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Forgive me for I am a math student and am quite ignorant in these topics. I would like to know, at least implicitly, how the current intensity of a lightbulb, for example, relates to the potential difference applied.

Say I have the following model for resistance as a function of temperature: $$R = R_0(1+\alpha (T-T_0))$$

where $R_0, T_0, \alpha$ are constants, and $T$ represents Temperature. Then, I know that at each voltage-current pair $(V,I)$ the dissipated energy is given by $$P = VI$$

If I assume that the dissipated energy $P$ is related to temperature change by $P=f(T-T_0)$ (is this true?) Then, if I can invert $f$, say $f^{-1} = g$, then I'll have $$R = R_0(1+\alpha g(P))\implies g(VI) = \frac 1\alpha(R-R_0)$$

Finally, by Ohm's law: $$g(VI) = \frac 1\alpha\left(\frac VI - R_0\right)$$

which is an implicit relation between $V$ and $I$ as I desired. The problem is I have no clue how dissipated energy in a circuit relates to temperature change, if it does. Can someone critique this reasoning of mine, and point me in the right direction?

I'm actually helping a high-school student out with a lab and I have the experimental data which looks an awful lot like a logistic curve (plotting $I$ against $V$ for a range of $V$). In the above I hoped to get an ODE that gave me this, but as you can see I haven't gotten very close, apparently. In the end I just want to know what sort of fit would be meaningful to the data.

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    $\begingroup$ @DanielSank Not irrelevant. SE questions are best answered with as much context given as possible, within reason. I think that one sentence is within reason. $\endgroup$ – GPerez Jun 14 '15 at 17:00
  • $\begingroup$ Of course context helps, but the style of the site favors objective questions with objective answers. As such I generally think that the question itself dictates the appropriate level of the answers. Therefore, explicit statements about the question's author's background tend to be more distracting than helpful. $\endgroup$ – DanielSank Jun 14 '15 at 17:02
  • $\begingroup$ When you say $T$ is the "total change in temperature" do you just mean that it is the temperature (in a sane unit system like Kelvin, i.e. no offset from zero)? Also, the equation $P=f(T)$ seems wrong. The energy dissipated by an electrical circuit ($P=IV$) is usually equal to the generated heat power but that's not necessarily the same as the change in the circuit's own temperature. $\endgroup$ – DanielSank Jun 14 '15 at 17:04
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    $\begingroup$ The dissipated power is simply $P=V^2/R$ but there is no way that you can estimate the temperature as a function of the dissipated power - that's system dependent. It's different for an ideal black body than it is for a grey body (not it becomes a non-trivial function of the body's spectral properties) and it's different, again, for an air cooled resistor. Unless you can tell us what your model assumptions are (I suppose it's black body radiation), we can't help you. $\endgroup$ – CuriousOne Jun 14 '15 at 17:39
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    $\begingroup$ @CuriousOne The lab instructions don't really ask for any of this. They just have to plot the data and say "yup, not linear". Certainly they have no knowledge of Stefan-Boltzmann! Still I'd like to find the models that describe the situation and give a decent explanation for the shape of the curve. The lab report is complete as is, at this point this is for understanding's sake (which sadly isn't provided by many high-school assignments). Also I have become interested in the matter myself and would like some closure! (...) $\endgroup$ – GPerez Jun 14 '15 at 18:09
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The temperature of the circuit can probably be crudely modeled by assuming that you have two channels: one for heat absorption, and one for heat loss.

Heat loss

Suppose you have something like a light bulb. The hot filament loses heat via radiation and conduction. Let's focus on radiation. The rate of heat radiated by an object of temperature $T$ depends strongly on the materials in the object. In general, the Stefan-Boltzmann law says that the total power radiated by an object of surface area $A$ and emissivity $\epsilon$, and temperature $T$ is $$P_{\text{radiated}} = A \epsilon \sigma T^4 $$ where $\sigma$ is a the Stefan-Boltzmann constant. The emissivity $\epsilon$ is highly material dependent. Let us just suppose it has some unknown value for the light bulb filament and combine all the constants in the Stefan-Boltzmann equation in to a constant $C$, giving us

$$P_{\text{radiated}} = C T^4 \, . $$

Heat absorption

Resistance in the light bulb filament happens roughly because electrons collide with the atoms in the metal, causing them to vibrate. This process, called Joule heating, causes the metal to absorb power given roughly by $$P_{\text{absorbed}} = I^2R \, . $$

Heating rates

Now we have two rates, $P_{\text{radiated}}$ and $P_{\text{absorbed}}$. Assuming that the temperature of the object is proportional to its internal energy $$T \propto E$$ and recalling that power is defined as the rate of energy change, we get \begin{align} \frac{dT}{dt} \propto \frac{dE}{dt} &= P_{\text{absorbed}} - P_{\text{radiated}} \\ &= I^2 R - C T^4 \\ &= I^2 R_0(1 + \alpha (T - T_0)) - C T^4 \, . \end{align} This says that when you turn on the light bulb (in this case with a constant current source), the temperature is time dependent. This means that the resistance and therefore the voltage are also time dependent. However, note the important fact that the right hand side goes to zero for a certain value of $T$. This means that the bulb will heat up as time passes, but will eventually reach an equilibrium temperature $T^*$. We can therefore assume that there exists a function $g$ such that $T^* = g(I)$.

At this point note that if we have a constant voltage source instead of a constant current source we would have had $$ \frac{dT}{dt} \propto \frac{V}{R_0(1 + \alpha (T - T_0))} - CT^4 $$ but the point about having an equlibrium temperature and a function $T^* = g(V)$ would still hold.

Comments on the equations in OP

The point is that your equation $$R = R_0(1 + \alpha g(P))$$ should probably be more like $$R = R_0(1 + \alpha g(I)) \qquad \text{or} \qquad R = R_0(1 + \alpha g(V)) $$ and this only holds when the system is in equilibrium. For a light bulb I'd guess that equilibrium is reached within a few seconds, but I don't really know for sure.

Anyway, in the equilibrium case $V$ is some implicit function of $I$ (or vice versa for a constant voltage source) so you can still write $$\text{unknown function}(IV) = \frac{1}{\alpha} \left( V/I - R_0 \right)$$ as you already did.

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  • $\begingroup$ Thank you, the equation $\frac{dT}{dt} \propto \frac{dE}{dt} =I^2 R - C T^4$ is very helpful. I think I will linearize near $T^*$ to find (an approximation of) $g(I)$ near this equilibrium, which I also assume is attained quite rapidly. What I don't understand is how you can assume we end up with a function of the product $IV$. Certainly $f(y(x))$ isn't always $h(x\cdot y(x))$? I would have stopped at $g(I(V)) = \frac 1\alpha(V/I - R_0)$. Still thanks a lot, this gives me something to go on! $\endgroup$ – GPerez Jun 14 '15 at 18:48
  • $\begingroup$ @GPerez assuming $T^*=g(I)$ then $R$ is also a function of $I$. Since $V = IR$ then V is also a function of I. Therefore, $T^*$ is implicitly some function of $IV$. Note that I wrote $\text{unknown function}$ instead of $g$. $\endgroup$ – DanielSank Jun 14 '15 at 18:51
  • $\begingroup$ I did accept, just after posting my comment though! And I think I understand now about $IV$, but I think $g(I)$ will also provide what I need. $\endgroup$ – GPerez Jun 14 '15 at 18:55

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