SCIENCE · INVESTIGATE AND EXPLAIN

Fit an offset and reserve a fourth record

A constant measurement offset bends a logarithm of the raw readings. The synthetic response is y=Ae^(−t/τ)+B, with amplitude, time constant and offset all unknown to the analyst. Three readings at independently exact times 0,1,2 s calibrate those three parameters; a fourth at 3 s must remain fresh.

Learning outcome: Fit only the first three readings before revealing the fourth. Record the offset-sensitive raw-log mistake and compare it with the difference-ratio solution. Perturb all three calibration readings within their bounds, preserve their shared roles in numerator and denominator, and retain the endpoint interval.

Starting recap

Review charge continuity, definite integrals, first-order differential equations, initial conditions and signed source work. Distinguish displayed time from physical time.

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Try

For τ=1 s and B=0.5 V, the three calibration readings are 4.5, approximately 1.971518 and 1.041341 V. Differences recover r=1/e, amplitude 4 V and offset 0.5 V. The fresh prediction at 3 s is 0.5+4e^(−3)≈0.699148 V; agreement tests the fitted response but cannot exclude every alternative mechanism.

Open the experiment

Predict

Which record is fresh validation after fitting A, B and τ from times 0,1,2 s?

Your prediction and reason

Experiment

Fit only the first three readings before revealing the fourth. Record the offset-sensitive raw-log mistake and compare it with the difference-ratio solution. Perturb all three calibration readings within their bounds, preserve their shared roles in numerator and denominator, and retain the endpoint interval.

Fit an offset without consuming the validation record

Worked teaching model · synthetic values · Synthetic reading y(t)=4 exp(−t/tau)+B. The analyst does not know amplitude, offset or tau: readings at exact times 0,1,2 s calibrate all three. The 3 s record is fresh validation. Independent bounded calibration-reading errors ±0.01 V, not random errors or confidence levels. No gain/clock uncertainty in this recipe.

True time constant: 1 s Constant reading offset: 0.5 V

Reset keeps your prediction, captures and practice history.

Three calibration times and one fresh time

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: fitted response, not new observations (solid).2: fresh observed record (open-circle marked location). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Three calibration times and one fresh time

fitted response, not new observations

time (s)	reading (V)
0	4.5
0.0375	4.352778
0.075	4.210974
0.1125	4.074389
0.15	3.942832
0.1875	3.816116
0.225	3.694065
0.2625	3.576505
0.3	3.463273
0.3375	3.354208
0.375	3.249157
0.4125	3.147973
0.45	3.050513
0.4875	2.95664
0.525	2.866221
0.5625	2.779131
0.6	2.695247
0.6375	2.614449
0.675	2.536626
0.7125	2.461666
0.75	2.389466
0.7875	2.319923
0.825	2.25294
0.8625	2.188422
0.9	2.126279
0.9375	2.066423
0.975	2.008769
1.0125	1.953238
1.05	1.899751
1.0875	1.848232
1.125	1.79861
1.1625	1.750814
1.2	1.704777
1.2375	1.660434
1.275	1.617724
1.3125	1.576585
1.35	1.536961
1.3875	1.498795
1.425	1.462034
1.4625	1.426626
1.5	1.392521
1.5375	1.359671
1.575	1.32803
1.6125	1.297554
1.65	1.2682
1.6875	1.239926
1.725	1.212692
1.7625	1.186461
1.8	1.161196
1.8375	1.13686
1.875	1.11342
1.9125	1.090843
1.95	1.069096
1.9875	1.04815
2.025	1.027975
2.0625	1.008543
2.1	0.9898257
2.1375	0.9717974
2.175	0.9544326
2.2125	0.937707
2.25	0.9215969
2.2875	0.9060798
2.325	0.8911338
2.3625	0.8767379
2.4	0.8628718
2.4375	0.8495161
2.475	0.836652
2.5125	0.8242613
2.55	0.8123267
2.5875	0.8008313
2.625	0.789759
2.6625	0.7790943
2.7	0.7688221
2.7375	0.7589279
2.775	0.7493979
2.8125	0.7402187
2.85	0.7313773
2.8875	0.7228613
2.925	0.7146588
2.9625	0.7067581
3	0.6991483

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

fresh observed record

time (s)	reading (V)
3	0.6991483

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Calibration reading at 0 s	4.5	V
Calibration reading at 1 s	1.971518	V
Calibration reading at 2 s	1.041341	V
Inferred decay ratio	0.3678794	ratio
Inferred time constant	1	s
Inferred amplitude	4	V
Inferred constant offset	0.5	V
Fresh three-second prediction	0.6991483	V
Fresh three-second observation	0.6991483	V
Fresh residual	0	V
Bounded time-constant lower endpoint	0.9712372	s
Bounded time-constant upper endpoint	1.030094	s

Successive differences remove a constant offset, but fitting three unknowns consumes three records.

For equally spaced times with separation 1 s, r=(y1−y2)/(y0−y1)=e^(−1/tau). Then tau=−1/ln r, A=(y0−y1)/(1−r), B=y0−A. The fourth record is validation, not another calibration equation. On this bounded grid every error-box corner has positive denominator and 0<r<1. A linear-fractional ratio takes its extrema on those corners; tau increases with r, giving deterministic endpoints, not a confidence interval.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Successive differences remove a constant offset, but fitting three unknowns consumes three records.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

State the current, source and measurement model

Derive the integral or initial-value solution

Write r=e^(−1/τ). Successive differences are A(1−r) and Ar(1−r), so r=(y1−y2)/(y0−y1), τ=−1/ln r, A=(y0−y1)/(1−r), B=y0−A. The admissible positive decay branch needs a nonzero first difference and r strictly between zero and one. Reusing y2 as validation after this inversion would double-count calibration evidence.

Test conservation, uncertainty and limits

The controls generate A=4 V and bounded τ, but the fitted formulas do not read the hidden generating parameters. For independent calibration-reading errors ±0.01 V, evaluate the eight box corners. Throughout the allowed grid, denominators stay positive and ratios stay inside (0,1); the linear-fractional ratio and monotone logarithm therefore give valid deterministic time-constant endpoints. These are not probability statements.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For exact calibration readings y0=5 V, y1=3 V, y2=2 V, predict y3 in volts.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic evidence checked 31 October 2026 against the primary explanation. No live circuit construction, energized switching, component rating or electrical handling instruction is requested. Preserve first predictions, critique, revision and fresh assigned work separately from public practice. Software agreement does not certify measurement technique or learning.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which initial state, calibration or constitutive assumption is needed for this conclusion, and what fresh evidence could test it?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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Previous: Retain initial current and the inductive energy cross term Next: Separate a physical time constant from a displayed one

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