24_Chap14NotesW12_2pgs

24_Chap14NotesW12_2pgs - Page 1 of 2 CHE 361 Frequency...

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Unformatted text preview: Page 1 of 2 CHE 361: Frequency Response Analysis Chap. 14 SEMD3 = “Frequency Response Analysis” = 3 “terms” involved Review Chap. 5.2.3 = 1st - order Sinusoidal Response, 5.4.2 = 2nd - order Sinusoidal Response, For physical systems = Examples 5.2 & 5.6 1 = "frequency" - physical interpretation is that the input to the process is changing with time as u '(t ) = A sin(ωt ) with ω in radians/time ω radians/time f = frequency in cycles/time = 2π radians/cycle P = period of oscillation, time per cycle in time units = 1/f 2 = "response" - for linear systems, if the input is a sine wave, the output (response) after a “long” time is also a sine wave with a shifted “argument” but the same frequency and in general a different amplitude. The shift and amplitude depend on the frequency of the input oscillation, thus the "frequency response" also implies a "frequency dependance". The term “long time” is long compared to the time constants of the system. 3 = "analysis" -looking at a range of frequencies all at once and drawing conclusions, extending the "physical" input/output ideas of oscillatory behavior into math concepts based on complex functions of complex variables. 14.1 - Sinusoidal Forcing of a First-order Process K = AR , φ = Phase Angle = − tan −1 (ωτ ) 22 ω τ +1 14.2 - Sinusoidal Forcing of an nth-order Process ( n negative real poles ) and "Shortcut" Method for Finding the Frequency Response of G(s) Model Substitute jω for s. Put resulting complex number into the form: = G ( jω ) Re (ω ) + Im (ω ) j Then amplitude ratio (AR) of G at ω = (Re2 + Im2)1/2 and the phase angle (PA) of G at ω = tan-1 (Im/Re) ... caution: pay attention to the quadrant of the complex number by examining the signs of Re and Im ! Two examples:14.1, 14.2 pgs 253-254 SEMD3 Page 2 of 2 14.3 Bode Diagrams = plots of AR and PA as functions of ( ω ) frequency First-order System (stable) = frequency response of a "pole" in LHP 5 steps include .... AR: low ω asymptote, AR = K with slope = 0 at ω = 1/τ, AR = 0.707 K (asymptotes meet at "corner" frequency) high ω asymptote, slope of AR( ω ) = - 1 PA: low ω asymptote, PA = 0 deg. at ω = 1/τ, PA = - 45 degrees (at "corner" or "break" frequency) high ω asymptote = -90 deg. Second-order System: given K , τ and ζ : AR = K (1 − ω τ ) + ( 2ζωτ ) 222 2 −2ζωτ and PA = φ = tan -1 22 1 − ω τ 2nd-order 14.3.3 = note two 1st-order systems in series = K x “pole1” x “pole2” AR overall (total) = product of ARi ( individual factors ) PA overall (total) = sum of PAi ( individual factors ) Process Zero 14.3.4 : AR & PA Since LHP zero can cancel LHP pole, that product ratio = 1, AR of zero at -1/τ = AR-1 of pole at -1/τ so that combined AR = 1 for all ω. Likewise, PA of zero at -1/τ = -PA of pole at -1/τ so that combined PA = 0 for all ω. For a zero in the RHP at -1/τz, where τz < 0, the AR is the same as if τz had the same size but different sign, while the PA has opposite sign for the two cases. Thus a RHP zero has a PA that starts at 0 and goes to -90 deg, similar to the PA for a LHP pole. Time delay 14.3.5: AR = 1 for all ω and PA = -(ωθ) radians (see Figure 13.6 or 14.4). Example 14.3: Bode plot for K = 5, time delay = 0.5, zero at -2.0, poles at -0.05 and -0.25. Table 14.2 Examples of G(s) and Bode Plots -- Also see CHE 361 Handout with plots from MATLAB Skip 14.4 = Frequency Response Characteristics of Feedback Controllers CHE 461 PID controller analysis 14.5 Nyquist Diagram of Gp(s) - plot of Re(ω) vs. Im(ω) on Cartesian coordinates for ω from 0 to ∞ and the "reflection" for ω from 0 to -∞. ...
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This note was uploaded on 03/01/2012 for the course CHE 361 taught by Professor Staff during the Winter '08 term at Oregon State.

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