= Fixed-temperature electrical substitution example
{title2=$P_{\rm in}=(2-1.005)/0.995\,\mathrm{mW}=1\,\mathrm{mW}$}
Stipulate a model receiver held at $T=5.4\,\mathrm K$ with a reference at $T_b=5.0\,\mathrm K$. Choose $G=5\,\mathrm{mW\,K^{-1}}$, so outward <thermal conduction> is $G(T-T_b)=2\,\mathrm{mW}$. Ignore other <heat transfer>, take $P_{\rm par}=0$ in both states, and stipulate perfect optical/electrical thermal equivalence. These values specify an original example; they are not measurements or claimed performance of a NIST instrument.
With the beam blocked, supply $P_{h,d}=2\,\mathrm{mW}$. With <light> admitted, the required heater <power> is $P_{h,l}=1.005\,\mathrm{mW}$. For stipulated <radiometric absorptance> $a=0.995$ at the chosen single <wavelength>,
$$
P_{\rm abs}=2-1.005=0.995\,\mathrm{mW},\qquad
P_{\rm in}=\frac{0.995\,\mathrm{mW}}{0.995}=1.000\,\mathrm{mW}.
$$
The illuminated receiver still gets $1.005+0.995=2\,\mathrm{mW}$ of heating. The remaining $0.005\,\mathrm{mW}$ of incident <radiant flux> escapes without heating it. Calling the measured heater reduction “incident <power>” would underestimate the stipulated incident <radiant flux> by $0.5\%$.
\Image[/electrical-substitution-dark-light-heater-balance.png]
{title=The same receiver <temperature> with different electrical and optical heating}
{id=electrical-substitution-dark-light-heater-balance}
{description=Original stipulated steady-state example, not experimental data. Both columns total $2\,\mathrm{mW}$ of deposited heating and balance the same outward <thermal conduction>. Incident <radiant flux> is $1\,\mathrm{mW}$, of which $0.995\,\mathrm{mW}$ is absorbed; the $0.005\,\mathrm{mW}$ that escapes is outside the heating column. No response time or measured uncertainty is implied.}
Two variations expose different pitfalls. First, if admitting the beam also adds $0.010\,\mathrm{mW}$ of stray heating while the same $0.995\,\mathrm{mW}$ is absorbed from the intended beam, the heater falls to $0.995\,\mathrm{mW}$. The apparent reduction is now $1.005\,\mathrm{mW}$; subtracting $\Delta P_{\rm par}=0.010\,\mathrm{mW}$ recovers the beam's absorbed <power>. Second, keep the original heater readings but stipulate a light-state receiver offset $\delta T=+1\,\mathrm{mK}$ at fixed reference <temperature>. The link carries an extra $G\delta T=0.005\,\mathrm{mW}$; the inferred absorbed <power> becomes $1.000\,\mathrm{mW}$, and incident <radiant flux> becomes $1.000/0.995\simeq1.005025\,\mathrm{mW}$. This second variation deliberately violates exact fixed-<temperature> substitution.
The heater can supply only nonnegative <power> in this model. Consequently an absorbed beam above the available dark heater <power> cannot be compensated at the chosen setpoint merely by reducing the heater. The operating point, link, or admitted <radiant flux> must change; zero heater demand is a range boundary, not evidence of greater <radiant flux> sensitivity.
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