Mechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter
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1 Time Respose & Frequecy Respose d -Order Dyamic System -Pole, Low-Pass, Active Filter R 4 R 7 C 5 e i R 1 C R R e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole, Low-Pass, Active Filter 1
2 Measuremets, Calculatios, Maufacturer's Specificatios Model Parameter ID Which Parameters to Idetify? What Tests to Perform? Physical System Physical Model Math Model Experimetal Aalysis Assumptios ad Egieerig Judgemet Physical Laws Model Iadequate: Modify Equatio Solutio: Aalytical ad Numerical Solutio Actual Dyamic Behavior Compare Predicted Dyamic Behavior Modify or Augmet Make Desig Decisios Model Adequate, Performace Iadequate Model Adequate, Performace Adequate Desig Complete Dyamic System Ivestigatio
3 Physical Model Ideal Trasfer Fuctio R 7 1 R eout 6 R1R3CC 5 ( s) = ei s s+ RC RC RC RRCC R 4 R 7 C 5 e i R 1 C R R e out 3
4 Zero-Order Dyamic System Model 4
5 1 st -Order Dyamic System Model τ = time costat K = steady-state gai t = τ Slope at t = 0 q o Kq = τ is t e τ Kq is q o t = = 0 τ 5
6 How would you determie if a experimetallydetermied step respose of a system could be represeted by a first-order system step respose? qo() t = Kqis 1 e o () q t Kq q Kq is () t is t o 1 = e τ Kq is = e () t τ t τ qo t t t log10 1 = log10 e = Kqis τ τ Straight-Lie Plot: q ( t) o log10 1 vs. t Kqis Slope = /τ 6
7 This approach gives a more accurate value of τ sice the best lie through all the data poits is used rather tha just two poits, as i the 63.% method. Furthermore, if the data poits fall early o a straight lie, we are assured that the istrumet is behavig as a first-order type. If the data deviate cosiderably from a straight lie, we kow the system is ot truly first order ad a τ value obtaied by the 63.% method would be quite misleadig. A eve stroger verificatio (or refutatio) of first-order dyamic characteristics is available from frequecyrespose testig. If the system is truly first-order, the amplitude ratio follows the typical low- ad highfrequecy asymptotes (slope 0 ad 0 db/decade) ad the phase agle approaches -90 asymptotically. 7
8 If these characteristics are preset, the umerical value of τ is foud by determiig ω (rad/sec) at the breakpoit ad usig τ = 1/ω break. Deviatios from the above amplitude ad/or phase characteristics idicate o-first-order behavior. 8
9 What is the relatioship betwee the uit-step respose ad the uit-ramp respose ad betwee the uit-impulse respose ad the uit-step respose? For a liear time-ivariat system, the respose to the derivative of a iput sigal ca be obtaied by differetiatig the respose of the system to the origial sigal. For a liear time-ivariat system, the respose to the itegral of a iput sigal ca be obtaied by itegratig the respose of the system to the origial sigal ad by determiig the itegratio costats from the zero-output iitial coditio. 9
10 Uit-Step Iput is the derivative of the Uit-Ramp Iput. Uit-Impulse Iput is the derivative of the Uit- Step Iput. Oce you kow the uit-step respose, take the derivative to get the uit-impulse respose ad itegrate to get the uit-ramp respose. 10
11 System Frequecy Respose 11
12 Bode Plottig of 1 st -Order Frequecy Respose db = 0 log 10 (amplitude ratio) decade = 10 to 1 frequecy chage octave = to 1 frequecy chage 1
13 d -Order Dyamic System Model dq0 dq = 0 i a a a q b q dt dt q0 = Kqi ω dq0 ζ dq0 dt ω dt a 0 ω = a 1 ζ = 0 0 K = steady-state gai 0 a a a b a udamped atural frequecy dampig ratio Step Respose of a d -Order System 13
14 dq0 ς dq0 dt ω dt 1 + ζ + q = Kq ω 0 i Step Respose of a d -Order System Uderdamped 1 ( ) ζω t q 1 o = Kqis 1 e si ω 1 ζ t + si 1 ζ ζ< 1 1 ζ Critically Damped ω ( ) t qo = Kq is 1 1+ωt e ζ = 1 Overdamped ( ζ+ ζ ) ζ+ ζ 1 1 ωt 1 e ζ 1 qo = Kq is ζ > 1 ζ ζ 1 ( ζ ζ 1) ωt + e ζ 1 14
15 Frequecy Respose of a d -Order System Laplace Trasfer Fuctio Q Q o i ( s) = s ω K ζs ω Siusoidal Trasfer Fuctio Qo K 1 ζ ( iω ) = ta Q i ω ω ω 4ζ ω 1 + ω ω ω ω 15
16 Frequecy Respose of a d -Order System 16
17 -40 db per decade slope Frequecy Respose of a d -Order System 17
18 Some Observatios Whe a physical system exhibits a atural oscillatory behavior, a 1 st -order model (or eve a cascade of several 1 st -order models) caot provide the desired respose. The simplest model that does possess that possibility is the d -order dyamic system model. This system is very importat i cotrol desig. System specificatios are ofte give assumig that the system is d order. For higher-order systems, we ca ofte use domiat pole techiques to approximate the system with a d - order trasfer fuctio. 18
19 Dampig ratio ζ clearly cotrols oscillatio; ζ < 1 is required for oscillatory behavior. The udamped case (ζ = 0) is ot physically realizable (total absece of eergy loss effects) but gives us, mathematically, a sustaied oscillatio at frequecy ω. Natural oscillatios of damped systems are at the damped atural frequecy ω d, ad ot at ω. ω d =ω 1 ζ I hardware desig, a optimum value of ζ = 0.64 is ofte used to give maximum respose speed without excessive oscillatio. 19
20 Udamped atural frequecy ω is the major factor i respose speed. For a give ζ respose speed is directly proportioal to ω. Thus, whe d -order compoets are used i feedback system desig, large values of ω (small lags) are desirable sice they allow the use of larger loop gai before stability limits are ecoutered. For frequecy respose, a resoat peak occurs for ζ < The peak frequecy is ω p ad the peak amplitude ratio depeds oly o ζ. ω p =ω 1 ζ peak amplitude ratio = ζ 1 ζ K 0
21 Badwidth The badwidth is the frequecy where the amplitude ratio drops by a factor of = -3dB of its gai at zero or low-frequecy. For a 1 st -order system, the badwidth is equal to 1/ τ. The larger (smaller) the badwidth, the faster (slower) the step respose. Badwidth is a direct measure of system susceptibility to oise, as well as a idicator of the system speed of respose. For a d -order system: BW = ω 1 ζ + 4ζ + 4ζ 4 1
22 As ζ varies from 0 to 1, BW varies from 1.55ω to 0.64ω. For a value of ζ = 0.707, BW = ω. For most desig cosideratios, we assume that the badwidth of a d -order all pole system ca be approximated by ω.
23 G(s) Kω s s + ςω +ω s = ςω ± iω 1 ς s = 1, = σ± iω 1, d σt σ y() t = 1 e cosω dt+ siωdt ωd 1.8 t r rise time ω 4.6 t s settlig time ςω πς ( ) = ζ < 1 ς Mp e 0 1 overshoot ζ 1 0 ζ ( ) Locatio of Poles Of Trasfer Fuctio Geeral All-Pole d -Order Step Respose 3
24 ω 1.8 t r σ 4.6 t s ( p ) ζ M 0 ζ 0.6 Time-Respose Specificatios vs. Pole-Locatio Specificatios 4
25 Experimetal Determiatio of ζ ad ω ζ ad ω ca be obtaied i a umber of ways from step or frequecy-respose tests. For a uderdamped secod-order system, the values of ζ ad ω may be foud from the relatios: πζ π T = ω d M 1 ζ p e = ζ = π log ( ) e M p ωd π ω d =ω 1 ζ ω = = 1 ζ T 1 ζ 5
26 Logarithmic Decremet δ is the atural logarithm of the ratio of two successive amplitudes. ( ) x() t ( + T) ζωt δ= l = l ( e ) =ζωt x t ζω π ζω π πζ ζ= = = = ω d ω 1 ζ 1 ζ δ= 1 l δ B B π +δ Free Respose of a d -Order System π T = ω d ζωt ( ) = ( ω +φ) x t Be si t d 6
27 If several cycles of oscillatio appear i the record, it is more accurate to determie the period T as the average of as may distict cycles as are available rather tha from a sigle cycle. If a system is strictly liear ad secod-order, the value of is immaterial; the same value of ζ will be foud for ay umber of cycles. Thus if ζ is calculated for, say, = 1,, 4, ad 6 ad differet umerical values of ζ are obtaied, we kow that the system is ot followig the postulated mathematical model. For overdamped systems (ζ > 1.0), o oscillatios exist, ad the determiatio of ζ ad ω becomes more difficult. Usually it is easier to express the system respose i terms of two time costats. 7
28 q Kq For the overdamped step respose: t o τ 1 τ ( ζ+ ζ ) ζ+ ζ 1 1 ωt 1 e ζ 1 qo = Kq is ζ > 1 ζ ζ 1 ( ζ ζ 1) ωt + e ζ 1 τ1 τ e e 1 = + τ τ τ τ is 1 1 t where τ 1 1 τ ζ ζ ω ζ + ζ ω ( ) ( )
29 To fid τ 1 ad τ from a step-fuctio respose curve, we may proceed as follows: Defie the percet icomplete respose R pi as: q o Rpi Kqis Plot R pi o a logarithmic scale versus time t o a liear scale. This curve will approach a straight lie for large t if the system is secod-order. Exted this lie back to t = 0, ad ote the value P 1 where this lie itersects the R pi scale. Now, τ 1 is the time at which the straight-lie asymptote has the value 0.368P 1. Now plot o the same graph a ew curve which is the differece betwee the straight-lie asymptote ad R pi. If this ew curve is ot a straight lie, the system is ot secod-order. If it is a straight lie, the time at which this lie has the value 0.368(P 1-100) is umerically equal to τ. Frequecy-respose methods may also be used to fid τ 1 ad τ. 9
30 Step-Respose Test for Overdamped Secod-Order Systems 30
31 Frequecy-Respose Test of Secod-Order Systems 31
32 Hardware Parameters R C R R =.4 kω = µ F, 0.00 µ F = 9.1 kω =.4 kω C = µ F, µ F R R = 10.0 kω = 10.0 kω 3
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