SCIENCE · INVESTIGATE AND EXPLAIN

Separate signed reactance, phase and dissipated power

In steady sinusoidal response, derivative and integral terms appear with different phases. With a cosine convention, series impedance is Z=R+i[omega L−1/(omega C)]. This packet fixes L=1 H,C=0.25 F and source RMS voltage 6 V, while resistance stays positive. The initial transient is assumed to have decayed.

Learning outcome: Compare frequencies below, at and above 2 rad/s for two positive resistances. Retain signed phase, RMS and peak current, real/reactive/apparent power and the power-triangle check. Explain why cancellation of net reactance does not imply zero instantaneous energy in L and C.

Starting recap

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

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

At omega=2 rad/s and R=2 Ω, net reactance is zero, RMS current is 3 A, peak current 3√2 A and average power 18 W. At omega=1 rad/s, X=−3 Ω, so current leads the source voltage; the negative reactive power does not mean the resistor produces energy.

Open the experiment

Predict

At omega=2 rad/s and R=2 Ω, what is average real power?

Your prediction and reason

Experiment

Compare frequencies below, at and above 2 rad/s for two positive resistances. Retain signed phase, RMS and peak current, real/reactive/apparent power and the power-triangle check. Explain why cancellation of net reactance does not imply zero instantaneous energy in L and C.

Use complex impedance and an RMS power account

Worked teaching model · synthetic values · Steady sinusoidal series L1 H,C.25 F,R selected,source RMS6 V with cos convention. Reactance X=omega L−1/(omega C). Initial transients have decayed. Positive phase is voltage leading current; no ideal zero-resistance driven resonance or real apparatus.

Angular frequency: 2 rad/s Resistance: 2 ohm

Reset keeps your prediction, captures and practice history.

Steady RMS current versus drive frequency

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: same positive resistance (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Steady RMS current versus drive frequency

same positive resistance

angular frequency (rad/s)	RMS current (A)
0.5	0.772988
0.54375	0.8450609
0.5875	0.9181896
0.63125	0.9924285
0.675	1.067822
0.71875	1.1444
0.7625	1.222179
0.80625	1.301156
0.85	1.381307
0.89375	1.462584
0.9375	1.544907
0.98125	1.628167
1.025	1.712214
1.06875	1.796862
1.1125	1.881875
1.15625	1.966973
1.2	2.051823
1.24375	2.136045
1.2875	2.219205
1.33125	2.300824
1.375	2.380384
1.41875	2.45733
1.4625	2.53109
1.50625	2.601083
1.55	2.666738
1.59375	2.727511
1.6375	2.782905
1.68125	2.832487
1.725	2.875905
1.76875	2.9129
1.8125	2.943317
1.85625	2.967106
1.9	2.984324
1.94375	2.995127
1.9875	2.999764
2.03125	2.998559
2.075	2.991898
2.11875	2.980214
2.1625	2.963971
2.20625	2.943646
2.25	2.919719
2.29375	2.89266
2.3375	2.862922
2.38125	2.83093
2.425	2.797081
2.46875	2.761739
2.5125	2.725233
2.55625	2.687858
2.6	2.649875
2.64375	2.611514
2.6875	2.572974
2.73125	2.534427
2.775	2.496021
2.81875	2.45788
2.8625	2.420107
2.90625	2.382789
2.95	2.345997
2.99375	2.309787
3.0375	2.274203
3.08125	2.23928
3.125	2.205044
3.16875	2.171513
3.2125	2.138698
3.25625	2.106606
3.3	2.075238
3.34375	2.044592
3.3875	2.014663
3.43125	1.985442
3.475	1.956921
3.51875	1.929087
3.5625	1.901927
3.60625	1.875428
3.65	1.849574
3.69375	1.82435
3.7375	1.799741
3.78125	1.775731
3.825	1.752303
3.86875	1.729442
3.9125	1.707132
3.95625	1.685356
4	1.664101

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
Net reactance	0	ohm
Impedance magnitude	2	ohm
RMS current	3	A
Peak current	4.242641	A
Voltage-leading-current phase	0	rad
Average real power	18	W
Signed reactive power	0	var
Apparent power	18	VA
Power factor	1	ratio
Power-triangle residual	0	VA²

The inductor and capacitor voltage phases oppose; subtract reactances before finding impedance magnitude.

With the cos convention, Z=R+i(omega L−1/(omega C)). Current lags voltage for positive X and leads for negative X. RMS quantities give average real power Irms²R; signed reactive power Irms²X describes reversible exchange, not net dissipated watts. At omega0=2 the reactances cancel and current is 6/R, finite because R is positive. Peak current is sqrt2 times RMS, not an interchangeable label.

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

The inductor and capacitor voltage phases oppose; subtract reactances before finding impedance magnitude.

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

The net reactance X=omega−4/omega can be positive, zero or negative. Current RMS magnitude is 6/sqrt(R²+X²), and voltage-leading-current phase is atan2(X,R). Positive X means current lags; negative X means it leads. Subtract the opposite reactances before taking the magnitude rather than adding all three voltage magnitudes.

Audit energy, uncertainty and domain

Average dissipated power is Irms²R. Signed reactive power Irms²X describes reversible exchange, not net heat, while apparent power is VrmsIrms. At omega=2 the reactances cancel, but current remains finite because R is nonzero. Peak current is sqrt2 times RMS; mixing those conventions creates a factor-of-two power error.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

At resonance with R=3 Ω and the same 6 V RMS source, find RMS current in amperes.

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.

Skipping retains experiments and awards no mastery.

Previous: Match three damping regimes without a singular formula Next: Fit signed frequency evidence without consuming validation

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.