Exact branch-transfer criterion for common-mode Thomson heat cancellation in thermoelectric couples

Abstract

Thermoelectric p- and n-type legs are commonly paired by matching their Seebeck magnitudes, although a cooler responds to heat transported through its complete electrical and thermal network. We decompose the leg coefficients into differential thermopower $\alpha =S_p-S_n$ and common thermopower $M=(S_p+S_n)/2$. In a connected steady-state scalar thermoelectric network, a temperature-independent co-shift applied to every electrically active segment is an exact terminal null. A temperature-dependent perturbation of the legs relative to fixed leads is instead physical. At fixed current and shared isothermal endpoints, its first-order cold-port response is the action of ${\Gamma }_m=Tdm/dT$ on the difference between the p- and n-branch oriented collection measures. We prove that every continuous ${\Gamma }_m$ cancels if and only if these measures are equal. In the constant-property, linear-common-mode limit, matching $R_i/K_i^{\mathrm{leg}}$ is sufficient and does not require identical legs. One- and two-dimensional calculations confirm the analytic reductions within their stated domains. For split thermal pads, the analysis gives the exact array law $\Delta Q_{c,\Sigma }={\sum }_j{C}_j{I}_j\Delta T_{c,j}$ and, for series elements with isothermal hot pairs, $I\Delta V_{\Sigma }=-\Delta Q_{c,\Sigma }$. A representative seven-pair model gives corresponding increments of 7.87 mW and $-2.80$ mV. Branch transfer and endpoint topology therefore provide distinct material-pairing and device-test criteria for common-mode Thomson heat.

38 pages, 5 figures