# Transfer Function, Newtonian Cooling

I have a system that is modeled by the following differential equation:

db/dt = j(t)ha + k(Ta-b(t))

where db/dt is the rate of temperature change, j(t) is an input, ha, k, Ta are all constants, and b(t) is an output. Note that this is newtonian cooling with a heating input, j(t)*ha.

I want to find the transfer function is the laplace domain, B(s)/J(s). Taking the laplacian of the equation of interest, assuming all IC's are 0, yields: s*B(s)+k*B(s)-(k*Ta)/s = ha*J(s)

What I can't figure out is the term k*Ta/s is not a function of J(s) or B(s) so I can not get a transfer function of purely B(s)/J(s). Does anyone know how to solve this equation or a better way to find the transfer function relating the input to the output?

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What about seeing B as a function of J and s: B(J,s). Then, you can look at the partial derivative of B w.r.t. J. –  Raskolnikov Nov 29 '10 at 19:18

When you take the Laplace transform and solve for B(s), you should consider both nonhomogeneous terms in the input:

$$B(s) = \frac{1}{s+k} \left (h_a J(s) + \frac{kT_a}{s} \right ).$$

The transfer function is just first factor, $H(s) = \displaystyle \frac{1}{s+k}$, and the solution $b(t)$ is given by the convolution of $h(t) = e^{-kt}$ with the forcing term $h_a j(t) + k T_a$.

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Thanks cch. Another approach a coworker recommended is to create a new time dependent term that includes all the constants:

c(t) = (ha/k)*j(t)+Ta

Note that ha, k, and Ta are all constant so c(t) and j(t) have no dynamics between themselves.

Then the differential equation simplifies to:

db/dt = k*(c(t)-b(t))

And the laplacian becomes:

sB(s) + kB(s) = kC(s)

And the transfer function becomes:

B(s)/C(s) = k/(s+k)

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