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 3 - + R 6 - + e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole, Low-Pass, Active Filter 1
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
Physical Model Ideal Trasfer Fuctio R 7 1 R eout 6 R1R3CC 5 ( s) = ei 1 1 1 1 s + + + s+ RC RC RC RRCC 3 1 4 3 4 5 R 4 R 7 C 5 e i R 1 C R 3 - + R 6 - + e out 3
Zero-Order Dyamic System Model 4
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
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 = 0.4343 Kqis τ τ Straight-Lie Plot: q ( t) o log10 1 vs. t Kqis Slope = -0.4343/τ 6
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
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
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
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
System Frequecy Respose 11
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
d -Order Dyamic System Model dq0 dq0 + 1 + 0 0 = 0 i a a a q b q dt dt 1 + + 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
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
Frequecy Respose of a d -Order System Laplace Trasfer Fuctio Q Q o i ( s) = s ω K ζs + + 1 ω Siusoidal Trasfer Fuctio Qo K 1 ζ ( iω ) = ta Q i ω ω ω 4ζ ω 1 + ω ω ω ω 15
Frequecy Respose of a d -Order System 16
-40 db per decade slope Frequecy Respose of a d -Order System 17
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
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
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 < 0.707. 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
Badwidth The badwidth is the frequecy where the amplitude ratio drops by a factor of 0.707 = -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
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.
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 0.6 0.6 ( ) Locatio of Poles Of Trasfer Fuctio Geeral All-Pole d -Order Step Respose 3
ω 1.8 t r σ 4.6 t s ( p ) ζ 0.6 1 M 0 ζ 0.6 Time-Respose Specificatios vs. Pole-Locatio Specificatios 4
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 1 + 1 ωd π ω d =ω 1 ζ ω = = 1 ζ T 1 ζ 5
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 1 + 1 Free Respose of a d -Order System π T = ω d ζωt ( ) = ( ω +φ) x t Be si t d 6
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
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 ( ) ( ) 1 1 1 8
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 1 100 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
Step-Respose Test for Overdamped Secod-Order Systems 30
Frequecy-Respose Test of Secod-Order Systems 31
Hardware Parameters R C R R 1 3 4 5 6 7 =.4 kω = 0.047 µ F, 0.00 µ F = 9.1 kω =.4 kω C = 0.00047 µ F, 0.001 µ F R R = 10.0 kω = 10.0 kω 3