= Thermopile thermal response
{title2=$\tau=C_{\rm th}/G_{\rm th}$}
A useful original model treats the <thermopile absorber> as one approximately uniform <temperature> $T_h(t)$ with <heat capacity> $C_{\rm th}$, connected to a fixed reference <temperature> $T_c$ by <thermal conductance> $G_{\rm th}$. Both parameters are positive constants here. Let $P_{\rm abs}(t)$ be an added absorbed <power> above the initial background. Linearize the outgoing <heat transfer> about that background. With $\Delta T=T_h-T_c$ measured from an initial state having $\Delta T=0$, <conservation of energy> gives the <first-order linear differential equation>
$$
C_{\rm th}\frac{d\Delta T}{dt}=P_{\rm abs}(t)-G_{\rm th}\Delta T.
$$
The left side is the rate of stored <energy>; the two terms on the right are added <power> and outgoing <heat transfer>. The effective <thermal conductance> can include several paths, rather than just <heat conduction> along the wires. The model assumes a stable reference, negligible electrical loading, a common absorber <temperature>, and changes small enough for constant parameters.
For a step from zero to constant $P_0$ at $t=0$, subtract the steady solution $\Delta T_\infty=P_0/G_{\rm th}$. The difference obeys <exponential decay>, so the initial condition fixes
$$
\tau=\frac{C_{\rm th}}{G_{\rm th}},\qquad
\Delta T(t)=\frac{P_0}{G_{\rm th}}\left(1-e^{-t/\tau}\right),\qquad
V_{\rm oc}(t)\approx N\alpha\frac{P_0}{G_{\rm th}}\left(1-e^{-t/\tau}\right).
$$
The <time constant> has units $(\mathrm{J\,K^{-1}})/(\mathrm{W\,K^{-1}})=\mathrm s$. With the same constructed $N=20$ and $\alpha=40\,\mu\mathrm{V\,K^{-1}}$, choose $C_{\rm th}=0.02\,\mathrm{J\,K^{-1}}$, $G_{\rm th}=0.01\,\mathrm{W\,K^{-1}}$ and $P_0=0.03\,\mathrm W$. Then
$$
\tau=2\,\mathrm s,\qquad \Delta T_\infty=3\,\mathrm K,\qquad
V_\infty=2.4\,\mathrm{mV}.
$$
At one <time constant>, the response is $1-e^{-1}\approx0.632$ of its final value: $V_{\rm oc}(2\,\mathrm s)\approx1.52\,\mathrm{mV}$. Reaching $95\%$ requires $t=-\tau\ln(0.05)\approx5.99\,\mathrm s$. A <voltmeter> read too soon therefore underestimates the steady <signal> even when its electrical calibration is perfect.
\Image[/thermopile-absorber-step-response.png]
{title=Calculated <thermopile> step response: <heat capacity>, <thermal conductance> and settling}
{description=Original model curves for added absorbed <power> $P_0=0.03\,\mathrm W$, $N=20$ and relative <Seebeck coefficient> $\alpha=40\,\mu\mathrm{V\,K^{-1}}$. The solid curve uses $C_{\rm th}=0.02\,\mathrm{J\,K^{-1}}$ and $G_{\rm th}=0.01\,\mathrm{W\,K^{-1}}$, giving <time constant> $2\,\mathrm s$ and final voltage $2.4\,\mathrm{mV}$. The dashed comparison halves only $G_{\rm th}$: final voltage doubles, but so does the <time constant>. Neither curve is an experimental observation.}
The steady ratio $V_\infty/P_0=N\alpha/G_{\rm th}$ is $0.080\,\mathrm{V\,W^{-1}}$ in this constructed model. Halving <thermal conductance> doubles this ratio but also doubles the <time constant> if <heat capacity> stays fixed. Reducing <heat capacity> alone speeds the response without changing its steady value. Increasing $N$ gives a larger <signal> only at fixed individual <temperature> differences: adding legs can also change <thermal conductance>, <heat capacity> and <electrical resistance>. The algebra identifies coupled design choices, rather than predicting a particular manufactured detector.
A finite <electrical resistance> produces <Thermal electrical noise>. For nearly uniform absolute <temperature> $T$ and effective measurement <signal bandwidth> $B$, its <root mean square> open-circuit voltage is
$$
v_{n,\rm rms}=\sqrt{4k_{\rm B}TR_{\rm pile}B}.
$$
Here $k_{\rm B}$ is the <Boltzmann constant>. Thermal electrical <noise> in the sensing element is an important contribution, and the <signal bandwidth> enters its voltage variance.<Thermopile detectors by Hamamatsu Photonics K.K.>[p. 3, §3-2 Noise, Eq. (3-2): element resistance, absolute temperature and measurement bandwidth]{ref} If identical pairs have fixed <electrical resistance> $r$ and fixed $\Delta T$, then $R_{\rm pile}=Nr$: the <signal> grows as $N$ while this <noise> grows as $\sqrt N$. The resulting <signal-to-noise ratio> grows only as $\sqrt N$ under those assumptions. Reference <temperature> drift, environmental <electromagnetic radiation>, and meter <noise> remain separate limits; reducing <signal bandwidth> trades fast information for less integrated electrical <noise>.
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