Control Systems Tutorial
Control Systems Useful Resources
Selected Reading
- State Space Analysis
- Control Systems - State Space Model
- Control Systems - Controllers
- Control Systems - Compensators
- Control Systems - Nyquist Plots
- Control Systems - Polar Plots
- Construction of Bode Plots
- Control Systems - Bode Plots
- Frequency Response Analysis
- Construction of Root Locus
- Control Systems - Root Locus
- Control Systems - Stability Analysis
- Control Systems - Stability
- Steady State Errors
- Time Domain Specifications
- Response of Second Order System
- Response of the First Order System
- Time Response Analysis
- Signal Flow Graphs
- Block Diagram Reduction
- Block Diagram Algebra
- Control Systems - Block Diagrams
- Electrical Analogies of Mechanical Systems
- Modelling of Mechanical Systems
- Mathematical Models
- Control Systems - Feedback
- Control Systems - Introduction
- Control Systems - Home
Control Systems Useful Resources
Selected Reading
- Who is Who
- Computer Glossary
- HR Interview Questions
- Effective Resume Writing
- Questions and Answers
- UPSC IAS Exams Notes
Construction of Bode Plots
Control Systems - Construction of Bode Plots
In this chapter, let us understand in detail how to construct (draw) Bode plots.
Rules for Construction of Bode Plots
Follow these rules while constructing a Bode plot.
Represent the open loop transfer function in the standard time constant form.
Substitute, $s=jomega$ in the above equation.
Find the corner frequencies and arrange them in ascending order.
Consider the starting frequency of the Bode plot as 1/10th of the minimum corner frequency or 0.1 rad/sec whichever is smaller value and draw the Bode plot upto 10 times maximum corner frequency.
Draw the magnitude plots for each term and combine these plots properly.
Draw the phase plots for each term and combine these plots properly.
Note − The corner frequency is the frequency at which there is a change in the slope of the magnitude plot.
Example
Consider the open loop transfer function of a closed loop control system
$$G(s)H(s)=frac{10s}{(s+2)(s+5)}$$
Let us convert this open loop transfer function into standard time constant form.
$$G(s)H(s)=frac{10s}{2left( frac{s}{2}+1 ight )5 left( frac{s}{5}+1 ight )}$$
$$Rightarrow G(s)H(s)=frac{s}{left( 1+frac{s}{2} ight )left( 1+frac{s}{5} ight )}$$
So, we can draw the Bode plot in semi log sheet using the rules mentioned earper.
Stabipty Analysis using Bode Plots
From the Bode plots, we can say whether the control system is stable, marginally stable or unstable based on the values of these parameters.
Gain cross over frequency and phase cross over frequency
Gain margin and phase margin
Phase Cross over Frequency
The frequency at which the phase plot is having the phase of -1800 is known as phase cross over frequency. It is denoted by $omega_{pc}$. The unit of phase cross over frequency is rad/sec.
Gain Cross over Frequency
The frequency at which the magnitude plot is having the magnitude of zero dB is known as gain cross over frequency. It is denoted by $omega_{gc}$. The unit of gain cross over frequency is rad/sec.
The stabipty of the control system based on the relation between the phase cross over frequency and the gain cross over frequency is psted below.
If the phase cross over frequency $omega_{pc}$ is greater than the gain cross over frequency $omega_{gc}$, then the control system is stable.
If the phase cross over frequency $omega_{pc}$ is equal to the gain cross over frequency $omega_{gc}$, then the control system is marginally stable.
If the phase cross over frequency $omega_{pc}$ is less than the gain cross over frequency $omega_{gc}$, then the control system is unstable.
Gain Margin
Gain margin $GM$ is equal to negative of the magnitude in dB at phase cross over frequency.
$$GM=20logleft( frac{1}{M_{pc}} ight )=20logM_{pc}$$
Where, $M_{pc}$ is the magnitude at phase cross over frequency. The unit of gain margin (GM) is dB.
Phase Margin
The formula for phase margin $PM$ is
$$PM=180^0+phi_{gc}$$
Where, $phi_{gc}$ is the phase angle at gain cross over frequency. The unit of phase margin is degrees.
The stabipty of the control system based on the relation between gain margin and phase margin is psted below.
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.