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AOSC 634 Air Sampling and Analysis Lecture 3 Measurement Theory Performance Characteristics of Instruments Dynamic Performance of Sensor Systems Response of a second order system to A step change A ramp change Copyright Brock et al. 1984; Dickerson 2015 1

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Dynamic Response Sensor output in response to changing input. Dynamic Characteristics of Second Order Systems EQ I Where n is the undamped natural frequency, a constant (s -1 ). is the damping ratio, a unitless constant. We must solve an initial value problem. 2

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Solving the differential equation 3 We will use the technique of variation of parameters to find complementary solutions. We must assume a time dependence of the form e rt and substitute this into Eq I. The characteristic equation is: Each root gives rise to a solution; there are four.

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Four roots of the characteristic equation leads to free oscillations X c (t) = C sin( n t + ) 2. 0 < leads to damped oscillations X c (t) = C exp (− n n t) sin( m t + ) Where m = n (1 – 2 ) ½ { n within 5% for < 0.3} leads to critically damped X c (t) = exp (− n t) (At + B) Where A and B are constants. 1 leads to an overdamped solution. 4

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5 1 leads to an overdamped solution. Where m = 1/ m And the characteristic time is 1/ m In dimensionless time n t = t” Critically damped systems are an ideal; in the real world only overdamped and underdamped systems exist. We will focus on underdamped systems such as the dew pointer (or a car with bad shocks). Overdamped systems lead to a “double first order”.

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Time Response of second order systems. Start with a system at rest where both the input and output are zero. X(0) = X I (0) = X 0 Their first derivatives are likewise zero at time zero. We will proceed as with the first order system assuming a step change. Using the dimension- less form. = 0, no damping. X’(t”) = 1 – cos (t”) 6

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Time Response of second order systems. < 1.0, underdamped. = 0, critically damped. X’(t”) = 1 – e -t” (t” + 1) 7

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Time Response of second order systems. > 1.0, overdamped. The damping number is With see Figure 2-11 of Brock et al. 8

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Response of a second order system to an a step increase of input. 9 Undamped Underdamped Critical Overdamped Dimensionless time t” = n t Output

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An example with doubly normalized time 10

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Notes on Figure 2-11. For all the final state is X I (t”) for t” > 0 – The slope is real, continuous, and near zero where t” << 1.0. Contrast with first order. For undamped systems, there is free oscillation at n. For 0 < there is damping at a frequency of: The modified (damped) natural freq in Hz. For there is large overshoot and a long time lag. 11

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Notes on Figure 2-11, continued. For there is large overshoot and a long time lag. – The amplitude of the oscillations decreases exponentially with a time constant of -1. The extrema can be found: Where the sub e represents extrema. The extrema come on time at t”. Where n is a positive integer. 12

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From the amplitude of the first extreme (assume here a maximum) we can calculate the damping ratio : 13 Practical application

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From the time (in units of t” of the first extreme (assume here a maximum) we can calculate the natural undamped frequency n : 14

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Note, the closer to is to unity, and the smaller n, the faster X’(t”) approaches X I. Example using Figure 2-11. Try this yourself with a mm ruler. Let’s check the curve with = 0.10 for the first maximum. Looking at a paper copy, X’(t”) max = 60 mm X’(t) final = X I = 35 mm Close to the 0.100 value in the book. If the max amplitude is twice the input then (2/1 – 1) is 1 and =0. 15

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Example using Figure 2-11, continued. Let’s look for the natural frequency, n. Let the time of the first max be 30 s, an arbitrary value. To get within e -1 of the final value requires: t” = 1/ = 10 = n t = 0.1t and t = 100 s! In general, the time to e -1 is ( n ) -1 for < 0.3. For > 0.3, use m. 16

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Summary Although less intuitive than first order systems, second order systems lend themselves to analysis of performance characteristics. A step change is in some ways a worst case scenario for overshoot. Any second order systems provide perfectly adequate temporal response in the real world where geophysical variables tend to show wave structure. 17

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References MacCready and Henry, J. Appl Meteor., 1964. Determination of the Dynamic Response of a Nitric Oxide Detector, K. L. Civerolo, J. W. Stehr, and R. R. Dickerson, Rev. Sci. Instrum., 70(10), 4078-4080, 1999. 18

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