How does ohmic heating reduce the useful electrical output of a thermoelectric generator?

In a thermoelectric power generator, ohmic heating—also known as Joule heating—occurs due to the electrical resistance of the semiconductor legs as current flows through the device.12

When an electrical load is connected to the generator, the Seebeck effect produces electric power. However, a portion of this generated power is dissipated as heat within the internal resistances of the device's legs rather than being delivered to the external load. Specifically, the heat power produced by this resistance is calculated as $\rho I^2(L/a)$, where $\rho$ is the electrical resistivity, $I$ is the current, $L$ is the leg length, and $a$ is the leg area.1

Because this internal dissipation consumes a fraction of the total generated electrical energy, it directly reduces the useful electrical power output available to the load resistance ($R_L$). To mitigate these losses, designers often optimize the geometry of the semiconductor legs, as ohmic heating is dependent on the inverse of the area-to-length ratio.1