{"id":50175,"date":"2025-11-14T12:19:20","date_gmt":"2025-11-14T11:19:20","guid":{"rendered":"https:\/\/www.gt-engineering.it\/?post_type=pubblicazione&p=50175"},"modified":"2026-02-09T13:03:01","modified_gmt":"2026-02-09T12:03:01","slug":"p5-functional-safety-pfd-calculation-second-part","status":"publish","type":"pubblicazione","link":"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p5-functional-safety-pfd-calculation-second-part\/","title":{"rendered":"P5: Functional Safety – PFD Calculation Second part"},"content":{"rendered":"\n\n \n \n \n
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\n Summary <\/h2>\n <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
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In the first part of the article we introduced the concepts of Unreliability function F(t) and of Reliability Function R(t). The former is the base for the parameter PFD(t) and the one used to indicate the reliability of a Safety Instrumented Function (SIR): PFD average or PFDavg<\/sub>.<\/h3>\n

In this second part, we show how to calculate, in practice, the PFDavg<\/sub>.<\/p>\n

 <\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
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\n PFDavg for different architecture <\/h2>\n <\/div>\n \n
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In low demand mode, the average PFD can be calculated by integrating the PFD(t) function over the test interval.<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

\n\n \n\n
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Let\u2019s considering a 1 out of 1 (1oo1) subsystem with Failure Rate \u03bb. This architecture consists of a single channel, where any dangerous failure leads to the failure of the safety function, when a demand arises.<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

\n\n \n\n
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starting from its Reliability function<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

\n\n \n\n
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the unreliability function can be written and approximated as<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

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therefore:<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

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In case of two 1oo1 Subsystems in series, the Average PFD is the sum of the average PFD of the single subsystem.<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

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Considering a 1oo2 subsystem, where each element has the same Failure Rate \u03bb, safety is guaranteed if at least one of them is functioning correctly<\/strong>. In other words, in case of one failure, the safety function is still guaranteed.<\/p>\n

This architecture consists of two channels connected in parallel, such that either channel can process the safety function. Thus there should be a dangerous failure in both channels before a safety function fails on demand.<\/p>\n

Supposing each element of the subsystem is, for example, a pressure transmitter; if one of them detects a dangerous situation, the subsystem triggers a shuts down.<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

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For devices connected in parallel, knowing their F(t), the total failure probability function Ftot<\/sub>(t) is obtained through this formula:<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

\n\n \n\n
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therefore :<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

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Considering a 1oo3 subsystem, where each element has the same Failure Rate \u03bb, safety is guaranteed if at least one of them is functioning correctly<\/strong>. In other words, in case of two faults, the safety function is still guaranteed.<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

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Supposing each element of the subsystem is, for example, a temperature sensor, if one of them detects a dangerous situation (for example a high temperature), the subsystem does not trigger the shutdown of the process. In this case two temperature sensors must detect a high temperature to trigger the system shut down. This architecture has a high level of both reliability and availability. In this case, we have to use a more general formula, able to express the R(t) function for architectures with redundancy, knowing \u201cn\u201d<\/em>, total number of devices\u201d and \u201ci\u201d, <\/em>number of devices that have to be functioning to ensure that the entire system is functioning correctly:<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

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This set of formulas can be used for different combination of devices connected in parallel, included the previous scenario. For the 2oo3 architecture:<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

\n\n \n\n
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Therefore, PFDavg <\/sub>can be calculated using the well-known formula:<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

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\n Summary Table <\/h2>\n <\/div>\n \n
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In case we consider Safety Critical Systems, only the dangerous part of the failure rate is significant. Moreover, considering non-repairable systems and no common cause failures<\/strong>, the formula for the Average PFD are summarised in the following table.<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

\n\n \n\n
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The value of PFDavg<\/sub>, in case of repairable systems and common cause failures for redundant subsystems (\u03b2\u22600), can be found in IEC 61508-6 Annex B.<\/p>\n

Just for a quick reference, for a 1oo2 architecture<\/strong>, in case only Common Cause Failures are taken into considerations, the formula for the PFD calculation becomes the following:<\/p>\n<\/div> <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n

\n\n \n
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\n Reliability of a Safety Function in Low Demand mode. <\/h2>\n <\/div>\n <\/div>\n \n \n
\n \n
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In low demand mode, the Reliability of a safety function is defined with the parameter PFDavg<\/sub>.<\/p>\n

IEC 61508 divides the requirements into four safety integrity levels, SIL1, SIL2, SIL3, and SIL4, with SIL4 being the most reliable and SIL 1 being the least reliable.<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
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However, it is not enough to have a safety instrumented function with a certain PFDavg<\/sub> value (for example, SIL 2), to decide that the SIF is a SIL 2 Safety instrumented Function.<\/p>\n

Consider a SIF that operates in low-demand mode and assume that we have determined the PFDavg<\/sub> to be 5\u00b710-3<\/sup>. Because this value is in the interval from 10-3<\/sup> to 10-2<\/sup>, the system may fulfil the requirements for SIL 2 if it also fulfils the requirements for<\/p>\n

    \n
  • Systematic Safety Integrity<\/li>\n
  • Hardware Safety integrity.<\/li>\n<\/ul>\n

    Please refer to Figure 2.13 {2.1.6.2}.<\/p>\n<\/div> <\/div>\n \n \n

    <\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
    \n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
    <\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
    \n\n \n \n \n
    \n \n
    \n

    A SIF will therefore not automatically fulfil the SIL 3 requirements for example, when the PFDavg<\/sub> is within the interval for SIL 3. Let\u2019s see an example for that.<\/p>\n<\/div> <\/div>\n \n \n

    <\/div>\n \n\n <\/div>\n <\/section>\n\n \n","protected":false},"excerpt":{"rendered":"

    In this article we show how to calculate, in practice, the PFDavg.<\/p>\n","protected":false},"author":1,"featured_media":43924,"parent":49606,"menu_order":4,"template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-50175","pubblicazione","type-pubblicazione","status-publish","has-post-thumbnail","hentry"],"acf":[],"yoast_head":"\nP5: Functional Safety - PFD Calculation Second part - Gt-Engineering<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p5-functional-safety-pfd-calculation-second-part\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"P5: Functional Safety - PFD Calculation Second part - Gt-Engineering\" \/>\n<meta property=\"og:description\" content=\"In this article we show how to calculate, in practice, the PFDavg.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p5-functional-safety-pfd-calculation-second-part\/\" \/>\n<meta property=\"og:site_name\" content=\"Gt-Engineering\" \/>\n<meta property=\"article:modified_time\" content=\"2026-02-09T12:03:01+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.gt-engineering.it\/wp-content\/uploads\/2023\/11\/2.webp\" \/>\n\t<meta property=\"og:image:width\" content=\"720\" \/>\n\t<meta property=\"og:image:height\" content=\"270\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/webp\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p5-functional-safety-pfd-calculation-second-part\/\",\"url\":\"https:\/\/www.gt-engineering.it\/en\/pubblications\/in-depth-articles-tuttomisure\/p5-functional-safety-pfd-calculation-second-part\/\",\"name\":\"P5: Functional Safety - 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