{"id":2060,"date":"2019-12-14T02:18:40","date_gmt":"2019-12-14T02:18:40","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=2060"},"modified":"2021-01-14T14:35:53","modified_gmt":"2021-01-14T14:35:53","slug":"10-6-evans-root-locus-construction-rule-4break-away-and-break-in-points","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/10-6-evans-root-locus-construction-rule-4break-away-and-break-in-points\/","title":{"raw":"10.6 Evans Root Locus Construction Rule # 4: Break-Away and Break-In Points","rendered":"10.6 Evans Root Locus Construction Rule # 4: Break-Away and Break-In Points"},"content":{"raw":"This Rule deals with break-away or break-in<\/strong> points. As the Gain [latex]K[\/latex] increases, the closed loop poles initially move along the Real axis since typically the open loop poles are real. When the two closed loop poles meet, the RL segments break away from the Real axis. The coordinate of that is called a breakaway point<\/strong>. In some systems, with multiple zeros, instead of a break-away point, there will be a break-in point<\/strong>, where the complex segments of the RL enter the Real axis.\r\n
\r\n\r\nRule 4:<\/strong> The break-away (break-in) points can be found by re-writing the closed loop characteristic equation as a function of gain [latex]K[\/latex], and taking a derivative of [latex]K[\/latex] w.r.t. s:\r\n\r\n\r\n\r\n
\r\n

[latex]\\frac{dK(s)}{ds}=0[\/latex] and solving for [latex]s_b[\/latex]<\/p>\r\n

Equation 10\u201112<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n<\/div>\r\nIn our example:\r\n

[latex]1+KG(s)=0[\/latex]<\/p>\r\n

[latex]1+K \\frac{10}{(s+10)(s+5)(s+2)}=0[\/latex]<\/p>\r\n

[latex]K=-1 \\cdot \\frac{(s+10)(s+5)(s+2)}{10}= -0.1 \\cdot (s^3+17s^2+80s+100)[\/latex]<\/p>\r\n

[latex] \\frac{dK(s)}{ds} =-(3s^2+34s+80)[\/latex]<\/p>\r\n

[latex]3s^2+34+80=0[\/latex]<\/p>\r\n

[latex]S_1=-8[\/latex]<\/p>\r\n

[latex]S_2=-3.33[\/latex]<\/p>\r\nWe have two values that could be our break-away point: [latex]s= -8[\/latex] and [latex]s =-3.33[\/latex], but the first one does not belong to the Root Locus (see Rule 2), hence our coordinate is [latex]-3.33[\/latex] - this is consistent with the Matlab results, as seen in Figure 10\u20116.","rendered":"

This Rule deals with break-away or break-in<\/strong> points. As the Gain [latex]K[\/latex] increases, the closed loop poles initially move along the Real axis since typically the open loop poles are real. When the two closed loop poles meet, the RL segments break away from the Real axis. The coordinate of that is called a breakaway point<\/strong>. In some systems, with multiple zeros, instead of a break-away point, there will be a break-in point<\/strong>, where the complex segments of the RL enter the Real axis.<\/p>\n

\n

Rule 4:<\/strong> The break-away (break-in) points can be found by re-writing the closed loop characteristic equation as a function of gain [latex]K[\/latex], and taking a derivative of [latex]K[\/latex] w.r.t. s:<\/p>\n\n\n\n
\n

[latex]\\frac{dK(s)}{ds}=0[\/latex] and solving for [latex]s_b[\/latex]<\/p>\n

Equation 10\u201112<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n<\/div>\n

In our example:<\/p>\n

[latex]1+KG(s)=0[\/latex]<\/p>\n

[latex]1+K \\frac{10}{(s+10)(s+5)(s+2)}=0[\/latex]<\/p>\n

[latex]K=-1 \\cdot \\frac{(s+10)(s+5)(s+2)}{10}= -0.1 \\cdot (s^3+17s^2+80s+100)[\/latex]<\/p>\n

[latex]\\frac{dK(s)}{ds} =-(3s^2+34s+80)[\/latex]<\/p>\n

[latex]3s^2+34+80=0[\/latex]<\/p>\n

[latex]S_1=-8[\/latex]<\/p>\n

[latex]S_2=-3.33[\/latex]<\/p>\n

We have two values that could be our break-away point: [latex]s= -8[\/latex] and [latex]s =-3.33[\/latex], but the first one does not belong to the Root Locus (see Rule 2), hence our coordinate is [latex]-3.33[\/latex] – this is consistent with the Matlab results, as seen in Figure 10\u20116.<\/p>\n","protected":false},"author":162,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2060","chapter","type-chapter","status-publish","hentry"],"part":1471,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2060","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/users\/162"}],"version-history":[{"count":7,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2060\/revisions"}],"predecessor-version":[{"id":2696,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2060\/revisions\/2696"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/parts\/1471"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2060\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=2060"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=2060"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=2060"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=2060"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}