Page 228 - Mechanics of Microelectromechanical Systems

4. Microtransduction: actuation and sensing 215
symmetrically with respect to the horizontal diameter of the circular loop,
two forces dF, which are equal and opposite according to Eq. (4.62), will
produce an elementary couple about the horizontal diameter, and the
corresponding moment is:

The total moment that will tend to rotate the loop about the horizontal
diameter, as indicated in Fig. 4.34, can be calculated as:

By comparing Eqs. (4.60) and (4.65) it can be seen that the moments for both
the rectangular and the circular loops can be written as:

where A is the area of the loop. This remark enables to generalize the
formulation of the mechanical moment produced in a loop carrying a current
when subject to an external magnetic field in the vector form:

where m is called the magnetic dipole moment – see Sadiku [4], for instance,
and is calculated as:

where is the direction perpendicular to the loop’s plane. In the case n
loops are used to increase the actuation/sensing capacity, the corresponding
bending moment of Eq. (4.67) will be n times larger.
One of the simplest implementations of using a loop carrying current for
actuation/sensing purposes in MEMS is to place the respective loop at the
free end of a cantilever beam, as discussed in the next example.

Example 4.8
A microcantilever is used to sense an external magnetic field whose
direction is known, as sketched in Fig. 4.35. Determine the value of the
magnetic field B, assuming that the geometry and the material properties of
the microcantilever are known, as well as the tip slope, which is measured
experimentally.