Variable reluctance (inductance) transducers

A variable-reluctance (inductance) transducer senses the measurand through a change in the inductance of a coil due to a change in the reluctance of its magnetic path.

Principle

The inductance of a coil of $N$ turns is

$$L = \frac{N^2}{\mathcal{R}}, \qquad \mathcal{R} = \frac{l_g}{\mu_0 A}\ \text{(air gap dominates)}$$

so $L = \dfrac{N^2\mu_0 A}{l_g}$. Changing the air gap $l_g$, the area $A$, or the permeability $\mu$ changes $L$.

Single-coil, change of air-gap type

A U- or E-shaped iron core carries the coil. A ferromagnetic armature is separated from it by a gap $l_g$. When the armature moves with the measurand, $l_g$ changes, so $L$ changes.

     +---[ coil ]---+
     |              |
   __|__          __|__   <- iron core
   |___|          |___|
      gap l_g (variable)
   ===================    <- movable armature
         ^ x

The relation $L \propto 1/l_g$ is nonlinear; it is approximately linear only for small gap changes. The iron armature also exerts a magnetic pull on the moving member.

Differential (push-pull) type

Two identical coils are placed on opposite sides of a common armature. When the armature moves up, the gap of one coil decreases ($L_1$ increases) and the other increases ($L_2$ decreases). The coils are placed in adjacent arms of an ac bridge, so the output is proportional to $L_1 - L_2$.

   [ coil 1 ]    gap 1
   ============  <- armature (moves)
   [ coil 2 ]    gap 2

Why differential is preferred

• Even-order nonlinearity cancels, giving better linearity and about twice the sensitivity.
• The magnetic pull forces on the armature cancel.
• Effects of temperature, supply variation and stray fields on both coils cancel in the bridge.

Typical uses: small displacement, pressure (diaphragm) and proximity sensing.