SCIENCE · INVESTIGATE AND EXPLAIN

Fit signed frequency evidence without consuming validation

Complex impedance measurement supplies both magnitude and phase; its imaginary part can therefore retain the sign of reactance. Here independently known resistance is 2 Ω, while inductance L and reciprocal capacitance K=1/C are unknown. Exact calibrated angular frequencies 1 and 2 rad/s give X1=L−K and X2=2L−K/2 in SI numerical units.

Learning outcome: Fit only the first two signed reactances and freeze the third-frequency prediction. Compare direct propagation through its observation weights with separately propagated parameter intervals. State what extra evidence would test the assumed frequency and phase calibration.

Starting recap

Review flux integrals, linear differential equations, initial conditions, complex numbers and vector derivatives. Keep source constraints, RMS conventions and calibration roles explicit.

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Try

For generated L=1 H and C=0.5 F, calibration reactances are −1 Ω at 1 rad/s and +1 Ω at 2 rad/s. They recover L=1 and K=2. The unused 3 rad/s prediction is 7/3 Ω. With each calibration error bounded by 0.02 Ω, its direct prediction bound is 7×0.02/3≈0.046667 Ω, before any fresh observation error is added.

Open the experiment

Predict

Which quantity is lost if only impedance magnitudes are retained?

Your prediction and reason

Experiment

Fit only the first two signed reactances and freeze the third-frequency prediction. Compare direct propagation through its observation weights with separately propagated parameter intervals. State what extra evidence would test the assumed frequency and phase calibration.

Fit two frequency records and reserve a third

Worked teaching model · synthetic values · Independently known series R2 ohm. Calibrated complex impedance supplies imaginary parts X1 at1 rad/s and X2 at2 rad/s; fit L and K=1/C only from them. A3 rad/s reactance is fresh validation. Independent imaginary-reading bounds ±.02 ohm; frequency and phase calibration are stipulated, not inferred.

Generating inductance: 1 H Generating capacitance: 0.5 F

Reset keeps your prediction, captures and practice history.

Two frequency calibrations and one unused reactance

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: fitted L and reciprocal C (solid).2: fresh three-radian-per-second record (open-circle marked location). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Two frequency calibrations and one unused reactance

fitted L and reciprocal C

angular frequency (rad/s)	imaginary impedance (ohm)
0.5	-3.5
0.53125	-3.233456
0.5625	-2.993056
0.59375	-2.774671
0.625	-2.575
0.65625	-2.391369
0.6875	-2.221591
0.71875	-2.063859
0.75	-1.916667
0.78125	-1.77875
0.8125	-1.649038
0.84375	-1.52662
0.875	-1.410714
0.90625	-1.300647
0.9375	-1.195833
0.96875	-1.095766
1	-1
1.03125	-0.9081439
1.0625	-0.8198529
1.09375	-0.7348214
1.125	-0.6527778
1.15625	-0.5734797
1.1875	-0.4967105
1.21875	-0.4222756
1.25	-0.35
1.28125	-0.2797256
1.3125	-0.2113095
1.34375	-0.1446221
1.375	-0.07954545
1.40625	-0.01597222
1.4375	0.04619565
1.46875	0.1070479
1.5	0.1666667
1.53125	0.2251276
1.5625	0.2825
1.59375	0.338848
1.625	0.3942308
1.65625	0.4487028
1.6875	0.5023148
1.71875	0.5551136
1.75	0.6071429
1.78125	0.658443
1.8125	0.7090517
1.84375	0.7590042
1.875	0.8083333
1.90625	0.8570697
1.9375	0.9052419
1.96875	0.952877
2	1
2.03125	1.046635
2.0625	1.092803
2.09375	1.138526
2.125	1.183824
2.15625	1.228714
2.1875	1.273214
2.21875	1.317342
2.25	1.361111
2.28125	1.404538
2.3125	1.447635
2.34375	1.490417
2.375	1.532895
2.40625	1.575081
2.4375	1.616987
2.46875	1.658623
2.5	1.7
2.53125	1.741127
2.5625	1.782012
2.59375	1.822666
2.625	1.863095
2.65625	1.903309
2.6875	1.943314
2.71875	1.983118
2.75	2.022727
2.78125	2.062149
2.8125	2.101389
2.84375	2.140453
2.875	2.179348
2.90625	2.218078
2.9375	2.256649
2.96875	2.295066
3	2.333333

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 three-radian-per-second record

angular frequency (rad/s)	imaginary impedance (ohm)
3	2.333333

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
Calibrating reactance at one radian per second	-1	ohm
Calibrating reactance at two radians per second	1	ohm
Inferred inductance	1	H
Inferred reciprocal capacitance	2	F⁻¹
Inferred capacitance	0.5	F
Fresh three-frequency prediction	2.333333	ohm
Fresh three-frequency observation	2.333333	ohm
Fresh residual	0	ohm
Inductance error bound	0.02	H
Reciprocal-capacitance error bound	0.04	F⁻¹
Direct third-prediction error bound	0.04666667	ohm

Phase-resolved reactance carries a sign; magnitude-only data would pose a different inverse problem.

At frequencies 1 and 2, X1=L−K and X2=2L−K/2 with K=1/C. Solving gives L=(2X2−X1)/3 and K=(2X2−4X1)/3. The third prediction is −5X1/9+16X2/9, so independent ±epsilon errors give 7epsilon/3. Separate L and K intervals share observations and are correlated. Frequency and phase calibration are explicit independent premises; a perfect fit cannot establish them.

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

Phase-resolved reactance carries a sign; magnitude-only data would pose a different inverse problem.

Observation to attach to a capture

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

Understand

State geometry, sources and initial data

Derive the field or circuit relation

Solve the two linear equations: L=(2X2−X1)/3 and K=(2X2−4X1)/3. Only those two records calibrate the two parameters. The third-frequency prediction at 3 rad/s is 3L−K/3, or directly −5X1/9+16X2/9. A fresh measurement at that frequency tests the fitted ideal series-element response.

Audit energy, uncertainty and domain

For independent imaginary-reading bounds ±epsilon, the third prediction bound is 7epsilon/3. Separate L and K errors share the same readings and are correlated, so treating their intervals as independent can overstate subsequent uncertainty. Frequency and phase calibration are stipulated independent premises; amplitude-only measurements would pose a different inverse problem with lost sign information.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

If X1=−3 Ω at 1 rad/s and X2=0 Ω at 2 rad/s, predict reactance at 3 rad/s in ohms.

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. Constants are rounded teaching values and are mutually consistent where Maxwell relations require it. No live circuit, high-voltage, energized-inductor, magnet, beam or optical exposure activity is requested. Retain first work, criterion-linked critique, revision and newly assigned evidence; public numerical feedback is not practical or learner certification.

Apply

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

Which source, initial-state, calibration or field-law premise makes the result conditional, and what unused evidence would challenge 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: Separate signed reactance, phase and dissipated power Next: Reconcile current surfaces without inventing gap conduction

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