Acoustic Sensors. Using the Excel data file that accompanies this assignment, co

Question

Acoustic Sensors. Using the Excel data file that accompanies this assignment, consider a pair of quartz thickness shear mode (TSM) sensors, where one sensor is used to detect proteins in a test sample, and the other is used as a reference. After establishing a baseline reading, the sensor is exposed sequentially to test samples containing 100, 200, and 300 ng/ml of analyte. a. Using the Sauerbrey equation (slide #14), what is the mass density (mass/area) detection resolution for this sensor? (Hint: Estimate f0 and find out frequency resolution f)

b. A thermometer was used to measure a temperature change of 1 degree Celcius. What is the thermal coefficient of resonant frequency for the sensor?

c. After using the reference sensor to correct for thermal drift, what is the uncompensated drift rate in the active sensor?

d. What is the limit of determination (LOD) for detected analyte concentration?

e. What is the mass density of analyte deposited on the sensor for the 100 ng/ml solution?

Time (sec) Active f0 (Hz) Time (sec) Reference f0 (Hz) Baseline 0.1 9999989.2 0.1 9800020.5 0.5 10000026.6 0.5 9800043.7 0.9 10000026.6 0.9 9800041.7 1.3 10000020.6 1.3 9800008.5 1.7 10000020 1.7 9800012.1 2.1 10000028.2 2.1 9800037.3 2.5 10000066.8 2.5 9800074.5 2.9 10000053.4 2.9 9800040.9 3.3 10000074.6 3.3 9800078.5 3.7 10000093.2 3.7 9800048.5 4.1 10000084.8 4.1 9800043.3 4.5 10000086.8 4.5 9800060.1 4.9 10000112.4 4.9 9800083.3 5.3 10000086.2 5.3 9800101.7 5.7 10000100.8 5.7 9800132.1 6.1 10000133.6 6.1 9800102.9 6.5 10000129 6.5 9800126.5 6.9 10000131.4 6.9 9800147.3 7.3 10000147.8 7.3 9800142.1 7.7 10000160.8 7.7 9800157.7 8.1 10000167.6 8.1 9800153.7 8.5 10000152.6 8.5 9800121.7 8.9 10000172.2 8.9 9800116.1 9.3 10000171.6 9.3 9800185.7 9.7 10000189.2 9.7 9800202.9 10.1 10000199.6 10.1 9800149.3 10.5 10000224.2 10.5 9800154.9 10.9 10000235.6 10.9 9800213.7 11.3 10000243.8 11.3 9800199.3 11.7 10000227.8 11.7 9800214.5 12.1 10000254.2 12.1 9800239.3 12.5 10000264 12.5 9800176.5 12.9 10000246.8 12.9 9800236.9 13.3 10000260.4 13.3 9800240.9 13.7 10000259.4 13.7 9800239.3 14.1 10000300.8 14.1 9800272.5 14.5 10000310 14.5 9800262.5 14.9 10000286.4 14.9 9800228.1 15.3 10000307.6 15.3 9800298.1 15.7 10000299.8 15.7 9800296.5 16.1 10000317.2 16.1 9800284.1 16.5 10000333.8 16.5 9800244.5 16.9 10000334 16.9 9800262.5 17.3 10000330.4 17.3 9800300.1 17.7 10000340.6 17.7 9800284.9 18.1 10000375.8 18.1 9800319.3 18.5 10000355.8 18.5 9800334.1 18.9 10000384 18.9 9800347.7 19.3 10000376.2 19.3 9800366.1 19.7 10000378.6 19.7 9800307.3 20.1 10000398.2 20.1 9800346.5 20.5 10000418.8 20.5 9800370.9 20.9 10000405.2 20.9 9800375.3 21.3 10000411.6 21.3 9800328.9 21.7 10000433 21.7 9800346.5 22.1 10000451.2 22.1 9800391.3 22.5 10000448 22.5 9800347.7 22.9 10000471.8 22.9 9800415.3 23.3 10000485.4 23.3 9800430.9 23.7 10000475.6 23.7 9800394.9 24.1 10000474 24.1 9800407.7 24.5 10000504 24.5 9800394.5 24.9 10000504.8 24.9 9800440.1 25.3 10000511.2 25.3 9800416.1 25.7 10000503.8 25.7 9800429.3 26.1 10000513.8 26.1 9800460.5 26.5 10000521.6 26.5 9800473.7 26.9 10000547.4 26.9 9800417.3 27.3 10000543.6 27.3 9800488.1 27.7 10000569 27.7 9800504.1 28.1 10000562.2 28.1 9800510.1 28.5 10000574.4 28.5 9800466.5 28.9 10000571.8 28.9 9800458.1 29.3 10000580.2 29.3 9800478.5 29.7 10000602.2 29.7 9800520.5 30.1 10000617.8 30.1 9800507.7 30.5 10000590 30.5 9800512.9 30.9 10000600 30.9 9800528.9 31.3 10000645.4 31.3 9800531.3 31.7 10000647.2 31.7 9800508.9 32.1 10000658.2 32.1 9800568.5 32.5 10000650.6 32.5 9800565.7 32.9 10000660.6 32.9 9800519.7 33.3 10000681.4 33.3 9800560.5 33.7 10000689.2 33.7 9800590.9 34.1 10000684 34.1 9800548.1 34.5 10000696 34.5 9800583.3 34.9 10000697.4 34.9 9800579.3 35.3 10000703 35.3 9800624.9 35.7 10000704.8 35.7 9800646.5 36.1 10000721.8 36.1 9800594.9 36.5 10000723.2 36.5 9800622.1 36.9 10000755 36.9 9800602.5 37.3 10000747.2 37.3 9800620.5 37.7 10000734.2 37.7 9800636.1 100 ng/ml 38.1 10001766 38.1 9800672.5 38.5 10001766.6 38.5 9800688.1 38.9 10001780.2 38.9 9800627.3 39.3 10001776.8 39.3 9800672.1 39.7 10001778.8 39.7 9800645.7 40.1 10001818.6 40.1 9800698.1 40.5 10001811 40.5 9800664.9 40.9 10001825 40.9 9800706.9 41.3 10001841.6 41.3 9800696.9 41.7 10001821.8 41.7 9800744.1 42.1 10001847.6 42.1 9800742.9 42.5 10001861.6 42.5 9800704.9 42.9 10001875 42.9 9800745.3 43.3 10001881.4 43.3 9800712.9 43.7 10001877.2 43.7 9800745.3 44.1 10001887.6 44.1 9800715.3 44.5 10001894.2 44.5 9800726.5 44.9 10001900.4 44.9 9800754.1 45.3 10001911.2 45.3 9800768.9 45.7 10001918.4 45.7 9800749.7 46.1 10001928.6 46.1 9800806.1 46.5 10001947.4 46.5 9800753.3 46.9 10001956 46.9 9800818.9 47.3 10001962.8 47.3 9800790.9 47.7 10001958.8 47.7 9800774.9 48.1 10001953.4 48.1 9800790.9 48.5 10001958.2 48.5 9800859.7 48.9 10001990.2 48.9 9800804.1 49.3 10001991.6 49.3 9800805.7 49.7 10001993.8 49.7 9800832.1 50.1 10001984.6 50.1 9800888.5 50.5 10002005.4 50.5 9800822.5 50.9 10002037 50.9 9800898.5 51.3 10002045 51.3 9800854.9 51.7 10002047.4 51.7 9800896.9 52.1 10002036.2 52.1 9800905.7 52.5 10002063.8 52.5 9800866.1 52.9 10002070 52.9 9800879.7 53.3 10002061.2 53.3 9800908.9 53.7 10002089.4 53.7 9800882.1 54.1 10002073 54.1 9800896.1 54.5 10002104.8 54.5 9800920.5 54.9 10002117 54.9 9800948.5 55.3 10002125.6 55.3 9800973.3 55.7 10002103.4 55.7 9800971.3 56.1 10002126 56.1 9800984.5 56.5 10002113 56.5 9800976.5 56.9 10002139.4 56.9 9800952.5 57.3 10002141.8 57.3 9800990.5 200 ng/ml 57.7 10003173.8 57.7 9800949.3 58.1 10003176.2 58.1 9800958.9 58.5 10003171.6 58.5 9800983.7 58.9 10003171.4 58.9 9800980.9 59.3 10003180.2 59.3 9800974.9 59.7 10003177.8 59.7 9801024.5 60.1 10003208 60.1 9800998.1 60.5 10003202.4 60.5 9801061.3 60.9 10003214.4 60.9 9801055.7 61.3 10003241 61.3 9801065.7 61.7 10003227 61.7 9801079.7 62.1 10003226.2 62.1 9801048.9 62.5 10003240 62.5 9801098.5 62.9 10003248.2 62.9 9801043.3 63.3 10003260.4 63.3 9801059.7 63.7 10003281 63.7 9801116.5 64.1 10003268.6 64.1 9801077.7 64.5 10003302.4 64.5 9801088.9 64.9 10003305.8 64.9 9801116.9 65.3 10003317.2 65.3 9801103.3 65.7 10003303.4 65.7 9801122.5 66.1 10003316.6 66.1 9801134.1 66.5 10003318 66.5 9801127.7 66.9 10003343.2 66.9 9801102.5 67.3 10003363.6 67.3 9801116.1 67.7 10003366.4 67.7 9801148.5 68.1 10003363.2 68.1 9801176.5 68.5 10003380.6 68.5 9801163.7 68.9 10003390.8 68.9 9801154.9 69.3 10003369 69.3 9801199.3 69.7 10003413.4 69.7 9801219.7 70.1 10003399.4 70.1 9801156.1 70.5 10003398 70.5 9801225.3 70.9 10003404 70.9 9801200.1 71.3 10003436.4 71.3 9801204.5 71.7 10003432.2 71.7 9801236.5 72.1 10003432.2 72.1 9801198.1 72.5 10003469 72.5 9801202.5 72.9 10003462.4 72.9 9801210.5 73.3 10003453 73.3 9801285.3 73.7 10003476.2 73.7 9801225.3 74.1 10003475.2 74.1 9801238.5 74.5 10003488.6 74.5 9801246.1 74.9 10003491.6 74.9 9801264.1 75.3 10003500.2 75.3 9801304.1 75.7 10003505 75.7 9801322.1 76.1 10003504.6 76.1 9801286.1 76.5 10003513.8 76.5 9801322.9 76.9 10003546 76.9 9801276.9 300 ng/ml 77.3 10004543.4 77.3 9801306.9 77.7 10004541 77.7 9801340.1 78.1 10004561.6 78.1 9801289.3 78.5 10004581.6 78.5 9801329.3 78.9 10004584.8 78.9 9801355.3 79.3 10004567 79.3 9801347.3 79.7 10004603.8 79.7 9801338.9 80.1 10004616.4 80.1 9801397.3 80.5 10004606 80.5 9801338.5 80.9 10004624.6 80.9 9801411.3 81.3 10004636 81.3 9801344.5 81.7 10004641.2 81.7 9801394.5 82.1 10004625.2 82.1 9801377.7 82.5 10004637.6 82.5 9801426.9 82.9 10004677.6 82.9 9801444.5 83.3 10004679.8 83.3 9801404.1 83.7 10004661.8 83.7 9801423.3 84.1 10004700.8 84.1 9801468.5 84.5 10004708.2 84.5 9801418.5 84.9 10004679.8 84.9 9801479.3 85.3 10004688.4 85.3 9801422.5 85.7 10004733.8 85.7 9801429.3 86.1 10004730.6 86.1 9801442.1 86.5 10004726.2 86.5 9801486.9 86.9 10004756.4 86.9 9801442.9 87.3 10004747.8 87.3 9801495.3 87.7 10004758.8 87.7 9801482.9 88.1 10004762.4 88.1 9801499.3 88.5 10004783.8 88.5 9801531.7 88.9 10004781 88.9 9801523.7 89.3 10004801.6 89.3 9801514.9 89.7 10004790.2 89.7 9801554.1 90.1 10004817.8 90.1 9801498.9 90.5 10004823.6 90.5 9801509.3 90.9 10004831 90.9 9801518.1 91.3 10004810 91.3 9801586.9 91.7 10004828.2 91.7 9801562.9 92.1 10004837.6 92.1 9801554.1 92.5 10004834.4 92.5 9801532.9 92.9 10004859.6 92.9 9801608.1 93.3 10004862.6 93.3 9801558.9 93.7 10004880.4 93.7 9801594.9 94.1 10004894 94.1 9801631.3 94.5 10004891.2 94.5 9801626.5 94.9 10004913.6 94.9 9801599.7 95.3 10004888.6 95.3 9801629.3 95.7 10004926.6 95.7 9801653.3 96.1 10004924 96.1 9801604.9 96.5 10004943.6 96.5 9801621.7

Explanation / Answer

A linear coefficient of thermal expansion, in which the length of some object is a function of temperature

L(T)=L0[1+(TT0)].L(T)=L0[1+(TT0)].

Similarly in basic circuits we define the temperature coefficients of resistance and resistivity

R(T)(T)=R0[1+(TT0)]=0[1+(TT0)],R(T)=R0[1+(TT0)](T)=0[1+(TT0)],

You'll see in the examples above that is a very common symbol for such parameters, though and also appear when there might be more than one such coefficient in a problem.

In principle there can be higher degree coefficients needed. You might see something like

X(T)=X0[1+(TT0)+(TT0)2],X(T)=X0[1+(TT0)+(TT0)2],

when more precision is needed. Here would be a "quadratic temperature coefficient of [quantity represented by XX]". You could add a cubic coefficient with a term like +(TT0)3+(TT0)3 inside the brackets.

It is no accident that the form of these expressions is reminiscent of a Taylor series; they come about exactly from approximating the desired quantity by a polynomial near some reference value.

Application to Your Case

The resonant frequency of some system depends on various physical parameters (possibly including both length and resistivity). So if you have measured the resonant frequency at, say, 20C20C, but need to use the device at 25C25C, you may need to assume a different resonant frequency. Having the temperature coefficient of that quantity will allow you to conveniently calculate the adjusted value.

Now, to compare the above to the expression you give in the question we need to transform the above. Starting a 'generic' temperature coefficient expression:

X(T)X(T)X(T)X(T)X0XXT1X0=.=X0[1+(TT0)]=X0[1+T]=X0+X0T=X0T=X0TX(T)=X0[1+(TT0)]X(T)=X0[1+T]X(T)=X0+X0TX(T)X0=X0TX=X0TXT1X0=.

So now we are able to interpret the symbols above: f0f0 is the resonance frequency at the reference temperature (equivalent to X(T)X(T)), f0f0 is the change in the resonance frequency (XX) and TT is the temperature difference between your working environment and the reference temperature. Finally that leaves TfTf as the temperature coefficient (a nasty bit of notation when you are using TTfor temperature in my opinion).

It is also worth unpacking a difference in the meaning of the 00 subscript in my examples and in the expression you found. In my examples it means "evaluated at the reference temperature", but it the expression you found it means "resonance"; the expression you found doesn't need an explicit notation for "at reference temperature" because all the uses of that symbol are buried in the

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