6. There are two TFs, and their bode magnitude plots are exactly the same, but t
ID: 2268619 • Letter: 6
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6. There are two TFs, and their bode magnitude plots are exactly the same, but their angle/phase plots are different. How can this happen? It's a one/two sentence answer. Because of the above reason, if you give me a Bode magnitude plot, the corresponding phase/angle plot may not be unique. Among all such phase plots possibilities, the one with overall smallest phase angles is called the minimum-phase system. Hence, I can reformulate the above question as: What causes a non-minimum phase behavior? It's a one/two sentence answer. Interesting Note: If you can assure me that a TF is minimum-phase, then just by looking by the magnitude plot, I can plot the phase plot. Because I know where all the poles and zeros are just from the Bode plots. Someone even came up with a fancy math formula for finding the phase angle LG(ja) just from the magnitude/gain information as a function of frequency: a. b. c. dlna From this formula, it seems that that phase plot of a minimum-phase system, only depends on the slope (in dB/decade) of the gain plot, and not on the actual gain value. Can you provide a simple reason why this may happen? It's a one/two sentence answer. d. Furthermore, and this is very important, it seems from the above formula that the phase angle LG(jo) at a frequency is a weighted integral of the slope of the magnitude plot (slope of IGlat a) over all the frequencies from negative to positive infinity. The weighting function is f (a). This weighting function is infinite at the frequency where the phase is calculated, f(a) = oo, when = a. what conclusions can you draw from this about G(ja) ? It's a one/two sentence answer.Explanation / Answer
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6a. The phase plots of any two systems will be different only if their delays are different. A delay element (eTs) in a transfer function does not add any magnitude, but adds phase. Hence two transfer functions having everything same except the delay (Value of T) will have differemt phase plots.
6b. A non-minimum phase is nothing but a maximum phase. Maximum phase is inserted into the system due to the presence of positive delay elements. These delays are caused due to presence of zeros on the right hand side of the s plane. Thus, a non-minimum phase system has zeros on the RHS of the s plane.
6c. The pahse angle contributed by a transfer function can be written as the sum of the angles contributed by the poles and zeros. Every pole adds a -20db/decade slope change to the Bode plot and every zero adds a +20db/decade slope change to the Bode plot. We also know that every pole adds -90 degrees (at infinite frequencies) and every zero adds + 90 degrees to the phase. Hence from the slope of the graph, we can calculate whether there is a pole or zero at that point in frequency, and hence determine the phase directly.
6d. Since the phase angle is a cummulative effect of all the poles and zeros, there is an integration operation performed over all frequencies. Further, the weighting factor denotes how much influence the pole or the zero at that frequency has on the phase at that frequency. A pole or zero will show maximum influence at later frequencies and not at starting. Thats the reason that the weighting factor is maximum at frequencies of calculation of phase.
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