# Bode Plot And Nyquist Plot Pdf

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Documentation Help Center Documentation. The plot displays the magnitude in dB and phase in degrees of the system response as a function of frequency.

## Relative Stability and Bode Plot – GATE Study Material in PDF

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.

As originally conceived by Hendrik Wade Bode in the s, the plot is an asymptotic approximation of the frequency response, using straight line segments. Among his several important contributions to circuit theory and control theory , engineer Hendrik Wade Bode , while working at Bell Labs in the s, devised a simple but accurate method for graphing gain and phase-shift plots. These bear his name, Bode gain plot and Bode phase plot.

Bode was faced with the problem of designing stable amplifiers with feedback for use in telephone networks. He developed the graphical design technique of the Bode plots to show the gain margin and phase margin required to maintain stability under variations in circuit characteristics caused during manufacture or during operation.

The Bode plot is an example of analysis in the frequency domain. This section illustrates that a Bode Plot is a visualization of the frequency response of a system. The response will be of the form. It can be shown [5] that the magnitude of the response is. A sketch for the proof of these equations is given in the appendix.

These quantities, thus, characterize the frequency response and are shown in the Bode plot. For many practical problems, the detailed Bode plots can be approximated with straight-line segments that are asymptotes of the precise response.

The effect of each of the terms of a multiple element transfer function can be approximated by a set of straight lines on a Bode plot. This allows a graphical solution of the overall frequency response function. Before widespread availability of digital computers, graphical methods were extensively used to reduce the need for tedious calculation; a graphical solution could be used to identify feasible ranges of parameters for a new design. This idea is used explicitly in the method for drawing phase diagrams.

The method for drawing amplitude plots implicitly uses this idea, but since the log of the amplitude of each pole or zero always starts at zero and only has one asymptote change the straight lines , the method can be simplified.

Given a transfer function in the form. In the case of an irreducible polynomial, the best way to correct the plot is to actually calculate the magnitude of the transfer function at the pole or zero corresponding to the irreducible polynomial, and put that dot over or under the line at that pole or zero.

To create a straight-line plot for a first-order one-pole lowpass filter, one considers the transfer function in terms of the angular frequency:. The above equation is the normalized form of the transfer function. The Bode plot is shown in Figure 1 b above, and construction of the straight-line approximation is discussed next. These two lines meet at the corner frequency. Frequencies above the corner frequency are attenuated — the higher the frequency, the higher the attenuation.

The frequency scale for the phase plot is logarithmic. Figures further illustrate construction of Bode plots. This example with both a pole and a zero shows how to use superposition. To begin, the components are presented separately. Figure 2 shows the Bode magnitude plot for a zero and a low-pass pole, and compares the two with the Bode straight line plots. The second Figure 3 does the same for the phase.

Figure 4 and Figure 5 show how superposition simple addition of a pole and zero plot is done. The Bode straight line plots again are compared with the exact plots. The zero has been moved to higher frequency than the pole to make a more interesting example. Notice in Figure 5 in the phase plot that the straight-line approximation is pretty approximate in the region where both pole and zero affect the phase.

Notice also in Figure 5 that the range of frequencies where the phase changes in the straight line plot is limited to frequencies a factor of ten above and below the pole zero location. Figure 2: Bode magnitude plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots.

Figure 3: Bode phase plot for zero and low-pass pole; curves labeled "Bode" are the straight-line Bode plots. Figure 4: Bode magnitude plot for pole-zero combination; the location of the zero is ten times higher than in Figures 2 and 3; curves labeled "Bode" are the straight-line Bode plots. Figure 5: Bode phase plot for pole-zero combination; the location of the zero is ten times higher than in Figures 2 and 3; curves labeled "Bode" are the straight-line Bode plots.

Bode plots are used to assess the stability of negative feedback amplifiers by finding the gain and phase margins of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by. The gain A OL is a complex function of frequency, with both magnitude and phase. Bode plots are used to determine just how close an amplifier comes to satisfying this condition. Key to this determination are two frequencies. The first, labeled here as f , is the frequency where the open-loop gain flips sign.

That is, frequency f is determined by the condition:. One measure of proximity to instability is the gain margin. Another equivalent measure of proximity to instability is the phase margin. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions minimum phase systems. Although these restrictions usually are met, if they are not another method must be used, such as the Nyquist plot.

Figures 6 and 7 illustrate the gain behavior and terminology. For a three-pole amplifier, Figure 6 compares the Bode plot for the gain without feedback the open-loop gain A OL with the gain with feedback A FB the closed-loop gain.

See negative feedback amplifier for more detail. In this vicinity, the phase of the feedback amplifier plunges abruptly downward to become almost the same as the phase of the open-loop amplifier. The amplifier is borderline stable. Figure 8 shows the gain plot. Figure 9 is the phase plot. Stability is not the sole criterion for amplifier response, and in many applications a more stringent demand than stability is good step response.

The Bode plotter is an electronic instrument resembling an oscilloscope , which produces a Bode diagram, or a graph, of a circuit's voltage gain or phase shift plotted against frequency in a feedback control system or a filter.

An example of this is shown in Figure It is extremely useful for analyzing and testing filters and the stability of feedback control systems, through the measurement of corner cutoff frequencies and gain and phase margins.

This is identical to the function performed by a vector network analyzer , but the network analyzer is typically used at much higher frequencies. Two related plots that display the same data in different coordinate systems are the Nyquist plot and the Nichols plot. These are parametric plots , with frequency as the input and magnitude and phase of the frequency response as the output. The Nyquist plot displays these in polar coordinates , with magnitude mapping to radius and phase to argument angle.

The Nichols plot displays these in rectangular coordinates, on the log scale. This section shows that the frequency response is given by the magnitude and phase of the transfer function in Eqs.

See also: Phase margin. Main articles: Nyquist plot and Nichols plot. A Nyquist plot. A Nichols plot of the same response. Unusual gain behavior can render the concepts of gain and phase margin inapplicable. Then other methods such as the Nyquist plot have to be used to assess stability.

Therefore, we could use the previous values from Figures 6 and 7. However, for clarity the procedure is described using only Figures 8 and 9. Rao Yarlagadda Analog and Digital Signals and Systems. AC, No 3. Quote: "Something should be said about his name.

To his colleagues at Bell Laboratories and the generations of engineers that have followed, the pronunciation is boh-dee. The Bode family preferred that the original Dutch be used as boh-dah. Retrieved Multivariable Feedback Control. Lee Analog design essentials. Dordrecht, The Netherlands: Springer. Categories : Plots graphics Signal processing Electronic feedback Electronic amplifiers Electrical parameters Classical control theory Filter frequency response.

Hidden categories: Articles needing additional references from December All articles needing additional references Commons category link is on Wikidata. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file.

## Relative Stability and Bode Plot – GATE Study Material in PDF

Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. A Bode Plot Characterization of All Stabilizing Controllers Abstract: In this technical note, we consider continuous-time control systems and present a new characterization of the Nyquist criterion in terms of Bode plots of the plant and the controller. This gives a nonparametric, and model independent characterization of arbitrary order stabilizing controllers.

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. As originally conceived by Hendrik Wade Bode in the s, the plot is an asymptotic approximation of the frequency response, using straight line segments. Among his several important contributions to circuit theory and control theory , engineer Hendrik Wade Bode , while working at Bell Labs in the s, devised a simple but accurate method for graphing gain and phase-shift plots. These bear his name, Bode gain plot and Bode phase plot. Bode was faced with the problem of designing stable amplifiers with feedback for use in telephone networks. He developed the graphical design technique of the Bode plots to show the gain margin and phase margin required to maintain stability under variations in circuit characteristics caused during manufacture or during operation.

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Along with these concepts, Nyquist Plot and Stability Criteria related to it forms some of the bedrock of Control Systems. Stability of a system depends on the location of roots of characteristic equation in the s-plane as we saw in Concept of Stability for Control System. The closed loop poles must lie in the left side of s-plane for the system to be stable. Here Nyquist Plot has its own Stability Criteria.

Apart from absolute stability, Nyquist Criteria can also be used for finding relative stability. Relative stability is calculated in terms of gain margin and phase margin. It is always carried out between stable systems.

Destiny 2 clear cache steam. Another widely used graphical representation of transfer functions is the Nyquist plot. It is a parametric plot of the real and imaginary part of a transfer function in the complex plane as the frequency parameter sweeps through a given interval. Nyquist plots are particularly helpful for 1. Stability analysis of control systems, design of closed-loop controllers by loop-shaping techniques, and performance tuning of controllers phase margin, gain margin, etc.

Step-by-step Nyquist plot example. For cases a and b we have the same analyses and conclusions.

#### Relative Stability & Bode Plot in PDF

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