{"id":2075,"date":"2019-12-14T03:07:41","date_gmt":"2019-12-14T03:07:41","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=2075"},"modified":"2021-01-14T16:16:37","modified_gmt":"2021-01-14T16:16:37","slug":"14-2-polar-plots-revisited","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/14-2-polar-plots-revisited\/","title":{"raw":"14.2 Polar Plots Revisited","rendered":"14.2 Polar Plots Revisited"},"content":{"raw":"The frequency response of a system is described by a complex frequency function, [latex]G(j \\omega)[\/latex]<\/span>. Any complex function can be represented in two different ways, using polar coordinates or rectangular coordinates. In general, consider the function to be represented in polar coordinates:\r\n\r\n \r\n\r\n\r\n\r\n
\r\n

[latex]G(j \\omega)=[\/latex] [latex] \\mid G(j \\omega) \\mid\u00a0 \\cdot e^{j \\angle G(j\\omega)} [\/latex]<\/span><\/p>\r\n

[latex]M( \\omega)= \\mid G(j \\omega) \\mid[\/latex]<\/p>\r\n

[latex] \\Phi ( \\omega) = \\angle G(j \\omega)[\/latex]<\/p>\r\n

[latex]G(j \\omega) = M( \\omega) \\cdot e^{j \\Phi( \\omega )}[\/latex]<\/p>\r\n<\/td>\r\n

Equation 14-1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn short-hand notation:\r\n\r\n\r\n\r\n
\r\n

[latex]G(j \\omega)= \\mid G(j \\omega) \\mid \\angle G(j \\omega)[\/latex]<\/span><\/p>\r\n

[latex]G(j \\omega)=M(\\omega) \\angle \\Phi( \\omega)[\/latex]<\/p>\r\n<\/td>\r\n

Equation 14-2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe two functions of frequency, magnitude function [latex]M( \\omega )[\/latex], and phase function [latex]\\Phi( \\omega )[\/latex] , can be computed and plotted, resulting in a familiar frequency response plot, also referred to as a Bode plot. The phase function [latex] \\Phi ( \\omega )[\/latex] is usually plotted using degrees vs. radian\/sec scale. The magnitude function [latex]M( \\omega )[\/latex], is usually plotted using the standard dB vs. radian\/sec scale. However, for some purposes, it may be more convenient to plot [latex] M( \\omega )[\/latex] using Volt\/Volt vs. radian\/sec scale.\r\n\r\nThe advantage of using the magnitude-phase representation of the frequency response is that both functions can be measured experimentally. This allows an empiric identification of the system transfer function [latex]G(s)[\/latex] based on the measured magnitude and phase plots.\r\n\r\nThe same frequency response function [latex]G(j \\omega )[\/latex] can be represented in rectangular coordinates:\r\n\r\n\r\n\r\n
\r\n

[latex]G(j \\omega) =[\/latex] [latex]Re \\{ G(j \\omega) \\} +jIm \\{ G(j \\omega) \\}[\/latex]<\/span><\/p>\r\n

[latex]Re( \\omega)= Re \\{ G(j \\omega) \\}[\/latex]<\/p>\r\n

[latex]Im( \\omega )=Im \\{ G(j \\omega ) \\}[\/latex]<\/p>\r\n

[latex]G(j \\omega)=Re( \\omega)+jIm( \\omega)[\/latex]<\/p>\r\n<\/td>\r\n

Equation 14-3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe two functions of frequency,[latex] Re ( \\omega ) [\/latex] and\u00a0 [latex] Im( \\omega )[\/latex], can be computed and plotted, but they cannot be measured experimentally. The relationship between [latex]Re ( \\omega )[\/latex] , [latex] Im( \\omega )[\/latex]<\/span> functions and [latex]M( \\omega )[\/latex] ,[latex]\\Phi( \\omega )[\/latex] functions, based on complex numbers algebra, is as follows:\r\n\r\n \r\n\r\n\r\n\r\n
\r\n

[latex]G(j \\omega)= M( \\omega) \\cdot e^{j \\Phi( \\omega)}=[\/latex] [latex]Re( \\omega )+[\/latex] [latex]jIm( \\omega )[\/latex]<\/p>\r\n

[latex]M( \\omega )=[\/latex] [latex]\\sqrt{Re( \\omega )^2+Im( \\omega )^2}[\/latex]<\/p>\r\n

[latex] \\Phi ( \\omega ) = tan^-1( \\frac{Im( \\omega )}{Re( \\omega )})[\/latex]<\/p>\r\n<\/td>\r\n

Equation 14-4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nInversely:\r\n\r\n\r\n\r\n
\r\n

[latex]G(j \\omega )=M( \\omega ) \\cdot e^{j \\Phi ( \\omega )} = Re( \\omega ) + jIm( \\omega )[\/latex]\r\n[latex] Re( \\omega ) = M( \\omega ) \\cdot cos( \\Phi ( \\omega ))[\/latex]\r\n[latex] Im( \\omega ) = M( \\omega ) \\cdot sin( \\Phi ( \\omega ))[\/latex]<\/p>\r\n<\/td>\r\n

Equation 14-5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nNote that in the above equations, the magnitude function is expressed in Volt\/Volt units, not in decibels. Functions [latex]Re( \\omega )[\/latex],[latex]Im( \\omega )[\/latex] can be plotted in rectangular coordinates (using Volt\/Volt units on both [latex]Re, Im[\/latex] axis) with frequency [latex] \\omega [\/latex] being a parameter along the curve, resulting in the Polar Plot<\/strong>.\r\n\r\nPolar Plots cannot be directly obtained from an experiment, and have to be computed based on magnitude-phase plots. Their application is mainly in determining the system stability in frequency domain (Gain and Phase Margin concepts and Nyquist Stability Criterion).","rendered":"

The frequency response of a system is described by a complex frequency function, [latex]G(j \\omega)[\/latex]<\/span>. Any complex function can be represented in two different ways, using polar coordinates or rectangular coordinates. In general, consider the function to be represented in polar coordinates:<\/p>\n

 <\/p>\n\n\n\n
\n

[latex]G(j \\omega)=[\/latex] [latex]\\mid G(j \\omega) \\mid\u00a0 \\cdot e^{j \\angle G(j\\omega)}[\/latex]<\/span><\/p>\n

[latex]M( \\omega)= \\mid G(j \\omega) \\mid[\/latex]<\/p>\n

[latex]\\Phi ( \\omega) = \\angle G(j \\omega)[\/latex]<\/p>\n

[latex]G(j \\omega) = M( \\omega) \\cdot e^{j \\Phi( \\omega )}[\/latex]<\/p>\n<\/td>\n

Equation 14-1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

In short-hand notation:<\/p>\n\n\n\n
\n

[latex]G(j \\omega)= \\mid G(j \\omega) \\mid \\angle G(j \\omega)[\/latex]<\/span><\/p>\n

[latex]G(j \\omega)=M(\\omega) \\angle \\Phi( \\omega)[\/latex]<\/p>\n<\/td>\n

Equation 14-2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The two functions of frequency, magnitude function [latex]M( \\omega )[\/latex], and phase function [latex]\\Phi( \\omega )[\/latex] , can be computed and plotted, resulting in a familiar frequency response plot, also referred to as a Bode plot. The phase function [latex]\\Phi ( \\omega )[\/latex] is usually plotted using degrees vs. radian\/sec scale. The magnitude function [latex]M( \\omega )[\/latex], is usually plotted using the standard dB vs. radian\/sec scale. However, for some purposes, it may be more convenient to plot [latex]M( \\omega )[\/latex] using Volt\/Volt vs. radian\/sec scale.<\/p>\n

The advantage of using the magnitude-phase representation of the frequency response is that both functions can be measured experimentally. This allows an empiric identification of the system transfer function [latex]G(s)[\/latex] based on the measured magnitude and phase plots.<\/p>\n

The same frequency response function [latex]G(j \\omega )[\/latex] can be represented in rectangular coordinates:<\/p>\n\n\n\n
\n

[latex]G(j \\omega) =[\/latex] [latex]Re \\{ G(j \\omega) \\} +jIm \\{ G(j \\omega) \\}[\/latex]<\/span><\/p>\n

[latex]Re( \\omega)= Re \\{ G(j \\omega) \\}[\/latex]<\/p>\n

[latex]Im( \\omega )=Im \\{ G(j \\omega ) \\}[\/latex]<\/p>\n

[latex]G(j \\omega)=Re( \\omega)+jIm( \\omega)[\/latex]<\/p>\n<\/td>\n

Equation 14-3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The two functions of frequency,[latex]Re ( \\omega )[\/latex] and\u00a0 [latex]Im( \\omega )[\/latex], can be computed and plotted, but they cannot be measured experimentally. The relationship between [latex]Re ( \\omega )[\/latex] , [latex]Im( \\omega )[\/latex]<\/span> functions and [latex]M( \\omega )[\/latex] ,[latex]\\Phi( \\omega )[\/latex] functions, based on complex numbers algebra, is as follows:<\/p>\n

 <\/p>\n\n\n\n
\n

[latex]G(j \\omega)= M( \\omega) \\cdot e^{j \\Phi( \\omega)}=[\/latex] [latex]Re( \\omega )+[\/latex] [latex]jIm( \\omega )[\/latex]<\/p>\n

[latex]M( \\omega )=[\/latex] [latex]\\sqrt{Re( \\omega )^2+Im( \\omega )^2}[\/latex]<\/p>\n

[latex]\\Phi ( \\omega ) = tan^-1( \\frac{Im( \\omega )}{Re( \\omega )})[\/latex]<\/p>\n<\/td>\n

Equation 14-4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Inversely:<\/p>\n\n\n\n
\n

[latex]G(j \\omega )=M( \\omega ) \\cdot e^{j \\Phi ( \\omega )} = Re( \\omega ) + jIm( \\omega )[\/latex]
\n[latex]Re( \\omega ) = M( \\omega ) \\cdot cos( \\Phi ( \\omega ))[\/latex]
\n[latex]Im( \\omega ) = M( \\omega ) \\cdot sin( \\Phi ( \\omega ))[\/latex]<\/p>\n<\/td>\n

Equation 14-5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Note that in the above equations, the magnitude function is expressed in Volt\/Volt units, not in decibels. Functions [latex]Re( \\omega )[\/latex],[latex]Im( \\omega )[\/latex] can be plotted in rectangular coordinates (using Volt\/Volt units on both [latex]Re, Im[\/latex] axis) with frequency [latex]\\omega[\/latex] being a parameter along the curve, resulting in the Polar Plot<\/strong>.<\/p>\n

Polar Plots cannot be directly obtained from an experiment, and have to be computed based on magnitude-phase plots. Their application is mainly in determining the system stability in frequency domain (Gain and Phase Margin concepts and Nyquist Stability Criterion).<\/p>\n","protected":false},"author":162,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2075","chapter","type-chapter","status-publish","hentry"],"part":2068,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2075","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":19,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2075\/revisions"}],"predecessor-version":[{"id":2731,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2075\/revisions\/2731"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/parts\/2068"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapters\/2075\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/media?parent=2075"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/pressbooks\/v2\/chapter-type?post=2075"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/contributor?post=2075"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/wp-json\/wp\/v2\/license?post=2075"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}