FreqResponse Analysis Design

2y ago
11 Views
2 Downloads
1.98 MB
121 Pages
Last View : 2m ago
Last Download : 2m ago
Upload by : Aydin Oneil
Transcription

Frequency Response Analysis & Design In conventional control-system analysis there aretwo basic methods for predicting and adjusting asystem’s performance without resorting to thesolution of the system’s differential equation.They are:– Root-Locus Method– Frequency-Response Method For the comprehensive study of a system byconventional methods it is necessary to use bothmethods of analysis.MechatronicsFrequency Response Analysis & DesignK. Craig1

Root-Locus Method– Precise root locations are known and actual timeresponse is easily obtained by means of the inverseLaplace Transform. Frequency-Response Method– Frequency response is the steady-state response of asystem to a sinusoidal input. In frequency-responsemethods, we vary the frequency of the input signal overa certain range and study the resulting response.– The design of feedback control systems in industry isprobably accomplished using frequency-responsemethods more often than any other, primarily because itprovides good designs in the face of uncertainty in theplant model.MechatronicsFrequency Response Analysis & DesignK. Craig2

– Many times performance requirements are given interms of frequency response and/or time response.– Noise, which is always present in any system, canresult in poor overall performance. Frequency responsepermits analysis with respect to this.– When the transfer function for a component isunknown, the frequency response can be determinedexperimentally and an approximate expression for thetransfer function can be obtained from the graph of theexperimental data.– The Nyquist stability criterion enables one toinvestigate both the absolute and relative stabilities oflinear closed-loop systems from a knowledge of theiropen-loop frequency-response characteristics.MechatronicsFrequency Response Analysis & DesignK. Craig3

– Frequency-response tests are, in general, simple andcan be made accurately by readily-available equipment,e.g., dynamic signal analyzer.– Correlation between frequency and transient responsesis indirect, except for 2nd-order systems.– In designing a closed-loop system, we adjust thefrequency-response characteristic of the open-looptransfer function by using several design criteria inorder to obtain acceptable transient-responsecharacteristics for the system.MechatronicsFrequency Response Analysis & DesignK. Craig4

For a stable, linear, time-invariant system, themathematical model is the linear ODE withconstant coefficients:dnqod n 1q odq oan a n 1 n 1 L a1 a0 qo ndtdtdtd m qid m 1q idqb m m b m 1 m 1 L b1 i b 0 qidtdtdt qo is the output (response) variable of the system qi is the input (excitation) variable of the system an and bm are the physical parameters of the systemMechatronicsFrequency Response Analysis & DesignK. Craig5

If the input to this system is a sine wave, the steadystate output (after the transients have died out) is alsoa sine wave with the same frequency, but with adifferent amplitude and phase angle.q i Qi sin(ω t) System Input: System Steady-State Output: qo Qo sin(ωt φ) Both amplitude ratio, Qo/Qi , and phase angle, φ,change with frequency, ω. The frequency response can be determinedanalytically from the Laplace transfer function:G(s)s iωMechatronicsFrequency Response Analysis & DesignSinusoidalTransfer FunctionM(ω) φ( ω)K. Craig6

A negative phase angle is called phase lag, and apositive phase angle is called phase lead. If the system being excited were a nonlinear ortime-varying system, the output might containfrequencies other than the input frequency and theoutput-input ratio might be dependent on the inputmagnitude. Any real-world device or process will only need tofunction properly for a certain range offrequencies; outside this range we don’t care whathappens.MechatronicsFrequency Response Analysis & DesignK. Craig7

SystemFrequencyResponseMechatronicsFrequency Response Analysis & DesignK. Craig8

When one has the frequency-response curves forany system and is given a specific sinusoidal input,it is an easy calculation to get the sinusoidal output. What is not obvious, but extremely important, isthat the frequency-response curves are really acomplete description of the system’s dynamicbehavior and allow one to compute the response forany input, not just sine waves. Every dynamic signal has a frequency spectrumand if we can compute this spectrum and properlycombine it with the system’s frequency response,we can calculate the system time response.MechatronicsFrequency Response Analysis & DesignK. Craig9

The details of this procedure depend on the natureof the input signal; is it periodic, transient, orrandom? For periodic signals (those that repeat themselvesover and over in a definite cycle), Fourier Series isthe mathematical tool needed to solve the responseproblem. Although a single sine wave is an adequate modelof some real-world input signals, the genericperiodic signal fits many more practical situations. A periodic function q i(t) can be represented by aninfinite series of terms called a Fourier Series.MechatronicsFrequency Response Analysis & DesignK. Craig10

a0 2 2 πnq i ( t ) a n cos T T n 1 T 2 πnt b n sin T t T2 2πn a n qi ( t ) cos t dtT T 2T2 2 πn b n q i ( t ) sin t dtT T FourierSeries2MechatronicsFrequency Response Analysis & DesignK. Craig11

q i (t)1.5Consider the Square Wave-0.010a0 T -0.50.01 0.5dt 0.01 1.5dt00.02 0.01 0.5 average value 2πn a n 0.5cos t dt 0.02 0.01 2πn 1.5cos 0.02 t dt 0 0 00.011 cos ( nπ ) 2πn 2πn b n 0.5sin t dt 1.5sin t dt 0.020.0250nπ 0.01044qi ( t ) 0.5 sin (100πt ) sin ( 300πt ) Lπ3π00.01MechatronicsFrequency Response Analysis & DesignK. Craig12t

The term for n 1 is called the fundamental orfirst harmonic and always has the same frequencyas the repetition rate of the original periodic waveform (50 Hz in this example); whereas n 2, 3, gives the second, third, and so forth harmonicfrequencies as integer multiples of the first. The square wave has only the first, third, fifth, andso forth harmonics. The more terms used in theseries, the better the fit. An infinite number givesa “perfect” fit.MechatronicsFrequency Response Analysis & DesignK. Craig13

21.51amplitudePlot oftheFourierSeries forthe squarewavethroughthe thirdharmonic0.50-0.5-1-0.01 -0.008 -0.006 -0.004 -0.00200.002 0.004 0.006 0.008time (sec)qi ( t ) 0.5 MechatronicsFrequency Response Analysis & Design0.0144sin (100πt ) sin ( 300πt )π3πK. Craig14

For a signal of arbitrary periodic shape (ratherthan the simple and symmetrical square wave), theFourier Series will generally include all theharmonics and both sine and cosine terms. We can combine the sine and cosine terms using:A cos ( ωt ) Bsin ( ωt ) C sin ( ω t α )C A 2 B2 1 Aα tanB Thusq i ( t ) Ai0 Ai1 sin ( ω1 t α1 ) Ai 2 sin ( 2ω1 t α 2 ) LMechatronicsFrequency Response Analysis & DesignK. Craig15

A graphical display of the amplitudes (Aik) and thephase angles (αk) of the sine waves in the equationfor qi(t) is called the frequency spectrum of qi(t). If a periodic qi(t) is applied as input to a systemwith sinusoidal transfer function G(iω), after thetransients have died out, the output q o(t) will be ina periodic steady-state given by:q o ( t ) Ao0 Ao1 sin ( ω1 t θ1 ) Ao2 sin ( 2ω1 t θ2 ) LA ok Aik G ( iωk )θk αk G ( iωk ) This follows from superposition and the definitionof the sinusoidal transfer function.MechatronicsFrequency Response Analysis & DesignK. Craig16

Review of Frequency-Response PerformanceSpecifications Let V be a sine wave (U 0) and wait for transients to dieout. Every signal will be a sine wave of the same frequency.We can then speak of amplitude ratios and phase anglesbetween various pairs of signals.CAG1G2 (iω)(iω) V1 G1G 2 H(iω)MechatronicsFrequency Response Analysis & DesignK. Craig17

The most important pair involves V and C. Ideally(C/V)(iw) 1.0 for all frequencies. Amplitude ratio and phase angle will approximate the idealvalues of 1.0 and 0 degrees for some range of lowfrequencies, but will deviate at higher frequencies.MechatronicsFrequency Response Analysis & DesignK. Craig18

Typical Closed-LoopFrequency-ResponseCurvesAs noise is generally in aband of frequencies abovethe dominant frequencyband of the true signal,feedback control systemsare designed to have adefinite passband in orderto reproduce the truesignal and attenuate noise.MechatronicsFrequency Response Analysis & DesignK. Craig19

The frequency at which a resonant peak occurs, ωr, is aspeed-of-response criterion. The higher ωr, the faster thesystem response. The peak amplitude ratio, Mp, is a relative-stabilitycriterion. The higher the peak, the poorer the relativestability. If no specific requirements are pushing thedesigner in one direction or the other, Mp 1.3 is oftenused as a compromise between speed and stability. For systems that exhibit no peak, the bandwidth is used fora speed of response specification. The bandwidth is thefrequency at which the amplitude ratio has dropped to0.707 times its zero-frequency value. It can of course bespecified even if there is a peak. It is the maximumfrequency at which the output of a system willsatisfactorily track an input sinusoid.MechatronicsFrequency Response Analysis & DesignK. Craig20

If we set V 0 and let U be a sine wave, we canmeasure or calculate (C/U)(iω) which shouldideally be 0 for all frequencies. A real systemcannot achieve this perfection but will behavetypically as shown.Closed-Loop Frequency Response to a Disturbance InputMechatronicsFrequency Response Analysis & DesignK. Craig21

Two open-loop performance criteria in common use tospecify relative stability are gain margin and phasemargin. The open-loop frequency response is defined as (B/E)(iω).One could open the loop by removing the summingjunction at R, B, E and just input a sine wave at E andmeasure the response at B. This is valid since (B/E)(iω) G1G2H(iω). Open-loop experimental testing has theadvantage that open-loop systems are rarely absolutelyunstable, thus there is little danger of starting up an untriedapparatus and having destructive oscillations occur beforeit can be safely shut down. The utility of open-loop frequency-response rests on theNyquist stability criterion.MechatronicsFrequency Response Analysis & DesignK. Craig22

Gain margin (GM) and phase margin (PM) are in thenature of safety factors such that (B/E)(iω) stays farenough away from 1 -180 on the stable side. Gain margin is the multiplying factor by which the steadystate gain of (B/E)(iω) could be increased (nothing else in(B/E)(iω) being changed) so as to put the system on theedge of instability, i.e., (B/E)(iω)) passes exactly throughthe -1 point. This is called marginal stability. Phase margin is the number of degrees of additional phaselag (nothing else being changed) required to createmarginal stability. Both a good gain margin and a good phase margin areneeded; neither is sufficient by itself.MechatronicsFrequency Response Analysis & DesignK. Craig23

Open-Loop Performance Criteria:Gain Margin and Phase MarginA system must have adequate stability margins.Both a good gain margin and a good phase marginare needed.Useful lower bounds: GM 2.5 PM 30 MechatronicsFrequency Response Analysis & DesignK. Craig24

Bode Plot View ofGain Margin and Phase MarginMechatronicsFrequency Response Analysis & DesignK. Craig25

It is important to realize that, because of modeluncertainties, it is not merely sufficient for a system to bestable, but rather it must have adequate stability margins. Stable systems with low stability margins work only onpaper; when implemented in real time, they are frequentlyunstable. The way uncertainty has been quantified in classicalcontrol is to assume that either gain changes or phasechanges occur. Typically, systems are destabilized wheneither gain exceeds certain limits or if there is too muchphase lag (i.e., negative phase associated with unmodeledpoles or time delays). As we have seen these tolerances of gain or phaseuncertainty are the gain margin and phase margin.MechatronicsFrequency Response Analysis & DesignK. Craig26

Frequency-Response Curves The sinusoidal transfer function, a complexfunction of the frequency ω, is characterized by itsmagnitude and phase angle, with frequency as theparameter. There are three commonly used representations ofsinusoidal transfer functions:– Bode diagram or logarithmic plot: magnitude of outputinput ratio vs. frequency and phase angle vs. frequency– Nyquist plot or polar plot: output-input ratio plotted inpolar coordinates with frequency as the parameter– Log-magnitude vs. phase plot (Nichols Diagram)MechatronicsFrequency Response Analysis & DesignK. Craig27

Bode Diagrams Advantages of Logarithmic Plots:– Rapid manual graphing is possible.– Wide ranges of amplitude ratio and frequency, both lowand high, are conveniently displayed.– Amplitude ratio exhibits straight-line asymptote regionsof definite slope. These are helpful in identifyingmodel type from experimental data.– Complex transfer functions are easily plotted andunderstood as graphical sums of simple (zero-order, 1storder, 2nd-order) basic systems since the dB(logarithmic) technique changes multiplication intoaddition and division into subtraction.MechatronicsFrequency Response Analysis & DesignK. Craig28

A sinusoidal transfer function may be representedby two separate plots:– Magnitude (dB) vs. frequency (log10)– Phase angle (degrees) vs. frequency (log10) The log magnitude (Lm) of a transfer function indB (decibel) is: 20log10 G ( iω) Frequency Bands: f2 2xwhere x # of octaves– Octavef1– An octave is a frequency band from f1 to f2 where f2/f1 2.f2x 10where x # of decades– Decadef1– A decade is a frequency band from f1 to f2 where f2/f1 10.MechatronicsFrequency Response Analysis & DesignK. Craig29

– As a number doubles, the dB value increases by 6 dB.– As a number increases by a factor of 10, the dB value0.01 40 dBincreases by 20 dB.Note that, when expressed in dB,the reciprocal of a number differsfrom its value only in sign.0.1 20 dB0.5 6 dB1.0 0 dB2.0 6 dB10.0 20 dB100.0 40 dB Generalized Form of the Sinusoidal TransferFunction:rG ( iω ) K (1 i ωT1 )(1 i ωT2 )( iω )mMechatronicsFrequency Response Analysis & Design 2ζ 1 2 (1 i ωT3 ) 1 iω 2 ( iω) ωn ωn K. Craig30

The log magnitude (Lm) of G(iω) is given by:Lm G ( iω ) Lm [ K ] Lm [1 i ωT1 ] ( r ) Lm [1 i ωT2 ] 2ζ 1 2 ( m ) Lm [i ω] Lm [1 i ωT3 ] Lm 1 iω 2 ( iω ) ωn ωn The phase angle is given by: G ( iω) K (1 i ωT1 ) ( r ) (1 i ωT2 ) 2ζ 1 2 ( m ) ( iω ) (1 i ωT3 ) 1 iω 2 ( iω ) ωn ωn Both the log magnitude and angle are functions offrequency.MechatronicsFrequency Response Analysis & DesignK. Craig31

When the log magnitude and angle are plotted asfunctions of log10(ω), the resulting curves arereferred to as Bode Plots. These two curves can be combined into a singlecurve of log magnitude vs. angle with frequencyas the parameter. This curve is called the NicholsDiagram. Drawing Bode Plots– The generalized form of a transfer function shows thatthe numerator and denominator have 4 basic types offactors: p 2ζ 1 m r2 K ( iω )(1 iωT ) 1 iω 2 ( iω ) ωn ωn MechatronicsFrequency Response Analysis & DesignK. Craig32

– The curves of log magnitude and angle vs. log10(ω) canbe drawn for each factor. Then these curves can beadded together graphically to get the curves for thecomplete transfer function. Asymptotic approximationsto these curves are normally used.– Gain K (positive)Lm [ K ] 20log10 ( K ) constant K 0 – Integral and derivative factors (iω) m m Lm ( iω ) m20log10 iω m20log10 ( ω ) m ( iω) m ( 90 ) constant The log magnitude curve is a straight line with a slope m(20)dB/decade m(6) dB/octave when plotted against log(ω). Itgoes through the point 0 dB at ω 1.MechatronicsFrequency Response Analysis & DesignK. Craig33

– 1st-Order Factors (1 iωT) 1 1Lm (1 i ωT ) 20log10 1 i ωT 20log10 1 ω2 T 2 0 dB for ω 1 1 (1 i ωT ) tan 1 ( ωT ) 20log10 ( ω T ) for ω 1ω 1/T: straight-line asymptote with zero slopeω 1/T: straight-line asymptote with 20 dB/decade slopeω 1/T: value is 0 dBωcf corner frequency 1/T frequency at which theasymptotes to the log magnitude curve intersect Phase angle straight-line asymptotes: 0 at ω 0.1ωcf, 45 atω ωcf, 90 at ω 10ωcf Angle curve is symmetrical about ωcf when plotted vs. log10(ω) MechatronicsFrequency Response Analysis & DesignK. Craig34

Bode Plotting of1st-OrderFrequencyResponseNote that varying thetime constant shifts thecorner frequency to theleft or to the right, butthe shapes of the curvesremain the same.MechatronicsFrequency Response Analysis & DesigndB 20 log10 (amplitude ratio)decade 10 to 1 frequency changeoctave 2 to 1 frequency changeK. Craig35

For the case where the exponent of the first-order term is r,the corner frequency is unchanged, and the asymptotes are stillstraight lines: the low-frequency asymptote is a horizontal lineat 0 dB, while the high-frequency asymptote has a slope of (20)r dB/decade. The error involved in the asymptoticexpressions is r times that for (1 iωT) 1. The phase angle is rtimes that of (1 iωT) 1 at each frequency point.– 2nd-Order Factors 2ζ 1 2 1 iω 2 ( iω ) ωn ωn 1 For ζ 1, the quadratic can be factored into two 1 st-orderfactors with real poles which can be plotted as described for a1st-order factor. For 0 ζ 1, the quadratic is plotted without factoring, as it isthe product of two complex-conjugate factors.MechatronicsFrequency Response Analysis & DesignK. Craig36

1222 2ζ 1 ω 2ζω 2 Lm 1 i ω 2 (i ω) 20log10 1 2 ωn ωn ωn ωn 2ζω 12 2ζ 1 ω2 n 1 iω 2 ( iω) tan 12ωωωn n 1 2ωn12For ω ωn: the log magnitude 0 dBFor ω ωn: the log magnitude 40 log 10 (ω/ωn) dBThe low-frequency asymptote is a horizontal line at 0 dB.The high-frequency asymptote is a straight line with a slope of 40dB/decade. The asymptotes, which are independent of ζ, cross at ωcf ωn.These are not accurate for a factor with low values of ζ. Phase angle: 0 at ω 0, 90 at ω ωn, 180 at ω MechatronicsFrequency Response Analysis & DesignK. Craig37

Frequency Responseof a2nd-Order SystemNote: The plots shown arefor a 2 nd-order term with anexponent of –1. For a 2ndorder term with an exponentof 1, the magnitudes of thelog magnitude and phaseangle are the same exceptwith a sign change.MechatronicsFrequency Response Analysis & DesignK. Craig38

Some Observations on 1st-Order Factors– Time Constant τ Time it takes the step response to reach 63% of the steady-statevalue– Rise Time Tr 2.2 τ Time it takes the step response to go from 10% to 90% of thesteady-state value– Delay Time Td 0.69 τ Time it takes the step response to reach 50% of the steady-statevalueMechatronicsFrequency Response Analysis & DesignK. Craig39

– Bandwidth The bandwidth is the frequency where the amplitude ratiodrops by a factor of 0.707 -3dB of its gain at zero or lowfrequency. For a 1 st -order system, the bandwidth is equal to 1/ τ. The larger (smaller) the bandwidth, the faster (slower) the stepresponse. Bandwidth is a direct measure of system susceptibility tonoise, as well as an indicator of the system speed of response.MechatronicsFrequency Response Analysis & DesignK. Craig40

Some Observations on 2nd-Order Factors– When a physical system exhibits a natural oscillatorybehavior, a 1st-order model (or even a cascade ofseveral 1st-order models) cannot provide the desiredresponse. The simplest model that does possess thatpossibility is the 2nd-order dynamic system model.– This system is very important in control design.– System specifications are often given assuming that thesystem is 2nd order.– For higher-order systems, we can often use dominantpole techniques to approximate the system with a 2ndorder transfer function.MechatronicsFrequency Response Analysis & DesignK. Craig41

– Damping ratio ζ clearly controls oscillation; ζ 1 isrequired for oscillatory behavior.– The undamped case (ζ 0) is not physically realizable(total absence of energy loss effects) but gives us,mathematically, a sustained oscillation at frequency ωn.– Natural oscillations of damped systems are at thedamped natural frequency ωd, and not at ωn.ωd ωn 1 ζ 2– In hardware design, an optimum value of ζ 0.64 isoften used to give maximum response speed withoutexcessive oscillation.– Undamped natural frequency ωn is the major factor inresponse speed. For a given ζ response speed is directlyproportional to ωn.MechatronicsFrequency Response Analysis & DesignK. Craig42

– Thus, when 2nd-order components are used in feedbacksystem design, large values of ωn (small lags) aredesirable since they allow the use of larger loop gainbefore stability limits are encountered.– For frequency response, a resonant peak occurs for ζ 0.707. The peak frequency is ωp and the peak amplituderatio depends only on ζ.K2ωp ωn 1 2ζpeak amplitude ratio 2ζ 1 ζ 2– The phase angle at the frequency where the resonantpeak occurs is given by:21 2ζφp tan 1ζMechatronicsFrequency Response Analysis & DesignK. Craig43

– Bandwidth The bandwidth is the frequency where the amplitude ratio dropsby a factor of 0.707 -3dB of its gain at zero or low-frequency. For a 1 st-order system, the bandwidth is equal to 1/τ. The larger (smaller) the bandwidth, the faster (slower) the stepresponse. Bandwidth is a direct measure of system susceptibility to noise,as well as an indicator of the system speed of response. For a 2 nd-order system:BW ωn 1 2ζ 2 2 4ζ 2 4ζ 4 As ζ varies from 0 to 1, BW varies from 1.55ωn to 0.64ωn. For avalue of ζ 0.707, BW ωn. For most design considerations,we assume that the bandwidth of a 2 nd-order all pole system canbe approximated by ωn.MechatronicsFrequency Response Analysis & DesignK. Craig44

Bode Plotting Procedure:– Rewrite the sinusoidal transfer function as a product ofthe four basic factors. p 2ζ 1 m r2 K ( iω )(1 iωT ) 1 iω 2 ( iω ) ωn ωn – Determine the value of 20log10(K) Lm(K) dB– Plot the low-frequency magnitude asymptote throughthe point Lm(K) at ω 1 with a slope 20(m) dB perdecade.– Complete the composite magnitude asymptotes Extend the low-frequency asymptote until the first frequencybreak point, then step the slope by r(20) or p(40),depending on whether the break point is from a 1 st-order or2nd-order term in the numerator or denominator. Continuethrough all break points in ascending order.MechatronicsFrequency Response Analysis & DesignK. Craig45

– Sketch in the approximate magnitude curve: Increasethe asymptote value by a factor of 3 dB at 1st-ordernumerator break points, and decrease it by a factor of -3dB at 1st-order denominator break points. At 2nd-orderbreak points, sketch in the resonant peak (or valley)using the relation that at the break point ω ωn: 1 2ζ 1 2 Lm 1 i ω 2 (i ω) 20log10 ( 2ζ ) ωn ωn – Plot the low-frequency asymptote of the phase curve, φ m(90 ).– As a guide, sketch in the approximate phase curve bychanging the phase by 90 or 180 at each breakpoint in ascending order.MechatronicsFrequency Response Analysis & DesignK. Craig46

– Locate the asymptotes for each individual phase curveso that their phase change corresponds to the steps inthe phase toward or away from the approximate curve.Sketch in each individual phase curve as indicated bythe detailed phase plots for the individual terms.– Graphically add each phase curve. Use grids if anaccuracy of about 5 is desired. If less accuracy isacceptable, the composite curve can be done by eye.Keep in mind that the curve will start at the lowestfrequency asymptote and end on the highest-frequencyasymptote and will approach the intermediateasymptotes to an extent that is determined by how closethe break points are to each other.MechatronicsFrequency Response Analysis & DesignK. Craig47

– Bode Plotting Examples s iω 2 1 2 1 2000 ( s 0.5 ) 0.5 0.5 s ( s 10 )( s 50 ) s s i ω i ω s 1 1 iω 1 1 10 50 10 50 102.5 2 s 2 0.2 s ( s 0.4s 4)s s 1 2 4 0.01 ( s 2 0.01s 1)2 s0.022s s 1 2 4 MechatronicsFrequency Response Analysis & DesignK. Craig48

Advantages of Working with Frequency Responsein terms of Bode Plots:– Bode plots of systems in series simply add, which isquite convenient.– Bode’s important phase-gain relationship is given interms of logarithms of phase and gain.– A much wider range of system behavior – from low- tohigh-frequency behavior – can be displayed.– Bode plots can be determined experimentally.– Dynamic compensator design can be based entirely onBode plots.MechatronicsFrequency Response Analysis & DesignK. Craig49

Why is it important for an engineer to know howto hand-plot frequency responses?– Allows engineer to deal with simple problems but alsoto check computer results for more complicated cases.– Often approximations can be used to quickly sketch thefrequency response and deduce stability as well asdetermine the form of the needed dynamiccompensations.– Hand plotting is useful in interpreting frequencyresponse data that have been generated experimentally.MechatronicsFrequency Response Analysis & DesignK. Craig50

Zero-Order Dynamic System ModelMechatronicsFrequency Response Analysis & DesignK. Craig51

Validation of a Zero-OrderDynamic System ModelMechatronicsFrequency Response Analysis & DesignK. Craig52

Example:RC Low-Pass FilterTime Response &FrequencyResponsei ine inioutRCe out ein RCs 1 R eout i Cs i 1 in out eout11 when i out 0ein RCs 1 τs 1MechatronicsFrequency Response Analysis & DesignK. Craig53

1st-Order Dynamic System ModelMechatronicsFrequency Response Analysis & DesignK. Craig54

Time Response to Unit Step Input10.90.8Amplitude0.70.6R 15 KΩC 0.01 µF0.50.40.30.20.10012345Time (sec)678x 10-4Time Constant τ RCMechatronicsFrequency Response Analysis & DesignK. Craig55

R 15 KΩC 0.01 µF00-5-20Phase (degrees)Gain dBFrequency Response-10-15-20-25210-40-60-80341010Frequency (rad/sec)105-100210341010Frequency (rad/sec)105Bandwidth 1/τeoutKiω ( )einiωτ 1K 0o( ωτ ) 1 tan ωτMechatronicsFrequency Response Analysis & Design22 1 K2ωτ 1( )2 tan 1 ωτK. Craig56

MatLab / Simulink DiagramFrequency Response for 1061 Hz Sine Inputτ 1.5E-4 sec1outputtau.s 1outputSine y Response Analysis & DesigntClocktimeK. Craig57

Amplitude Ratio 0.707 -3 dBPhase Angle -45 Response to Input 1061 Hz Sine -0.8-100.51MechatronicsFrequency Response Analysis & Design1.52time (sec)2.533.54x 10-3K. Craig58

Example:Time Response & Frequency Response2-Pole, Low-Pass, Active FilterR4R1e inC2R3R7C5 MechatronicsFrequency Response Analysis & DesignR6 e outK. Craig59

Physical Model Ideal Transfer Function R 7 1 R R R C C eout 6 1 3 2 5 s()ein 111 12s s R 3 C2 R1C2 R 4 C2 R 3 R4 C2 C5R4R1e inC2R3R7C5 R6 e outMechatronicsFrequency Response Analysis & DesignK. Craig60

2nd-OrderDynamicSystem Modelωn @a0 undamped natural frequencya2a1ζ@ damping ratio2 a 2a 0d 2q 0dqa 2 2 a1 0 a0 q0 b0 q ibdtdtK @ 0 steady-state gaina01 d 2q 0 2ζ dq 0 q 0 Kq i22ωn dtωn dtStep Responseof a2nd-Order SystemMechatronicsFrequency Response Analysis & DesignK. Craig61

1 d 2q 0 2ς dq 0 ζ q 0 Kq i22ωn dtωn dtStep Responseof a2nd-Order SystemUnderdamped 1q o Kqis 1 e ζω n t sin ωn 1 ζ 2 t sin 1 1 ζ 21 ζ2 () ζ 1Critically Damped q o Kqis 1 (1 ωn t ) e ωn t ζ ζ 2 1 ( ζ ζ 2 1) ωn te 1 2Over2 ζ 1 damped q o Kqis ζ ζ 2 1 ( ζ e 22ζ 1 MechatronicsFrequency Response Analysis & Designζ 1 2ζ 1) ωn t K. Craig62ζ 1

Kω2nG(s) 2s 2 ςωn s ω2ns1,2 ςωn iωn 1 ς 2s1,2 σ iωdy (t ) 1 e σtLocation of PolesOfTransfer Function σ cos ωd t ω sin ωd t d 1.8rise timeωn4.6ts settling timeςωntr πςMp e1 ς2( 0 ζ 1)ζ 1 0.6 overshoot( 0 ζ 0.6)MechatronicsFrequency Response Analysis & DesignGeneral All-Pole2nd-OrderStep ResponseK. Craig63

ωn 1.8trσ ζ 0.6 (1 M p )4.6ts0 ζ 0.6Time-Response Specifications vs. Pole-Location SpecificationsMechatronicsFrequency Response Analysis & DesignK. Craig64

Frequency Responseof a2nd-Order SystemQoK(s ) 2Qis2ζs 12ωn ωnLaplace Transfer FunctionSinusoidal Transfer FunctionQo( iω ) QiMechatronicsFrequency Response Analysis & DesignK ω 1 ωn 2 2 4ζ 2ω2 2ω n tan 12ζ ω ωn ω ω n K. Craig65

Frequency Responseof a2nd-Order SystemMechatronicsFrequency Response Analysis & DesignK. Craig66

-40 dB per decade slopeFrequency Responseof a2nd-Order SystemMechatronicsFrequency Response Analysis & DesignK. Craig67

Minimum-Phase and Nonminimum Phase Systems– Transfer functions having neither poles nor zeros in theRHP are minimum-phase transfer functions.– Transfer functions having either poles or zeros in theRHP are nonminimum-phase transfer functions.– For systems with the same magnitude characteristic, therange in phase angle of the minimum-phase transf

response is easily obtained by means of the inverse Laplace Transform. Frequency-Response Method – Frequency response is the steady-state response of a system to a sinusoidal input. In frequency-response methods, we vary the frequency of the input signal over a certain

Related Documents:

Present ICE Analysis in Environmental Document 54 Scoping Activities 55 ICE Analysis Analysis 56 ICE Analysis Conclusions 57 . Presenting the ICE Analysis 59 The ICE Analysis Presentation (Other Information) 60 Typical ICE Analysis Outline 61 ICE Analysis for Categorical Exclusions (CE) 62 STAGE III: Mitigation ICE Analysis Mitigation 47 .

Research Design: Financial Performance Analysis In this study, financial performance analysis will be used. The analysis is based on three types of analysis methods which are horizontal analysis, trend analysis and ratio analysis. All data analysis is based on the items on the financial statement. A financial statement is a written record

Design and Analysis Software v2 file used as a template to contain primary analysis and secondary analysis settings. This option can be used for the analysis of legacy EDS or SDS files when a specific set of primary and secondary analysis settings are needed. The primary and secondary analysis settings will be used when the analysis (-a) option .

Legal Design Service offerings Legal Design - confidential 2 Contract design Litigation design Information design Strategy design Boardroom design Mastering the art of the visual Dashboard design Data visualization Legal Design What is especially interesting in the use of visual design in a p

Module 7: Fundamental Analysis (NCFM Certification) 1. Introduction of Fundamental Analysis What is Fundamental & Technical Analysis? Difference between technical & fundamental analysis Features & benefits of Fundamental analysis 2. Top-Down Approach in Fundamental Analysis Economic Analysis Industry Analysis Company analysis 3.

Qualitative analysis, quantitative analysis, non-financial indicator analysis, financial indicator analysis, internal performance analysis, external performance analysis, project-orientated analysis, organization-orientated analysis 8 [36] Area-based Knowledge measurement in products and processes,

Oasys GSA Contents Notation 8 Degrees of freedom 10 Active degrees of freedom 10 Degrees of Freedom with no Local Stiffness 11 Analysis Options 13 Static Analysis 13 Static P-delta Analysis 13 Modal Analysis 14 Modal P-delta Analysis 14 Ritz Analysis 15 Modal Buckling Analysis 16 Model Stability Analysis 17 Non-linear Static Analysis 18

Introduction to Groups, Rings and Fields HT and TT 2011 H. A. Priestley 0. Familiar algebraic systems: review and a look ahead. GRF is an ALGEBRA course, and specifically a course about algebraic structures. This introduc-tory section revisits ideas met in the early part of Analysis I and in Linear Algebra I, to set the scene and provide motivation. 0.1 Familiar number systems Consider the .