Controls - gain margin question

[LTME]

Homework Statement

Find the Bode Plot, and the phase and gain margins for the uncompensated system.

Homework Equations

G = (3.6s+6)/(s(.1s^2+.7s+1))

The Attempt at a Solution

I understand the phase and gain margin ideas, but the gain margin hinges on the phase crossing -180 degrees, and in this case it never crosses. So am I doing something wrong here? If not, how do I find the gain margin in this case? Is it infinite? Thanks for any help.

 

[berkeman]

Homework Statement

Find the Bode Plot, and the phase and gain margins for the uncompensated system.

Homework Equations

G = (3.6s+6)/(s(.1s^2+.7s+1))

The Attempt at a Solution

View attachment 92373

I understand the phase and gain margin ideas, but the gain margin hinges on the phase crossing -180 degrees, and in this case it never crosses. So am I doing something wrong here? If not, how do I find the gain margin in this case? Is it infinite? Thanks for any help.

Phase margin is measured in degrees, and gain margin is measured in dB. What do you get for this problem for each?

 

[LTME]

The phase margin is approximately 46 degrees. The gain margin, on the other hand, is the gain needed to increase gain to 0 dB when phase is equal to -180 degrees. Well in this case, the Phase is never equal to -180 degrees so how do I establish the gain margin? That is the entire point of this posting. I don't understand how to find gain margin when the phase never crosses -180 degrees.

 

[berkeman]

The phase margin is approximately 46 degrees. The gain margin, on the other hand, is the gain needed to increase gain to 0 dB when phase is equal to -180 degrees. Well in this case, the Phase is never equal to -180 degrees so how do I establish the gain margin? That is the entire point of this posting. I don't understand how to find gain margin when the phase never crosses -180 degrees.

If you extend the frequency plot out another decade or two, the phase should get pretty close to -180 degrees. You have a good point about what is the gain margin if the phase asymptotically approaches -180 degrees with the gain still falling...

 

[LvW]

Yes - this leads to the intuitive - but unrealistic - answer: The gain margin is infinite.
This applies to the given function - however, one should know that each real system will exhibit additional poles and, hence, the phase will cross the 180 deg line ar a finite frequency.

 

[gfd43tg]

By the way, in MATLAB with Bode plots you can left click the plot, press characteristics, and click show all stability margins to get your gain and phase margins.

 

[rude man]

Yes - this leads to the intuitive - but unrealistic - answer: The gain margin is infinite.
This applies to the given function - however, one should know that each real system will exhibit additional poles and, hence, the phase will cross the 180 deg line ar a finite frequency.

Doen't matter if the gain is below 0dB at that point.

 

[LvW]

Doen't matter if the gain is below 0dB at that point.

The question was about the gain margin (and not if it "matters").

 

[rude man]

The question was about the gain margin (and not if it "matters").

It does not "matter" so long as the gain is < 0 dB when the phase hits -180 deg or intreger multiples of +/-180 deg.

 

[rude man]

Yes - this leads to the intuitive - but unrealistic - answer: The gain margin is infinite.
This applies to the given function - however, one should know that each real system will exhibit additional poles and, hence, the phase will cross the 180 deg line ar a finite frequency.

No such thing as "infinite gain margin" unless the gain is absolute zero (-∞dB). The gain margin is the number of dB below 0 dB when the phase shift is an integer number of π. It can never be infinite unless you have a short circuit!

 

[LvW]

No such thing as "infinite gain margin" unless the gain is absolute zero (-∞dB). The gain margin is the number of dB below 0 dB when the phase shift is an integer number of π. It can never be infinite unless you have a short circuit!

rude man, if you read my answer carefully, you will notice that I spoke in my post#5 about an "unrealistic" case which never will happen.
I am familiar with the stability criterion (general and simplified form) and the definition of the stability margins.

It does not "matter" so long as the gain is < 0 dB when the phase hits -180 deg or intreger multiples of +/-180 deg.

But the title of this thread is "gain margin question" . Therefore, the value of the loop gain <0 dB matters because it gives the margin, OK?