# Relationship between brushless coil inductance and back EMF as a function of phase angle

In general, does the inductance across the coils of a brushless motor have the same waveform as the back EMF? That is, if back EMF is sinusoidal, should I expect the inductance to be sinusoidal [with twice the frequency]?

My understanding is that the inductance of a coil in the stator changes with rotor position when the nearby permeable material in the rotor magnets changes the effective permeability of the inductor depending on their proximity to the rotor magnets. I am assuming that the small test signal to probe inductance doesn't saturate the magnetic material, so it stays linear and reacts identically independent of the pole orientation. However, it's beyond my introductory understanding of magnetism whether the geometry that produces a certain waveform for back EMF also varies the inductance in the same way.

So far, I've taken a crude look at the inductance of a motor with a sinusoidal back EMF signal:

The measurement circuit

I see a vaguely sinusoidal dependence on rotor angle, but this preliminary glance was exceptionally ad hoc, using the ADC of a cheap microcontroller and a tabletop function generator to get a quick look into the system. I'm seeking a theoretical relationship between the inductance and back EMF before moving much further with the idea.

• I believe it is $$V(t)=L*dI/dt+I(t)*dL/dt$$ for incremental rotation within 1/2 phase or converted to trig for sinusoidal motion – Tony Stewart Sunnyskyguy EE75 Mar 30 '18 at 19:22
• On steppers you can generate a DC supply voltage when power is off, as I have done and miniature fans start to turn. – Tony Stewart Sunnyskyguy EE75 Mar 30 '18 at 19:30
• @TonyStewart.EEsince'75 Hmm, does that equation suggest that in general, voltage, current and inductance won't be the same waveform if inductance is constrained to half the frequency as current and voltage? For sinusoids, certainly one of them ought not to be sinusoidal I believe. – concat Mar 30 '18 at 20:20
• I was assuming during commutation interval. But I'm not sure on sin relationships of dI(t) and dL(t) if in phase or not. I suspect it is. where rotation speed increases voltage from flux rate of change but not dL(t) – Tony Stewart Sunnyskyguy EE75 Mar 30 '18 at 20:57