handbook of modern sensors

46 3 Physical Principles of Sensing
(A)
(B)
Fig. 3.5. Cylindrical capacitor (A); capacitive displacement sensor (B).
Formula (3.20) is important for the capacitive sensor’s design. It establishes a relationship
between the plate area and the distance between the plates. Varying either
of them will change the capacitor’s value, which can be measured quite accurately
by an appropriate circuit. It should be noted that Eqs. (3.19) and (3.20) hold only for
capacitors of the parallel type. A change in geometry will require modified formulas.
The ratio A/d may be called a geometry factor for a parallel-plate capacitor.
A cylindrical capacitor, shown in Fig. 3.5A, consists of two coaxial cylinders of
radii a and b and length l. For the case when l ≫ b, we can ignore fringing effects
and calculate capacitance from the following formula:
C = 2πε 0l
ln(b/a) . (3.21)
In this formula, l is the length of the overlapping conductors (Fig. 3.5B) and
2πl[ln(b/a)] −1 is called a geometry factor for a coaxial capacitor. A useful displacement
sensor can be built with such a capacitor if the inner conductor can be
moved in and out of the outer conductor. According to Eq. (3.21), the capacitance of
such a sensor is in a linear relationship with the displacement, l.
3.2.2 Dielectric Constant
Equation (3.20) holds for a parallel-plate capacitor with its plates in vacuum (or air,
for most practical purposes). In 1837, Michael Faraday first investigated the effect of
completely filling the space between the plates with a dielectric. He had found that
the effect of the filling is to increase the capacitance of the device by a factor of κ,
which is known as the dielectric constant of the material.
The increase in capacitance due to the dielectric presence is a result of molecular
polarization. In some dielectrics (e.g., in water), molecules have a permanent dipole
moment, whereas in other dielectrics, molecules become polarized only when an external
electric field is applied. Such a polarization is called induced. Both cases, either
permanent electric dipoles or those acquired by induction, tend to align molecules