= Thermoelectric couple figure of merit
{title2=$Z_{\rm pair}$}
For a constant-property <thermoelectric module> consisting of two material legs, let $\alpha_{np}=S_p-S_n$ be their <Seebeck coefficient> difference, $R$ their total series <electrical resistance>, and $K$ their total parallel <thermal conductance>. The couple's <thermoelectric figure of merit> is
$$
Z_{\rm pair}=\frac{\alpha_{np}^2}{RK}.
$$
Its units are $\mathrm{K^{-1}}$; multiply by an absolute <temperature> to form a dimensionless figure. This expression assumes that contact losses are absent or consistently included in $R$ and $K$.
For equal leg <lengths> $l$ and <areas> $A$, with equal <electrical resistivity> $\rho$ and <thermal conductivity> $\kappa$, one has $R=2\rho l/A$ and $K=2\kappa A/l$. Consequently $Z_{\rm pair}=\alpha_{np}^2/(4\rho\kappa)$. The factor four follows from combining two series resistances and two parallel conductances; it disappears only if the symbols have already been defined as pair properties. With opposite equal <Seebeck coefficients> $S_p=-S_n=S$, the result reduces to $S^2/(\rho\kappa)$.
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