Coupled resonators and mode splitting

Take two identical series LC loops with negligible resistance. Choose current directions and coil dot orientations so that the mutual terms enter both loop equations with positive M. The loop charge coordinates then share an inductance matrix rather than independent inductances. Reciprocity makes that matrix symmetric in this ideal model. Positive magnetic energy requires the magnitude of mutual inductance to be smaller than self-inductance; treating the coupling coefficient as arbitrarily large would produce an unphysical storage matrix.

These equations assume a constant, instantaneous mutual inductance and electrically small loops. They neglect radiative coupling, frequency dependent materials, parasitic capacitance, and additional coil modes. A capacitor connecting two resonators produces a different coupling matrix and need not preserve the same uncoupled frequency shift. Reporting only a coupling number without specifying its definition, topology, sign convention, and units is therefore insufficient. The mode calculation must begin from the selected energy and circuit model.

Adding and subtracting the loop equations separates equal-charge and opposite-charge motion. In the symmetric mode the inductive contribution is L plus M, while in the antisymmetric mode it is L minus M. For positive M the symmetric mode has the lower frequency because the effective inductance is larger. Swapping a coil's orientation changes which coordinate pattern is called symmetric, but the unordered pair of physical frequencies remains the same when the sign of M changes.

For weak coupling, expand the square roots to first order in M divided by L. The separation of angular frequencies is approximately the isolated angular frequency multiplied by the magnitude of that dimensionless ratio. This approximation is not the exact square-root expression, and it loses accuracy for strong coupling. Using ordinary frequencies in one part of a calculation and angular frequencies in another creates an extra factor of two pi, so specify the convention before converting a measured separation into a coupling rate.

If only the first capacitor starts charged and both currents initially vanish, its charge is the average of the two modal cosines, while the second charge is their difference. Trigonometric identities expose a fast carrier multiplied by a slow beat factor. The charge amplitude can migrate between loops; a zero charge envelope does not itself imply zero current or uniquely partition the mutual magnetic energy. Unequal bare frequencies bias the modes toward individual resonators, so an excitation may exchange only part of its energy. Dissipation reduces the amplitude available for later returns.

A measured spectrum also depends on which mode the source excites and which coordinate the detector observes. A symmetric drive may fail to excite the antisymmetric mode even when it exists. If damping widths are comparable with splitting, two eigenfrequencies need not appear as two distinct maxima. Conversely, interference can create a sharp dip between broad responses. Distinguish the mathematical eigenvalue problem from the driven transfer function and state the linewidth and excitation assumptions when interpreting an avoided crossing.