How does the area-to-length ratio of semiconductor legs affect electrical resistance heating?

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thermoelectric power generator Components of a thermoelectric generator.

In a thermoelectric power generator, the ohmic heating—or electrical resistance heating—that occurs within the semiconductor legs is determined by the formula $ρ I^{2} (L / a)$ , where $ρ$ is the electrical resistivity, $I$ is the electric current, $L$ is the length of the leg, and $a$ is the cross-sectional area.1

Because the heat power produced by electrical resistance is proportional to the inverse of the area-to-length ratio ( $L / a$ ), the relationship functions as follows:

Increasing the area-to-length ratio ( $a / L$ ): This reduces the electrical resistance of the legs, thereby decreasing the amount of power dissipated as ohmic heating.1
Decreasing the area-to-length ratio ( $a / L$ ): This increases the electrical resistance, leading to higher ohmic heating losses within the legs.1

While increasing the area-to-length ratio helps minimize resistance heating, it simultaneously increases the thermal conduction power. Consequently, an optimal design typically utilizes relatively long and thin legs to balance these competing factors.1