Gafar Barazin Nyquist: Me ke Nauyi? (Da Kuma Yadda ake Gafa)

Maa Nini Shaida Nyquist

Shaida Nyquist (ko da kuma Diagram Nyquist) yana nuna shaida na tashin rawa wanda ake amfani da ita a kula takamakawa da prosesin alama. Ake amfani da shaidar Nyquist don bincike zafiya ta takamakawa mai sabbin bayanin adadin rawa. A cikin koordinatin Cartesian, kafin da dukkan da na transfer function ya zama a x-axis, kuma kafin da mulkin da na transfer function ya zama a y-axis.

An gudanar da adadin rawa a matsayin paramita, wanda ya haɗa da shaida da ake tsara a kan adadin rawa. Zan iya bayyana shaidar Nyquist daidai a cikin koordinatin polar, inda zan iya bayyana gain da na transfer function a kan radial coordinate, kuma fase da na transfer function a kan angular coordinate.

Binciken zafiya ta takamakawa mai sabbin bayanin adadin rawa tana nuna a kan in tabbatar da abubuwan da suka samu wurin bayanan s-plane.

Idan wurare suka zaune a hagu na biyu na s-plane, ana ce takamakawa yana da zafiya. Ana iya tabbatar da zafiyan da na takamakawa a cikin yanayi a fadada adadin rawa – kamar shaidar Nyquist, Nichols plot, da Bode plot.

Nyquist stability criterion an amfani da ita don tabbatar da in wurare suka zaune a hagu na biyu na s-plane.

Don in fahimtar shaidar Nyquist, muna buƙata da wasu terminologiyoyi. Tabbacin da ke cikin complex plane ana kiran contour.

Nyquist Path ko Nyquist Contour

Nyquist contour yana nuna contour da ke ciki a s-plane wanda ya kula hagu na biyu na s-plane duka.

Don in kula hagu na biyu na s-plane, ana kira semicircle path mai inganci da diameter a jω axis da center a origin. Radius da semicircle ya zama Nyquist Encirclement.

Nyquist Encirclement

Ana ce point ya kula contour idan ana samun shi a cikin contour.

Nyquist Mapping

Inganci wanda ake amfani da ita don in tabbatar da point a s-plane zuwa point a F(s) plane ana kiran mapping and F(s) ana kiran mapping function.

Yadda Ake Gargajiya Shaidar Nyquist

Za a iya gargajiya shaidar Nyquist a cikin yanayin:

• Rukun 1 – Tambayata poles da G(s) H(s) a jω axis sama da wannan da a origin.

• Rukun 2 – Zabi Nyquist contour daidai – a) Kula hagu na biyu na s-plane tare da semicircle da radius R da R yana ci gaba zuwa infinity.

• Rukun 3 – Bayyana segments da ke cikin contour a kan Nyquist path

• Rukun 4 – Gargajiya segment da segment tare da substitution da equation da ke cikin segment a kan mapping function. Amma, za a iya sketsh polar plots da ke cikin segment.

• Rukun 5 – Mapping da ke cikin segments suna zo ne mirror images da mapping da ke cikin path da +ve imaginary axis.

• Rukun 6 – Semicircular path wanda ya kula hagu na biyu na s plane yana ci gaba zuwa point a G(s) H(s) plane.

• Rukun 7- Haɗa da mapping da ke cikin segments masu sauƙi don in tabbatar da Nyquist diagram daidai.

• Rukun 8 – Tabbar matafiya da ke cikin encirclements a kan (-1, 0) da kuma tabbatar da zafiya a kan N = Z – P

itace Open loop transfer function (O.L.T.F)

itace Closed loop transfer function (C.L.T.F)
N(s) = 0 itace open loop zero and D(s) itace open loop pole
Daga ma'anar zafiya, ba za a iya samun closed loop poles a hagu na biyu na s-plane. Characteristics equation 1 + G(s) H(s) = 0 itace closed-loop poles .

Idan 1 + G(s) H(s) = 0, don haka q(s) ya kamata a ci zero.

Saboda haka, daga ma'anar zafiya, zeroes da ke cikin q(s) ba za a iya samun a hagu na biyu na s-plane.
Don in tabbatar da zafiya, an kula hagu na biyu na s-plane duka. Ana kira semicircle wanda ya kula abubuwan da ke cikin hagu na biyu na s-plane tare da radius da semicircle R yana ci gaba zuwa infinity. [R → ∞].

Rukun na farko don in fahimtar amfani da Nyquist criterion a cikin tabbatar da zafiya na takamakawar control systems shine mapping daga s-plane zuwa G(s) H(s) – plane.

s an kiran independent complex variable and value da G(s) H(s) yake da dependent variable wanda ake sa a complex plane da ke kira G(s) H(s) – plane.

Saboda haka, har point da ke cikin s-plane, akwai corresponding point a G(s) H(s) – plane. A cikin ingancin mapping, independent variable s an yi variya a kan specified path a s-plane, and corresponding points a G(s)H(s) plane an haɗa. Wannan ya kula ingancin mapping daga s-plane zuwa G(s)H(s) – plane.

Nyquist stability criterion ya ce N = Z – P. Inda, N itace total no. da ke cikin encirclement a kan origin, P itace total no. da ke cikin poles and Z itace total no. da ke cikin zeroes.
Case 1: N = 0 (ba a ci encirclement), don haka Z = P = 0 and Z = P
Idan N = 0, P ya kamata a ci zero, saboda haka takamakawa yana da zafiya.
Case 2: N > 0 (clockwise encirclement), don haka P = 0, Z ≠0 and Z > P
A cikin case 2, takamakawa ba da zafiya.
Case 3: N < 0 (counter-clockwise encirclement), don haka Z = 0, P ≠0 and P > Z
Takamakawa yana da zafiya.

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