# Bode Plot : Table, Differences,Stability, Advantages, Disadvantages & Its Applications

Bode plots are fundamental tools in control systems and signal processing, offering a graphical representation of a system’s frequency response. Named after engineer Hendrik Wade Bode, these plots consist of two separate graphs: one for magnitude and one for phase, both plotted against frequency. Even though, these plots provide a simple technique to determine system stability. Unlike the Nyquist stability criterion, they cannot handle transfer functions through right half-plane singularities. To understand Bode plots, understanding phase & gain margins are essential. This article provides brief information on the bode plot and its applications.

## What is a Bode Plot in a Control System?

A graphical representation that is used to determine the control system stability is known as the Bode Plot. The Bode plot describes the frequency response of linear time-invariant systems and it applies to the smallest phase transfer function. These are very helpful in representing the gain & phase of a control system like a function of frequency which is known as frequency domain performance of a system.

A Bode plot in the control system is very useful for mapping the frequency response with two graphs; a Gain plot and a phase plot. The gain plot represents the control system’s magnitude response as a frequency function. So this graph is plotted above a logarithmic scale. The phase plot represents the frequency-dependent phase shift of the control system’s output signal as compared to its input signal. This graph is plotted over a logarithmic scale.

#### Gain Margin

The sum of gain that can be enhanced or reduced without making the control system unstable is known as the gain margin. The gain margin is expressed generally as a magnitude within dB. If the Gain Margin is higher, the control system’s stability will be greater.

Generally, the gain margin is directly read from the Bode plot as shown below. This can be done by measuring the vertical distance between the magnitude curve & the x-axis at the frequency wherever the Bode phase plot is 180° which is called the phase crossover frequency.

The Gain Margin formula can be simply expressed as GM = 0 – G dB.

Where ‘G’ is the gain and the magnitude in dB is read from the magnitude plot’s vertical axis at the phase crossover frequency. In the above plot diagram, the Gain is 20. So by using the above gain margin formula, it is equivalent to 0 – 20 dB => -20 dB which is unstable. Here, it is important to note that the Gain Margin and Gain are not similar things because the Gain Margin is the (-) negative of the gain.

#### Phase Margin

The phase margin is the sum of phases, which can be reduced or enhanced without making the control system unstable. Generally, it is expressed as a phase within degrees. When the Phase Margin is greater, the control system stability will be greater.

Generally, the phase margin is read from the Bode plot directly which is shown in the below diagram. This can be done by measuring the vertical space between the x-axis and phase curve at the frequency wherever the Bode magnitude plot is equal to 0 dB, so this point is called the gain crossover frequency.

It is significant to understand that the Phase Margin and phase lag are not the same. The phase margin formula can be simply expressed as PM = Φ – (- 180 degrees)

Where ‘Φ’ in the above equation is the phase lag and is read at the gain crossover frequency from the vertical axis of the phase plot. The phase lag in the above plot is -189°. Thus using the above phase margin formula, the phase margin is equivalent to -189° – (-180°) => -9° which is unstable.

### Bode Plot Table

The bode plot tabular form is shown below.

 Term Type Slope dB/Sec G(jω)H(jω) Magnitude (dB) Phase Angle in Degrees Constant 0 K 20logK 0 ‘0’ at Origin 20 jω 20logω 90 ‘n’ zeors at origin 20n (Jω)^n 20nlogω 90n Pole at origin -20 1/Jω -20logω -90/270 ‘n’ poles at origin -20n 1/(Jω)^n -20nlogω -90n/270n Simple ‘0’ 20 1+Jωr ‘0’ for ω < 1/r 20logωr for ω > 1/r. ‘0’ for ω < 1/r 90 for ω > 1/r. Simple Pole -20 1/1+Jωr 0 for ω < 1/r −20logωr for ω > 1/r 0 for ω < 1/r −90 or 270 for ω > 1/r 2nd order derivative term 40 ω^2n(1−ω^2/ω^2+2jδω/ωn) 40 log ωn for ω < ωn 20 log (2δω2n) for ω = ωn 40 logω for ω > ωn 0 for ω < ωn 90 for ω = ωn 180 for ω > ωn. 2nd order integral term -40 1/ω^2n(1−ω^2/ω^2n+2jδω/ωn) −40 logωn for ω < ωn −20log(2δω2n) for ω = ωn −40 logω for ω > ωn −0 for ω < ωn − 90 for ω = ωn −180 for ω > ωn

## How to Draw Bode Plot?

The step-by-step procedure to draw a Bode plot is discussed below.

The first step is to get the transfer function of the control system because it signifies the main relationship between the input & output signals of the control system. Generally, the transfer function is written as a polynomial ratio within the Laplace domain.

After that, need to decide the system’s magnitude response by using the transfer function’s magnitude logarithm which is expressed in dB. Usually, this magnitude response is plotted above a logarithmic frequency axis.

Establish the control system’s phase response by discovering the argument of the T.F. within the Laplace domain. Generally, this phase response is plotted above a linear frequency axis.

After that, need to plot the magnitude response over a logarithmic frequency axis where the x-axis signifies the frequency logarithm & the y-axis signifies the magnitude response within decibels.

Draw the phase response above a linear frequency axis. So, the x-axis signifies the frequency, whereas the y-axis signifies the phase response within degrees.

Plot the asymptotes for the magnitude response by plotting the lines that signify the system performance at higher & lower frequencies. These asymptotes can be measured from the poles & zeros of the T.F.

Spot the break frequencies above the bode plot wherever the performance of the control system varies. So these break frequencies can be measured from the poles & zeros of the T.F.

Plot a smooth curve throughout the plotted points to estimate the real magnitude & phase response of the control system.

A Bode plot gives a complete understanding of the frequency response of a control system by drawing both the magnitude as well as phase responses. It can be used for designing and analyzing a control system.

#### Bode Plot Stability

The criterion list relevant for drawing Bode plots like gain margin, phase margin, gain cross-over frequency, phase cross-over frequency, corner frequency, resonant frequency, factors, slope, and angle are discussed below.

Gain Margin

When the gain margin is higher, the system stability will be higher. It is the amount of gain that can be enhanced or reduced without making the control system unstable. Generally, it is expressed in decibels.

Phase Margin

When the phase margin is greater, then control system stability will be higher. It is the phase that can be enhanced or decreased without making the control system unstable. Generally, it is expressed in phase.

Gain Crossover Frequency

This is the frequency upon which the magnitude curve will cut the zero decibels axis within the bode plot.

Phase Crossover Frequency

In the bode plot, this is a frequency at which point the phase curve will cut the negative times of the 180-degree axis.

Corner Frequency

The frequency at which point the two asymptotes meet or cut each other is called break frequency.

Resonant Frequency

The frequency value at which point the G (jω) modulus has a peak value is called the resonant frequency.

Factors

Each loop’s transfer function is the product of different factors like Integral factors (jω), constant term K, and first-order factors (1 + jωT)(± n) wherever ‘n’ is an integer, quadratic factor, or second-order.

Slope

The slope in the bode plot corresponds to every factor and is expressed in the decibels for each decade.

Angle

The angle in the bode plot corresponds to every factor & the angle for every factor is expressed simply in the degrees.

## Difference B/W Bode Plot and Nyquist Plot

The difference between the Bode plot and the Nyquist plot includes the following.

 Bode Plot Nyquist Plot A Bode plot is a graphical representation of the frequency response of a control system. The Nyquist plot is an extension of the polar plot used to find the stability of a control system. It shows the phase & magnitude of the T.F. as a frequency function. It shows the complex T.F. of the system within the complex plane. Bode plot is useful for both analysis & designing of a control system. It is mainly used for stability analysis of control systems. This plot shows the response of the control system for different frequencies. This plot shows the system stability by specifying the number of uneven poles within the complex plane. There are two plots where one is for magnitude or gain and the other plot is for phase. This plot combines both the gain & phase in the complex plane into a single plot.

### Advantages & Disadvantages

The advantages of a bode plot include the following.

• It identifies phases & gains margins with the least calculation.
• This plot helps in calculating the transfer function of the system.
• It is useful to design the filters.
• The bode plot covers both the low and high frequencies.
• This can be drawn for both the open-loop & closed-loop systems.
• This plot changes the magnitude multiplication into addition.
• It has very little information loss.
• The Bode plot has two graphs which helps eliminate the main confusion between the phase & magnitude plots.
• The frequency dependence in the bode plot is visible very clearly.

The disadvantages of the bode plot include the following.

• Bode plot applies only to linear time-invariant systems.
• It does not apply to the system with very low or high frequencies.
• Bode plot focuses mainly on the frequency response without believing the transient time effect.
• It is not responsive to the changes happening within the measuring system.

### Applications

The applications of the bode plot include the following.

• The Bode plot is used to identify the constancy of the control system.
• This is a graphical representation of the frequency response of the control system used in electrical and control engineering.
• This plot is used to get insight into the transfer function’s frequency response.
• It is used to design & analyze the control system.
• This plot gives the relative stability of the control system in gain & phase margins.
• It is used for measuring the frequency response angle & magnitude of an SISO system.
• This is very helpful for both the design and analysis of circuit & control systems for a better understanding of their performance.
• This plot is used in the control systems & analysis of the frequency domain to signify the gain as well as phase response of a time-invariant and linear system relating to frequency.

#### What Does a Bode Plot Tell You?

Bode plot tells you a simple & common method of explaining the frequency response of an LTI system.

#### What is a Polar Plot?

It is a graphical representation of the frequency response of a control system within the frequency domain

#### Why Polar Plots are Preferred Over Bode Plots?

Because they show the control system’s frequency response characteristics over the whole frequency range within a single plot.

#### Why Bode Plot is Drawn for Open Loop Transfer Function?

The Bode plot is plotted for the open-loop transfer function because it provides a very convenient method to decide the stability of a control system. An open loop system’s bode plot provides the gain, phase margin & gain margin so that we can design easily the feedback system to meet the requirements of the system.

#### Does Bode Plot use Open-Loop or Closed-Loop?

Bode plots show us phase and magnitude only for an open loop control system.

#### Which Type of Loop Transfer Function is used for the Bode Stability Criterion?

An open-loop transfer function is used for the bode stability criterion due to its less frequency response than one amplitude ratio for all frequencies.

Thus, this is an overview of the Bode Plot, its working, and its applications. A Bode plot is a frequency response plot including two graphs like phase and magnitude. Generally, it is drawn for the open-loop control system because it decides conveniently the stability as well as other associated parameters. This plot assists us in determining the stability of the system & provides us with a method to develop that stability. Here is a question for you, what is a control system?