Studio Exercise Time Response & Frequency Response 1st-Order Dynamic System RC Low-Pass Filter
i i in R out Assignment: Perform a Complete e e Dynamic System Investigation in C out of the RC Low-Pass Filter
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 1 Measurements, Calculations, Which Parameters to Identify? Model Manufacturer's Specifications What Tests to Perform? Parameter ID
Physical Physical Math System Model Model
Assumptions Physical Laws Equation Solution: Experimental and Analytical Analysis Engineering Judgement Model Inadequate: and Numerical Modify Solution
Actual Predicted Dynamic Compare Dynamic Behavior Behavior
Make Design Design Modify Model Adequate, Model Adequate, Complete or Decisions Performance Inadequate Performance Adequate Augment Dynamic System Investigation
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 2 Zero-Order Dynamic System Model
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 3 Validation of a Zero-Order Dynamic System Model
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 4 1st-Order Dynamic System Model
τ = time constant K = steady-state gain
t = τ
Slope at t = 0
t − Kq Kqis τ is qe = q o = o τ t0= τ Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 5 • How would you determine if an experimentally- determined step response of a system could be represented by a first-order system step response?
t − τ Straight-Line Plot: qtoi()=−Kqs1e qto ( ) t log10 1− vs. t qt()− Kq − Kq ois=−e τ is Kqis t Slope = -0.4343/τ qt() − 1e−=o τ Kqis
qto () tt log10 1−=−log10 e =−0.4343 Kqis τ τ
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 6 – This approach gives a more accurate value of τ since the best line through all the data points is used rather than just two points, as in the 63.2% method. Furthermore, if the data points fall nearly on a straight line, we are assured that the instrument is behaving as a first-order type. If the data deviate considerably from a straight line, we know the system is not truly first order and a τ value obtained by the 63.2% method would be quite misleading. – An even stronger verification (or refutation) of first-order dynamic characteristics is available from frequency- response testing. If the system is truly first-order, the amplitude ratio follows the typical low- and high- frequency asymptotes (slope 0 and –20 dB/decade) and the phase angle approaches -90° asymptotically.
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 7 – If these characteristics are present, the numerical value of τ is found by determining ω (rad/sec) at the
breakpoint and using τ = 1/ωbreak. Deviations from the above amplitude and/or phase characteristics indicate non-first-order behavior.
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 8 • What is the relationship between the unit-step response and the unit-ramp response and between the unit-impulse response and the unit-step response? – For a linear time-invariant system, the response to the derivative of an input signal can be obtained by differentiating the response of the system to the original signal. – For a linear time-invariant system, the response to the integral of an input signal can be obtained by integrating the response of the system to the original signal and by determining the integration constants from the zero-output initial condition.
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 9 • Unit-Step Input is the derivative of the Unit-Ramp Input. • Unit-Impulse Input is the derivative of the Unit- Step Input. • Once you know the unit-step response, take the derivative to get the unit-impulse response and integrate to get the unit-ramp response.
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 10 System Frequency Response
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 11 Bode Plotting of 1st-Order Frequency Response
dB = 20 log10 (amplitude ratio) decade = 10 to 1 frequency change octave = 2 to 1 frequency change
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 12 i i Analog Electronics: in R out RC Low-Pass Filter Time Response & e e Frequency Response in C out
ein RCs +1 −R eout = iin Cs −1 iout
eout 11 = == when iout 0 ein RCs +τ1 s +1
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 13 Time Response to Unit Step Input
1
0.9
0.8
0.7
0.6 ude t i 0.5 pl R = 15 KΩ m A 0.4 C = 0.01 µF 0.3
0.2
0.1
0 0 1 2 3 4 5 6 7 8 Time (sec) -4 x 10 Time Constant τ = RC
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 14 • Time Constant τ – Time it takes the step response to reach 63% of the steady-state value
• Rise Time Tr = 2.2 τ – Time it takes the step response to go from 10% to 90% of the steady-state value
• Delay Time Td = 0.69 τ – Time it takes the step response to reach 50% of the steady-state value
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 15 R = 15 KΩ Frequency Response C = 0.01 µF
0 0
-5 -20 ) s e
-10 e -40 r B g d e d in Ga
-15 ase ( -60 h P
-20 -80
-25 -100 2 3 4 5 2 3 10 10 10 10 10 10 Frequency (rad/sec) Frequency (ra Bandwidth = 1/τ e KK∠0D K out ()itω= = = ∠−an−1 ωτ eiωτ +12221− 2 in ()ωτ +1t∠ anωτ ()ωτ +1
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 16 • Bandwidth – The bandwidth is the frequency where the amplitude ratio drops by a factor of 0.707 = -3dB of its gain at zero or low-frequency. – For a 1st -order system, the bandwidth is equal to 1/ τ. – The larger (smaller) the bandwidth, the faster (slower) the step response. – Bandwidth is a direct measure of system susceptibility to noise, as well as an indicator of the system speed of response.
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 17 MatLab / Simulink Diagram Frequency Response for 1061 Hz Sine Input
τ = 1.5E-4 sec
1 output tau.s+1 output First-Order Sine Wave Plant
input t input Clock time
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 18 Amplitude Ratio = 0.707 = -3 dB Phase Angle = -45°
Response to Input 1061 Hz Sine Wave 1
Input 0.8
0.6
0.4
0.2 ude t i 0 pl am -0.2 Output -0.4
-0.6
-0.8
-1 0 0.5 1 1.5 2 2.5 3 3.5 4 time (sec) -3 x 10
Mechatronics K. Craig Studio Exercise – 1st-Order Dynamic System 19