{"id":218057,"date":"2025-05-23T06:58:49","date_gmt":"2025-05-23T06:58:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=218057"},"modified":"2025-05-23T06:58:52","modified_gmt":"2025-05-23T06:58:52","slug":"mason-gain-formula-for-find-out-transfer-function-of-a-given-signal-flow-graph","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/23\/mason-gain-formula-for-find-out-transfer-function-of-a-given-signal-flow-graph\/","title":{"rendered":"Mason gain formula for find out transfer function of a given signal flow graph"},"content":{"rendered":"\n

Mason gain formula for find out transfer function of a given signal flow graph<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

\n\n\n\n

Mason\u2019s Gain Formula<\/strong><\/h3>\n\n\n\n

$$
T = \\frac{Y(s)}{X(s)} = \\frac{\\sum_{k=1}^N P_k \\Delta_k}{\\Delta}
$$<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • $T$ = overall transfer function (output\/input)<\/li>\n\n\n\n
  • $P_k$ = gain of the $k^{th}$ forward path from input to output<\/li>\n\n\n\n
  • $\\Delta$ = determinant of the graph, defined as: $$
    \\Delta = 1 – (\\text{sum of all individual loop gains}) + (\\text{sum of gain products of all possible non-touching loops taken two at a time}) – (\\text{sum of gain products of all possible non-touching loops taken three at a time}) + \\cdots
    $$<\/li>\n\n\n\n
  • $\\Delta_k$ = value of $\\Delta$ for the graph with all loops touching the $k^{th}$ forward path removed (loops touching the forward path are excluded)<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    <\/h3>\n\n\n\n

    Mason\u2019s Gain Formula is a systematic method used in control systems engineering and signal processing to find the overall transfer function<\/strong> of a system modeled by a signal flow graph (SFG)<\/strong>. The SFG represents the relationships between system variables using nodes and directed branches with associated gains.<\/p>\n\n\n\n

    The transfer function, $T = \\frac{Y(s)}{X(s)}$, expresses how the output $Y(s)$ is related to the input $X(s)$ in the Laplace domain. When a system is represented by a complex network of paths and loops, directly writing the transfer function can be difficult. Mason\u2019s Gain Formula simplifies this by accounting for all possible paths and loops systematically.<\/p>\n\n\n\n

    Key elements:<\/strong><\/p>\n\n\n\n

      \n
    • Forward paths $P_k$:<\/strong> These are paths from the input node to the output node without passing through any node twice.<\/li>\n\n\n\n
    • Loops:<\/strong> Closed paths that start and end at the same node without visiting any node twice.<\/li>\n\n\n\n
    • Non-touching loops:<\/strong> Loops that do not share any common node.<\/li>\n<\/ul>\n\n\n\n

      The denominator $\\Delta$ reflects the interaction of loops within the graph. It starts with 1, then subtracts the sum of all individual loop gains (since loops tend to reduce system gain), then adds back gains of all pairs of non-touching loops (as their combined effect can increase the gain), subtracts for triples, and so forth, alternating signs for higher-order combinations.<\/p>\n\n\n\n

      Each forward path has its own $\\Delta_k$, which is the same as $\\Delta$ but excludes loops that touch the forward path because those loops directly affect that path\u2019s gain.<\/p>\n\n\n\n

      By plugging all this into Mason\u2019s formula, you get the total transfer function incorporating all feedback and feedforward effects, accurately describing the system\u2019s behavior.<\/p>\n\n\n\n

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

      Mason gain formula for find out transfer function of a given signal flow graph The Correct Answer and Explanation is: Mason\u2019s Gain Formula $$T = \\frac{Y(s)}{X(s)} = \\frac{\\sum_{k=1}^N P_k \\Delta_k}{\\Delta}$$ Where: Mason\u2019s Gain Formula is a systematic method used in control systems engineering and signal processing to find the overall transfer function of a system […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-218057","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218057","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=218057"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/218057\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=218057"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=218057"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=218057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}