![]() So, in this video, I want to focus solely on transfer functions that have a right half plane zero, the systems that exhibit this behavior where the response initially moves in the "wrong" direction before correcting itself. We have other videos that talk about transport delay and I’ve linked to those videos in the description. OK, these are the two ways that we can create non-minimum phase systems, with a pure transport delay and with right half plane zeros. So, both non-minimum phase systems produce what looks like a delayed response and this makes sense because there is additional phase lag in both of these systems. The interesting thing here is that the right half plane zero causes the step response to dip in the wrong direction first before recovering back to the same steady state value as the other two. The step response for the minimum phase system is the blue line on the far left, and the delayed response the green line, predictably just shifts the response to the right by the amount of the delay. OK, now let’s look at the step response for each of these transfer functions and see how the additional phase lag impacts the system in the time domain. I suppose a system with infinite time delay, one that just blocks the signal completely would be a maximum phase system but clearly that’s not useful. There is no maximum phase shift since we could just delay the signal by any amount of time we want. ![]() Interestingly, even though there is only one minimum phase system for a given magnitude response, there are an infinite number of non-minimum phase systems, which is why you never hear the term maximum phase system. The transfer function (s+2)/(s^2 + 3s + 1) is the minimum phase system for this particular magnitude response, and the other two transfer functions are non-minimum phase systems, which hopefully is obvious from the fact that there is additional phase. And with this we can talk about the concept of minimum phase and non-minimum phase systems. So, we have three different transfer functions, and each have the same frequency response magnitude, but different phase. And check this out, this transfer function also has the same magnitude, but again not the same phase shift. For this transfer function, we can move the zero from s = -2 over into the right half plane at s = +2. The other way is to move one or more of the transfer function zeros from the left half plane over into the right half plane. So, adding additional delay into the system is one way to affect just the phase. If we input a 1 rad/s sine wave into this function, the phase will be shifted about -122 degrees. The jaggedness of the phase plot at higher frequencies is the result of how I’m wrapping the signal when I plot it so that it stays between + and - 180 degrees. Notice that the magnitude response is the same, but we’ve added additional phase. And we can see this is the case when we graph the frequency response. If we delay the signal by some amount of time - like the 1 second that I’m doing here, that won’t impact the size of the signal, it’ll just shift it in time affecting the phase. So, the top graph will be the same but the bottom graph will be different.įor example, we could simply add a time delay into our system. ![]() We can create any number of transfer functions that produce this but that all have different phase responses. This isn’t the only transfer function that can create this specific magnitude response. Or in other words, the input signal would be delayed by about 1/6th of a cycle and the amplitude would be a little bit lower. ![]() And from this we can see that if we input a 1 rad/s sine wave into this system the output signal would have a magnitude around 2.6 dB lower and the signal would be shifted about -64 degrees. The top graph shows how the magnitude of the input signal is altered for each frequency in the spectrum and the bottom graph shows how the phase is shifted for each frequency. For example, the Bode plot for (s+2)/(s^2 + 3s + 1) looks like this. We can graph the frequency response of a transfer function with a Bode plot. I’m Brian, and welcome to a MATLAB Tech Talk. So, in this video, we’re going to talk about what minimum phase means, what causes a non-minimum phase system, and how that impacts the system behavior. In a similar way, we can glean some additional information about how our system will behave if we know whether it’s a minimum phase or non-minimum phase system. We gain some insight into the system if we know, for example, that we’re dealing with a type 1, second-order transfer function. We like to categorize transfer functions into groups and label them because it helps us understand how a particular system will behave simply by knowing the group that it’s part of.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |