Homework 10, due Wednesday May 2

[goodwine]

1. Sketch the Bode plot for the low pass filter circuit in the following figure if R=100 and C=100.
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2. Sketch the Bode plot for the high pass filter circuit in the following figure if R=10 and C=10.
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3. If we connect the output of the high pass filter to the input of the low pass filter (or vice-versa) we should have a band pass filter. Determine the transfer function and sketch the Bode plot for the circuit in the following figure. Use the same parameter values as in the previous problems. Hint: determining the transfer function is trivial.
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   What sort of engineering application would use a band pass filter?
4. How would you modify the band pass filter to make the band narrower or wider? Do so and use Matlab to plot the resulting Bode plot.
5. How would you modify the band pass circuit to make the band pass section sharper, i.e., a steeper transitions? Do so and use Matlab to sketch the resulting Bode plot.
6. Consider
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   1. Sketch the root locus plot for this transfer function.
   2. For the closed loop transfer function (putting G(s) in the following block diagram), use the Routh array to determine the values of k for which the closed loop transfer function is stable.
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   3. Use your root locus plot from the first problem to verify the value of k for which the closed loop system transitions from stable to unstable.
   4. Use Matlab to construct the Bode plot for k G(s) (multiply G(s) by k) for the case where k is much less, equal to and much greater than the k value where it switches from stable to unstable. In fact, do not use the bode() command, but rather use the margin() command.
      • What is the effect of increasing or decreasing k on the Bode plots? Do the shapes of the curves change at all?
      • Critical points on a Bode plot the frequencies where the magnitude curve goes through 0 dB and the phase goes through -180 degrees. When you used the k value that was the boundary between stable and unstable response, what was the frequency where the magnitude when through 0 dB? What was the frequency where the phase went through -180 degrees? Explain why the magnitude will be 0 dB and the phase will be -180 degrees when the system is exactly at the boundary of stability. (Hint: use the 1+kG(s)=0 equation in the case where s=iw (w=omega)).
      • The gain and phase margins which are marked using the margin() command in Matlab give a measure of stability of a system. Explain the intuition behind these measures.

[alam2]

are we supposed come up with a transfer function from output voltage to input voltage for the given circuit and sketch the bode plots from there?

[goodwine]

are we supposed come up with a transfer function from output voltage to input voltage for the given circuit and sketch the bode plots from there?

Yes, that's exactly right.

[pschluet]

For problem 6b, you ask us to determine a specific k value (using the Routh Array) where the system becomes unstable. In class you told us that the Routh Array only tells us how many poles are in the RH Plane. How do we find a specific k value?

[pschluet]

Problem 5: It seems to be that multiplying the transfer function by 1/s would make the band pass section sharper, but I do not know what this would physically correspond to in the system. Is this a sufficient answer (multiply G(s) by 1/s), or do I need to explain what would happen physically in the system? If I need to explain it in regards to exactly what I am doing to the system, could you give me a hint as to what multiplying by 1/s would correspond to?

[goodwine]

For problem 6b, you ask us to determine a specific k value (using the Routh Array) where the system becomes unstable. In class you told us that the Routh Array only tells us how many poles are in the RH Plane. How do we find a specific k value?

If there are any poles in the RHP, then it's unstable. Hence, find the range of k values for which there are no RHP poles.

[goodwine]

Problem 5: It seems to be that multiplying the transfer function by 1/s would make the band pass section sharper, but I do not know what this would physically correspond to in the system. Is this a sufficient answer (multiply G(s) by 1/s), or do I need to explain what would happen physically in the system? If I need to explain it in regards to exactly what I am doing to the system, could you give me a hint as to what multiplying by 1/s would correspond to?

Well, the way to check that would be to do it and plot the Bode plot to see if it makes it sharper. It's not exactly the answer I was looking for, but perhaps that works. I don't know myself, so you'll have to plot it to find out.

[awulz]

I know this question is a bit late with the homework due tomorrow, but in general:

In HW Question #1, I arrive at a transfer function (w/ substituted values) of
G(s) = 1 / (10000s + 1). Is the coefficient of 10000 what drives the Magnitude of G(s) to have its critical value at 10^-4 (on the bode plot). If so, why is that? What is governing the critical point?

Also, on Question #3, it appears that for the magnitude plot, the maximum point is the line from 10^-4 to 10^-2. What governs this point to be at a magnitude of -40db versus at 0db?

Thanks,
Alex

Edit: Mistyped the exponent

[goodwine]

I know this question is a bit late with the homework due tomorrow, but in general:

In HW Question #1, I arrive at a transfer function (w/ substituted values) of
G(s) = 1 / (10000s + 1). Is the coefficient of 10000 what drives the Magnitude of G(s) to have its critical value at 10^-4 (on the bode plot). If so, why is that? What is governing the critical point?

If you substitute s=iw (w = omega) then you have
1/(10000 w i + 1)

What governs the critical point (actually it's called the "break point" although I didn't emphasize that in class) is the frequency at which the iw is either smaller or bigger than what being added to it. In this case, that's when w=10^(-4).

Also, on Question #3, it appears that for the magnitude plot, the maximum point is the line from 10^-4 to 10^-2. What governs this point to be at a magnitude of -40db versus at 0db?

It's just the relative magnitudes of the two break points. If they were a lot farther apart the maximum could be higher. If the band were narrower, it would be lower.