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So far, we have looked at the asymptotic behavior of first order constructs, like pure integrators or single bode plot single pole and zeros. Once you start working with typical dynamic systems, it is very likely that you will have to deal with higher order polynomial expressions. The trick to dealing with those is remembering that any polynomial, no matter the order, can always be factored into a bunch of first order constructs which will correspond to the real roots, and a number of second order constructs which will correspond to complex conjugate pairs of roots.
Typical examples of second order systems are mass spring dampers and RLC circuits.
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Both of this, depending on the ratio bode plot single pole either the damping to the mass or the resistance to the inductance, will have a pair of complex conjugate roots.
Anyways, if we calculate the roots of that quadratic polynomial using the standard formula-- minus b plus minus square root of whatever-- we find that the complex conjugate pair of roots will have this form.
Note that these roots will only be a complex conjugate pair as long as the damping ratio zeta is less than 1. Anything greater than 1, and both roots will become real numbers, which means that the system will behave as the product of two first order poles. As we did before to calculate the frequency response, we replace s by jw in the transfer function.
Understanding Bode Plots, Part 3: Simple Systems
So both the magnitude and the phase will be approximately 0. Finally, when the frequency w is much larger than the natural frequency the quadratic term will dominate. When taking the log, the square will come out and multiply the 20, so the magnitude will asymptotically approach a straight line with a slope of dBs per decade.
The phase will go to degrees because G will now fall bode plot single pole the negative real axis. This adjusted value is what is called a damped natural frequency. And note that, in that case, the magnitude of the resonant peak will go towards infinity.
A small value of the damping ratio means a higher and sharper resonant peak as well as a sharper shift in the phase. You can see that as we increase zeta, the magnitude of the resonant peak comes down and the phase transition becomes smoother. Here I want to highlight the damping ratio of 0.
This damping makes the magnitude -3 dBs at the natural frequency. At this point let me go back to our interactive design tool because there are a couple of additional things I want to highlight.
Understanding Bode Plots, Part 4: Complex Systems
First, let me bring in a pair of complex conjugate poles, and I am going to place them close to 10 radians per second. Let me just make sure that I set the natural frequency to exactly I notice that, because I am starting with a damping ratio of 1, my polynomial is the product of two real roots.
As soon as I change the damping to any value lower than 1, let's say 0. If I choose a smaller damping ratio, what you're going to see is a sharper and higher magnitude peak.
We just saw how a function like Bode in MATLAB can quickly and easily create a frequency response plot directly from the dynamic equations of, or the input, output transfer functions of our system. The key as control engineers is not just to be able to create those plots. The important thing is having a good understanding of what those magnitude and phase traces are telling us about our system behavior and stability. Bode plots were originally developed by Dr. Hendrik Bode, hence the name, while he was working for Bell Labs in the s, just before World War II.
In this particular case, I picked 0.