Homework 11

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goodwine
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Homework 11

Post by goodwine »

Reading: Chapters 1 - 12.

Exercises: 9.28-9.32.
Bill Goodwine, 376 Fitzpatrick
pat

Re: Homework 11

Post by pat »

For 9.30, how do we find the steady state value? Also, it says for a unit step response, but aren't Bode plots for sinusoidal inputs?
akulczyc

Re: Homework 11

Post by akulczyc »

pat wrote:For 9.30, how do we find the steady state value? Also, it says for a unit step response, but aren't Bode plots for sinusoidal inputs?
Bode, root locus, and Nyquist are tools to help you know more information about the system. They don't care which input you use, since they are plotted with information from the transfer function only. With a sinusoidal input you can infer additional information from the plot, but the input does not have to be sinusoidal.
One way of finding the steady-state value is to find a transfer function that would produce that Bode plot (or something quite similar) and then use the final value theorem to determine the steady state value.
smcshane

Re: Homework 11

Post by smcshane »

I think I understand how the bode phase plot changes, but I am confused about finding the initial and final angles. What is the method for determining these?
jconcelm

Re: Homework 11

Post by jconcelm »

For 9.30, I'm confused as to how to get the information from the bode plot that would be needed to sketch a step response. Are we supposed to be able to determine the transfer function from the bode plot? If we are supposed find the transfer function, is there a way other than guess and check to determine the transfer function?
akulczyc

Re: Homework 11

Post by akulczyc »

smcshane wrote:I think I understand how the bode phase plot changes, but I am confused about finding the initial and final angles. What is the method for determining these?
jconcelm wrote:For 9.30, I'm confused as to how to get the information from the bode plot that would be needed to sketch a step response. Are we supposed to be able to determine the transfer function from the bode plot? If we are supposed find the transfer function, is there a way other than guess and check to determine the transfer function?
These two kinda go together, since the angles help you figure out what the transfer function is. Any zero adds 90 degrees to the plot and any pole subtracts 90 degrees. So a transfer function with two zeros and three poles will have an ending angle of -90 degrees. The beginning angle will be zero (you may have to start the plot at 0.00001 to see it), unless there are poles or zeros at the origin. If this is the case, then you do the same thing for the origin, but only count the poles and zeros at the origin.

For example (s+1)/(s^2(s+10)) will start at -180 degrees (two poles at the origin) and end at -180 degrees. But note that the Bode plot will not be a straight line since the zero will increase the angle and the pole will bring it back down to -180.

From the magnitude portion of the bode plot, you can determine the gain of the transfer function (if any) and more precisely where the zeros and poles are located. The gain is determined by looking at the starting value. If it is 0, it means that when you plug in s = 0, the transfer function equals 1. Recall that the slope will change either +20db/decade with a zero and -20db/decade with a pole. It may help to draw in the dashed lines shown in Figure 9.63. This way you have a corner point, which is the location of your pole or zero.

So from looking at the Bode diagram you can determine the transfer function (or at least a close approximate). This is the route I went to solve the problem and to sketch the step response shown in the homework solution.
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