Motor theory is based on frames of reference. Synchronous motors have stator and rotor synchronous reference frames from which to express mechanical and electrical field activity. Induction motors have three; the rotor magnetic field vector is not tied to the rotating mechanical rotor, but "slips" with respect to it. Reference frames also apply in simple form to unmoving magnetic circuits. The result can be either a useful application of motor thinking to power conversion magnetics, or it can help in understanding motor reference-frame theory.

Transformer circuit quantities can be expressed in terms of the primary or secondary circuit. When the secondary circuit is referred to the primary (some engineers say "reflected," by referred is the preferred terminology), the secondary element values are transformed by the turns ratio, n, (or for some quantities, by n²), which is the referral ratio. The primary circuit, like a motor stator circuit, offers a view that is electrically equivalent to a secondary-referred equivalent circuit, though from the primary side. In synchronous or induction motors, the rotor windings are the secondary of a transformer in which the secondary windings move. Motors are transformers with a moving winding.

The simplest case of a magnetic device is that of an inductor. Inductors usually have electrical windings with multiple turns. The various magnetic and electrical relations interplay in a way that suggests a circuit reference-frame and a magnetic field reference-frame. The terminal electrical quantities are in the circuit frame. For instance, current, i, flowing through the inductor from its terminals does not "appear" to the field as i but as N× i, for each of the N loops of current adds to the total field flux an amount f , of flux. To the field flux, f , the voltage across each turn is v/N, but at the inductor terminals, the applied voltage is v, the sum of the per-turn voltages. The table summarizes the two frames of reference.

From the terminals (circuit frame), the flux referred to the circuit is the flux linkage,

This circuit quantity relates the circuit current and inductance:

The corresponding equation, referred to the field, is the "magnetic Ohm's Law" (MWL):

f  = L× (N× i)

The field flux, f , is l referred to the per-turn field reference-frame, the permeance, L, is the per-turn-squared inductance, and N× i is the field-referred current, or "MMF."

The two equations are equivalent but expressed in circuit and field reference-frames. The relating quantity is the number of turns, N. By multiplying MW L by N and substituting,

N× f  = N× L× (N× i)

or

l  = N^2× L× i = L× i

and

L = N^2× L

The permeance symbol in magnetics manufacturers' core data is typically the non-mnemonic symbol, A_L. It is the field-referred inductance.

The field current of N× i and turn-voltage of v/N transform the circuit impedance to a "field impedance" of

Each turn has an applied or induced voltage of v/N while enclosing a field caused by a loop with an equivalent current of N× i. Current viewed in the field frame is increased while voltage is decreased relative to the inductor terminals.

The familiar n² turns-ratio transform of transformer impedances relates to N² for single-winding inductors. This circuit-field transform of reference-frames is more basic and general than for transformers alone, and appears as a factor relating circuit and field quantities in inductors.

Ó Dennis L. Feucht, 2001