{"id":2013,"date":"2019-12-08T01:35:21","date_gmt":"2019-12-08T01:35:21","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&#038;p=2013"},"modified":"2021-01-14T14:37:09","modified_gmt":"2021-01-14T14:37:09","slug":"10-8-evans-root-locus-construction-rule-6-angles-of-departures-arrivals-at-complex-poles-zeros","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/10-8-evans-root-locus-construction-rule-6-angles-of-departures-arrivals-at-complex-poles-zeros\/","title":{"raw":"10.8 Evans Root Locus Construction Rule #6: Angles of Departures\/Arrivals at Complex Poles\/Zeros","rendered":"10.8 Evans Root Locus Construction Rule #6: Angles of Departures\/Arrivals at Complex Poles\/Zeros"},"content":{"raw":"This Rule deals with angles of departure (arrival) from\/to complex poles (zeros).\r\n\r\n&nbsp;\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>Rule 6:<\/strong> If [latex]G(s)[\/latex] has a pole <strong>p<\/strong> of multiplicity <strong>r<\/strong>, then <strong>r <\/strong>branches of the root locus depart from <strong>p<\/strong>. The angle of departure of these root loci from <strong>p<\/strong> are described by this equation:\r\n<table style=\"width: 100%;border-collapse: collapse\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 100%\">\r\n<p style=\"text-align: right\"><span>[latex]\\theta_{dep}[\/latex]= [latex] \\frac{ \\angle [G(s)(s-p)^r]_{s=p}+(2k+1) \\cdot 180^{\\circ}}{r}[\/latex]\u00a0 [latex]k=0,1,2,...,r-1[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><span>Equation 10-13<\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSimilarly, if [latex]G(s)[\/latex] has a zero <strong>z<\/strong> of multiplicity <strong>r<\/strong>, then <strong>r <\/strong>branches of the root locus arrive at <strong>z<\/strong>. The angle of arrival of these root loci to <strong>z<\/strong> are described by this equation:\r\n<table style=\"width: 100%;border-collapse: collapse\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 100%\"><span>[latex] \\theta_{arr} = \\frac{-\\angle[\\frac{G(s)}{(s-z)^r}]_{s=z}+(2k+1) \\cdot 180^{\\circ}}{r} k=0,1,2,...,r-1[\/latex]<\/span>\r\n<p style=\"text-align: right\">Equation 10-14<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\nSee examples in the next section for illustration of this rule.","rendered":"<p>This Rule deals with angles of departure (arrival) from\/to complex poles (zeros).<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox shaded\">\n<p><strong>Rule 6:<\/strong> If [latex]G(s)[\/latex] has a pole <strong>p<\/strong> of multiplicity <strong>r<\/strong>, then <strong>r <\/strong>branches of the root locus depart from <strong>p<\/strong>. The angle of departure of these root loci from <strong>p<\/strong> are described by this equation:<\/p>\n<table style=\"width: 100%;border-collapse: collapse\">\n<tbody>\n<tr>\n<td style=\"width: 100%\">\n<p style=\"text-align: right\"><span>[latex]\\theta_{dep}[\/latex]= [latex]\\frac{ \\angle [G(s)(s-p)^r]_{s=p}+(2k+1) \\cdot 180^{\\circ}}{r}[\/latex]\u00a0 [latex]k=0,1,2,...,r-1[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><span>Equation 10-13<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Similarly, if [latex]G(s)[\/latex] has a zero <strong>z<\/strong> of multiplicity <strong>r<\/strong>, then <strong>r <\/strong>branches of the root locus arrive at <strong>z<\/strong>. The angle of arrival of these root loci to <strong>z<\/strong> are described by this equation:<\/p>\n<table style=\"width: 100%;border-collapse: collapse\">\n<tbody>\n<tr>\n<td style=\"width: 100%\"><span>[latex]\\theta_{arr} = \\frac{-\\angle[\\frac{G(s)}{(s-z)^r}]_{s=z}+(2k+1) \\cdot 180^{\\circ}}{r} k=0,1,2,...,r-1[\/latex]<\/span><\/p>\n<p style=\"text-align: right\">Equation 10-14<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<p>See examples in the next section for illustration of this rule.<\/p>\n","protected":false},"author":162,"menu_order":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2013","chapter","type-chapter","status-publish","hentry"],"part":1471,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2013","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":16,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2013\/revisions"}],"predecessor-version":[{"id":2698,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2013\/revisions\/2698"}],"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\/2013\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=2013"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=2013"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=2013"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=2013"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}