{"id":1999,"date":"2019-12-07T02:13:14","date_gmt":"2019-12-07T02:13:14","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&#038;p=1999"},"modified":"2019-12-19T01:14:40","modified_gmt":"2019-12-19T01:14:40","slug":"10-4evans-root-locus-construction-rule-2-segments-of-root-locus-on-real-axis","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/10-4evans-root-locus-construction-rule-2-segments-of-root-locus-on-real-axis\/","title":{"raw":"10.4\tEvans Root Locus Construction Rule # 2: Segments of Root Locus on Real Axis","rendered":"10.4\tEvans Root Locus Construction Rule # 2: Segments of Root Locus on Real Axis"},"content":{"raw":"This Rule deals with points on the Real axis. Consider the angle criterion:\r\n<table style=\"border-collapse: collapse;width: 100%;height: 42px\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 14px\">\r\n<td style=\"width: 50%;height: 14px\">[latex] \\angle G(s) = 180^{\\circ} \\pm n\\cdot360^{\\circ} [\/latex]<\/td>\r\n<td style=\"width: 50%;height: 14px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"width: 50%;height: 14px\">[latex]\\angle(s-z_1)+\\angle(s-z_2)+...+\\angle(s-z_m)-... [\/latex]<\/td>\r\n<td style=\"width: 50%;height: 14px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px\">\r\n<td style=\"width: 50%;height: 14px\">[latex] \\angle(s-p_1)-\\angle(s-p_2)-...-\\angle(s-p_n)=180^{\\circ}\\pm n360^{\\circ} [\/latex]<\/td>\r\n<td style=\"width: 50%;height: 14px;text-align: right\">Equation 10-6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTake some test points for [latex] s^* [\/latex] along the Real axis and see what satisfies the angle criterion.\r\n<p style=\"text-align: center\">[latex] \\angle G(s^*)=\\sum\\theta_z - \\sum\\theta_p [\/latex]<\/p>\r\nWhere must Real axis segments of the root locus lie? Rule 2 answers that question.\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>Rule 2:<\/strong> A point s on the Real axis belongs to the root locus if and only if it is to the left of the ODD number of open loop singularities (a singularity is either a pole or a zero).\r\n\r\n<\/div>\r\nAs an example, consider the RL plot shown in Figure 10\u20114, with Real Axis segments as shown in Figure 10\u20115.\r\n\r\n[caption id=\"attachment_2006\" align=\"aligncenter\" width=\"692\"]<img src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_5.png\" alt=\"Figure 10\u20115 Segments of a Root Locus Plot on Real Axis\" width=\"692\" height=\"550\" class=\"wp-image-2006 size-full\" \/> Figure 10\u20115 Segments of a Root Locus Plot on Real Axis[\/caption]","rendered":"<p>This Rule deals with points on the Real axis. Consider the angle criterion:<\/p>\n<table style=\"border-collapse: collapse;width: 100%;height: 42px\">\n<tbody>\n<tr style=\"height: 14px\">\n<td style=\"width: 50%;height: 14px\">[latex]\\angle G(s) = 180^{\\circ} \\pm n\\cdot360^{\\circ}[\/latex]<\/td>\n<td style=\"width: 50%;height: 14px\"><\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"width: 50%;height: 14px\">[latex]\\angle(s-z_1)+\\angle(s-z_2)+...+\\angle(s-z_m)-...[\/latex]<\/td>\n<td style=\"width: 50%;height: 14px\"><\/td>\n<\/tr>\n<tr style=\"height: 14px\">\n<td style=\"width: 50%;height: 14px\">[latex]\\angle(s-p_1)-\\angle(s-p_2)-...-\\angle(s-p_n)=180^{\\circ}\\pm n360^{\\circ}[\/latex]<\/td>\n<td style=\"width: 50%;height: 14px;text-align: right\">Equation 10-6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Take some test points for [latex]s^*[\/latex] along the Real axis and see what satisfies the angle criterion.<\/p>\n<p style=\"text-align: center\">[latex]\\angle G(s^*)=\\sum\\theta_z - \\sum\\theta_p[\/latex]<\/p>\n<p>Where must Real axis segments of the root locus lie? Rule 2 answers that question.<\/p>\n<div class=\"textbox shaded\">\n<p><strong>Rule 2:<\/strong> A point s on the Real axis belongs to the root locus if and only if it is to the left of the ODD number of open loop singularities (a singularity is either a pole or a zero).<\/p>\n<\/div>\n<p>As an example, consider the RL plot shown in Figure 10\u20114, with Real Axis segments as shown in Figure 10\u20115.<\/p>\n<figure id=\"attachment_2006\" aria-describedby=\"caption-attachment-2006\" style=\"width: 692px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/pressbooks.library.ryerson.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_5.png\" alt=\"Figure 10\u20115 Segments of a Root Locus Plot on Real Axis\" width=\"692\" height=\"550\" class=\"wp-image-2006 size-full\" srcset=\"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_5.png 692w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_5-300x238.png 300w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_5-65x52.png 65w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_5-225x179.png 225w, https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-content\/uploads\/sites\/75\/2019\/12\/figure_10_2_5-350x278.png 350w\" sizes=\"auto, (max-width: 692px) 100vw, 692px\" \/><figcaption id=\"caption-attachment-2006\" class=\"wp-caption-text\">Figure 10\u20115 Segments of a Root Locus Plot on Real Axis<\/figcaption><\/figure>\n","protected":false},"author":156,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1999","chapter","type-chapter","status-publish","hentry"],"part":1471,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1999","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\/156"}],"version-history":[{"count":9,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1999\/revisions"}],"predecessor-version":[{"id":2339,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/1999\/revisions\/2339"}],"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\/1999\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=1999"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=1999"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=1999"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=1999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}