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AstroFreak
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Homework Statement
The stator of the magnetic circuit of the above figure has a coil of turns 200, which carries a current of 1A. The rotor has a radius of 30mm, a height of 30mm and an axial depth of 20mm. The material for the stator and rotor has infinite permeability. The air gap between the stator and the rotor is 0.1mm and increases rapidly outside the overlap.
a) Determine the max. flux in the air gap and obtain an expression for the inductance of the stator coil as a function of \theta.
b) Find the max. field energy in each of the air gaps and the corresponding value of \theta.
c) Calc. the max. torque of that this device can produce.
Homework Equations
Ni = H_c l_c + H_g l_g
B = \mu_0 H
The Attempt at a Solution
To find the max flux density in the air gap, I use Ampere's Law:
Ni = H_c l_c + H_g l_g
Since the material of the core has infinite permeability, H_c = 0
Therefore, Ni = H_g l_g and since B = \mu_0 H
Ni = \frac{B}{\mu_0} \times 2 l_g
I then arrive at this expression:
B = \frac{Ni\mu_0}{2l_g}
Sub. the value of \mu_0 and the length of the air-gap (0.1 mm) and the number of turns (200) and I arrive at a value of 1.26T.
The value seems reasonable to me (i have no answers which I can check against).
The next part of the question has me stumped. How do I calc. the inductance of the machine?
I know that there is going to be an average inductance (lets call that L_0 and a max. one L_{max}
The inductance as a function of \theta is simply:
L(\theta) = L_0 + L_{max}cos(2\theta)
To compute the expression, I need to find the average inductance (which is simply \frac{L_{max} - L_{min}}{2}) and the max one. How can I go about doing this?
Any help?
I could use max. flux density (found above), with flux linkage as
\lambda = NBA and L = \frac{\lambda}{i}
But this only provides me with L_{max}. How can I go about finding L_{min} in order to find the average inductance and hence the expression?