# Reluctance

UNDER CONSTRUCTION

I wanted to create a reluctance simulator but was stunned about the dearth of material concerning reluctance sensors or rather reluctance detectors and their operation. My thought is to use a gyrator to simulate an inductor, which would then be used for my reluctance simulator which in turn would interface with the provided black box reluctance detector.

So the first order of business is to establish a relationship between reluctance and inductance.

Assuming a magnetic circuit with a uniform flux and field inside an element, then the magnetic motive force, mmf, between the ends of the element is giving by equation 1a). As an aside notice the similarity of the mmf with the electro-motive force, emf, $latex emf=\mathcal{E}=E L$.

1a)                    $latex mmf=\mathcal{F}=H L_m$

1b)                    $latex mmf=\mathcal{F}=\frac{B L_m}{\mu}$

1c)                    $latex mmf=\mathcal{F}=\frac{L_m}{A_c\mu}\Phi$

Now given the equation for the magnetic circuit is

1d)                    $latex mmf=\mathcal{F}=R\Phi$

Therefore the reluctance is

2)                     $latex R=\frac{L_m}{A_c\mu}$

The next step is to determine the relationship between reluctance and inductance. The same assumptions used above still apply. The inductance is a terminal characteristic, therefore both Ampere’s and Faraday’s law is requied.

Starting with equation 1a) from above and using Ampere’s law

3a)                   $latex mmf=\mathcal{F}=H L_m= i$

Equation 3) is valid for a single turn coil. For multiple turn coils, equation 3a) is modified

3b)                   $latex mmf=\mathcal{F}=H L_m=N_c i$

The terminal voltage is given by Faraday’s law

3c)                   $latex v=N_c\frac{d\Phi}{d t}$

3d)                   $latex v=N_c A_c\frac{d B}{d t}$

3e)                   $latex v=N_c A_c\mu\frac{d H}{d t}$

3f)                   $latex v={N_c}^2\frac{A_c\mu}{L_m}\frac{d i}{d t}$

Now the terminal voltage is given by

3g)                  $latex v=L\frac{d i}{d t}$

Therefore the inductance is given by

4)                    $latex L={N_c}^2\frac{A_c\mu}{L_m}$

The inductance is scaled inverse reluctance, also known as permeance.