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In my previous posts, we have seen how a system is represented in the frequency domain by replacing s by jω and the implementation of ‘Bode plots‘ for frequency response analysis. In Bode plots, the magnitude and the phase were drawn separately on a semilog paper. In Polar and Nyquist plots we plot the magnitude and phase using polar coordinates. Each point on the polar plot is a phasor of magnitude M and phase angle ‘φ’. The distance from the origin is proportional to magnitude G(jω) i.e |G(jω)| and the angle represents the phase of G(jω) i.e ∠ G(jω) . Now let us see some of the advantages of Polar plots: It depicts the frequency response characteristics over the entire frequency range in a single plot.(there is single graph to represent both phase and magnitude , unlike in Bode plot). Much easier to determine both and . Here we will have to work with open loop transfer function G(s)H(s) (and not with closed loop transfer function and unlike Bode plot we need not required to convert G(s)H(s) to the time constant form). Now we will take examples to understand the use of Polar plots for analysis, Polar Plot of a 1st Order Pole () : So, we need to draw polar plot of the given transfer function G(s)H(s) = Step 1 : The first step would be to convert this transfer function to the frequency domain. This can be done by converting ‘s’ by ‘jω’ G(jω)H(jω) = , Step 2 : We now find the magnitude and phase |G(jω)H(jω)| = ∠ G(jω)H(jω) = = ∠ G(jω)H(jω) = Step 3 : Vary ‘ω’ from 0 to ∞ Now instead of taking different values of ω, we simply take two extreme values of ω i.e ω = 0 and ω = ∞ At ω = 0, |G(jω)H(jω)| = = 5 ∠ G(jω)H(jω) = = 0 At ω = ∞ , |G(jω)H(jω)| = = 0 ∠ G(jω)H(jω) = = -90 Now these two points are sufficient to draw the polar plot. At ω = 0 since the magnitude is +5 and angle is 0,we draw it on the right-se horizontal axis. At ω = ∞ , the magnitude is 0 while angle is -90 , hence we draw it as dot(zero magnitude) on the -90 axis. The direction is represented by an arrow(refer below diagram) . We can now generalize that the polar plot of a 1st order pole ( ) will always have a shape shown above. Effect of adding more Simple Poles: Let us add one more simple pole to above example, Then we have, G(s)H(s) = The given system has two poles i.e s = -2 and s = -4 , Step 1 : The first step would be convert this transfer function to the frequency domain. This can be done by converting ‘s’ by ‘jω’ G(jω)H(jω) = We don’t need to convert this to time constant form. Step 2 : We now find the magnitude and phase |G(jω)H(jω)| = , ∠ G(jω)H(jω) = = ∠ G(jω)H(jω) = Step 3 : Vary ‘ω’ from 0 to ∞ Now instead of taking different values of ω, we simply take two extreme values of ω i.e ω = 0 and ω = ∞ At ω = 0, |G(jω)H(jω)| = = 1.25 ∠ G(jω)H(jω) = = 0 , At ω = ∞ , |G(jω)H(jω)| = = 0 ∠ G(jω)H(jω) = = -90 -90 = -180 From the above two point now we can draw the polar plots, We can note here that the starting and ending pint will not change as long as system has two simple poles on the LHP. Thus, we can generalize that G(jω)H(jω) = will have a polar plot as shown below: From comparing both the above polar plots, we can make an important conclusion that, the addition of simple pole to system stretches the polar plot by -90 . Effect of adding pole at origin: Now we shall see what happens when we add pole at origin, Now we have, G(s)H(s) = The given system has two poles i.e s = -2 and s = 0 , Step 1 : The first step would be convert this transfer function to the frequency domain. This can be done by converting ‘s’ by ‘jω’ G(jω)H(jω) = We don’t need to convert this to time constant form. Step 2 : We now find the magnitude and phase |G(jω)H(jω)| = , ∠ G(jω)H(jω) = = ∠ G(jω)H(jω) = – 90 Step 3 : Vary ‘ω’ from 0 to ∞ Now instead of taking different values of ω, we simply take two extreme values of ω i.e ω = 0 and ω = ∞ At ω = 0, |G(jω)H(jω)| = = , ∠ G(jω)H(jω) = – 90 = – 90 , At ω = ∞ , |G(jω)H(jω)| = 0 ∠ G(jω)H(jω) = – 90 = -90 – 90 = – 180 From the above two points now we can draw the polar plot, We can note here that the at ‘ω’ = 0 , magnitude is ∞ , we draw line parallel to the -90 axis assuming it will touch -90 axis at ∞. We note that adding a pole at the origin: Shifts the polar plot by -90 at ω = 0 and. Magnitude at ω = 0 becomes infinity. What will happen if two poles present at the origin? As expected the polar plot will get shifted by -180 (since two poles present at origin) So we have seen effect of addition of poles to the system, so what will be the effect of addition of zeros? Result of addition of zeros is exactly opposite to that of poles. A pole adds -90 to the plot, while a zero adds +90 to the polar plot. Stability on Polar plots: We can check stability using polar plot in very simple manner. Imagine yourself walking on the polar plot, along the direction of the arrow i.e from = 0 to ∞ . The entire area to the right of you upto the real axis represented by an infinite radius is sa to be enclosed by the polar plot. The portion inse the shaded area is sa to be enclosed by the polar plot. A system is stable if the (-1,0) point is not enclosed by the polar plot. Polar plots are simple method to check the stability of the system. They are however not the preferred choice when the system has poles on the right half plane.(In such cases we use Nyquist plots which are an extension of polar plots and will be seen in my next post.So stay tuned…) report this ad
What is polar plot in control systems?
Definition: The plot that represents the transfer function of the system G(jω) on a complex plane, constructed in polar coordinates is known as Polar Plot. The polar plot representation shows the plot of magnitude versus phase angle on polar coordinates with variation in ω from 0 to ∞.
How do you know if a polar plot is stable?
…
Polar Plot and its Analysis
- Put s = jω in the transfer function.
- Get expression for |G(jω)| and ∠G(jω)
- For ω = 0 and ω = ∞, calculate both |G(jω)| and ∠G(jω)
Why polar plots are preferred over Bode plots?
Now let us see some of the advantages of Polar plots: It depicts the frequency response characteristics over the entire frequency range in a single plot. (there is single graph to represent both phase and magnitude , unlike in Bode plot).
What is a polar plot and its advantages?
The angular frequency (ω) varies from zero to infinity, while the Nyquist plot varies from a negative value of infinity to positive infinity. The primary advantage of the polar plot is that it depicts the frequency response characteristics of a system over the entire frequency range in a single plot.
How do you read a polar plot?
If an angle is negative, then the direction of the measurement of the angle is opposite to the direction of the graph. In the below polar graph, 150° and −210° reach the same position. The angle −210° means the angle is measured in the opposite direction to the direction of the graph, which is anticlockwise.
How do you plot a polar diagram?
- Polar Plot – Free Template Download.
- Getting started.
- Step #1: Set up a helper table.
- Step #2: Compute the Angle (theta) values.
- Step #3: Compute the Radius values.
- Step #4: Copy the last Radius values into the helper row.
- Step #5: Calculate the x- and y-axis values for each company.
What is gain margin and phase margin?
The gain margin is the factor by which the gain must be multiplied at the phase crossover to have the value 1. The phase crossover occurs at 0.010 Hz and so the gain margin is 1.00/0.45=2.22. The phase margin is the number of degrees by which the phase angle is smaller than −180° at the gain crossover.
Is polar plot and Nyquist plot same?
The frequency in the case of the Nyquist plot varies from -infinity to infinity. The primary difference between the polar plots and the Nyquist plot is that the polar plots are based on frequencies range from zero to infinity, while the Nyquist plot also deals with negative frequencies.
What is Bode plot in control system?
In electrical engineering and control theory, a Bode plot /ˈboʊdi/ is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude (usually in decibels) of the frequency response, and a Bode phase plot, expressing the phase shift.
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Why do we use polar plots? See some more details on the topic Control System – Nyquist plot – …
Control Systems – Polar Plots – Tutorialspoint
The Polar plot is a plot, which can be drawn between the magnitude and the phase angle of G(jω)H(jω) by varying ω from zero to ∞. The polar graph sheet is …
Learn Control Systems Polar Plots in Control Systems Tutorial …
Let us draw the polar plot for this control system using the above rules. Step 1 − Substitute, s=jω in the open loop transfer function.
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