Where does the maximum torque equation come from?

[CoolDude420]

Homework Statement

Homework Equations

The Attempt at a Solution

I'm quite new to these chapters and I can't seem to get my head around the theory and especially this question. That's the solution above. Where does the T_max = 1/8... equation come from? I understand that the mechanical work done in this system is the torque. And the derivative of work is power. I'm quite confused

 

[rude man]

I got an answer by:
computing difference in stored magnetic energy as i^/2/2 x (L_max - L_min))
and equating it to work done by torque over 90 degrees of rotor angle.
To compute i I used v = iωL.
I used the average value of L over the 90 degrees.
I can't really justify using averaged L, nor why the result should give max. torque rather than averaged. But maybe you can use this as a guide. My answer was 12.5% high.

(Were you given some kind of formula relating max. torque to Φ_max and delta reluctance? You could compute Φ_max from v = N dΦ/dt = ωNΦ. Realize that all these formulas are approximations).

 

[rude man]

I think the 1/8 factor is some kind of average number pulled out of the air by your professor.

It can be fairly accurately computed that
∫₀^π/2 T(θ)dθ = N²Φ²_max(1/L₁ - 1/L₂)
= Φ²_max(R₁ - R₂)
where R is reluctance and T is torque;
but the question is how to apportion T(θ) between θ=0 and θ=π/2. Depending on how you do I imagine the factor 1/8 could be conjured up.