7 351 1st order system, part 4

7 351 1st order system, part 4 - 1st-Order System: Time...

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1 1 1st-Order System: Time Response If a square wave of t/ τ = 0.5 is applied to a first order system, sketch its response. initial final Output 00 . 5 2 1st-Order System: Time Response 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 t/ ¬± Amplitude t/ τ = 10 t/ τ = 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 t/ ⎠? Notice the transient to the steady state. What average value is the response approaching? If a square wave of t/ τ = 0.5 is applied to a first order system, sketch its response.
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2 3 1st-Order System: Time Response 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 t/ ¬± Amplitude t/ τ = 10 t/ τ = 0.5 Notice the transient to the steady state. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 t/ ±⎭ Does this make sense? 4 1st-Order System: Time Response What would the response look like if instead put in a triangle, cosine, sawtooth, or other periodic wave of t/ τ = 0.5? The response will be broadly similar, but with the exact shapes of the transient curve and ripples modified. For a sine wave, the ripples will be sinusoidal, for example, and the transient less steep.
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3 5 1st-Order System: Time Response If a triangle wave of period t/ τ = 100 is applied to a first order system, sketch its response. initial final Output T = 100 τ , so f = 1/100 τ , ω = 2 π /100 τ , and ωτ = 2 π /100 = 0.063 20 40 60 80 100 Input (units) triangle input 6 1st-Order System: M( ω ) ωτ = 0.063
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4 7 1st-Order System: Time Response If a triangle wave of period t/ τ = 100 is applied to a first order system, sketch its response. initial final Output 20 40 60 80 100 Input (units) Output (units) A K A The response follows the input. The system responds faster than the input changes. 8 Discussion Why is there no phase shift for ωτ < 0.1? Discuss in terms of when/why the response goes up, down. 1st-Order System: Sine Input Phase Response, > 10 ωτ Æ The input is slow enough that the sensor is basically as steady state all the time, and perfectly follows the input.
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5 9 Discussion Why is the maximum lag 90 o ? Discuss in terms of when/why the response goes up, down. 1st-Order System: Sine Input Phase Response, ωτ > 10 ωτ 10 The input is driving the system higher than its current magnitude essentially the whole time that the input is positive. When the input goes negative, then the system begins decreasing its response. 1st-Order System: Sine Input Phase Response, > 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 024681 0 t/ ∠≥ Amplitude 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t/
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6 11 Sensor Response Example Q: What is the steady-state response of the sensor to an input of T(t) = C 2 + A 2 sin( ω t ), where = 20 rad/sec, C 2 = 26 ° C, A 2 = 1.5 ° C?
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7 351 1st order system, part 4 - 1st-Order System: Time...

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