Nyquist Plot: Yeh ne ku ye? (An Jêrîn Yek Bicyan Bike)

Nyquist Plot çi ye?

Yek Nyquist plot (an jî Nyquist Diagram) yek grafik bingehkirî ye ku di control engineering û signal processing de bikar î. Nyquist plots bi serastî bikar inin ji bo daxwazkirina êzikînê ya cihêrên control system û feedback. Di koordinatên Cartesian de, parçeyên girtî yên transfer function bi X-axis re hatine nîşan kirin, û parçeyên imajiner bi Y-axis re.

Girîngî wek parametrekî tîne veqetand, ku hûn grafik bi girîngî re tîne wergerandin. Yek Nyquist plot din dikarin bi koordinatên polar tîne şînin, ku gaina transfer function radîal coordinate ye, û fazaya transfer function angular coordinate ye.

Analîza êzikînê ya cihêrên feedback control system li gorîdena lokasyonê yên roots yên characteristic equation di s-plane de.

Sistem êzik e, ger roots di sîtên çepa s-plane de be. Stabiliteya relative ya sistem bi karîna metoda frequency response - wêçî Nyquist plot, Nichols plot, û Bode plot.

Nyquist stability criterion bikar înin ji bo daxwazkirina nekinan roots yên characteristic equation di navcheya xweyê de s-plane.

Ji bo fahmîna Nyquist plot, pirmezin hewce ne ku eweriyên terminologîyan bigerin. Bînin ku contour yek path bi vebijarkirî ye di complex plane de.

Nyquist Path an Nyquist Contour

Nyquist contour yek contour bi vebijarkirî ye di s-plane de ku heta sîtên rastên s-plane derbas bike.

Bi tenê ku heta sîtên rastên s-plane derbas bike, yek semicircle path mezin dibaxîn bi diameter di jω axis de û center di origin de. Radiusi semicircle dibaxîn wek Nyquist Encirclement.

Nyquist Encirclement

Pîv pêjirî ye ku contour bi serastî tîne encircled, ger bi serastî di contour de be.

Nyquist Mapping

Prosesa ku di vê procesê de pîv di s-plane de tîne transform kirin bi pîv di F(s) plane de nameke mapping û F(s) nameke mapping function.

Ji Dê Wekhevî Nyquist Plot

Nyquist plot bi karîna paşeyên vê dibava dê wekhevin:

• Paşa 1 – Daxwazkirin poles of G(s) H(s) di jω axis de, kanîs di origin de.

• Paşa 2 – Pîvîna Nyquist contour – a) Sîtên rastên s-plane heta bi drawing a semicircle of radius R bi R tends to infinity.

• Paşa 3 – Identify the various segments on the contour with reference to the Nyquist path

• Paşa 4 – Perform the mapping segment by segment by substituting the equation for the respective segment in the mapping function. Basically, we have to sketch the polar plots of the respective segment.

• Paşa 5 – Mapping of the segments are usually mirror images of mapping of the respective path of +ve imaginary axis.

• Paşa 6 – The semicircular path which covers the right half of s plane generally maps into a point in G(s) H(s) plane.

• Paşa 7- Interconnect all the mapping of different segments to yield the required Nyquist diagram.

• Paşa 8 – Note the number of clockwise encirclements about (-1, 0) and decide stability by N = Z – P

is the Open loop transfer function (O.L.T.F)

is the Closed loop transfer function (C.L.T.F)
N(s) = 0 is the open loop zero and D(s) is the open loop pole
From a stability point of view, no closed loop poles should lie on the RH side of the s-plane. Characteristics equation 1 + G(s) H(s) = 0 denotes closed-loop poles .

Now as 1 + G(s) H(s) = 0 hence q(s) should also be zero.

Therefore, from the stability point of view zeroes of q(s) should not lie in the RHP of the s-plane.
To define the stability entire RHP (Right-Hand Plane) is considered. We assume a semicircle that encloses all points in the RHP by considering the radius of the semicircle R tends to infinity. [R → ∞].

Paşa yekêm ji bo fahmîna application of the Nyquist criterion li gorînen stabilitیاt control systems û mapping from the s-plane to the G(s) H(s) – plane.

s bi independent complex variable tekerdiye û value corresponding ya G(s) H(s) bi dependent variable tekerdiye di another complex plane de nîşan kirin wek G(s) H(s) – plane.

Naha, ji bo her pîv di s-plane de, pîv corresponding da di G(s) H(s) – plane de ye. Li dema process of mapping, independent variable s bi serastî di vebijarkirî path de di s-plane de tîne veqetand, û pîv corresponding da di G(s)H(s) plane de hatine connect kirin. Ev process of mapping from the s-plane to the G(s)H(s) – plane complete bike.

Nyquist stability criterion says that N = Z – P. Where, N is the total no. of encirclement about the origin, P is the total no. of poles and Z is the total no. of zeroes.
Case 1: N = 0 (no encirclement), so Z = P = 0 and Z = P
If N = 0, P must be zero therefore the system is stable.
Case 2: N > 0 (clockwise encirclement), so P = 0, Z ≠0 and Z > P
In both cases the system is unstable.
Case 3: N < 0 (counter-clockwise encirclement), so Z = 0, P ≠0 and P > Z
The system is stable.

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