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Frequency Response Analysis in Control System by Engineering funda, #FrequencyResponse
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We took a detailed look at timing analysis (in my previous series) where we made various inputs to the system such as B. steps, ramps, parabolas, etc. All of these inputs are functions of time (not frequency). Frequency response refers to the steady-state response of the system when subjected to a sinusoal input. Using this method, we vary the input frequency within a specific range and examine the resulting response. Imagine a linear system, we put the system into a sinusoal input (linear), r(t) = A sin(t). In steady state, the output c(t) of the system has the form: c (t) = B sin (t + φ ) The phase relationship between the sinusoal input and the steady state output is called the frequency response. Method used in frequency response: We have two plots: magnitude and phase. In both cases the variable is the angular frequency ” . The following techniques are commonly used: Bode plot: There are two separate plots: (i) magnitude and (ii) phase, both logarithmic to frequency. Polar and Nyquist plot: It is a single plot of magnitude versus phase angle, as ω varies from zero to infinity in polar coordinates. Disadvantages of the frequency response method: Compared to the extensive methods developed for modeling and simulation, the method consered to obtain the frequency response is outdated. These techniques are mainly applied to linear systems Derivation of Resonant Frequency and Resonant Peak Frequency Two important specifications of the frequency response are the resonant peak ( ) and the resonant frequency ( ). We will now derive them. Conser a second-order criterion system. Here, G(s) = , H(s) = 1 The closed loop transfer function is T(s) = C(s)/R(s) = We convert the equation to a frequency response by replacing s with j. So, T(j) = C(j)/R(j) = = , dive the numerator and denominator by and combine the real part C(j)/R(j) = (where = u ) , the concept is complicated so With magnitude and phase, magnitude (M) = , we know that the resonant frequency is maximum at the magnitude. Using the maximal theorem of derivatives, we get the maximum magnitude when the first term of = 0 , = = 0 , = [ (-1/2) ]  = 0 cannot be zero. So, = 0 , solve we get u = 0 or = 0 then, = u cannot be zero, so, = 0 , therefore, u = , also, = u > 0 , so, u = , u also = = then = (here we have) So, = == Resonant Frequency Then, the resonant peak can be easily found by substituting the value of u into the equation for M, solving we get == . from the two equations d. H. And it can be concluded that as ” approaches zero, approaches ∞ (infinity) and approaches . for 0 < < 0.707 greater than 1, and < . for > 0.707 there is no formant. Bandwth: In addition to resonant frequency and peaking, there is a very important specification of frequency response – bandwth. It is the frequency range up to the cutoff frequency (), beyond which the amplitude (M) drops 3 dB (0.707) below the zero frequency level. Let us derive the formula for bandwth. From the above derivation, we can directly write the value of size “M”: Size(M) = |T(j)| = , now |T(j)| The 3 dB drop is calculated as: 20 log|T(j )| = -3 , 20log = -3 , -20 log() = -3 When taking the antilog of both ses, = 1.995 (can be rounded down to 2) = 0 Taking the square root (using the quadratic formula), We take the square root of both ses, u = (since u > 0 we only take positive values) Now u = = , , = , here is the cutoff frequency. Therefore, the bandwth is 0 – . Therefore, the frequency response of the controller method reveals the nature of the process when the signal is input. So we generally do this by analyzing “magnitude” and “phase” separately. The magnitude response of the machine gives us an ea of the gain/loss the system imparts at a given frequency. The step response tells us how much the system can delay/advance a specific frequency portion of the signal. In my next article, we will see different ways to analyze the frequency response of any system in detail. First we’ll learn about the “Bode plot” technique, then polar plots and Nyquist plots. So stay tuned. report this ad
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