The Effect of a Zero on a Second Order System's Performance Copyright 2010, Kevin Walker
<Text-field style="Heading 1" layout="Heading 1">Form of the System</Text-field> A second order system can be described using its natural frequency, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIm5GJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1, its damping ratio, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, and its gain, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, which we will assume to be unity. Maple will assume that all variables are complex unless it is told otherwise. A few hints to Maple on the domain of these parameters will often result in more compact and intuitive solutions later. We consider only positive frequencies, 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 print(); # input placeholder and set the damping ratio between 0 and 1 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 If the damping ratio were greater than 1, then the second order system would simplify to two first order poles, so for the time being we will assume that it is less than one. The transfer function of the system is (with unit gain LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=): 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiYpJkknb21lZ2F8aXJHNiI2I0kibkdGJiIiIyIiIiwoKiQpSSJzR0YmRilGKkYqKipGKUYqSSR4aXxpckdGJkYqRiRGKkYuRipGKiokRiNGKkYqISIi To introduce a zero into the system at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8=, we multiply the numerator of the transfer function by LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXJvd0dGJDYoLUkjbWlHRiQ2I1EhRictRiw2KUYuLUYsNihGLi1GLDYmRi4vJStleGVjdXRhYmxlR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGLkY4RjtGPi1JJm1mcmFjR0YkNigtRi82JVEic0YnLyUnaXRhbGljR0Y6L0Y/USdpdGFsaWNGJy1GLDYmLUYvNiVRImFGJ0ZHRklGOEY7Rj4vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlUvJSliZXZlbGxlZEdRJmZhbHNlRictSSNtb0dGJDYtUSIrRidGPi8lJmZlbmNlR0ZaLyUqc2VwYXJhdG9yR0ZaLyUpc3RyZXRjaHlHRlovJSpzeW1tZXRyaWNHRlovJShsYXJnZW9wR0ZaLyUubW92YWJsZWxpbWl0c0dGWi8lJ2FjY2VudEdGWi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRmlvRjhGO0Y+LUkjbW5HRiQ2JEZSRj5GOEY7Rj5GPg==. Since this term is zero when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GUy1GLDYlUSJhRidGL0YyRjk=, therefore the transfer function also goes to zero (and hence the name "zero"). We choose to use 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 instead of the more canonical form of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJzRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIitGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTi1GMTYlUSJhRidGNEY3RjdGNy1GMTYjUSFGJ0Y3 to keep the DC gain of the system at unity. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KigpJkknb21lZ2F8aXJHNiI2I0kibkdGJiIiIyIiIiwoKiQpSSJzR0YmRilGKkYqKipGKUYqSSR4aXxpckdGJkYqRiRGKkYuRipGKiokRiNGKkYqISIiLCYqJkYuRipJImFHRiZGMkYqRipGKkYq Note that 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, which means that the zero has less and less effect as it moves further away from the origin. The fraction may be expanded into two terms: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEpbG9uZ0Zvcm1GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSdleHBhbmRGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUTRzZWNvbmRPcmRlcldpdGhaZXJvRidGL0YyRjlGOS8lK2V4ZWN1dGFibGVHRj1GOQ== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYqKikmSSdvbWVnYXxpckc2IjYjSSJuR0YnIiIjIiIiLCgqJClJInNHRidGKkYrRisqKkYqRitJJHhpfGlyR0YnRitGJUYrRi9GK0YrKiRGJEYrRishIiJGL0YrSSJhR0YnRjNGKyomRiRGK0YsRjNGKw== The second term is the standard second order system as given in . 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Ly0mSSJYRzYiNiNJI3NvR0YmNiNJInNHRiYqJikmSSdvbWVnYXxpckdGJjYjSSJuR0YmIiIjIiIiLCgqJClGKkYxRjJGMioqRjFGMkkkeGl8aXJHRiZGMkYtRjJGKkYyRjIqJEYsRjJGMiEiIg== The first term is the perturbation caused by the zero: 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Ly0mSSJYRzYiNiNJI3pwR0YmNiNJInNHRiYqKikmSSdvbWVnYXxpckdGJjYjSSJuR0YmIiIjIiIiLCgqJClGKkYxRjJGMioqRjFGMkkkeGl8aXJHRiZGMkYtRjJGKkYyRjIqJEYsRjJGMiEiIkYqRjJJImFHRiZGOQ==
<Text-field style="Heading 2" layout="Heading 2">Relationship between the Two Terms</Text-field> Recalling the Laplace transform property, 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, and assuming zero initial conditions, we can see that the inverse transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3pwRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEic0YnRjJGNS9GNlEnbm9ybWFsRidGSEZI is the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3NvRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEidEYnRjJGNS9GNlEnbm9ybWFsRidGSEZI scaled by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUubGluZXRoaWNrbmVzc0dGMS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjI=. That is, 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. This equivalence can be illustrated through the time-domain step responses. These are obtained by first multiplying the transfer functions by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRInNGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUubGluZXRoaWNrbmVzc0dGMS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUY4NiNRIUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Yy, which applies a unit step to the system at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, then by taking the inverse Laplace transform. That is, we will show the equivalence of 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. Beginning with the right-hand-side of this equation, 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYiIiJGIyooKSwmRiNGIyomLCYqJClJJHhpfGlyRzYiIiIjRiMhIiJGI0YjRi5GKkYjRiMjRiNGLUYjLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YsNiMsJCooSSJ0R0YsRiNGK0YjJkknb21lZ2F8aXJHRiw2I0kibkdGLEYjRi5GIy1JJGNvc0dGMjYjLCYqKClGKEYvRiNGOEYjRjlGI0YjLUknYXJjdGFuR0YyNiMqJkZCRi5GK0YjRi5GI0Yu 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiwpLCYqJClJJHhpfGlyRzYiIiIjIiIiISIiRipGKiNGKkYpRismSSdvbWVnYXxpckdGKDYjSSJuR0YoRipJImFHRihGKy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjLCQqKEkidEdGKEYqRidGKkYtRipGK0YqLUkkc2luR0Y0NiMqKEYjRipGOkYqRi1GKkYq LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= The left-hand-side of the equation can be obtained by: 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiwpLCYqJClJJHhpfGlyRzYiIiIjIiIiISIiRipGKiNGKkYpRismSSdvbWVnYXxpckdGKDYjSSJuR0YoRipJImFHRihGKy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKDYjLCQqKEkidEdGKEYqRidGKkYtRipGK0YqLUkkc2luR0Y0NiMqKEYjRipGOkYqRi1GKkYq Since and are identical, therefore we have shown that the inverse transform of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3pwRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEic0YnRjJGNS9GNlEnbm9ybWFsRidGSEZI is the time derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRI3NvRidGMkY1RjJGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEidEYnRjJGNS9GNlEnbm9ybWFsRidGSEZI scaled by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21pR0YkNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRidGO0Y+LyUubGluZXRoaWNrbmVzc0dGMS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZELyUpYmV2ZWxsZWRHUSZmYWxzZUYnLUY4NiNRIUYnRjI=.
<Text-field style="Heading 2" layout="Heading 2">Time-domain Step Response</Text-field> The time-domain step response of the second order system with a zero is 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCgiIiJGIyomLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqKCksJiokKUkkeGl8aXJHRioiIiNGIyEiIkYjRiMjRiNGMkYjSSJ0R0YqRiMmSSdvbWVnYXxpckdGKjYjSSJuR0YqRiNGIy1JJGV4cEdGJzYjLCQqKEY1RiNGMUYjRjZGI0YzRiNGMyosLUkkc2luR0YnRitGI0Y6RiNGLUYzLCYqJkYxRiNJImFHRipGI0YzRjZGI0YjRkRGM0Yj Since the arguments of the sine and cosine functions are identical, we can convert them into a single trigonometric function with a new magnitude and phase offset. This gives a more intuitive form of the step response. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYiIiJGIyooKSwmRiNGIyooLCYqJClJJHhpfGlyRzYiIiIjRiMhIiJGI0YjRi4pLCYqJkYrRiNJImFHRixGI0YuJkknb21lZ2F8aXJHRiw2I0kibkdGLEYjRi1GIylGMkYtRi5GIyNGI0YtRiMtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiw2IywkKihJInRHRixGI0YrRiNGM0YjRi5GIy1JJGNvc0dGOzYjLCYqKClGKEY4RiNGQUYjRjNGI0YjLUknYXJjdGFuR0Y7NiQqKEZHRi5GMEYjRjJGLkYuRi5GI0Yj LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Interactive Exploration</Text-field> Click and drag the poles or the zero in the S-plane to see the effect on the time domain response on the right. S-Plane (Frequency Domain) use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. notDragging := true; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. current_x := GetProperty('PolesPlot', 'clickx'); current_y := GetProperty('PolesPlot', 'clicky'); SetDragTarget(); UpdatePlots(); end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. current_x := GetProperty('PolesPlot', 'endx'); current_y := GetProperty('PolesPlot', 'endy'); SetDragTarget(); UpdatePlots(); end use; 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Step Response (Time Domain) 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The time-domain step response:
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