl the modfed T' of all groups s determned and then determne the system's tme to falure from the modfed T's and compare t to the system's msson lfe. 6) Step (2) s repeated untl a statstcally adequate samplng sze s obtaned. The flow chart for the methodology s shown n Fgure 6. PC f PC2 PC3 PC PC5 PC6 Fgure.5 Relablty Block Dagram of the Low Nose Amplfer

11 Select the desred number N of smulatons to be conduced, and start smulaton and set = Identfy the falure dstrbutons functon for each ndependent component. Generate a unformly dstrbuted random number [ 0 - ] for each component. Transform the generated random number to the correspondng tme to falure based on ts dstrbuton. Identfy the frst faled component and check whch group of s-dependent components belongs to, and set the component tme to falure to be T. Modfy the remanng tmes to falure of the survvng components n the same group wth faled component by W(n, n f) no Fnsh modfcaton? yes Determne the system s tme to falure as the modfed T of the fnal faled component Compare the modfed T wth the desred performance to evaluate the MMIC s whether the smulaton s a success or a falure. no Is = N? yes Calculate the relablty and MTTF by Equatons (.8) and (.9), and error by Equaton (.20). Fgure 6 Flow chart for calculaton of MTTF. III. VALIDATION OF MMIC RELIABILITY MODEL The two crcut examples have been smulated for both cases. For Case, the correlatons between s of both TIA and the LNA have been estmated by SPICE crcut analyss, and the Monte Carlo relablty smulatons for both MMICs have also been performed. For Case 2, the LNA and power amplfer have been analyzed for valdaton. LNA and TIA Hgh Temperature Analyss The assumptons for the relablty analyss are: ) The relatonshp between channel temperature (T j ) and medan lfe (t m ) s gven by Arrhenus equaton and s gven as: & Ea # t m = tmo exp $! (30) $ % k( T j + 273)! "

12 Where, t mo = for power type or for the LNA, and k = ev/ K 2) The medan lfe tm at temperature T m can be estmated by gven actvaton energy (Ea), test temperature (T o ) and medan lfe (t o ) t m = t mo & exp $ Ea $ %, k * + + T0 Tm )# '! (!" (3) The overall actvaton energy was calculated to be.6ev for each of the ndvdual s. (3) Tme to falure data of the MMIC components tested by prevously by the manufacturer most closely fts a lognormal dstrbuton. Therefore lognormal dstrbutons are used for all s. The lognormal probablty dstrbuton functon f(t) s gven as: & 2,. # m ) f ( t) = exp $ * lnt lnt '! (.32) t - 2/ $ % (!" where σ (standard devaton) and t m (medan lfe) are two parameters should be gven to determne operatonal lfetme t. 4) The nteractons between s can estmated by applyng weghtng factor, W j = /( - r j ) to modfy the tme to falure of the survvng components as shown n Fgure.4. (5) The lfe performance of passve components can be neglected. The computatonal schematc for Monte Carlo technque appled to the TIA and the LNA MMIC relablty analyss s shown n Fgure 7, and ts algorthm s the followng: INPUT N (the desred samplng sze) Whle number of samplng n <= N {For each samplng {Input number NC of components of the system and Group them nto dependence or ndependence groups ndvdually Whle <= NC {Input sgma s and medan lfe tm Select a random number x Transform random number x to random tme to falure TTF based on ts lfe dstrbuton} Determne the component whch s faled frst and let ts tme to falure be T. Whle j <= NC - {Modfy the tme to falure of all survvng components wth a weghng factor w(n, nj) based on ther correlated relatons.} Determne system tme to falure T Compute relablty, MTTF and error

13 start n = N s = N f = Mttf = 0 = call random (seed, rndnum) = + Convert rndnum to T() = I n = n + Determne Mtrnd (refer to next page) Mtrnd > Mt? yes no N f = N f + N s = N s + Mttf = Mttf + mtrnd no yes n = N? MTTF = Mttf/N R = N s/n Error = 200.*SQRT[R/N*( - R)] stop LNA and Power Amplfer Relablty Analyss Fgure 7 Flow Chart for Calculaton of MMIC MTTF. The relablty analyss of both the amplfers s smlar as n the prevous case except that the s-dependent groups must be dentfed and weghtng factors must be estmated by Equaton (3). Wth some mnor modfcatons, the algorthm and computer program for both TIA and LNA are stll applcable for both the LNA and the power amplfer, (see Fgure 8).

14 no If the frst faled component belong to s-dependence groups? yes Modfy the tmes to falure T of the survvng components by T = T + ) T/W(n, n f ) no Fnsh modfcaton? yes Determne the system s tme to falure Smulaton Results Fgure 8 The subroutne to estmate the modfed MMIC tme to falure The results of the relablty smulaton for TIA, and the LNA and power amplfer based on dscrete component data are shown n Fgures 9 to. The smulatons by Monte Carlo technques for both dependent (modfed by a weghtng factor) and ndependent (based on Ml-HDBK method) cases have been performed. The results show that the estmaton of MMICs' lfe ncludng nteractons between s s closer to expermental data than the estmaton wthout takng nto account the nteractons. The results also ndcate that nterdependences between devces s an mportant consderaton and cannot be gnored Expermental Monte Carlo - Dep. MIL-HDBK-Indep Temperature (EC) Fgure 9 MTTF versus Temperature for TIA

15 Experm ental M onte C arlo - Dep. M IL-HD BK -Indep Temperature (EC) Fgure 0 MTTF versus Temperature for the LNA EG -600 E xp. EG -600 M.C. EG E xp. EG M.C Temperature (EC) Fgure MTTF versus Temperature for the LNA and Power Amplfer Fgures 9 to show that the smulatons gve a conservatve estmaton of the MTTF. The excellent agreement even holds for the temperature range of 225 C through 325 C, thus ndcatng that the smulaton technque s applcable for hgh temperature smulatons, where large non-lneartes exst n the crcut s materal propertes. Ths nvestgaton has therefore presented the smulaton methodology for analog crcuts operatng n mcrowave systems such as MMICs. The approach outlned n ths paper may be used for analog type crcuts where the correlaton coeffcents have been dentfed.

16 Conclusons In the case of a complex MMIC crcut, t s not plausble to attan the analytcal relablty by the Markov approach for constant falure rate, whch perhaps s the best and most straghtforward analytcal approach to computatons n systems wth dependence. The equatons become numerous and out of control for a large MMIC system, and the Markov method may break down when falure rates become non-constant. We have shown that the Monte Carlo technque s the approprate methodolgy for predctng relablty of such complex crcuts. We have successfully establshed a new relablty smulaton model for MMICs and have shown that t has a wde applcablty to analog crcuts n general. The relablty model wll be applcable over a wde temperature range and hence may be used for mcrowave systems. REFERENCES. MIL-HDBK-27F, Relablty Predcton of Electronc Equpment, pp. 5-7, Sec. 5.4, Yonglu Deng and Shbn Song, "Relablty Analyss of A Multcomponent System n A Multstate Markovan Envronment," Mcroelectron. Relab. Vol. 33, No. 9, pp , T. Aven, "Relablty Evaluaton of Multstate Systems wth Multstate Components," IEEE Trans. on Relablty, Vol. R-34, No. 5, pp , J. Y. Ln, "A Monte Carlo Smulaton to Determne Mnmal Cut Sets and System Relablty," Proceedngs of Annual. R&M Symp. pp , C. Km and H. K. Lee, "A Monte Carlo Smulaton Algorthm for Fndng MTBF," IEEE Trans. on Relablty, Vol. 4, No. 4, pp , M. M. Aldrs, "A Smulaton Approach for Computng Systems relablty," Mcroelectron. Relab. Vol. 27, No. 3, pp , S. J. Kamat and M. W. Rley, "Determnaton of Relablty Usng Event-Based Monte Carlo Smulaton," IEEE Trans. on Relablty, Vol. R-24, No., pp , S. J. Kamat and W. E. Franzmeer, "Determnaton of Relablty Usng Event-Based Monte Carlo Smulaton Part II," IEEE Trans. on Relablty, pp , Oct C. L. Hwang, F. A. Tllman, and M.H. Lee, "System-Relablty Evaluaton Technques for Complex/Large Systems A Revew," IEEE Trans. on Relablty, Vol. R-30, No. 5, pp , A. Gandn, "Importance and Senstvty Analyss n Assessng System Relablty," IEEE Trans. on Relablty, R-39, pp6-70, L. B. Page and J. E. Perry, "A Model for System Relablty wth Common Cause Falures," IEEE Trans. on Relablty, R-38, pp406-40, S. S. Rao, Relablty-Based Desgn, pp , 2nd ed., McGraw-Hll, New York, H. Ref, "Generalzed Monte Carlo Perturbaton Algorthm for Correlated Samplng and A Second-order Taylor Seres Approach," Annals of Nuclear Energy, vol, pp , J. Yuan, M. T. La and K. L. Ko,"Evaluaton of System Relablty wth Common Cause Falures by a Pseudo-envronments Model," IEEE Trans. on Relablty, R-38, pp , J. T. H.Lee The Desgn of RF Integrated Crcuts Cambrdge U.K. Cambrdge Unversty Press, S. S. Naseh and S.H. Deen, RF CMOS Relablty, pp , Vol., No