Distinguish actuator capacity from an energy criterion

A stronger request cannot create a larger heater

Select the heater limited to 8 W scenario. The standard enclosure still starts at 20°C in a 20°C ambient, has heat capacity 100 J/K and conductance 1 W/K, and targets 30°C. Only the actuator ceiling changes. Predict whether increasing gain from 2 to 12 W/K can meet the lower tail-temperature requirement of 29°C.

At the initial ten-kelvin error, gain 2 requests 30 W and gain 12 requests 130 W. Both apply 8 W. Each first interval therefore supplies 80 J, not the requested amount times ten seconds. The command ledger records the shortage of actuator capacity without treating undelivered power as stored energy.

Compare capacity with the required operating point

At a true temperature of 29°C, the nominal outward heat-transfer rate would be G(T − Ta) = 1 × (29 − 20) = 9 W. The maximum input is only 8 W. At the 30°C target the required steady input would be 10 W. Thus this actuator cannot maintain either the lower tail boundary or the target under the stipulated standard-plant conditions.

With constant maximum input, equilibrium satisfies 8 = 1 × (T − 20), giving T = 28°C. From the 20°C start, the exact one-node response approaches 28°C from below. The time constant is C/G = 100 seconds. After 120 seconds, T = 20 + 8[1 − exp(−1.2)] ≈ 25.590446°C. A larger heat capacity would change the approach rate, not this constant-input equilibrium.

The controller is still requesting more than 8 W throughout each allowed limited-case run. Consequently all three gains produce the same applied-power and true-temperature traces. Capture gains 2 and 12 and compare the request columns with the identical output. This is a capacity limitation, not evidence that gain has no effect in every scenario.

Close the whole-run energy account

Twelve ten-second intervals at 8 W supply 8 × 120 = 960 J. The stored-energy increase is CΔT = 100 × (25.590446 − 20) ≈ 559.044630 J. Ambient transfer over the run is the difference, approximately 400.955370 J. The sum closes to 960 J before display rounding.

Do not compute ambient energy as the final outward rate multiplied by the whole duration. The body started at ambient temperature, so outward transfer started at zero and grew as the body warmed. The exact interval ledger accounts for that change. The thermal-capacitance reference provides the one-temperature modeling context; these numbers are original synthetic calculations.

The 960 J total is comfortably below the supplied 2,000 J ceiling, but the temperature service is insufficient. “Uses less energy” is not enough to establish a successful design. Record the failed tail band and the passing energy criterion separately. The fixed three-token hardware cost is yet another indicator; it is not electrical energy or a heater-power rating.

A criterion is not an operating rule

The lab checks input energy against 2,000 J after accounting for the run. It does not enforce an automatic energy cutoff. If a designer wanted shutdown at a particular cumulative energy, that would change the operating rule and temperature trajectory. A paper statement that a design exceeds an allowance cannot retroactively remove energy from its recorded run.

Likewise, this model does not include a depleted battery, changing voltage, heater efficiency curve or active cooling. The 8 W describes applied thermal input to the modeled body. It is not a measured wall-plug consumption or an efficiency claim for actual equipment. Keep the source of the resource limitation explicit.

Transfer: a changed ambient on paper

Suppose a different stipulated ambient were 22°C, while conductance remained 1 W/K and maximum input 8 W. Compute the constant-input equilibrium. Would equality with the 30°C target prove the original finite-time tail requirement is met?

Check the ambient-capacity transfer

Equilibrium would be 22 + 8/1 = 30°C. That removes the steady capacity shortfall at the target, but does not establish the temperature at 90–120 seconds. Starting conditions and the time-dependent solution still matter. The widget fixes ambient at 20°C; this is a paper calculation, not another captured scenario or a proof of finite-time compliance.

Return to power and energy or the thermal energy balance if the accounting is unclear. Continue to controller redesign with a clear distinction: some failures can be improved by changing the response rule, while a hard capacity shortfall needs a different design condition.