# Deriving the transfer functions of a heating system

I'm tying to develop a model for a heated system that consists of a small steel block (~25 sq. in.) with a heating element embedded in it. The block acts as a sort of hot-plate that is used to deform other materials. There is a temperature sensor embedded in the block as well for feedback. I'm only trying to model the temperature of the block, not the temperature of the deformed materials.

I've used Newton's Law of Cooling to determine the time constant for the steel block cooling with open air convection:

$${dT(t) \over dt} = -k \left(T(t) - T_{\rm ambient}\right)$$

where $k$ is the time constant (and is ${\approx 0.00228} sec^{-1}$). Taking the Laplace Transform of this gives me:

$$H(s) = {T(0) \cdot s - k \cdot T_{\rm ambient} \over s - k}$$

I've also been able to determine the heating capacity of the heating element coupled with the steel block:

$${dT(t) \over dt} = -k \left(T(t) - T_{\rm ambient}\right) + Q$$

where $Q$ is the heating (and is ${\approx 0.412}$ $°C/sec^{-1}$). Taking the Laplace Transform of this gives me:

$$H(s) = {T(0) \cdot s - k \cdot T_{\rm ambient} + Q \over s - k}$$

I've simulated the responses of both of these transfer functions under step inputs, and they match closely with experimental results. But when I attempt to add closed loop control, I get simulation results that do not make sense to me. The transfer function I've been calculated for simple negative feedback is:

$$H(s) = {{T(0) \cdot s - k \cdot T_{\rm ambient} + Q} \over {(T(0)+1) \cdot s - k \cdot T_{\rm ambient} + Q - k}}$$

Simulating this transfer function with a step input produces a system that goes to the target temperature almost immediately. I'm expecting to see an exponential response starting from the $T(0)$ point and increasing to the target temperature, similar to what I see when simulating the open loop responses.

I've gone back to my old textbooks and searched the internet for additional resources. I've solved a number of mass-spring and RC circuit problems, but I can't seem to make the connection between those problems and this one. So I'm not sure where I'm going wrong.

• Shouldn't the transfer function relate the input and output, so $Q$ and $T(t)$? – fibonatic Aug 21 '16 at 23:35
• The transfer-function should not include the initial conditions. – Arnfinn Sep 14 '16 at 11:46
• you can start with a change of variable $\delta T(t) = T(t) - T_{amb}$ then take the derivative and obtain $\dot{\delta T}(t)$ then you get an exponential behavior. – percusse Dec 21 '16 at 2:46