= Thermoelectric cooling current limit
{title2=$I_*=\alpha T_c/R$}
At fixed face <temperatures>, <differentiation> of the <thermoelectric cooler energy balance> gives
$$
\frac{dQ_c}{dI}=\alpha T_c-RI,\qquad
\frac{d^2Q_c}{dI^2}=-R<0,\qquad
I_* =\frac{\alpha T_c}{R},\qquad
Q_{c,\max}=\frac{\alpha^2T_c^2}{2R}-K\Delta T.
$$
Thus current chosen for maximum cooling <power> is not automatically current for maximum <coefficient of performance>. Cooling exists only when $(\alpha T_c)^2>2RK\Delta T$, and then only between the two positive roots
$$
I_-<I<I_+,\qquad
I_\pm=\frac{\alpha T_c\pm\sqrt{(\alpha T_c)^2-2RK\Delta T}}{R}.
$$
At equality the parabola merely touches $Q_c=0$. For the toy module the interval is approximately $0.767<I<10.433\,\mathrm A$, with $I_*=5.60\,\mathrm A$ and $Q_{c,\max}=23.36\,\mathrm W$. At $12\,\mathrm A$, the object is heated even in the nominal cooling polarity: $Q_c=-17.6\,\mathrm W$. This mathematical interval supplies no permission to exceed an actual device's current or <temperature> ratings.
\Image[/peltier-cooling-current-balance.png]
{title=Chosen <thermoelectric cooling> model: useful cooling and current-reversal separation}
{description=Original calculated curves, not experimental data. Left: positive <Peltier effect> transport competes with <Joule heating> and the fixed leak, leaving a bounded cooling interval. Right: holding the same two face <temperatures>, the half-difference of $Q_c(+I)$ and $Q_c(-I)$ isolates $\alpha T_c I$; the half-sum is $-I^2R/2-K\Delta T$. Subtracting $Q_c(0)$ isolates $-I^2R/2$. The toy parameters are those in <thermoelectric cooler energy balance>.}
The fixed-face constraint matters. With fixed $T_h$ but a freely cooling object, $T_c$ must be solved from its <heat> load and leaks rather than held at $280\,\mathrm K$ while claiming an arbitrary <temperature> drop. For zero load and optimal current, the same model gives
$$
\frac{\alpha^2T_c^2}{2R}=K(T_h-T_c),\qquad
T_{c,\min}=\frac{\sqrt{1+2ZT_h}-1}{Z},\qquad
Z=\frac{\alpha^2}{RK}.
$$
Here $Z$ has units $\mathrm{K^{-1}}$ and is a module parameter. The toy model gives $Z=0.0020\,\mathrm{K^{-1}}$ and $T_{c,\min}\approx241.62\,\mathrm K$ when $T_h=300\,\mathrm K$. This is an ideal zero-load limit with stipulated constant properties; cooling a real load requires $Q_c$ above zero and reduces the attainable drop. Contact <thermal resistance>, a warming <heat sink>, external leaks and temperature-dependent coefficients further change the result. Electrical control is convenient, but <conservation of energy> still requires a thermal path that removes the extracted <heat> plus the electrical input.
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