3. (20 pts). The Calrod heating element transfers heat largely by a radiation mechanism. If the rate of electrical energy input to the heater is Q and the rot temperature and ambient temperatures are T and T_a, respectively, then an appropriate unsteady-state model for the system is:

Find the transfer functions relating T' to Q' and T' to T_a'. (Be sure they are both in standard form.)

The first thing we realize here is that the equation is non-linear. We will have to linearize the equation to get the transfer functions into standard form.

No linearize the equation:

Invert into the Laplace domain:

Rearrange:

Now the transfer functions can be written down directly: