How to Draw Polar Plot in Control System TUTORIAL

Control

MIRCEA IVANESCU , in Mechanical Engineer's Handbook, 2001

8.i The Polar Plot Representation

The polar plot representation is the graphical representation of Y(jω). The polar plot is obtained as the locus of the real and imaginary parts of Y(jω) in the polar plane. The coordinates of the polar plot are the real and imaginary parts of Y(jω). For example, nosotros reconsider the open up-loop arrangement for a translational machinery (5.13), Y ₁(southward) = k_Yard/south(ms + k_f ). The transfer function can be rewritten equally

(8.7) $Y_1(s)=\frac{\mathrm{1000}}{s(\tau \mathrm{south}+\mathrm{i})},$

where

(viii.viii) $\begin{array}{l}\mathrm{1000}=\frac{k_M}{k_f}\\ \tau =\frac{m}{{\mathrm{one\; thousand}}_f}.\end{array}$

The frequency transfer function Y _one(jω) will be

(eight.9) $Y_1(\mathrm{southward}){|}_{\mathrm{south}=j\omega }=Y_1(j\omega )=\frac{\mathrm{thousand}}{-{\omega }^2+j\omega }.$

From Eqs. (8.iii)–(8.half dozen), nosotros obtain

(8.10) $\begin{array}{l}{P}_{\mathrm{i}}(\omega )=\frac{-g{\omega }^2\tau }{{\omega }^4{\tau }^2+{\omega }^2}\\ Q_1(\omega )=\frac{-\omega \mathrm{1000}}{{\omega }^4{\tau }^2+{\omega }^2}\end{array}$

and

(viii.eleven) $M_{\mathrm{ane}}(\omega )=\frac{k}{{({\omega }^{\mathrm{iv}}{\tau }^2+{\omega }^2)}^{1/2}}$

(8.12) ${\Psi }_{\mathrm{one}}(\omega )=-{tan}^{-1}\left(\frac{-1}{\omega \tau }\right).$

We discover several values of M(ω) and Ψ(ω):

For ω = 0,

$\begin{array}{l}{M}_{\mathrm{one}}(0)=\infty \\ \Psi (0)=-\frac{\pi }{2};\end{array}$

For ω = 1/τ,

$\begin{array}{l}{M}_{\mathrm{ane}}\left(\frac{1}{\tau }\right)=\frac{\sqrt{2}}{2}k\tau \\ \Psi \left(\frac{1}{\tau }\right)=-\frac{3\pi }{4};\end{array}$

For ω = ∞,

$\begin{array}{l}{M}_{\mathrm{ane}}(\infty )=0\\ \Psi (\infty )=-\pi .\end{array}$

The polar plot of Y _one(jω) is shown in Fig. eight.1.

Figure eight.ane. Polar plot for open-loop organization of translational machinery.

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Nyquist Diagrams

William Bolton , in Instrumentation and Command Systems (Tertiary Edition), 2021

12.ii The Polar Plot

The polar plot of the frequency response of a system is the line traced out every bit the frequency is changed from 0 to infinity by the tips of the phasors whose lengths represent the magnitude, i.east. aamplitude gain, of the organization and which are drawn at angles corresponding to their phase ϕ (Figure 12.two).

Figure 12.2. Polar plot with the plot as the line traced out by the tips of the phasors equally the frequency is inverse from zero to infinity.

Example

Draw the polar diagram for the following frequency response data.

Freq. rad/s	1.4	two.0	2.6	three.2	three.viii
Magnitude	ane.six	1.0	0.6	0.4	0.2
Stage deg.	−150	−160	−170	−180	−190

Note that the to a higher place data gives the phases with negative signs. This means they are lagging behind the 0° line by the amounts given. Effigy 12.3 shows the polar plot obtained.

Effigy 12.iii. Example.

12.2.1 Nyquist Diagrams

The term Nyquist diagram is used for a diagram of the line joining the serial of points plotted on a polar graph when each indicate represents the magnitude and phase of the open-loop frequency response respective to a particular frequency. To plot the Nyquist diagram from the open-loop transfer function of a organization we need to determine the magnitude and the phase as functions of frequency.

Example

Decide the Nyquist diagram for a kickoff-social club system with an open-loop transfer office of 1/(i+ τs).

The frequency response is:

$\frac{1}{1+\mathrm{j}\omega \tau }=\frac{\mathrm{one}}{\mathrm{ane}+\mathrm{j}\omega \tau }\times \frac{1-\mathrm{j}\omega \tau }{\mathrm{ane}-\mathrm{j}\omega \tau }=\frac{1}{1+{\omega }^2{\tau }^2}-\mathrm{j}\frac{\omega \tau }{\mathrm{i}+{\omega }^{\mathrm{two}}{\tau }^2}$

The magnitude is thus:

$\text{Magnitude}=\frac{1}{\sqrt{1+{\omega }^{\mathrm{ii}}{\tau }^{\mathrm{ii}}}}$

and the phase is:

$\text{Phase}=-{\tan }^{-1}\omega \tau$

At nix frequency the magnitude is i and the stage 0°. At infinite frequency the magnitude is zero and the phase is −90°. When ωτ=i the magnitude is one/√2 and the phase is −45°. Substitution of other values leads to the event shown in Figure 12.4 of a semicircular plot.

Figure 12.4. Nyquist diagram for a first-social club arrangement.

Example

Determine the Nyquist plot for the system having the open-loop transfer function of ane/s(south+i).

The frequency response is:

$G(\mathrm{j}\omega )=\frac{\mathrm{one}}{\mathrm{j}\omega (\mathrm{j}\omega +1)}=\frac{1}{\mathrm{j}\omega -{\omega }^{\mathrm{two}}}=\frac{\mathrm{ane}}{\mathrm{j}\omega -{\omega }^2}\times \frac{-\mathrm{j}\omega -{\omega }^{\mathrm{two}}}{-\mathrm{j}\omega -{\omega }^{\mathrm{ii}}}$

$=\frac{-{\omega }^2}{{\omega }^2+{\omega }^4}-\mathrm{j}\frac{\omega }{{\omega }^2+{\omega }^4}$

the magnitude and phase are thus:

$\text{Magnitude}=\frac{1}{\omega \sqrt{{\omega }^{\mathrm{two}}+1}}$

$\text{Phase}={\tan }^{-\mathrm{one}}\frac{-1}{-\omega }=180°+{\tan }^{-\mathrm{one}}\frac{\mathrm{ane}}{\omega }$

When ω=∞ so the magnitude is 0 and the phase is 0°. As ω tends to 0 then the magnitude tends to infinity and the phase to 270° or −xc°. Effigy 12.5 shows the polar plot.

Figure 12.5. Example.

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Plotting

Brent Maxfield P.E. , in Essential PTC Mathcad® Prime® iii.0, 2014

Creating a Simple Polar Plot

Creating a simple polar plot is like to creating a simple XY plot. Blazon CTRL+7 , or select Polar Plot from the Insert Plot control on the Plots tab (Plots>Traces>Insert Plot> Polar Plot).

Click on the bottom (angular axis) placeholder. This is where you lot type the angular variable. Unless you specify otherwise, PTC Mathcad assumes the variable to exist in radians. Blazon the name of a previously undefined variable. The variable can be any PTC Mathcad variable name. Next, click on the right side radial-axis placeholder and type an expression using the angular variable divers on the angular-centrality. This sets the backdrop of the radial axis. Click outside of the plot region to view the plot. For every angle from 0 to 2π, PTC Mathcad plots a radial value. PTC Mathcad automatically selects the radial range. Come across Figure 8.i.

FIGURE eight.1. Polar plot of functions

You may also use a previously defined function in a unproblematic polar plot. Begin the polar plot region by typing CTRL+seven . Type the name of the angular variable in the bottom placeholder. Type the proper name of the function on the right placeholder using the athwart variable name equally the argument of the office. Meet Figure viii.2.

Figure 8.2. Polar plot of functions

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Fundamentals of Tire Beliefs

Joop P. Pauwelussen , in Essentials of Vehicle Dynamics, 2015

Remarks

1.

The polar plot of F _x versus F _y , every bit depicted in Figure 2.30 for the empirical Magic Formula, tin can too be derived for the physical brush model. Considering the strength characteristics based on the castor model saturate without disuse for large slip, this polar plot will be similar to the one in Figure two.29. Nosotros determined this polar plot for the castor model for F _z =4000   [N] (run across Effigy two.44). When nosotros plot the aligning torque versus F _x , expression (two.82) leads to a plot that is symmetric in F _x , unlike Figure 2.33. This can be corrected past adding simple carcass flexibility to the brush model. This means that the entire carcass is pinned to the projected center of the wheel through springs interim in lateral and longitudinal direction with unlike stiffness values. Just as in the discussion on Figure two.33, the resulting carcass deflections will so contribute to the aligning torque, leading to the loss of symmetry, as indicated in Figure 2.33. We refer to Ref. [32] for farther details.

Figure ii.44. Polar diagram of F ₁₀ versus F _y for abiding slip angle for F _z =4000 [N] and μ=1.0 based on the brush model.

ii.

The combined equations (two.79a) and (2.79b) evidence that the explicit expression of contact shear strength versus slip does not alter if we move from pure skid to combined slip. The simply departure is that ρ equals α or −κ/(1+κ) in case of pure lateral or longitudinal slip, or the total theoretical slip according to Eq. (2.76). That means that expression (2.78) tin be interpreted as

$\left(\begin{array}{c}{F}_x({\rho }_x,{\rho }_y)\\ F_y({\rho }_x,{\rho }_y)\end{array}\right)=\left(\begin{array}{c}{\rho }_x\\ {\rho }_y\end{array}\right)·\frac{F_{\mathrm{pure}}(\rho )}{\rho }$

Thus far, we discussed an isotropic castor model. The preceding expression has been the inspiration for the approximation (2.71), which gave accurate results that were sufficient in many cases, especially if qualitative analysis (trends, sensitivity) is the objective.

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Feedback Control Systems

Revised by William R. Perkins , in Reference Data for Engineers (Ninth Edition), 2002

Proceeds-Phase Plot

The proceeds-phase plot is the straight transfer of the polar plot from polar coordinates to rectangular coordinates. The ordinate is the gain in decibels, and the abscissa is the phase angle in degrees.

The gain and phase angle of the 2 functions G(jω) and −[i/N(ω, X)] are, for M(jω)

$\begin{array}{cc}\text{Gain}& 20\log \left|G\left(j\omega \right)\right|\\ \text{Phase angle}& \angle \mathrm{Chiliad}\left(j\omega \right)\end{array}$

and for N(X, ω)

$\begin{array}{cc}\text{Gain}& -\mathrm{xx}{\log }_{\mathrm{x}}\left|N(\omega ,X)\right|\\ \text{Phase angle}& -{180}^{\circ }-\angle \mathrm{Northward}(\omega ,\mathrm{10})\end{array}$

Typical gain-phase plots for various types of N(ω, X) are given in Fig. 53.

Fig. 53. Typical gain-stage plots of various N(ω, X).

The arrangement is stable if the −(one/Northward) locus does not intersect with the Grand(jω) plot. If the −(1/Northward) locus intersects with the G(jω) plot, the system has a sustained oscillation (Fig. 54).

Fig. 54. Stability criteria for proceeds-stage plot.

In the case of sustained oscillation, there may exist more than than one point of intersection, as shown in Fig. 55. Points A and C are stable points (stable limit cycle), and bespeak B is an unstable point (unstable limit wheel). The stability of the limit cycle is determined in a fashion similar to that of the polar plot, except that if the −[ane/N(X)] locus in the direction of increasing X crosses the G(jω) locus point in the management of increasing frequency from left to right, the limit cycle is stable; if it crosses from correct to left, the limit cycle is unstable. Fig. 56 is a typical gain-phase plot of Grand(jω) and −[ane/North(ω, 10)], where Due north(ω, X) is a office of ω. The family of −(ane/Northward) plots are the abiding-frequency loci. Betoken A is the location for sustained oscillation.

Fig. 55. Proceeds-phase plot of stable and unstable limit cycles.

Fig. 56. Typical gain-phase plot of Thousand(jω) and −[1/N(ω, Ten)] equally a function of ω.

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Drape measurement technique using manikins with the help of epitome analysis

Awadhesh Kumar Choudhary , Payal Bansal , in Manikins for Textile Evaluation, 2017

8.8.4.1 Image assay of fabric and garment drape

A monochrome (black and white/binary) prototype was converted into a polar plot ( µ, r), as the discrete points making up the image'due south contour were converted into polar coordinates, where the 10-axis presented the angle µ of each point in degrees (from 0±to 360±) from the horizontal line passing through the center, and Y-axis presented distance r of each point from the eye, in centimeters (Fig. 8.19). The polar plot was converted into Cartesian plot (x, y), where 10 was the angle of each coordinate's position and y was the radius. This plot was called the shape signature equally it presented the original distinctive wave of each prototype. The ideal (reconstructed) wave shape was recomposed from the determined average moving ridge values measured; namely wave length (WL), amplitude, and height (Jiang et al., 2010; Cassidy, Cassidy, Cassie, & Arkison, 1991).

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Nichols-Krohn-Manger-Hall Chart

Yazdan Bavafa-Toosi , in Introduction to Linear Control Systems, 2019

8.5 M- and N-Contours

As in the case of drawing the Nyquist diagram, the problem associated with the polar plots and the use of Grand- and N-circles is that a modify in the proceeds (i.e., multiplication or division of the gain by a constant) results in a certain amount of deformation of the polar plot. There nosotros saw that this problem was rectified by the apply of log magnitude in place of magnitude. This idea was as well used by Nichols and Krohn who suggested the Lm versus phase diagram. They originally used the proper name ψ-contours ⁱⁱⁱ instead of N-contours. However, and then the community well-nigh unanimously adopted the name N-contours, to become with M-contours, and we do and so equally well. Moreover, they directly started from ψ-contours, i.east., they did not introduce ψ-circles, although they did introduce M-circles. Anyway, by this transformation of the vertical axis, the M- and N-circles are also transformed. The results are called the M- and N-contours, depicted in the sequel Figs. 8.6 and eight.seven. The effigy repeats every $360\mathrm{deg}$ and is symmetric with respect to either of the multiples of $\pm 180\mathrm{deg}$ line. It looks similar Fig. viii.six. In this figure $M_{\mathrm{one}}<{\mathrm{1000}}_2<0<M_3$ and $-{\alpha }_i$ is used instead of $360-{\alpha }_i$ for simplicity of notation—they refer to the same point; the same for $-{\beta }_i,-{\gamma }_i$ . If these contours are superimposed on the open up-loop Lm versus phase grid of the system then the resulting plane is the Nichols chart which may also exist chosen the NKMH due to the contributions of all. An expansion of the $[-320°0°]$ interval of the NKMH chart is delineated in Fig. 8.7. Note that once upon a fourth dimension before the MATLAB® era such sheets were commercially available.

Figure 8.vi. M- and N-contours.

Figure eight.7. Expansion of part of the NKMH chart plane.

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Plotting

Brent Maxfield , in Essential Mathcad for Applied science, Science, and Math (2nd Edition), 2009

Chapter seven volition:

▪

Review how to create simple 2D X-Y plots and Polar plots.

▪

Show how to ready plot ranges.

▪

Instruct how to graph multiple functions in the same plot.

▪

Discuss the use of range variables to control plots.

▪

Tell how to plot data points.

▪

Describe the steps necessary to format plots, including the use of log calibration, filigree lines, scaling, numbering, and setting defaults.

▪

Discuss the utilise of titles and labels.

▪

Prove how to get numeric readout of plotted coordinates.

▪

Show how the employ of plots tin can help find the solutions to various engineering science problems.

▪

Hash out plotting over a log scale.

▪

Innovate 3D plotting.

▪

Use engineering examples to illustrate the concepts.

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Charts

Bernard Liengme , Keith Hekman , in Liengme'due south Guide to Excel® 2016 for Scientists and Engineers, 2020

Exercise 9: Polar (Radar) Chart

Excel does not provide a true polar chart pick but in simple cases, one may use a radar nautical chart to plot polar information.

(a)

On Sheet8 of Chap7.xlsx begin by entering the text shown in rows ane and iii of Fig. 7.29.

Fig. 7.29.

(b)

Fill A4:A363 with values 0 through 359. You can enter 0 in A4 and with A4 selected use Dwelling house / Editing / Fill / Series… (Home / Fill / Series… On a Mac) with options shown in Fig. 7.30 to quickly do this.

Excel's radar charts care for the beginning cavalcade as categories, non numeric values so it is essential to use a abiding increment in that cavalcade when plotting a function

Fig. 7.thirty.

(c)

In B4 enter =   1   +   SIN(RADIANS(A4)) and fill this down to B363 by double-clicking the make full handle.

(d)

Select a cell such as A4 and create a radar chart. You may exist surprised by the effect. Considering we have numeric values in the first column, Excel thinks we want two information series. On a PC right-click (on a Mac hold the

key and click) the chart and open the Select Data dialog. Two operations are needed: (i) remove the first series (Cardioid θ), and (two) edit the Category Axis Labels specifying the range Sheet8!$A$4:$A$363.

(due east)

The large number (360) of labels effectually the chart's category axis makes them unreadable. Double-click the cardioid in the chart and open the Format Data Series dialog. In the Serial Selection section uncheck the box Category labels.

(f)

Now nosotros will add a dummy series to become neater labels. Enter the series 0 through 345 in D4:D27. In E3 enter the formula =   MAX($B$3:$B$363) and drag this down to E27.

(yard)

Select D3:E7 and, using the techniques from Do 4, add the dummy data equally a 2d series on the nautical chart. Format this data series: (i) specify it is to use the secondary axis, and (two) specify no line and no makers.

(h)

Salve the workbook.

The reader may be wondering why nosotros used 360 information points for the radar chart. Return to Sheet7 and, post-obit step (d) listed previously, make a radar chart of the range A3:B19. A chart with few information points is totally unacceptable. Be aware that the method shown in this practice works just for elementary curves; mostly, information technology is more satisfying to use the parametric equation approach of Exercise 7.

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An Introduction to PTC® Mathcad Prime® 3.0

Brent Maxfield P.E. , in Essential PTC Mathcad® Prime number® 3.0, 2014

Plots Tab

The Plots tab is shown beneath. The Plots tab allows yous to quickly insert 2-dimensional 10-Y plots, Polar plots, and three-dimensional plots. Plotting will exist discussed at length in Affiliate seven, but let'south take a quick wait at how to create some unproblematic plots.

To create a unproblematic X-Y plot, click the Insert Plot control in the Traces group on the Plots tab, then select XY Plot (Plots>Traces>Insert Plot>XY Plot).

You may likewise blazon CTRL+ii . This places a blank X-Y plot operator on the worksheet.

Click on the bottom center placeholder. This is where y'all blazon the x-centrality variable. Type the name of a previously undefined variable. The variable is allowed to be "x" simply can be any PTC Mathcad variable proper name. Side by side, click on the eye right placeholder, and type an expression using the variable named on the x-centrality. Click exterior the operator to view the X-Y plot. PTC Mathcad automatically selects the range for both the x-axis and the y-centrality. Run across Figure 1.12.

Effigy 1.12. X-Y plot examples

Another way to create a plot is to define a user-defined role prior to creating the plot. Open the X-Y Plot operator by typing CTRL+ii . Click the bottom placeholder and type a variable name for the x-axis. This variable name does non need to be the same one used as the argument to define the office. On the correct placeholder, type the name of the function. Use the variable name from the ten-centrality equally the argument of the part. Here again, PTC Mathcad selects the range for both the x-centrality and the y-centrality. Run into Figure 1.13.

Effigy 1.xiii. 10-Y plot of functions

If you use a previously defined variable, then PTC Mathcad will not plot a graph over a range of values. It will only plot the value of the variable used. In some cases, this may only be a single point. For a plot, it is important to use only undefined variables. Nosotros will talk over the utilise of range variables in plots in Chapter 7. This is a case where a previously divers variable can be used.

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How to Draw Polar Plot in Control System TUTORIAL

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