Inductance matrices and magnetic energy

Flux linkage is the sum of oriented magnetic fluxes through every turn of a winding. If each turn sees the same flux, it is turn count times that single-turn flux. For fixed geometry and a linear material response, each current contributes proportionally to each linkage, defining an inductance matrix measured in henries. Leakage flux reduces coupling between coils without necessarily reducing each coil's self-inductance by the same fraction.

Choose current references entering dotted terminals and voltage drops from dot to undotted terminal. For windings whose dotted current references produce aiding flux, the mutual entry is positive. Reverse one winding's reference direction and its linkage reference consistently: both mutual entries change sign, while the self entries remain unchanged. The sign is therefore a convention tied to orientations, whereas the physical fields and total energy are unchanged by a consistent relabelling.

With negligible resistance, source power is the sum of current times linkage rate. For a constant symmetric inductance matrix this integrates to one half of the current vector contracted with its linkage. The mutual term appears once without a factor one half because two equal cross terms occur in the quadratic form. Dropping it ignores the energy of the shared field and predicts the same energy for aiding and opposing currents.

A passive system's energy cannot be negative for any current vector. Positive self-inductances therefore require the magnitude of mutual inductance to be no larger than the geometric mean of the self entries. Equality is an ideal perfectly coupled limit with a zero-energy current combination; ordinary leakage makes the determinant positive. Reciprocity can fail in a biased nonreciprocal medium, and hysteresis or saturation can invalidate this constant-matrix energy expression even when the circuit still uses the word inductance.

Connecting two windings in series imposes equal physical current magnitudes, but the signed current vector depends on their relative connection. Aiding references give an effective inductance equal to the sum of self terms plus twice the mutual entry. Opposing references subtract that cross contribution. With incomplete coupling the opposing connection generally retains positive inductance because some field energy resides in flux that fails to cancel.

The matrix is a reduced description of a distributed field. It assumes the circuit is small enough for propagation delays to be neglected and the selected current distributions are well defined. Eddy currents can make an inferred inductance frequency dependent; core nonlinearity can make differential inductance differ from linkage divided by current. Check the operating regime before applying a low-frequency constant value to a fast pulse or a saturating transformer.