cel, etc, not by hand!$)$ with proper formatting $($axis labels, etc$)$ but be sure to clearly show all analysis by hand.

$1.$ You are using a sensing system that consists of a pressure $($gauge$)$ sensor$,$ a low pass filter, and an DAQ board $($which will be ignored for most of this analysis$).$ The sensor has a $2$ nd$-$order response and the filter has a $1$st$-$order response. DAQ boards can be modeled as zeroth order instruments and perfect sensitivity$.$

The sensor data sheet describes its static sensitivity as $\backslash (1\backslash$mathrm$\{$mV$\}/\backslash$mathrm$\{$Pa$\}\backslash ),$ and that its peak output is $8$ mV at $10$ kHz for a nominal $2$ Pa peak gauge pressure at each frequency $($see plot below$).$

You construct the filter out of a resistor$-$capacitor design, choosing $\backslash ($ R$=100\backslash$Omega $\backslash )$ and $\backslash ($ C$=0\mathrm{.}4\backslash$mu $\backslash$mathrm$\{$~F$\}\backslash ).$ The cutoff frequency for such a filter is $\backslash (\backslash$omega$\_\{$c$\}=\backslash$frac$\{1\}\{\backslash$tau$\}\backslash ),$ and $\backslash (\backslash$tau$=$R $\backslash$cdot C $\backslash ).$ The filter describes its sensitivity as $1$ V$/$V$.$ Neither instrument has an offset.

a$.)$ Draw a block diagram of this measurement system.

b$.)$ Determine the damping ratio and natural frequency of the sensor

c$.)$ Determine the equations and Bode plots $($magnitude and phase, as functions of frequency $\backslash ($ f $\backslash ))$ for the steady$-$state response as a function of frequency, from $\backslash (0-45\backslash$mathrm$\{$kHz$\}\backslash )$ for:

$-$ Stage I $($separately$)$

$-$ Stage II $($separately$)$

$-$ Stage I and II combined $($the entire system$)$

$-$ On the appropriate plot, confirm that the equation for the cutoff frequency given above is correct; show this result and write a short sentence that supports your analysis.

$-$ Determine the value for the relevant Quality factor $\backslash ($ Q $\backslash ).$

$-$ Discuss and identify the range of frequencies $("$bandwidth$")$ for which this system would remain in the "Transmission band" for the magnitude ratio and still remain within a half cycle of the input signal; identify this frequency range.

d$.)$ Determine the equation for the output voltage of this system $($as a function of time$)$ for a pressure $\backslash ($ p$($t$)=2\backslash$sin $\backslash$left$(2\backslash$pi $*9\backslash$times $10^\{3\}$ t$\backslash$right$)\backslash$mathrm$\{$kPa$\}\backslash ).$ It might be helpful to plot this as a function of time, as you think through this $\backslash$& the next step.

e$.)$ Determine the peak voltage that would be input to the DAQ board. Determine which DAQ board input range would be most appropriate: $\backslash (\backslash$pm $0\mathrm{.}5$ ; $\backslash$pm $1,\backslash$pm $2\mathrm{.}5,\backslash$pm $5\backslash ),$ or $\backslash (\backslash$pm $10\backslash$mathrm$\{$~V$\}\backslash )$