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## Control System Lectures – Bode Plots, Introduction

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A Bode plot is a frequency domain technique (used for frequency response analysis) in which magnitude and phase are plotted one below the other. So we have two graphs, Figure 1: Magnitude (M) of the transfer function as a function of frequency (ω) and Figure 2: Phase angle (φ) as a function of frequency (ω). The open-loop transfer function GH(jω) is plotted in decibels on the y-axis, while the input frequency is plotted on the x-axis. In Figure 2, the phase angle of GH (jω) is plotted on the y-axis and the x-axis has the input frequency on a log scale (similar to Figure 1). Both of the above features are plotted on a piece of semi-logarithmic paper, and both plots use the same X-axis scale. Standard form of GH (jω): The frequency transfer function can be written as the generalized quotient of a polynomial. GH(s) = GH(s) = this needs to be decomposed to get the general type, GH(s) = we convert the above equation to time constant form, GH(s) = we now set s = jω to convert the transfer function Converted to the frequency domain, GH (jω) = this is the gain constant for the Bode gain. The above equation is in standard form or Bode form. Below is a list of possible factors for GH(jω) , Bode gain or constant factor. The pole at the origin of order k. The zero point at the origin of order g. (1+) or finite simple poles at first-order poles. . A finite simple zero or zero of first order at (1+). The second-order pole is located at . second-order zero at . Frequency Response Specifications: Gain Span: . It is a measure of relative stability, the higher the gain margin, the more stable the machine, defined as the reciprocal of the open loop transfer function evaluated at the phase crossover frequency, which is -180 for phase. That is, the gain span: where , ∠ GH = -180 , where “” is the phase transition frequency. PhaseMargin(): . The higher the phase margin, the higher the stability of the system. Phase margin is a measure of relative stability, defined as 180 plus the phase angle of the open-loop transfer function at unity gain. That is, = [ 180 + arg GH() ] degrees GH() | = 1 is the crossover frequency of the gain, as shown above. bandwth:. The bandwth of the system is the frequency range in which the system responds satisfactorily. This performance depends on the application and characteristics of the specific system. Typically, the bandwth of a system is defined as the range of frequencies within which the amplitude ratio is within -3 dB of its value at a given frequency. In many feedback control systems, this frequency is zero. In this case, the bandwth is equal to the cutoff frequency. Cut-off frequency():. The frequency at which the magnitude of the closed-loop response is 3dB below its zero frequency value is called the “cutoff frequency”. Cutoff rate: . The cutoff frequency is the rate at which the amplitude ratio drops above the cutoff frequency. Resonance Peak Frequency(): . The formant is a measure of relative stability and is the maximum value of the amplitude of the closed-loop frequency response. Resonance frequency():. The resonant frequency is the frequency at which the “” appears. Gain Crossover Frequency: . This is the point on the magnitude plot where the magnitude of GH(jω) is 1. It is measured in frequency. ContactAngle(): . This angle is 180° plus the negative triangle angle of the transfer function at the gain crossover frequency. Phase transition frequency: . This is the frequency at which the GH has a phase of -180. Relative Stability: The relative stability of the system is specified by the gain and phase margins relative to the open-loop frequency response of the system. Furthermore, these parameters can be easily determined from the Bode plot of GH(jω). In most cases, a positive gain and a defined phase margin ensure the stability of the control loop system. For a stable system D. H. GM and PM = positive for an unstable system d. H. 2. When both GM and PM are negative The system is absolutely stable when GM +∞, and marginally stable when both GM and PM are zero. In my next article, we’ll see the complete steps to solving a Bode plot (along with an example). See you all in my next article. report this ad

## What are the steps to plot a Bode plot?

**Key Concept –**

**To draw Bode diagram there are four steps:**

- Rewrite the transfer function in proper form.
- Separate the transfer function into its constituent parts.
- Draw the Bode diagram for each part.
- Draw the overall Bode diagram by adding up the results from part 3.

## 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.

## How do you make a Bode plot in Matlab?

In Matlab for a bode plot, the bode inbuilt function is available. For using these inbuilt bode function, we need to **create one transfer function on a Matlab**; for that, we can use a tf inbuilt function which can be available on Matlab.

## What is gain margin and phase margin in Bode plot?

**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.

## Why is Bode plot used?

A Bode Plot is a useful tool that **shows the gain and phase response of a given LTI system for different frequencies**. Bode Plots are generally used with the Fourier Transform of a given system. An example of a Bode magnitude and phase plot set.

## How many graphs are required to complete Bode plot?

Bode analysis consists of plotting **two graphs**: the magnitude of Φ_{0}(s) with s = jω, and the phase angle of Φ_{0}(s) with s = jω, both plotted as a function of the frequency ω. Log scales are usually used for the frequency axis and for the magnitude of Φ_{0}(jω).

## How do you find the stability of a Bode plot?

**If both the gain margin GM and the phase margin PM are positive, then the control system is stable**. If both the gain margin GM and the phase margin PM are equal to zero, then the control system is marginally stable. If the gain margin GM and / or the phase margin PM are/is negative, then the control system is unstable.

## How do you draw a Bode plot with zero at origin?

Key Concept: Bode Plot for Zero at Origin

The plots for a zero at the origin are like those for the pole but mirrored about 0dB or 0°. For a simple zero at the origin **draw a straight line with a slope of +20 dB per decade and going through 0 dB at 1 rad/ sec**. The phase plot is at +90°.

## What are poles and zeros in Bode plot?

For a = 0—that is, a pole or a zero at s = 0—the plot is simply **a straight line of 6 dB/octave slope intersecting the 0-dB line at ω = 1**. In summary, to obtain the Bode plot for the magnitude of a transfer function, the asymptotic plot for each pole and zero is first drawn.

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### Bode Plot Notes Step by Step – Slideshare

BEE-502 Automatic Control Systems Unit-4, Bode Plot Supplementary Notes … Easy language and step by step procedure for drawing Bode Plots.

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