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Full thermoelectric equations
Often, more than one of the above effects is involved in the operation of a real thermoelectric device. The Seebeck effect, Peltier effect, and Thomson effect can be gathered together in a consistent and rigorous way, described here; the effects of Joule heating and ordinary heat conduction are included as well. As stated above, the Seebeck effect generates an electromotive force, leading to the current equation


 * $$\mathbf J = \sigma (-\boldsymbol \nabla V - S \nabla T).$$

To describe the Peltier and Thomson effects the flow of energy must be considered. To start, the dynamic case where both temperature and charge may be varying with time can be considered. The full thermoelectric equation for the energy accumulation, $$\dot e$$, is


 * $$\dot e = \nabla \cdot (\kappa \nabla T) - \nabla \cdot (V + \Pi) \mathbf J + \dot q_\text{ext},$$

where $$\kappa$$ is the thermal conductivity. The first term is the Fourier's heat conduction law, and the second term shows the energy carried by currents. The third term, $$\dot q_\text{ext}$$, is the heat added from an external source (if applicable).

In the case where the material has reached a steady state, the charge and temperature distributions are stable, so one must have both $$\dot e = 0$$ and $$\nabla \cdot \mathbf J = 0$$. Using these facts and the second Thomson relation (see below), the heat equation then can be simplified to


 * $$-\dot q_\text{ext} = \nabla \cdot (\kappa \nabla T) + \mathbf J \cdot \left(\sigma^{-1} \mathbf J\right) - T \mathbf J \cdot \nabla S.$$

The middle term is the Joule heating, and the last term includes both Peltier ($$\nabla S$$ at junction) and Thomson ($$\nabla S$$ in thermal gradient) effects. Combined with the Seebeck equation for $$\mathbf J$$, this can be used to solve for the steady-state voltage and temperature profiles in a complicated system.

If the material is not in a steady state, a complete description will also need to include dynamic effects such as relating to electrical capacitance, inductance, and heat capacity.