Bode diagram and the relative stability of a feedback system
Bode diagram
Bode diagram is one of the methods of studying the stability of a linear feedback system. In this, we plot the loop transfer function for s=jw in the form of two separate graphs. One graph plots the magnitude of L(jw) in decibels against the logarithm of w. The other graph plots the phase of L(jw) in degrees against the logarithm of w.
The most useful features of the Bode diagram are as follows:
- It can be used to perform the necessary calculations for different frequencies with ease and it is also quicker.
- It helps in developing engineering intuition regarding the effective placement of pole-zero on the frequency response.
A low-frequency asymptote consisting simply of a 0 dB line, and the high-frequency asymptote consisting of a slope of 20 dB/decade helps approximate the pole factor to the gain response. These two asymptotes intersect at a point, known as the corner or the break frequency. The approximation error is the difference between the actual gain response and the approximate gain response. It attains a maximum value of 3 dB at break frequency.
The relative stability of the feedback system
Now that we are familiar with the Bode diagram, we can better understand the relative stability of the feedback system. The relative stability is determined according to how close the loop transfer function of the system is to the critical point. As the Bode diagram consists of both the graphs, it consists of both the most commonly used measures of relative stability.
The first measure is known as the gain margin, it can be defined as the number of decibels by which the magnitude of L(jw) in decibels against the logarithm of w, must be changed to keep the system stable.
The frequency at which the magnitude of L(jw) against the logarithm of w is -180 degrees, is known as the phase crossover frequency.
The second measure of relative stability is the phase margin. It is expressed in degrees. Phase margin can be defined as the magnitude of the minimum angle by which the phase of L(jw) must be changed to keep the system stable. The frequency at which the magnitude of the phase of L(jw) is 1, is known as the gain crossover frequency.
From our understanding till now, we can make the following two observations:
- In the case of a stable system, both the gain margin and the phase margin are positive. Therefore, the phase crossover must be larger than the gain crossover frequency.
- If the gain margin is negative or the phase margin is negative then the system is unstable.
Reference
Bode diagram and the relative stability of a feedback system