Only in this case, because the 0 is in the numerator, the phase would shift +90 degrees, and the magnet to slope would be +20 dBs per decade.Īt this point, I would like to give you a feel for how all of this works in a more interactive fashion. Using the same approach, we can see that a single 0 will result in a similar trace. So if you want a little more accuracy on the angle, we can assume that the phase drops 45 degrees per decade before and after the value of a pole. From the plot, we can also see that the phase angle takes about two decades to shift 90 degrees. Obviously, we are going to see the biggest difference around the cut-off value of 1/tau. Notice that the actual Bode plot deviates very little from our asymptotic approximation. Meaning that the phase will be close to -90 degrees, and the log of the magnitude will be close to a straight line, rolling off at -20 degrees per decade and crossing zero, where w equals to 1/tau. When the frequency is way higher than the pole, tau*w will become dominant, in which case, G becomes close to a negative, purely imaginary vector. Meaning that the phase will be very close to 0, and the log of its magnitude will be very close to 0, too. Note that this makes G become close to 1/1, which will be a vector on the real axis. But if you think of that expression, on an asymptotic manner, and break the diagram in two parts- when the frequency is well below the pole, in this case, below 1/tau, radiance per second, tau multiplied by w will become very small and the number 1 will dominate the expression. On first impression this looks like it is going to be very hard to draw. The magnitude of that vector will be log of 1, which goes to 0, minus 20 times the log of the magnitude of the denominator. Once again, if we want to look at the frequency response, we need to replace s by jw. Now let's move on to our first art of construct like a single pole with a time constant of tau. And the phase will be a constant positive 90 degrees. In this case, the magnitude will be a line going up with a slope of +20 dBs per decade. The phase remains constant and -90 degrees, and is independent of the frequency.Ĭonversely, if we look at a pure differentiator, which corresponds to just s in the Laplace domain, because w is in the numerator now. Notice that this line has a slope of -20 dBs per unit, which is a frequency decade in this case. So that part goes away and magnitude trace of the Bode plot becomes just a line, because we are plotting it on a horizontal axis that represents log of w. In terms of dBs, the log of a fraction is the log of the numerator, in this case, 1, minus the log of the denominator, in this case, w. Notice that as the frequency omega goes from 0 to infinity, the magnitude of our vector would go from infinity to 0. This vector has a constant phase angle of -90 degrees, and a magnitude of 1/w. Negative because we are multiplying the numerator and denominator by the square root of -1. If we replace s by jw, our function G becomes a vector on the negative imaginary axis. The simplest construct I can start with is a pure integrator, which corresponds to 1/s in the Laplace domain. And they will help us gain a better understanding of how those plots are actually built. The ideas behind the asymptotic approach are quite simple, but extremely powerful. Remember that this was way before computers, so I guess engineers at the time were stuck with slide rulers, using them to manually calculate all those logarithms. This guy was a brilliant controls engineer, and he came up with what, at the time, was the groundbreaking idea of using asymptotic magnitude and phase plots to facilitate stability analysis and control system design in the frequency domain. Hendrik Bode, hence the name, while he was working for Bell Labs in the 1930s, just before World War II. The important thing is having a good understanding of what those magnitude and phase traces are telling us about our system behavior and stability.īode plots were originally developed by Dr. The key as control engineers is not just to be able to create those plots. 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.
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