# Obtain a Transfer Function from Bode Plot shown

I want to obtain a transfer function from the Bode Plots I already got the corner frequencies (approximates) and all the terms for my T.F. Then when I try to check my results in MATLAB my plot drops to -40 dB instead of -20 dB like in the original plot. • Looks like you have two zeroes at ~30 rad/s, not one at 30 rad/s and one at 40 rad/s Jan 4, 2019 at 21:08
• Look at the high-frequency gain of your Bode plot -- the one you need to match is 0dB, yours is -5dB. If you assume that your poles are in the right spots, then you have three constraints for the three numerator terms: the DC gain, the high-frequency gain, and the gain at the bottom of the trough. You should be able to solve for the three coefficients directly, without worrying about the actual zero positions. Jan 4, 2019 at 21:17

A quick look at this shows a common factor of 10 between the first low-frequency pole (1.3 rad/s) and the first zero at 13 rads/s (slope is now 0) then the second zero (slope is +1) with a final second pole to set the slope back to 0. The below curves seem ok then: You can see the first pole when the phase crosses -45°, then the zero is not far otherwise the phase would further drop.

Edit

The question is how did I get the first pole and the zero? I read the curve starting from the flat portion (0 dB) and then tried to identify the -3-dB point on the magnitude graph or the -45° on the phase plot. It is an approximation here and implies that the pole and the zero are well spread. Then, once the pole is obtained, I determined the magnitude of a pole-zero pair. I then solved for the zero position which brings a valley in the magnitude plot at -19.7 dB roughly read from the magnitude curve. Voilà ! You wrote the transfer function the right way, by keeping the poles and zeros well factored. This is what is called a low-entropy format in which a leading term would indicate what the gain is for $$\s=0\$$ in your case (1 or 0 dB). Writing these transfer functions the right way is part of the fast analytical circuits techniques or FACTs introduced many years ago. They describe how to determine transfer functions in a swift and efficient manner and how to format a transfer function so that poles, zeros and gain immediately show up when looking at the equation.

• sorry for asking, but I'm new to this Bode stuff, where did you get the first low-frequency pole value of 1.3 rad/s and the first zero at 13 rads/s? Jan 5, 2019 at 3:18
• @DavidDM, I've added a section in my answer. Let me know if you need further details. Jan 5, 2019 at 8:27

I disagree with your initial values of corner frequencies. I would say they are more likely to be closer to these values (roughly drawn at the lower and upper 3 dB points): - 