$2.$ You are tasked with measuring the temperature in a system that undergoes periodic heating due to convective heat loss. Your thermocouple has a time constant $\backslash (\backslash$tau$=8\backslash$mathrm$\{$~s$\}\backslash )$ and its specification sheet tells you that it's static sensitivity is $\backslash (1\backslash$mathrm$\{$mV$\}/\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ and no offset. You can assume it has a $1$st$-$order response. You've monitored your sensor long enough so you can safely assume that its transient response has died out.

The average temperature $\backslash (=25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ the max temperature $\backslash (=45^\{\backslash$circ$\}\backslash$mathrm$\{$C$\},\backslash$& $\backslash )$ the period of oscillation $\backslash (=30\backslash$mathrm$\{$~s$\}\backslash ).$

a$.)$ Determine the expressions that describes the input $\backslash ($ T$($t$)\backslash )$ and output $\backslash ($ V$($t$)\backslash ).$

b$.)$ Using first$-$order analysis, calculate the phase for this system and input.

c$.)$ Determine and plot $($on the same plot, using either the basic 'plot' function or the dual$-$axis 'plotyy' function if you're in Matlab$)$ the input signal and predicted output signal from the sensor for approximately $5$ cycles. Report the equation form of the output $\backslash ($ y $\backslash )$ as a function of time; the only symbol in this equation should be that for time.

d$.)$ Calculate the dynamic error of this measurement. Should this error change with time or frequency? Explain your answer.

e$.)$ Based on the graph that you've produced, use the Matlab cursor tool to determine the time difference $\backslash (\backslash$Delta t $\backslash )$ of the relative phase difference between the two waves. From this $\backslash (\backslash$Delta t $\backslash ),$ calculate what the corresponding $\backslash (\backslash$Delta $\backslash$Phi $\backslash )$ should be and compare it to the $\backslash (\backslash$Phi $\backslash )$ result you got in the first step. If a difference exists, explain why.