Pràctica 1 (2013)Trabajo Inglés
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< CSL; Linear Circuits and Systems (Practice 1) >
Characterization of a system of second order
Authors: Castán, Anna and Sanchez, Andrea
2.266Ω < kR
The purpose of this practice is to characterize the behavior of a system of
2nd order from studying their response time. In response to an excitation
system must distinguish two components corresponding to the forced response
with a shape that is determined by the characteristics of the input signal
(although its amplitude depends on the system that processed) and the
corresponding reply or natural free form and the existence of which depends
on the transfer function of the system (although its amplitude and phase
depends on the excitation). The study of the natural response thus allows to
obtain relevant information about the transfer function and therefore on the
characteristics of the analyzed system.
·Canonical expression of the corresponding transfer function and the transfer function: The system studied in this practice is a circuit 2nd order (Sallen-Key filter) that can control the damping coefficient regulating the value of a variable resistor. (Picture 1).
1/3<k<5,82 updamped kr < 39.633,30 ξ=0 k=1/3 kR = 2.266,66Ω ξ<0 k<1/3 kR < 2.266,66Ω Oscillator Unstable 2.4 Calculate, for k = 14 the length of the transition edge triggered after each [The transient steady linear system can be seen after a period equal to 5 times extinguished constant time corresponding to 4the dominant pole].
2 − ξW0 + W0 ξ − 1= -1,6· 10 Rad/s √ 2 4 − ξW0 − W0 ξ − 1 = − 2,7·10 Rad/s √ 4 f=1÷T T >1÷1.6·10 Rad/s =62,5µs f =16.000Hz 2.5 What should be, for this value of k, the maximum frequency of the signal applied to the input square for the transient caused after each flank could be considered extinct before the next transition.
T >10·τ→F < 1÷10τ =1.6kHz 2.6 Locate and enter the expression to calculate the poles of a 2nd order system with natural response from “overdamped” ξ and W 0.
2 2 −ξW0±W0 ξ −1 → |ξW0| > |W0 ξ −1| √ √ Outline Sallen-key circuit 2.PRELIMINARI STUDY 2.1 Write an expression to calculate, from R, C k, the damping coefficient and the2 natural pulsation system characterized by H (s).
W0 = 2/RC→ W0 = √2/RC 2ξW0 = [3−1/k] → ξ = (1/2W0RC)[3−1/k] = (1/2√2)[3−1/k] 2.2 If R=6.8k(Ohm) and C=10nF, Calculate value of the natural frequency Wo.
2.7 Locate and expression to calculate the poles of a 2nd order system with natural response from “updamped” ξ and W 0..
2 P =ξW +j·W 1−ξ →|P | =|P2|=W 1 0 0√ 1 0 2 P2 = −ξW0+jW0 1−ξ √ 3. CIRCUIT ASSEMBLY W0 = √2÷RC → W0 = √2÷(6.8·(10^3)·10·(10^-9)= 20.797,25Hz≈ 25,80kHz 2.3 For the same values of R and C, calculate the range of values of the constant k resulting in a natural response corresponding to each type of damping possible. Complete a table.
Regardless Regardless of Regardless of the Deadening of the values the values of values of kR of ξ k ξ >1 k > 5,82 kR > 39.633,30Ω Regardless of the values of kR Regardless of the values of kR ξ=1 k = 5,82 kR = 39.633,30 Ω Two views of the circuit < CSL; Linear Circuits and Systems (Practice 1) > 4.“OVERDAMPED” AND “UPDAMPED” Check with the oscilloscope input The two responses: Overdamped and Updamped ·OVERDAMPED ξ>1 time constant 1/(dominant pole) = ·UPDAMPED 0< ξ<1 frequency 1/T = 1/(500x10^-6)= 2.000Hz = 2kHz 5.CALCULATIONS -For the answer overdamped.
5.1- How much is the dominant pole? P = 1/Tmax (Tau) = 1/(4·Ttransitorio) = 1/(4·500x10^-6) = 13,15x10^3 5.2- Use written expression in section 2.6 and the value of the natural frequency calculated in section 2.2 to calculate, from this measure, the damping coefficient and the value of the poles of the function transfer.
-For the answer underdamped.
5.3- How much is the imaginary part of the poles of the transfer function? T(tau) = 1/P P = 1/T(Tau) = 1/0,4ms= 2.500 5.4- Use written expression in section 2.7 and the value of the natural frequency calculated in section 2.2 to calculate, based on this measure, the damping coefficient and the value of the poles of the transfer function.
6.CONCLUSIONS During this practice we have learned to collect data to measure the time constant of the transient and frequency using the oscilloscope display while adjusting the potentiometer. We have seen the difference between an overdamped response and damped response. We had a small problem with the breadboard and prevented us visualize but we have to finish the job with satisfaction with other peer group.