Meteorological Instrument Performance Characteristics

Dynamic Response » Response to Specific Input Functions » Transfer Function Exercise

Use the image below to answer the following question.

Step impulse with a =1 and =1 response.

Step impulse with a 𝜔 = 1 and 𝛄 = 1 response. Image by NCAR/EOL.

Question

A temperature sensor on an aircraft has first-order dynamic response, and you need to make measurements that respond to abrupt (i.e., step-function) changes by reaching 90% of the final value in 100 m of flight path, while flying 200 m/s. What is the required time constant of the sensor? Answer in seconds (within 10%).

The correct answer is c.

The answer is c, 0.22 seconds or 0.5 seconds / 2.3, where 2.3 is obtained using the following procedure. Follow blue line (first-order response) to find the time corresponding to M1 = 0.90; the desired time constant is 0.5 s (i.e. 100 m/200 m s-1) divided by this value. Answer image key is shown below.

Answer image key corresponding to figure above, except that the solution for x = 0.90 is 2.3 is shown.

Answer image key corresponding to figure above, showing the solution for x = 0.90 is 2.3.

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Use the image below to answer the following question.

Ramp input with omega equal to 1 and gamma equal to 1. Use this figure to determine the natural frequency required to get a second-order time constant equal to the first-order time constant at critical damping at longer times.

Ramp input with 𝜔 = 1 and 𝛄 = 1 . Use this figure to determine the natural frequency 𝜔 required to get a second-order time constant equal to the first-order time constant at critical damping at longer times.

Question

For a ramp input with slope such that x = t/𝛕, at long times a first-order system will lag behind that input by 𝛕 (in these plots, 1 s), so this is a possible way to find the time constant and to provide corrections for such a system. For a second-order system, the response is more complicated and depends on both the characteristic angular frequency and the damping. However, for critical damping the system (at long time) again lags behind the input signal by a fixed amount, so the lag again can be used to find the characteristic angular frequency ω of this system. What value of 𝛚 is needed so that the lag of a second-order critically damped system is also 1?

The correct answer is b.

The answer is 2.06 s-1 (~ 2.1 s-1). To find the characteristic angular frequency of the system so that it is equal to the first-order system, the second-order response must overlap the first order response at a time after the transient effects have disappeared. For a second-order system with critical damping (𝛄 = 1) to have the same time constant as a first-order system, its characteristic frequency would have to be approximately twice that of a first order system. See the answer image key below.

This image shows that in order to overlap the responses of a second-order system and first-order system so that they would have the same time constant, the characteristic frequency of the second-order system would have to double.

This image shows that in order to overlap the responses of a second-order system and first-order system so that they would have the same time constant, the characteristic frequency of the second-order system would have to double.

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Use the image below to answer the following questions.

System response to an impulse (step) function with damping.

System response to an impulse (step) function with damping.

To determine the type of response characteristic of an unknown system, it is often revealing to determine the system response to an impulse function. Suppose you have a measuring system with unknown characteristics, and you find it responds to an impulse as shown.

Question 1

Is this measuring system first-order or second-order? (Answer this question before continuing to the next two questions.)

The correct answer is b.

The response is not simple-exponential motion toward x. This is evident in the behavior of the x in time.

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