en>AnomieBOT: Dating maintenance tags: {{Dn}}

2013-12-27T02:43:16Z

Dating maintenance tags: {{Dn}}

← Older revision		Revision as of 04:43, 27 December 2013
Line 1:		Line 1:
	~~Hello buddy~~. Let ~~me introduce myself~~. ~~I am Luther Aubrey~~. ~~What he really enjoys performing is to perform handball but he is struggling to discover time for it~~. ~~Bookkeeping has been his day occupation for a while~~. ~~Some time in~~ the ~~past he selected~~ to ~~live~~ in ~~Kansas.~~<br><br>~~Here is my weblog;~~ [~~http:~~//~~Www.sceltic.at~~/~~index~~.~~php?mod~~=~~users&action~~=~~view&id~~=~~19873 extended car warranty~~]		{{Unreferenced\|date=December 2009}}
			{{Context\|date=October 2009}}
			The '''magnitude condition''' is a constraint that is satisfied by the locus of points in the [[s-plane]] on which [[closed-loop poles]] of a system reside. In combination with the [[angle condition]], these two mathematical expressions fully determine the [[root locus]].

			Let the [[characteristic equation]]{{dn\|date=December 2013}} of a system be <math>1+\textbf{G}(s)=0</math>, where <math>\textbf{G}(s)=\frac{\textbf{P}(s)}{\textbf{Q}(s)}</math>. Rewriting the equation in [[polar form]] is useful.

			<math>e^{j2\pi}+\textbf{G}(s)=0 </math>

			<math>\textbf{G}(s)=-1=e^{j(\pi+2k\pi)}</math> where
			<math>(k=0,1,2,...)</math> are the only solutions to this equation. Rewriting <math>\textbf{G}(s)</math> in [[factored form]],

			<math>\textbf{G}(s)=\frac{\textbf{P}(s)}{\textbf{Q}(s)}=K\frac{(s-a_1)(s-a_2)\cdots(s-a_n)}{(s-b_1)(s-b_2)\cdots(s-b_m)}</math>,

			and representing each factor <math>(s-a_p)</math> and <math>(s-b_q)</math> by their [[vector (geometric)\|vector]] equivalents, <math>A_pe^{j\theta_p}</math> and <math>B_qe^{j\phi_q}</math>, respectively, <math>\textbf{G}(s)</math> may be rewritten.

			<math>\textbf{G}(s)=K\frac{A_1 A_2 \cdots A_ne^{j(\theta_1+\theta_2+\cdots+\theta_n)}}{B_1 B_2 \cdots B_m e^{j(\phi_1+\phi_2+\cdots+\phi_m)}}</math>

			Simplifying the characteristic equation,

			<math>e^{j(\pi+2k\pi)}=K\frac{A_1 A_2 \cdots A_ne^{j(\theta_1+\theta_2+\cdots+\theta_n)}}{B_1 B_2 \cdots B_m e^{j(\phi_1+\phi_2+\cdots+\phi_m)}}=K\frac{A_1 A_2 \cdots A_n}{B_1 B_2 \cdots B_m}e^{j(\theta_1+\theta_2+\cdots+\theta_n-(\phi_1+\phi_2+\cdots+\phi_m))}</math>,

			from which we derive the magnitude condition:

			<math>1=K\frac{A_1 A_2 \cdots A_n}{B_1 B_2 \cdots B_m}</math>.

			The [[angle condition]] is derived similarly.

			{{DEFAULTSORT:Magnitude Condition}}
			[[Category:Control theory]]

en>Mandarax: /* Finding closed-loop poles */Typo patrol, typos fixed: preceeding → preceding using AWB (7571)

2011-02-13T09:02:47Z

Finding closed-loop poles: Typo patrol, typos fixed: preceeding → preceding using AWB (7571)

New page

Hello buddy. Let me introduce myself. I am Luther Aubrey. What he really enjoys performing is to perform handball but he is struggling to discover time for it. Bookkeeping has been his day occupation for a while. Some time in the past he selected to live in Kansas.<br><br>Here is my weblog; [http://Www.sceltic.at/index.php?mod=users&action=view&id=19873 extended car warranty]

Closed-loop pole - Revision history

en>AnomieBOT: Dating maintenance tags: {{Dn}}

en>Mandarax: /* Finding closed-loop poles */Typo patrol, typos fixed: preceeding → preceding using AWB (7571)