Yes, I do not mean "domains", I am talking about the magnetic "domain" as compared to the electric "domain".

I'm glad and that's clear now.

BTW: I dislike analyzing inductors in the electric domain because they are predominantly current/flux devices at low frequencies.

However I like to analyze capacitors in the electric domain.

So the net flux through the coil is zero, the classical superconducting loop expelling flux.

If we begin by shorting such coil with zero flux penetrating it, then yes.

But it is possible to start with non-zero flux and then such coil will freeze this flux. This flux can do real work, e.g. attract soft iron piston from afar.

But we are not talking about an ideal shorted coil, we are dealing with a real load resistor, albeit of low value.

I view the resistance as inserting an imperfection into the coil. The moment the resistance is introduced, the energy stored in the coil starts to leak out.

It is like putting a hole in the bottom of the bucket and poring water into it. If you pour in more than leaks out through the hole you get the illusion that the water level depends on the speed with which new water is arriving. Without the hole, the level depends merely on the quantity of water arriving.

This was an analogy to di/dt and Δi

And I meant the induced current and flux that it creates is larger than the resultant flux through the coil, in other words close to the situation for your ideal shorted coil where the resultant flux is zero.

That's hard to understand without the words "absolute value" while disregarding signs or direction of the fluxes ...but now I know what you mean.

Again I am talking about a PM generator which in its simplest form could be a magnet rotating within a coil.

Your paper did not have a diagram of that rotating arrangement and this is a "piston project" thread that implies a reciprocating linear motion and linear forces.

Normally the peak induced voltage hence also coil current occurs when the magnet axis is at 90 degrees to the coil axis. This is also the position where that same coil current creates maximum torque on the magnet.

With a magnet rotating inside the coil like in a Joseph Newman arrangement, yes.

When the coil current is shifted by almost 90 degrees, peak current then occurs at a magnet position where torque is near zero.

Do you mean the coil's current shifted 90º in reference to the magnet's angle or in reference to something else?

Linear force on a magnetic dipole is proportional to the gradient of the flux density. The gradient does not appear in the torque formula, see below.

<snip>

A magnetic dipole of moment mu in a field B exhibits a torque T of magnitude T = mu*B*sin(theta) where theta is the angle between the dipole axis and the field B. I think you have confused this with the linear force F on a dipole that has a magnitude of F = mu*(dB/dx)*cos(theta) where B lies along the x axis.

With a magnet rotating inside the coil, your are correct about the torque. I thought you were referring to a linear force because this is a reciprocating piston thread.

Clever animation. But the real issue here is whether a magnet bouncing over a coil that is not quite shorted has any potential for OU operation, that electrical output can exceed mechanical input. You have not stated whether you agree with the magnetic domain phase shifts as indicated by my phasor diagram (although your ideal shorted coil analysis where the self-flux exactly cancels the applied flux would suggest that you do agree).

I don't know yet. We have to discuss it more.

Please elaborate on your statement in the paper:

"

*Because the current creating the Lenz reaction is at 90° phase with respect to the resultant flux in the coil...*"

First of all, I'd like to confirm, that you are referring to a temporal phase shift (not a spatial phase shift).

IMO any resistance in the coil's circuit is just an energy leak.

How can an energy that is leaking out make more energy?

But it is true that in the frequency domain the phase shift in a series RL circuit is dependent on L, f and R.

The currents in the power supply, in the resistance and in the inductor are always in phase, because it is a series circuit.

We have to be clear what phase shift we are discussing in an RL circuit, e.g. between:

1) Driving voltage and inductor's current

2) Driving voltage and inductor's voltage

3) Driving voltage and resistor's current

4) Driving voltage and resistor's voltage

5) Resistor's voltage and inductor's current

6) Resistor's voltage and inductor's voltage