= Thermopile series voltage
{title2=$V_{\rm oc}=\sum_{i=1}^N E_i$}
Use the A-positive terminal order of <Seebeck effect> for each <thermocouple>. Its <thermocouple measuring junction> has <temperature> $T_{h,i}$ and its reference connections have <temperature> $T_{c,i}$. Define the difference of <Seebeck coefficients> as $\alpha_i(T)=S_{A,i}(T)-S_{B,i}(T)$. Connect the B reference end of one <thermocouple> to the A reference end of the next at a common local <temperature>, so the terminal <electric potential differences> add:
$$
E_i=\int_{T_{c,i}}^{T_{h,i}}\alpha_i(T)\,dT,
\qquad V_{\rm oc}=\sum_{i=1}^{N} E_i.
$$
The first A reference terminal is the positive meter input; the final B reference terminal is the negative input. Following this connected path and adding the individual <Seebeck effect> <integrals> proves the result. Matched meter leads with common endpoint <temperatures> cancel as in an <isothermal terminal block>. The manufacturer's sum formula also allows unequal individual contributions.<Thermopile detectors by Hamamatsu Photonics K.K.>[p. 2, Eq. (2-1): total output is the sum of individual thermocouple outputs]{ref}
\Image[/thermopile-series-junction-geometry.png]
{title=<Thermopile> geometry: series junctions, absorber and reference support}
{description=Original idealized wiring, showing three of $N$ <thermocouples>. Each A leg rises from reference <temperature> $T_c$ to absorber <temperature> $T_h$, and each B leg returns. B-to-A connections at $T_c$ join adjacent pairs; the thermally shared absorber and support do not short the electrically distinct junctions. A-positive and B-negative meter connections match the <Seebeck effect> convention. Arrows indicate absorbed <electromagnetic radiation> and escaping <heat>, not <electric current>.}
If all pairs share $T_h,T_c$ and the same nearly constant difference $\alpha$ of <Seebeck coefficients> throughout that <temperature> <interval>, then
$$
V_{\rm oc}\approx N\alpha\Delta T,\qquad \Delta T=T_h-T_c.
$$
This $\alpha$ is the pair's relative <Seebeck coefficient>, not either material's individual coefficient or a universal device constant. As an original constructed example, choose $N=20$, $\alpha=40\,\mu\mathrm{V\,K^{-1}}$ and $\Delta T=5\,\mathrm K$. Each <thermocouple> contributes $0.20\,\mathrm{mV}$ and
$$
V_{\rm oc}=20(40\times10^{-6}\,\mathrm{V\,K^{-1}})(5\,\mathrm K)=4.0\,\mathrm{mV}.
$$
Reversing the meter connections changes the sign. Exchanging $T_h$ and $T_c$ reverses the <integrals> and hence the <electromotive force>; this reversible sign response does not make a detector with <heat conduction> a reversible thermodynamic process. Reversing the wiring of one pair instead gives $(N-2)\alpha\Delta T$, because one contribution is subtracted rather than added. For this example the result is $3.6\,\mathrm{mV}$, and at equal junction <temperatures> every contribution vanishes.
The open-circuit condition matters. If the array's total <electrical resistance> is $R_{\rm pile}$ and the meter input has <electrical resistance> $R_m$, the elementary <voltage divider> model gives $V_m=V_{\rm oc}R_m/(R_m+R_{\rm pile})$. Thus $R_m\gg R_{\rm pile}$ is needed to suppress <voltmeter loading>. Appreciable <electric current> also permits <Joule heating> and the thermoelectric <heat transfer> to disturb the assumed junction <temperatures>.
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