Infer a decay time separately from a voltage offset

Define topology, recipient and initial conditions

A discharge trace can appear to approach a nonzero voltage because of a fixed instrument offset. The supplied true signal is 8exp(−t/2) V, but the indicated signal is Aexp(−t/τ)+o with A equal to instrument gain times true initial voltage. We deliberately keep the source generator visible for teaching while asking which parameters the observations alone identify.

Derive the quantitative response

For readings at equally spaced times 0,1,2 s, let D01=R0−R1 and D12=R1−R2. The common offset cancels, and D12/D01=exp(−1/τ). The natural logarithm is the inverse of exp, so τ=−1/ln(D12/D01) seconds. Then A=D01/[1−exp(−1/τ)] and o=R0−A. This is an algebraic inversion of a supplied exponential, not a required differential-equation solution.

Check evidence and the limits of the model

Even an exact fit cannot separate gain from true initial voltage because only their product enters A. It also does not uniquely attribute a changed τ to capacitance: resistance, leakage or other unmodeled paths can alter the effective time constant. A fourth withheld time checks the single-exponential model, while an independent voltage reference or static resistance/capacitance observation tests different assumptions. Small differences and ratios near one can amplify reading errors, so a neat displayed result does not imply precise physical identification.

Worked original case

With gain1.1 and offset .5 V, the indicated amplitude is 8.8 V, but differences retain decay factor exp(−.5)≈.606531 and recover τ=2 s. The fit gives offset .5 V and amplitude8.8 V, not independent proof of gain1.1 or true initial8 V.

Investigation

Compare offsets −1,0,+1 V at one gain and show which differences remain invariant. Then change gain and check that inferred τ remains the same while fitted amplitude changes. Freeze a third-second prediction and propose an independent reference to distinguish gain from initial voltage.

Changed-case practice

Sources and evidence boundary

Original numerical cases and prose checked 30 October 2026 against the relevant primary explanation. All records are synthetic and all diagrams mathematical. No practical wiring, battery testing, capacitor handling, mains measurement or equipment safety rating is requested. Public feedback does not verify independent mastery; retain first predictions, contrasting recipes and criterion-based critique in the separate course evidence workspace.