Temperature-NonThermal Relationship - Revision history

Sharon Honecker: /* Conditional Reliability Function */ changed R(T,t,U,V) to R((t|T),U,V)

2017-02-08T21:15:02Z

Conditional Reliability Function: changed R(T,t,U,V) to R((t|T),U,V)

← Older revision		Revision as of 21:15, 8 February 2017
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	The T-NT Weibull conditional reliability function at a specified stress level is given by:		The T-NT Weibull conditional reliability function at a specified stress level is given by:

	::<math>R(T,t,U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac{{{e}^{-{{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}}\,\!</math>		::<math>R((t\|T),U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac{{{e}^{-{{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}}\,\!</math>

	or:		or:

	::<math>R(T,t,U,V)={{e}^{-\left[ {{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }} \right]}}\,\!</math>		::<math>R((t\|T),U,V)={{e}^{-\left[ {{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }} \right]}}\,\!</math>

	===Reliable Life===		===Reliable Life===

Sharon Honecker: /* Conditional Reliability */ changed R(T,t,V,U) to R((t|T),V,U)

2017-02-08T21:14:13Z

Conditional Reliability: changed R(T,t,V,U) to R((t|T),V,U)

← Older revision		Revision as of 21:14, 8 February 2017
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	The conditional reliability function for the T-NT exponential model is given by,		The conditional reliability function for the T-NT exponential model is given by,

	::<math>R(T,t,U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}\,\!</math>		::<math>R((t\|T),U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}\,\!</math>

	====Reliable Life====		===Reliable Life===

	For the T-NT exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}\,\!</math>, is given by:		For the T-NT exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}\,\!</math>, is given by:

Nicolette Young: /* Acceleration Factor */

2015-12-17T17:53:10Z

Acceleration Factor

← Older revision		Revision as of 17:53, 17 December 2015
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	[[Image:ALTA10.3.png\|center\|~~400px~~\|Acceleration Factor vs. Temperature at a fixed voltage level.]]		[[Image:ALTA10.3.png\|center\|450px\|Acceleration Factor vs. Temperature at a fixed voltage level.]]


	[[Image:ALTA10.4.png\|center\|~~400px~~\|Acceleration Factor vs. Voltage at a fixed temperature level.]]		[[Image:ALTA10.4.png\|center\|450px\|Acceleration Factor vs. Voltage at a fixed temperature level.]]

	=T-NT Exponential=		=T-NT Exponential=

Richard House: /* A look at the Parameters B and n */

2012-09-21T22:31:50Z

A look at the Parameters B and n

← Older revision		Revision as of 22:31, 21 September 2012
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	[[Image:ALTA10.2.png\|400px\|center\|Life vs. Voltage plot at a fixed temperature level.]]		[[Image:ALTA10.2.png\|400px\|center\|Life vs. Voltage plot at a fixed temperature level.]]

	===A look at the Parameters B and n===		===A look at the Parameters ''B'' and ''n''===
	Depending on which stress type is kept constant, it can be seen from the linearized T-NT relationship that either the parameter <math>B\,\!</math> or the parameter <math>n\,\!</math> is the slope of the resulting line. If, for example, the non-thermal stress is kept constant then <math>B\,\!</math> is the slope of the life line in a Life vs. Temperature plot. The steeper the slope, the greater the dependency of the product's life to the temperature. In other words, <math>B\,\!</math> is a measure of the effect that temperature has on the life and <math>n\,\!</math> is a measure of the effect that the non-thermal stress has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the temperature. Similarly, the larger the value of <math>n,\,\!</math> the higher the dependency of the life on the non-thermal stress.		Depending on which stress type is kept constant, it can be seen from the linearized T-NT relationship that either the parameter <math>B\,\!</math> or the parameter <math>n\,\!</math> is the slope of the resulting line. If, for example, the non-thermal stress is kept constant then <math>B\,\!</math> is the slope of the life line in a Life vs. Temperature plot. The steeper the slope, the greater the dependency of the product's life to the temperature. In other words, <math>B\,\!</math> is a measure of the effect that temperature has on the life and <math>n\,\!</math> is a measure of the effect that the non-thermal stress has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the temperature. Similarly, the larger the value of <math>n,\,\!</math> the higher the dependency of the life on the non-thermal stress.

Richard House: /* A look at the Parameters B\,\! and n\,\! */

2012-09-21T22:29:31Z

A look at the Parameters B\,\! and n\,\!

← Older revision		Revision as of 22:29, 21 September 2012
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	[[Image:ALTA10.2.png\|400px\|center\|Life vs. Voltage plot at a fixed temperature level.]]		[[Image:ALTA10.2.png\|400px\|center\|Life vs. Voltage plot at a fixed temperature level.]]

	===A look at the Parameters ~~<math>~~B~~\,\!</math>~~ and ~~<math>~~n~~\,\!</math>~~===		===A look at the Parameters B and n===
	Depending on which stress type is kept constant, it can be seen from the linearized T-NT relationship that either the parameter <math>B\,\!</math> or the parameter <math>n\,\!</math> is the slope of the resulting line. If, for example, the non-thermal stress is kept constant then <math>B\,\!</math> is the slope of the life line in a Life vs. Temperature plot. The steeper the slope, the greater the dependency of the product's life to the temperature. In other words, <math>B\,\!</math> is a measure of the effect that temperature has on the life and <math>n\,\!</math> is a measure of the effect that the non-thermal stress has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the temperature. Similarly, the larger the value of <math>n,\,\!</math> the higher the dependency of the life on the non-thermal stress.		Depending on which stress type is kept constant, it can be seen from the linearized T-NT relationship that either the parameter <math>B\,\!</math> or the parameter <math>n\,\!</math> is the slope of the resulting line. If, for example, the non-thermal stress is kept constant then <math>B\,\!</math> is the slope of the life line in a Life vs. Temperature plot. The steeper the slope, the greater the dependency of the product's life to the temperature. In other words, <math>B\,\!</math> is a measure of the effect that temperature has on the life and <math>n\,\!</math> is a measure of the effect that the non-thermal stress has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the temperature. Similarly, the larger the value of <math>n,\,\!</math> the higher the dependency of the life on the non-thermal stress.

Richard House at 22:28, 21 September 2012

2012-09-21T22:28:35Z

Show changes

Richard House at 03:04, 30 August 2012

2012-08-30T03:04:42Z

Richard House: /* Approximate Confidence Bounds for the T-NT Lognormal */

2012-08-23T02:38:35Z

Approximate Confidence Bounds for the T-NT Lognormal

← Older revision		Revision as of 02:38, 23 August 2012
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	==Approximate Confidence Bounds for the T-NT Lognormal==		==Approximate Confidence Bounds for the T-NT Lognormal==
	===Bounds on the Parameters===		===Bounds on the Parameters===
	Since the standard deviation, <math>{{\widehat{\sigma }}_{{{T}'}}}</math> , and <math>\widehat{C}</math> are positive parameters, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})</math> and <math>\ln (\widehat{C})</math> are treated as normally distributed and the bounds are estimated from:		Since the standard deviation, <math>{{\widehat{\sigma }}_{{{T}'}}}\,\!</math> , and <math>\widehat{C}\,\!</math> are positive parameters, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})\,\!</math> and <math>\ln (\widehat{C})\,\!</math> are treated as normally distributed and the bounds are estimated from:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	The lower and upper bounds on <math>B</math> and <math>n</math> are estimated from:		The lower and upper bounds on <math>B\,\!</math> and <math>n\,\!</math> are estimated from:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	The variances and covariances of <math>B</math> , <math>C,</math> <math>n,</math> and <math>{{\sigma }_{{{T}'}}}</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B},</math> <math>\widehat{C},</math> <math>\widehat{n}</math> , <math>{{\widehat{\sigma }}_{{{T}'}}})</math> as follows:		The variances and covariances of <math>B\,\!</math> , <math>C,\,\!</math> <math>n,\,\!</math> and <math>{{\sigma }_{{{T}'}}}\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B},\,\!</math> <math>\widehat{C},\,\!</math> <math>\widehat{n}\,\!</math> , <math>{{\widehat{\sigma }}_{{{T}'}}})\,\!</math> as follows:

	::<math>\left( \begin{matrix}		::<math>\left( \begin{matrix}
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	::<math>R({T}',U,V;B,C,n,{{\sigma }_{{{T}'}}})=\int_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})+\widehat{n}\ln ({{U}_{i}})-\tfrac{\widehat{B}}{{{V}_{i}}}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt</math>		::<math>R({T}',U,V;B,C,n,{{\sigma }_{{{T}'}}})=\int_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})+\widehat{n}\ln ({{U}_{i}})-\tfrac{\widehat{B}}{{{V}_{i}}}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt</math>

	Let <math>\widehat{z}(t,U,V;B,C,n,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})+\widehat{n}\ln (U)-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.</math>		Let <math>\widehat{z}(t,U,V;B,C,n,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})+\widehat{n}\ln (U)-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},\,\!</math> then <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.\,\!</math>
	For <math>t={T}'</math> , <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{C})+\widehat{n}\ln (U)-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}</math> , and for <math>t=\infty ,</math> <math>\widehat{z}=\infty .</math>		For <math>t={T}'\,\!</math> , <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{C})+\widehat{n}\ln (U)-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}\,\!</math> , and for <math>t=\infty ,\,\!</math> <math>\widehat{z}=\infty .\,\!</math>

	The above equation then becomes:		The above equation then becomes:
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	::<math>R(\widehat{z})=\int_{\widehat{z}({T}',U,V)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<math>R(\widehat{z})=\int_{\widehat{z}({T}',U,V)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

	The bounds on <math>z</math> are estimated from:		The bounds on <math>z\,\!</math> are estimated from:

	::<math>\begin{align}		::<math>\begin{align}
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	::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',U,V)}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',U,V)}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

	The next step is to calculate the variance of <math>{T}'(U,V;\widehat{B},\widehat{C},\widehat{n},{{\widehat{\sigma }}_{{{T}'}}})</math> :		The next step is to calculate the variance of <math>{T}'(U,V;\widehat{B},\widehat{C},\widehat{n},{{\widehat{\sigma }}_{{{T}'}}})\,\!</math> :

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Confidence Bounds on Time */

2012-08-23T02:35:28Z

Confidence Bounds on Time

← Older revision		Revision as of 02:35, 23 August 2012
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	where <math>\widehat{u}=\ln \widehat{T}.</math>		where <math>\widehat{u}=\ln \widehat{T}.</math>

	The upper and lower bounds on <math>u</math> are estimated from:		The upper and lower bounds on <math>u\,\!</math> are estimated from:

	::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>		::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>

Richard House: /* Confidence Bounds on Reliability */

2012-08-23T02:34:55Z

Confidence Bounds on Reliability

← Older revision		Revision as of 02:34, 23 August 2012
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	::<math>\widehat{R}(T,U,V)={{e}^{-e\widehat{^{u}}}}</math>		::<math>\widehat{R}(T,U,V)={{e}^{-e\widehat{^{u}}}}</math>

	The next step is to find the upper and lower bounds on <math>u</math> :		The next step is to find the upper and lower bounds on <math>u\,\!</math> :

	::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>		::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}</math>

← Older revision		Revision as of 03:04, 30 August 2012
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	=T-NT Exponential=		=T-NT Exponential=
	By setting <math>m=L(U,V)\,\!</math>, the exponential ~~<math>~~pdf~~</math>~~ becomes:		By setting <math>m=L(U,V)\,\!</math>, the exponential ''pdf'' becomes:

	::<math>f(t,U,V)=\frac{{{U}^{n}}}{C}{{e}^{-\tfrac{B}{V}}}\cdot {{e}^{-\tfrac{{{U}^{n}}}{C}\left( {{e}^{-\tfrac{B}{V}}} \right)t}}</math>		::<math>f(t,U,V)=\frac{{{U}^{n}}}{C}{{e}^{-\tfrac{B}{V}}}\cdot {{e}^{-\tfrac{{{U}^{n}}}{C}\left( {{e}^{-\tfrac{B}{V}}} \right)t}}</math>
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	=T-NT Lognormal=<!-- THIS SECTION HEADER IS LINKED TO: Temperature-Nonthermal_Relationship_Example. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->		=T-NT Lognormal=<!-- THIS SECTION HEADER IS LINKED TO: Temperature-Nonthermal_Relationship_Example. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
	The ~~<math>~~pdf~~</math>~~ of the lognormal distribution is given by:		The ''pdf'' of the lognormal distribution is given by:

	::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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	::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>		::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>

	The T-NT lognormal model ~~<math>~~pdf~~</math>~~ can be obtained by setting <math>\breve{T}=L(V)\,\!</math>. Therefore:		The T-NT lognormal model ''pdf'' can be obtained by setting <math>\breve{T}=L(V)\,\!</math>. Therefore:

	::<math>\breve{T}=L(V)=\frac{C}{{{U}^{n}}}{{e}^{\tfrac{B}{V}}}</math>		::<math>\breve{T}=L(V)=\frac{C}{{{U}^{n}}}{{e}^{\tfrac{B}{V}}}</math>
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	::<math>{{\overline{T}}^{\prime }}=\ln (C)-n\ln (U)+\frac{B}{V}</math>		::<math>{{\overline{T}}^{\prime }}=\ln (C)-n\ln (U)+\frac{B}{V}</math>

	Substituting the above equation into the lognormal ~~<math>~~pdf~~</math>~~ yields the T-NT lognormal model ~~<math>~~pdf~~</math>~~ or:		Substituting the above equation into the lognormal ''pdf'' yields the T-NT lognormal model ''pdf'' or:

	::<math>f(T,U,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T,U,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)+n\ln (U)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>