Bode Plots in Control System

Bode plots describe the linear time-invariant systems’ frequency response (change in magnitude and phase as a function of frequency). It helps in analyzing the stability of control system. It applies to the minimum phase transfer function i.e. (poles and zeros should be in the left half of the s-plane).

In this article, we are going to learn what is Bode Plot and Types of Bode Plots and how to draw Blode plot and parameters of Bode plot, we are going to learn what is Phase and Gain Margin and what are the advantages and disadvantages of Bode plots in Control System.

A graph is called as Bode plot which is frequently used in control system engineering to assess a control system’s stability. Two graphs, the Bode phase plot (which expresses the phase shift in degrees) and the Bode magnitude plot (which expresses the magnitude in decibels), are used to map the frequency response of the system.

Hendrik Wade Bode first introduced Bode plots in the 1930s while he was employed by Bell Laboratories in the United States. Bode plots, unlike the Nyquist stability criterion, can handle transfer functions with right half plane singularities, despite being a reasonably straightforward approach for calculating system stability.

It represents the magnitude response of the system as a function of frequency. It is plotted on a logarithmic scale.

It depicts the frequency-dependent phase shift of the system’s output signal compared to its input signal. It’s also drawn on a logarithmic scale.

Bode plot representation for the open loop system is:

20 log|G(jω)|

Step 1: Write the given transfer function in the standard form.

Transfer function:

[Tex]G(s)= \frac{(s+a)(s+b)}{(s+p)(s+q)} [/Tex] —– (1)

Standard form of equation 1:

[Tex]G(s) = \frac{ab(1+ \frac{s}{a})(1+\frac{s}{b})}{pq (1+\frac{s}{p})(1+\frac{s}{q})} [/Tex]

 Take [Tex]\frac{ab}{pq} [/Tex] as a constant k.

Step 2: Identify the slope of the first line for the bode plot. The slope of the first line is based on poles and zeros at the origin. Refer to the following table.

 Step 3: Find the gain of 1^st line at ω=1 rad/sec

Gain|_ω=1 = 20 log k

Where k = [Tex]\frac{ab}{pq} [/Tex]

Step 4: Write all the corner frequencies in ascending order and define the slope of each line

Step 5: Write the phase equation and make a table of phase and frequency.

Φ = tan^-1([Tex]\frac{w}{a} [/Tex]) + tan^-1([Tex]\frac{w}{b} [/Tex]) – tan^-1([Tex]\frac{w}{p} [/Tex]) – tan^-1([Tex]\frac{w}{q} [/Tex])

Blode plots show the frequency response, that is, the changes in magnitude and phase as a function of frequency.

Parameters of Bode Plot

Figure 1 shows the gain and phase plot. The gain cross over frequency (w_pc) and phase crossover frequency (w_pc) can be calculated using gain plot and phase plot respectively.

W_gc is the value at 0dB whereas W_pc is the value at -180^o.

Phase margin and gain can be calculated by extending the graph as shown in the figure.

Stability by bode plot:

 ωpc > ωgc ->System is stable  ωpc < ωgc ->System is unstable ωpc = ωgc ->System is marginally stable

The phase margin indicates how much more phases shift we may put in the open loop transfer function before our system becomes unstable.

It can be calculated from the phase at the gain cross-over frequency.

Phase Margin (PM) = 180^∘+∠G(jω)H(jω)∣_ω=ωgc
= -180∘

Gain crossover frequency: 

It is the frequency at which the magnitude of G(s) H(s) is unity as seen in the figure 1.

|G(jω)H(jω)|_ω=ωgc = 1

The gain margin is the amount of open loop gain that can be increased before our system becomes unstable.

It can be calculated from the gain at the phase cross-over frequency.

Gain Margin (GM): [Tex]\frac{1}{|G(jw)H(jw)|}_{w=w_{pc}} [/Tex]

Phase crossover frequency:

It is the frequency where the phase angle of G(s) H(s) is -180 degrees as seen in the figure 1.

∠G(jω)H(jω)∣_ω=ωpc= -180^∘

1. It helps in identifying the stability of the system.
2. It helps in identifying phases and gaining margins with minimum calculation.
3. It can be used to calculate the system’s transfer function.
4. It can show the amplification and attenuation in the gain plot which is helpful in designing the filters.

1. It is only applicable to LTI (linear time-invariant) system.
2. It is not suitable for the system having extremely high or low frequencies.
3. It focuses on the frequency response without considering the transient time effect.

 Example: Find the transfer function from the bode plot given in the figure?

Solution:

Step 1: Corner frequency: ω= 1, 10, 100

Step 2: Calculation of the slope

Therefore, if the slope is increasing then it is a zero else it is a pole.

ω = 1 (zeros) ω = 1 (pole) ω = 1 (pole)

Step 3:  Calculating the gain of the first line at ω = 1

Gain = 20 log(k) + (slope of 1st line) log(ω) -20    = 20log(k) + 0 k       = 0.1

Step 4: Writing the transfer function

[Tex]T(s) = \frac{k(1+\frac{s}{1})}{(1+\frac{s}{10})(1+\frac{s}{100} )} [/Tex]

k= 0.1

[Tex]T(s) = \frac{0.1(1+\frac{s}{1})}{(1+\frac{s}{10})(1+\frac{s}{100} )} [/Tex]

[Tex]T(s) = \frac{100(s+1)}{(s+10)(s+100)} [/Tex]

In Conclusion, The Bode plots are similar to the asymptotic Bode plots since they display the magnitude and also the phase plots as straight lines. The Bode plots will only differ in that they will use simple curves rather than straight lines. For other terms of the open loop transfer function that are listed in the table, you can also create the required Bode graphs.

Which graph paper is used for Bode plots?

Semi-Logarithmic graph paper.

What is the Bode plot’s stability?

According to the bode plot, system is said to be stable when gain margin is positive and the phase margin should be greater than the gain margin. 

What is the unit of Bode Plot?

The bode magnitude plot measures the system Input/Output ratio in special units called decibels. The Bode phase plot measures the phase shift in degrees (typically, but radians are also used).

What are decibels (dB)?

It is a unit of measurement which is used to calculate the logarithmic ratio of one value of a power or field quantity to another. The logarithmic quantity is known as the power level or field level, respectively.