me368.engr.wisc.edume368.engr.wisc.edu/.../final_exam_practice_2012.docx · Web vieworder low pass Butterworth filter with a cutoff frequency of 1 Hz, what will be the standard deviation

ME368 practice final Distributed May 2012

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Each numbered problem is weighted equally

(1) Consider the following (x,y) data pairs: (0,0); (1,0); (2,3). Find the equation for the best-fit line through the data. Show your work.

(2) Determine the natural frequency [rad/s] of the system that produced the data shown in Figure 4.7 Dunn.

(3) A digitizer behaves as though it has an internal low-pass 2nd order Butterworth filter with a cutoff frequency of 12 Hz. Data recorded by the digitizer exhibits a standard deviation of 10 -6 V. If this data is digitally filtered with a 2nd order low pass Butterworth filter with a cutoff frequency of 1 Hz, what will be the standard deviation of the filtered data [V]?

(4) The length, L and width, w of a rectangle are measured to be 2 m and 3 m, respectively. The instrument used exhibits a single-measurement precision of ±0.04 m (P=95%) and a bias of ±0.05 m. Report the area A of the rectangle along with its random uncertainty, its systematic uncertainty, and its combined uncertainty.

(5) Consider the following table of digitizer specifications

a) Fill in the missing cells in rows 1 – 3b) Assume sample compression is applied to the row 3 digitizer, to bring the effective sample

rate down to the value shown in row 4. What value goes in the missing cell?


(6) Imagine you have 1 strain gage (unstrained resistance = 350 Ω), a box of fixed resistors (each 350 Ω), a power supply supplying exactly 5V, and a single voltmeter.

a) sketch a voltage divider circuit that could be used to monitor the resistance of the strain gage. Determine the voltage change that will be measured when the gage is strained to the point that its resistance becomes 351 Ω.

b) sketch a wheatstone bridge circuit (see Chapter 2 Dunn) that could be used to monitor the resistance of the strain gage. Determine the voltage change that will be measured when the gage is strained to the point that its resistance becomes 351 Ω.

(7) Design an RC low-pass filter that can be built in lab. Your filter must attenuate a sine wave at f = 10 kHz to 1 % of it’s amplitude. Some theoretical filter performance curves are shown below and can help you in your design. Sketch the circuit diagram for your electrical RC filter, and include the value of R and the value of C in your sketch.

(8) A bang-bang controller operates on the circuit used in lab 12 with R = 1 MΩ and C = 1µF. The set point has been at 0 V for some time, and the system output has stabilized at 0 V. The set point is changed to 5 V.

a) Estimate t1%, the time for the system output to reach within 1 % of the new setpoint.

b) Which of the following would decrease t1%? (select all that apply)

1. reducing the capacitance, C


2. changing the maximum (ao0) channel output from 5 V to 10 V.

3. using a PID controller with gains chosen based on the Ziegler-Nichols recipe instead of the bang-bang controller.

4. using a PID controller with manually optimized gains instead of the bang-bang controller.

(9) Three different thermometers (A, B, and C) are placed in ice-water baths. Four temperatures are recorded for each. Data is shown below:

temperatures [deg C]

A B C

trial 1 1.1 1.1 0

trial 2 -2 1 0.4

trial 3 2 0.9 0.1

trial 4 -1 1 0.3

a) Based only on the data shown, which thermometer has the best precision? Compute and present its precision.

b) Based only on the data shown, which thermometer has the best accuracy? Compute and present its accuracy.

c) Which thermometer has the best combined uncertainty? (see equation 7.9) Compute and present its RMS error.

(10) Three amplitude vs. frequency plots are given below. For each, sketch the corresponding voltage vs. time plot. Make your sketches as quantitative as possible; you may wish to write some words to help us understand what you’re trying to sketch if you think your sketches are unclear.


(11) Examine this controller block diagram:

a) Which coefficient is the integral gain?

b) Which coefficient is the derivative gain?


c) Should the integral gain be positive or negative? Support your answer with a sketch that includes an error vs time plot. Also explain why the integral gain is useful.

d) Should the derivative gain be positive or negative? Support your answer with a sketch that includes an error vs. time plot. Also explain why the derivative gain is useful.

(12) a) Shown below are system responses for P controllers for a set point change from 2 to 4. Which has the higher KP? How do you know? (10 words or less is fine)

b) Shown below are system responses for a P and a PD controller. Which one includes D control? How do you know? (10 words or less is fine)

#1

#2


c) Shown below are system responses for a P and a PI controller to a set point change from 2 to 3. Which one includes I control? How do you know? (10 words or less is fine)

#3

#4


(13) A large (10 mm diameter) and a small (1 mm diameter) thermocouple (TC) are transitioned from a room-temperature water bath to a hot water bath. The results are plotted.

a) Identify which (red circles or blue squares) is the large TC.

b) Estimate the time constant of the small TC, τ1mm

c) Estimate τ10mm

d) compute the ratio of the two τ values. Does it agree with theory? Explain.

(14) Consider a small TC with τ = 0.01 s and a large TC with τ = 100 s. Describe the expected response of each of these TCs when immersed in a water bath whose temperature varies sinusoidally between 0 °C and 100 °C at 1 Hz according to Tbath(t) = 50°C + 50°C sin (ωt) with ω = 2πf (f = 1 Hz).


(15) During a Ziegler-Nichols tuning exercise the following data was obtained.

Compute the Ziegler-Nichols gain coefficients according to:

Kp = 0.6 Ku; Ti = 0.5 Tu; Td = 0.125 Tu; KI = KP/TI ; KD = KP * TD

Ku is the “ultimate gain”

Tu is the period of the oscillations


(16) Gaussian white noise is input to the circuit shown below and, in turn, monitored by a data acquisition system. Analysis of the measured data reveals a sample standard deviation SX of 0.03 V. Predict SX for the case where C is reduced to 10 nF. Explain your reasoning.

solutions to select problems begin on the following page


solutions to select problems

4

The nominal value of A is A = L w = 2 m x 3 m = 6 m2.

The random (precision) and systematic (bias) uncertainties are calculated in the same way

And the combined uncertainty is calculated as:

Summarizing:

A = 6.00 ± 0.14 m2 (random @ P = 95%)

± 0.18 m2 (systematic)

± 0.23 m2 (combined)


5

Consider the following table of digitizer specifications

a) Fill in the missing cells in rows 1 – 3Those values, shown in the table below, are obtained using Q=E_FSR/((2^M) – 1)

b) Assume sample compression is applied to the row 3 digitizer, to bring the effective sample rate down to the value shown in row 4. What value goes in the missing cell?

The noise will scale down by the square root of the number of points averaged during the sampling, i.e., noise will be reduced by a factor of sqrt (1e5/14.2857), resulting in the value shown in the table below.


6

Imagine you have 1 strain gage (unstrained resistance = 350 Ω), a box of fixed resistors (each 350 Ω), a power supply supplying exactly 5V, and a single voltmeter.

c) sketch a voltage divider circuit that could be used to monitor the resistance of the strain gage. Determine the voltage change that will be measured when the gage is strained to the point that its resistance becomes 351 Ω.

Vvoltmeter = Vsupply x (Rgage / Rgage + Rfixed) = 2.5 V (unstrained) = 2.5036 V (strained).

ΔV = 3.6 mV.

d) sketch a wheatstone bridge circuit (see Chapter 2 Dunn) that could be used to monitor the resistance of the strain gage. Determine the voltage change that will be measured when the gage is strained to the point that its resistance becomes 351 Ω.

Using equation 2.29, Vvoltmeter = Vsupply x [δR /R

4+2δR /R ]= Vsupply x (0.00286/4.00571)= Vsupply x 0.000713 = 3.6 mV as above.


7

Design an RC low-pass filter that can be built in lab. Your filter must attenuate a sine wave at f = 10 kHz to 1 % of it’s amplitude. Some theoretical filter performance curves are shown below and can help you in your design. Sketch the circuit diagram for your electrical RC filter, and include the value of R and the value of C in your sketch.

Simple RC filter behaves like 1st order Butterworth (only the red curve in the plot is needed)

1st order Butterworth attenuates sine wave by 1% when f / f_cutoff = 100 (plot reading, and knowing that 1% amplitude corresponds to 1% voltage)

since f / f_cutoff = 100 @ f = 10 kHz, f_cutoff = 100 Hz.

choose R and C satisfying 1 / (2πRC) = 100 Hz; e.g., R = 1592 Ω, C = 1 μF.

sketch circuit


8

A bang-bang controller operates on the circuit used in lab 12 with R = 1 MΩ and C = 1µF. The set point has been at 0 V for some time, and the system output has stabilized at 0 V. The set point is changed to 5 V.

a) Estimate t1%, the time for the system output to reach within 1 % of the new setpoint.

It’s a first order system. The controller will command 5 V so all we care about is the response of a first order system to a step input. Solve equation 4.21 Dunn with delta = 1% = 0.01, tau = 1 s. Get t = 4.6s.

b) Which of the following would decrease t1%? (select all that apply)

1. reducing the capacitance, C

2. changing the maximum 9263 (ao0) channel output from 5 V to 10 V.

3. using a PID controller with gains chosen based on the Ziegler-Nichols recipe instead of the bang-bang controller.

4. using a PID controller with manually optimized gains instead of the bang-bang controller.

1 and 2


9 Three different thermometers (A, B, and C) are placed in ice-water baths. Four temperatures are recorded for each. Data is shown below:

temperatures [deg C]

A B C

trial 1 1.1 1.1 0

trial 2 -2 1 0.4

trial 3 2 0.9 0.1

trial 4 -1 1 0.3

a) Based only on the data shown, which thermometer has the best precision? Compute and present its precision.

B. From appropriate equation in Table 9.1, standard deviation SX= 0.082 degrees C. Precision = t3,95%SX = 0.26 degrees C (P=95%)

b) Based only on the data shown, which thermometer has the best accuracy? Compute and present its accuracy.

A. Accuracy = Average – true value = 0.025 degrees C.

c) Which thermometer has the best combined uncertainty? (see equation 7.9) Compute and present its combined uncertainty.

C. Just looking at the data, it is pretty obvious that C will have the best combined uncertainty. This can be proved by calculation according to equation 7.9.


10 Three amplitude vs. frequency plots are given below. For each, sketch the corresponding voltage vs. time plot on the axes provided. Make your sketches as quantitative as possible; you may wish to write some words to help us understand what you’re trying to sketch if you think your sketches are unclear.

sketch resembles a simple sine wave, with mean = 0, with ~ 4 cycles / s

sketch has an obvious sinusoidal component near 2 cycles / s, sketch includes noise (weak content at many frequencies), mean ~ 0

(nonzero) DC or very low frequency signal


11 Examine this controller block diagram:

a) Which coefficient is the integral gain?

K_3

b) Which coefficient is the derivative gain?

K_2

c) Should the integral gain be positive or negative? Support your answer with a sketch that includes an error vs time plot. Also explain why the integral gain is useful.

Positive. Consider the case when the measured temperature is stabilizing above the set temperature as shown.


The integral action should reduce the heat in this case to eliminate the steady-state error. The shaded area in the sketch above is the signal that is multiplied by K_3 at the bottom-right of the block diagram in the problem statement. If K_3 is positive, the integral action will be negative, tending to reduce the heat, as required to reduce the steady-state error. System sketches with and without integral gain are shown:

d) Should the derivative gain be positive or negative? Support your answer with a sketch that includes an error vs. time plot. Also explain why the derivative gain is useful.

Positive. Consider the case where the measured temperature is falling toward the set temperature as shown.


The derivative action should add more heat to the system in this case so the measured temperature does not undershoot as badly. The derivative is positive so if K_D is positive the controller action will be positive, or in the direction of increasing heating, as desired.


12

a) Shown below are system responses for P controllers for a set point change from 2 to 4. Which has the higher KP? How do you know (10 words or less is fine)?

#2 has the higher KP. One way to know is that the steady-state error is less (see example problem from lecture Monday 4/19). Another way to know is that it approaches the set point faster and overshoots.

b) Shown below are system responses for a P and a PD controller. Which one includes D control? How do you know (10 words or less is fine)?

#1

#2


Left has D control. Properly-tuned D control reduces overshoot.

c) Shown below are system responses for a P and a PI controller to a set point change from 2 to 3. Which one includes I control? How do you know (10 words or less is fine)?

#3 has I control. Properly-tuned I control reduces steady-state error.

#3

#4


13

A large (10 mm diameter) and a small (1 mm diameter) thermocouple (TC) are transitioned from a room-temperature water bath to a hot water bath. The results are plotted.

a) Identify which (red circles or blue squares) is the large TC.

b) Estimate the time constant of the small TC, τ1mm

c) Estimate τ10mm

d) compute the ratio of the two τ values. Does it agree with theory? Explain.

a. Red circles.

b. 0.4 s.

c. 4 s.

d. ratio = 10. It does agree with theory: Dunn pg. 110 states that τ = mC/hA. Simplifying, mC/hA = k1m/A=k2V/A=k3d where m is mass, C is heat capacity, h is convection coefficient, A is the surface area of the TC, V is the volume of the TC, d is the diameter of the TC, and the k’s are constants. Since τ is proportional to diameter, the ratio of diameters should equal the ratio of time constants, and it does.


14

Consider a small TC with τ = 0.01 s and a large TC with τ = 100 s. Describe the expected response of each of these TCs when immersed in a water bath whose temperature varies sinusoidally between 0 °C and 100 °C at 1 Hz according to Tbath(t) = 50°C + 50°C sin (ωt) with ω = 2πf (f = 1 Hz).

The small TC will track the bath temperature very well and the large TC will read a nearly steady value of 50 C. Justification: according to eq. 4.38 (or Figure 4.3) the magnitude ratios are 0.998 and 0.00159. Additional note: eq 4.35 or Figure 4.4 can be used to determine that the phase shift is only a few degrees in the case of the small TC and is nearly 90 degrees for the large TC.

15.

During a Ziegler-Nichols tuning exercise the following data was obtained.

Compute the Ziegler-Nichols gain coefficients according to:

Kp = 0.6 Ku; Ti = 0.5 Tu; Td = 0.125 Tu; KI = KP/TI ; KD = KP * TD


Ku is the “ultimate gain”

Tu is the period of the oscillations

Choose Ku 25. Compute Kp = 15. From plot, find Tu = 50 s. Compute Ti = 25 s, Td = 6.25 s. Compute Ki = 0.6, Kd=94


16.

Gaussian white noise is input to the circuit shown below and, in turn, monitored by a data acquisition system. Analysis of the measured data reveals a sample standard deviation SX of 0.03 V. Predict SX for the case where C is reduced to 10 nF. Explain your reasoning.

[x/2] fc α 1/C for this low-pass filter fc is increased by a factor of 100

[x/2] SX α sqrt(fc), so SX is increased by a factor of 10 to SX = 0.3 V

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me368.engr.wisc.edume368.engr.wisc.edu/.../final_exam_practice_2012.docx · Web vieworder low pass Butterworth filter with a cutoff frequency of 1 Hz, what will be the standard deviation