Warm for the Holidays $(14$ points$)$

Winter is coming, and both you and your roommate are in desperate need of electric heating eye pads to

avoid overly dry eyes this holiday. Tragically the circuit for your eye pads broke, yet fortunately you've

taken EECS$16$A and have come up with a clever fix by designing a voltage divider and a comparator circuit!

$($a$)(4$ points$)$ First you build a circuit that converts temperature change to voltage change. Your design is

shown in Fig. $17.$ In your design you use two temperature dependent resistors, whose values are given

by R$\_(0)+\backslash$alpha T$,$R$\_(0)-\backslash$alpha T$,$ where R$\_(0)$ is the resistor value at $0$ degrees centigrade, $\backslash$alpha is a thermal coefficient,

and T is the temperature of your eye pads.

What is the temperature dependent output voltage, V$\_($T$),$ of this circuit, as a function of V$\_($s$),$R$\_(0),\backslash$alpha $,$

and T $?$ Is V$\_($T$)$ a linear function of T $?$ Clearly show all your work.

Figure $17$: Temperature Sensing Circuit

$($b$)(4$ points$)$ We want to use a comparator to turn the heat ON and OFF, and you set up the circuit in Fig.

You process the V$\_($T$)$ to make Volts