Reject a coupling matrix that permits negative field energy
Signed mutual inductance depends on the reference orientations of two circuits. For a reciprocal linear pair, flux linkages are L1I1+MI2 and MI1+L2I2. The model fixes L1=1 H and L2=4 H, while a proposed M may be positive or negative. Changing the reference direction can change the sign of M without making self-inductance negative.
Learning outcome: Test both signs of M inside and outside the bound. Retain the determinant before evaluating a chosen state, and explain why the mutual term can be negative while the total admissible energy stays nonnegative.
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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For M=1 H, I1=1 A and I2=−1 A, energy is 0.5+2−1=1.5 J. For M=3 H the determinant is 4−9=−5 H², so the proposal is rejected even if a particular trial current happened to give positive energy. At I2=−1 A its formal value is −0.5 J, exposing the nonpassive quadratic form.
Open the experimentPredict
Which condition makes this proposed pair admissible as a passive linear energy matrix?
Experiment
Test both signs of M inside and outside the bound. Retain the determinant before evaluating a chosen state, and explain why the mutual term can be negative while the total admissible energy stays nonnegative.
Reject an energetically impossible proposed coupling
Worked teaching model · synthetic values · Linear reciprocal coupled inductors L1=1 H,L2=4 H and fixed first current1 A. Proposed M is signed by the declared reference orientations. A passive field-energy matrix requires M²≤L1L2. Invalid proposals report the violated condition but withhold physical energy/flux predictions.
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Energy for the admissible coupled proposal
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: positive-semidefinite quadratic form (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Energy for the admissible coupled proposal
positive-semidefinite quadratic form
| second current (A) | total energy (J) |
|---|---|
| -2 | 6.5 |
| -1.95 | 6.155 |
| -1.9 | 5.82 |
| -1.85 | 5.495 |
| -1.8 | 5.18 |
| -1.75 | 4.875 |
| -1.7 | 4.58 |
| -1.65 | 4.295 |
| -1.6 | 4.02 |
| -1.55 | 3.755 |
| -1.5 | 3.5 |
| -1.45 | 3.255 |
| -1.4 | 3.02 |
| -1.35 | 2.795 |
| -1.3 | 2.58 |
| -1.25 | 2.375 |
| -1.2 | 2.18 |
| -1.15 | 1.995 |
| -1.1 | 1.82 |
| -1.05 | 1.655 |
| -1 | 1.5 |
| -0.95 | 1.355 |
| -0.9 | 1.22 |
| -0.85 | 1.095 |
| -0.8 | 0.98 |
| -0.75 | 0.875 |
| -0.7 | 0.78 |
| -0.65 | 0.695 |
| -0.6 | 0.62 |
| -0.55 | 0.555 |
| -0.5 | 0.5 |
| -0.45 | 0.455 |
| -0.4 | 0.42 |
| -0.35 | 0.395 |
| -0.3 | 0.38 |
| -0.25 | 0.375 |
| -0.2 | 0.38 |
| -0.15 | 0.395 |
| -0.1 | 0.42 |
| -0.05 | 0.455 |
| 0 | 0.5 |
| 0.05 | 0.555 |
| 0.1 | 0.62 |
| 0.15 | 0.695 |
| 0.2 | 0.78 |
| 0.25 | 0.875 |
| 0.3 | 0.98 |
| 0.35 | 1.095 |
| 0.4 | 1.22 |
| 0.45 | 1.355 |
| 0.5 | 1.5 |
| 0.55 | 1.655 |
| 0.6 | 1.82 |
| 0.65 | 1.995 |
| 0.7 | 2.18 |
| 0.75 | 2.375 |
| 0.8 | 2.58 |
| 0.85 | 2.795 |
| 0.9 | 3.02 |
| 0.95 | 3.255 |
| 1 | 3.5 |
| 1.05 | 3.755 |
| 1.1 | 4.02 |
| 1.15 | 4.295 |
| 1.2 | 4.58 |
| 1.25 | 4.875 |
| 1.3 | 5.18 |
| 1.35 | 5.495 |
| 1.4 | 5.82 |
| 1.45 | 6.155 |
| 1.5 | 6.5 |
| 1.55 | 6.855 |
| 1.6 | 7.22 |
| 1.65 | 7.595 |
| 1.7 | 7.98 |
| 1.75 | 8.375 |
| 1.8 | 8.78 |
| 1.85 | 9.195 |
| 1.9 | 9.62 |
| 1.95 | 10.055 |
| 2 | 10.5 |
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.
| Quantity | Value | Unit |
|---|---|---|
| Proposed mutual inductance | 1 | H |
| Inductance-matrix determinant | 3 | H² |
| Passive coupling condition satisfied | 1 | flag |
| First flux linkage | 0 | Wb turn |
| Second flux linkage | -3 | Wb turn |
| Total coupled magnetic energy | 1.5 | J |
| Mutual energy term | -1 | J |
The signed mutual term can lower the total without making a passive quadratic form negative.
Reciprocity gives linkage vector [L1 I1+M I2, M I1+L2 I2]. Integrating source work gives U=(L1 I1²+2M I1I2+L2 I2²)/2. Nonnegative energy for every current pair requires positive self inductances and M²≤L1L2. At equality an ideal perfect-coupling null direction is possible; this does not certify a realizable zero-energy apparatus. A rejected diagnostic curve is explicitly not a physical storage prediction.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The signed mutual term can lower the total without making a passive quadratic form negative.
Reciprocity gives linkage vector [L1 I1+M I2, M I1+L2 I2]. Integrating source work gives U=(L1 I1²+2M I1I2+L2 I2²)/2. Nonnegative energy for every current pair requires positive self inductances and M²≤L1L2. At equality an ideal perfect-coupling null direction is possible; this does not certify a realizable zero-energy apparatus. A rejected diagnostic curve is explicitly not a physical storage prediction.
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