Optical-electrical heating non-equivalence

{title2=$P_{\rm abs}=P_{h,d}-P_{h,l}$}

An explicit lumped model makes the cancellation understandable. Treat the receiver as having uniform <temperature> $T$, <heat capacity> $C$, heater <power> $P_h$ deposited in the receiver, net parasitic input $P_{\rm par}$, and outward <heat transfer> $Q_{\rm out}(T,T_b)$ to a reference at <temperature> $T_b$. With incident <radiant flux> $P_{\rm in}$ and <radiometric absorptance> $a$, <conservation of energy> gives
$$
C\frac{dT}{dt}=P_h+aP_{\rm in}+P_{\rm par}-Q_{\rm out}(T,T_b).
$$
The dark state has no admitted beam. Compare settled dark and light states, denoted $d$ and $l$; each has $dT/dt=0$. Subtracting their balances gives
$$
P_{\rm abs}=aP_{\rm in}=\Delta P_h+\Delta Q_{\rm out}-\Delta P_{\rm par},\qquad
\Delta P_h=P_{h,d}-P_{h,l},
$$
where $\Delta Q_{\rm out}=Q_{{\rm out},l}-Q_{{\rm out},d}$ and $\Delta P_{\rm par}=P_{{\rm par},l}-P_{{\rm par},d}$. At identical $T$ and $T_b$, with unchanged surroundings and parasitic heating, both correction differences vanish. Thus $P_{\rm abs}=\Delta P_h$, and $P_{\rm in}=\Delta P_h/a$. Notice the signs: extra stray <thermal radiation> admitted with the beam also reduces heater <power>, so it must be subtracted from that reduction before attributing the result to the intended beam.

Measure the heater's own terminal voltage $V_h$ and <electric current> $I_h$: for steady electrical heating, $P_h=V_hI_h=I_h^2R_h$, where $R_h$ is its <electrical resistance>. A supply voltage that includes wire voltage drops counts <Joule heating> in the leads too; only the portion deposited in the receiver belongs in this balance. Changes in that deposited lead <heat> belong in $\Delta P_{\rm par}$. <Optical-electrical heating non-equivalence> adds a further instrument-specific correction when the uniform-<temperature> model is insufficient.<Spectroradiometric Detector Measurements: Ultraviolet, Visible, and Near-Infrared Detectors for Spectral Power by Thomas C. Larason and others>[§5.2, printed p. 30: heater wires, window transmission and optical/electrical non-equivalence]{ref}


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