Determining of the extent of long-term reliability tests

Microelectronics and Reliability

P e r g a m o n Press 1971. Vol. 10, pp. 285-286.

P r i n t e d in G r e a t Britain

TECHNICAL NOTE Determining

of the Extent of Long-Term

Reliability Tests

A b s t r a c t - - T o d e t e r m i n e t h e r a n g e of t h e tested p o p u l a t i o n a n d t h e d u r a t i o n of t h e reliability test u n d e r n o m i n a l operating conditions for a n exponential distribution of t h e t i m e b e t w e e n failures, a n original n o m o g r a m is presented.

IN considering t h e feasibility of reliability tests, we m u s t d e t e r m i n e h o w m a n y i t e m s are r e q u i r e d to be s u b j e c t to testing, for h o w l o n g a t i m e period t r or to h o w m a n y failures r to obtain t h e reliability p a r a m e t e r s (e.g. M T B F , ~,) of t h e basic p o p u l a t i o n w i t h an error n o t exceeding a specified value. F o r example, t h e p u r p o s e of t h e test m a y be to define t h e confidence interval o f reliability p a r a m e t e r s at a specified confidence level. {x~ T h e tests t h a t are m o s t f r e q u e n t l y e n c o u n t e r e d are reliability tests u n d e r n o m i n a l operating conditions.* T h e results of t h e s e tests p e r m i t e s t i m a t i o n of t h e reliability o f c o m p o n e n t s a n d s y s t e m s , b u t are u s e d for verifying t h e results of s h o r t e d or accelerated tests too. N o w , we constrict ourselves to t h e reliability tests p e r f o r m e d u n d e r n o m i n a l operating conditions a n d a s s u m e t h a t t h e n o n - f a i l u r e operating t i m e follows a n exponential law. It follows t h a t t h e occurrence of r failures w i t h i n a t i m e interval (0, t r ) is s u b j e c t to P o i s s o n d i s t r i b u t i o n w i t h an average value :~~ p = T~

p is to be d e t e r m i n e d for t h e specified values of ( 1 - =) a n d r. F o r this purpose, for example, an analytic expression m a y be derived f r o m t h e x 2 - d i s t r i b u t i o n / ~ T h e distribution f u n c t i o n x 2 with k degrees of f r e e d o m m a y be w r i t t e n as =h

1

kk_ 1

x

.e --2dz (3)

0

w h e r e I'(k/2) is t h e g a m m a f u n c t i o n of a r g u m e n t k/2. T h e r a n d o m variable z c o m p l y with t h e e q u a t i o n

z = x~.

(4)

T h e value of

zh

= z~(k)

(5)

is a kvantil of t h e x~-distribution for k degrees of f r e e d o m at t h e p probability level. F o r

(1)

w h e r e 9, is t h e failure rate of t h e basic population of t h e specific type of p r o d u c t s a n d T is t h e a c c u m u l a t e d operating time. F o r p l a n n i n g t h e range of reliability tests, t h e n u m b e r of s p e c i m e n s to be tested n a n d t h e test interval tT are

f

P ( z n > z) = F(zn) = p -- 2k/2 ~(k/2) J =2

k = 2(c+1)

(6)

w h e r e c is an integer a n d y =~

yn = T

(7)

will be 2Yh

oO

1

¢

P= F(2yn)=l f yce-Udy = 1--c[f Yce-udy

=

2yine-Un. 1-i'

2y h

(8)

i~ 0

By c o m p a r i n g t h e relations (2) a n d (8), we obtain

r e q u i r e d to be d e t e r m i n e d so that, in testing, we o b t a i n at least r * ~ r failures w i t h a probability of ( 1 - =). W e u s u a l l y choose 0 < = < 0 . 2 , b u t m o s t f r e q u e n t l y , = 0"1 a n d = = 0'05.

(9)

F(2yh) = 1 -for c=r--1

yn=

~

p = 1--=.

(10)

F r o m E q u a t i o n s (5), (7), (9), (10) it follows

Thus, from the equation r--1

P(r*~r) = 1--P(r*
X21-=(2r) ~.I

-- 1 - - ~

P--

(2)

2

(11)

L e t u s consider that, for tests w i t h r e p l a c e m e n t o f t h e i t e m s that have failed, we have 7" = n ' t r (12)

i=0 * N a m e l y , l o n g - t e r m reliability tests. 285

286

TECHNICAL

NOTE

whereas for tests w i t h o u t r e p l a c e m e n t we obtain

n×fT =

r T = X

t,+(n--r)tT.

(13)

If, in b o t h cases, we consider E q u a t i o n (1) a n d a s s u m e that an inequality n>>r is valid, we obtain t h e following expression for p l a n n i n g the r a n g e of reliability tests x Z l _ ~ (2r) T

=

Z0 is t h e considered operating u n d e r t h e F o r a = 0-1 a n d enables t h e range determined.

n't~

--

(14)

102

REFERENCES 1. K. F. TOML~EK. Microelectronics and Reliability, 10, (1971).

2. S. R. CALABRO, Reliability Principles and Practices. M c G r a w - H i l l , N e w York (1962).

3. A. C. AITKEN, Statistical Mathematics. Oliver and Boyd (1947).

I02

02

105~ IO-5--T-- fO- a

B-

8

~2 104~

52"_5 I0~ =

c

10._4~ -- i0_ 8

Example

Research Institute of Telecommunication Prague 1O, Czechoslovakia

1 0 - ! i I0-ro

tT i--

XO

[ I/hd

[hr]

--6

2Z0 failure rate of t h e basic p o p u l a t i o n specified conditions. ~ = 0"05, t h e n o m o g r a m in Fig. 1 of reliability tests to be quickly

D e t e r m i n e t h e d u r a t i o n of reliability test of 200 pieces of transistors, T y p e 2N706A, so that, for probability of = 0"1, at least r = 2 failures occur if, for t h e specified operating conditions a n d failure criteria, we can consider a failure rate of 4 x 10-e/hr. I n the n o m o g r a m in Fig. 1 we subtract t~ = 4900 hr. K. F. TOM~EK

2X

--4

r

e ~0.05 a - O . I

i=1 (t~ - - t i m e to failure of t h e i th item)

Z ~ - ~ ) (2,r) ,

p

2~a

--I0

~--103

--8 6-------6

4

2~ 2

--4

z-

l_ 2---2

zZF--2 I0O~ ~-104

--2

--4

--iO

-. - 6

~-rO 3

--8 ~o-i04

E-2

?-2

I0O~ o

5

IO ~ _~[0 ~ 5--~5

Key :

io-3

5 I0- 7

Iio

FIG. 1. N o m o g r a m for d e t e r m i n i n g t h e range of l o n g t e r m reliability tests. (Work with the s a m e range o f scales only.)