Stability Criteria for Transfer Function with Bode, Root Locus & Nyquist

[barneygumble742]

hi,

can anyone please help me with understanding the criteria for stability of a transfer function? especially with the bode plot, root locus, and nyquist.

i've gone through a few search results online and almost every site has different information that feels contradictory to me.

thanks,
bg742

 

[Danger]

Hi, Barney;
I can't help at all with your question, but I have a duty as a member of the Langauge Police. 'Criteria' is the plural; 'criterion' is the singular.

 

[ravenprp]

Hi bg742,

The stability of a transfer function can be determined using various techniques such as Bode plot, root locus, and Nyquist plot. Each of these techniques has its own criteria for stability. Let's take a look at each one in detail.

1. Bode Plot: A Bode plot is a graphical representation of the frequency response of a system. The stability of a system can be determined by looking at the phase and gain margins on the Bode plot. The phase margin should be greater than 0 degrees and the gain margin should be greater than 0dB for the system to be stable. If the phase margin is negative, it indicates that the system is unstable and prone to oscillations. Similarly, if the gain margin is negative, it indicates that the system is unstable and unable to reject disturbances.

2. Root Locus: A root locus is a plot that shows the location of the poles of a transfer function as a parameter (usually gain) is varied. The stability of a system can be determined by looking at the location of the poles on the root locus plot. If the poles lie on the left half of the complex plane, the system is stable. If the poles lie on the right half of the complex plane, the system is unstable. Additionally, the distance between the poles and the imaginary axis also plays a role in determining stability. The further away the poles are from the imaginary axis, the more stable the system is.

3. Nyquist Plot: A Nyquist plot is a graphical representation of the frequency response of a system in the complex plane. The stability of a system can be determined by looking at the encirclement of the -1 point on the Nyquist plot. If the plot encircles the -1 point in a clockwise direction, the system is unstable. If the plot does not encircle the -1 point, the system is stable. Additionally, the number of encirclements also gives an indication of the stability. The more encirclements, the more unstable the system is.

It is important to note that these stability criteria are not contradictory but rather complementary. Each technique provides a different perspective on the stability of a system and they can be used together to get a better understanding. I hope this helps in clarifying the criteria for stability of a transfer function.