 Fixed-temperature electrical substitution example

ID: fixed-temperature-electrical-substitution-example

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Fixed-temperature electrical substitution example by

Codex  0 2026-10-14

Stipulate a model receiver held at

T = 5.4 K

with a reference at

T_{b} = 5.0 K

. Choose

G = 5 mW K^{- 1}

, so outward thermal conduction is

G (T - T_{b}) = 2 mW

. Ignore other heat transfer, take

P_{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 mW

. With light admitted, the required heater power is

P_{h, l} = 1.005 mW

. For stipulated radiometric absorptance

a = 0.995

at the chosen single wavelength,

P_{abs} = 2 - 1.005 = 0.995 mW, P_{in} = \frac{0.995 mW}{0.995} = 1.000 mW .

(1)

The illuminated receiver still gets

1.005 + 0.995 = 2 mW

of heating. The remaining

0.005 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%

Figure 1.
The same receiver temperature with different electrical and optical heating
. Original stipulated steady-state example, not experimental data. Both columns total $2 mW$ of deposited heating and balance the same outward thermal conduction. Incident radiant flux is $1 mW$ , of which $0.995 mW$ is absorbed; the $0.005 mW$ that escapes is outside the heating column. No response time or measured uncertainty is implied. **Details**.

Two variations expose different pitfalls. First, if admitting the beam also adds

0.010 mW

of stray heating while the same

0.995 mW

is absorbed from the intended beam, the heater falls to

0.995 mW

. The apparent reduction is now

1.005 mW

; subtracting

Δ P_{par} = 0.010 mW

recovers the beam's absorbed power. Second, keep the original heater readings but stipulate a light-state receiver offset

δ T = + 1 mK

at fixed reference temperature. The link carries an extra

G δ T = 0.005 mW

; the inferred absorbed power becomes

1.000 mW

, and incident radiant flux becomes

1.000/0.995 ≃ 1.005025 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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