Bode plots & What are they used for?

Bode plot is a very important tool in control system. 

Let's have a short look on history of Bode Plot so that it sets the plot for the concept.

Bode plot was was developed by Hendrik Wade Bode in the 1930's while he was designing feedback for use in telephone networks which showed the gain and phase margins required to maintain stability in a Linear Time invariant systems. 

So what is a Bode Plot? 

It is a graph of response of a Linear Time-Invariant system to sinusoidal inputs in the frequency domain. 

It is a powerful tool for carrying out characterizing an LTI system or designing  a controller.

An LTI system responds to a sinusoidal input in a very similar manner. The output of a LTI system sinusoid is also a sinusoid with different amplitude and phase. But THE FREQUENCY OF THE SIGNAL REMAINS SAME.

Magnitude

For a given frequency (ω) we define the magnitude of an LTI system as the ratio of the magnitude of the sinusoidal output (y) to the magnitude of the sinusoidal input (u):

Magnitude is always given in decibels (db) while working in Bode plots. 

For converting magnitude to decibels use the following equation.

Phase 

Phase is a measure of angular lag (or Lead) between the input and the output signals of an LTI system. It is typically given in degrees. In the Bode plot shown above, at a frequency of 1 rad/s, the output signal lags behind the input signal by approximately 45 degrees. As frequency increases so does the amount of phase lag.

Gain Crossover Frequency

Gain Crossover frequency is the frequency at which the magnitude of the system is 0 dB, or 1. 

In magnitude plot above, the gain crossover frequency is approximately 3.5 Hz.

Phase Crossover frequency

Phase crossover frequency is the frequency at which the system produces a phase lag of -180 degrees. 

In the bode plot above the gain crossover frequency is 5.5 Hz.

Gain Margin

When looking at the Bode plot of an open-loop system, gain margin is a measure of how much gain is required to cause the closed-loop system to go unstable. The gain margin in the Bode magnitude plot shown below is approximately +7 dB. In this example, the closed-loop system gain can be increased by +7 dB before it becomes marginally stable.

Determining gain margin

In your Bode plot, start by drawing the horizontal lines which cross the 0 dB and

-180 deg marks in the magnitude and phase plots respectively. In the phase plot, identify the location where the -180 degrees line intersects with the phase plot. Then draw a vertical line from the intersection all the way up to the 0 dB line. The segment of the vertical line which lies between the 0 dB line and the magnitude plot is your gain margin. So why was the gain margin in the above example positive? If the line segment falls below the 0 dB line, then you have a positive gain margin, otherwise you have a negative gain margin.

Phase Margin

When looking at the Bode plot of an open-loop system, phase margin is a measure of how much phase change is required to cause the closed-loop system to go unstable. The phase margin in the Bode magnitude plot shown above is approximately +40 degrees. In this example, the system phase can be increased by +40 degrees before the closed-loop system becomes marginally stable.

Determining phase margin

In your Bode plot, start by drawing the horizontal lines which cross the 0 dB and -180 deg marks in the magnitude and phase plots respectively. In the magnitude plot, identify the location where the 0 dB line intersects with the magnitude plot. Then draw a vertical line from the intersection all the way down to the -180 degrees line. The segment of the vertical line which lies between the -180 degrees line and the phase plot is your phase margin. So why was the phase margin in the above example positive? If the line segment falls above the -180 degrees line, then you have a positive phase margin, otherwise you have a negative phase margin.

Same figure repeated for convenience of understanding Gain and Phase Margins. 

Useful Applications of a Bode Plot

Bode plots have many practical applications. Bode plots can be used to experimentally model a first-order system by determining the gain and time constant from the Bode magnitude plot. 

Natural Frequency and Damping

You can use a Bode plot to get an idea of how damped your system is, as well as estimating its natural frequency.

For example, the Bode plot on the left compares the responses of three second-order systems all of which have the same natural frequency (ωn = 10 rad/s), but different damping ratios (ζ = 0.01, ζ= 0.1, ζ = 1). As you can see from the graph, the system that has the least amount of damping (ζ = 0.01) has a sharper more definitive magnitude peak. In other words, damping is inversely related to the amplitude of the peak.

The Bode plot on the right compares the responses of three second-order systems all of which have the same damping ratio (ζ= 0.01), but different natural frequencies (ω = 1, ω = 10, and ω= 20). It's evident from the graph that the frequency at which the peak magnitude occurs is the natural frequency of the system.

Estimating the Order of a System

The Bode phase plot is a good predictor of the order of a system. If the phase drops below -90 degrees, the system must be a second or higher-order system. This is shown in the Bode plot below. It's evident from the graph that the phase of the second-order system drops below -90 degrees while the phase of the first-order system approaches (but never drops below) -90 degrees.

System Stability

One can determine the stability of a closed-loop system by looking at the Bode plot of the open-loop system. As mentioned earlier, the gain and phase margins of an open-loop system are measures of how much additional gain and phase shift is required to bring the closed-loop system to the brink of instability. A large gain margin means a closed-loop system can withstand greater changes in the system parameters before going unstable. A large phase margin means the system will be more tolerant of time delays. This is particularly important given that most controllers are now implemented digitally, which means the controller's sampling rate will impact its performance. By taking advantage of these margins, control engineers are able to design controllers that are better able to respond disturbances.

So how can you determine the stability of a system by looking at a Bode plot? There are a few things you can examine. For example, a positive phase and gain margin indicate that the closed-loop system is stable, while negative values indicate an unstable system.

Another thing to look for is whether the phase plot crosses the -180 degrees line. A phase plot that does not cross the -180 degrees line indicates that the system has an infinite gain margin, i.e. even an infinitely large gain will not cause the system to go unstable.

So, that's it for this article

Thank you & see you in next one...