Page 658 - The Mechatronics Handbook

0066_Frame_C20.fm Page 128 Wednesday, January 9, 2002 1:48 PM

Thus, the expressions for the electromagnetic force and torque components are derived. These equa-
tions provide one with a clear perspective on how to model, analyze, and control the electromagnetic
forces and torques changing the applied magnetic field because the terms

∂B
∂------ i
∂j
B ij = ∂B i and B ij()k = ---------
--------
∂j ∂k
can be viewed as the control variables. It must be emphasized that the electromagnetic field (B ij and B (ij)k )
is controlled by regulating the current in the planar microwindings and designing the microwindings
(or other radiating energy microdevices). As was discussed, the derived forces and torques must be used
in the torsional-mechanical equations of motion for the microactuator, and, in general, the six-degree-
of-freedom microactuator mechanical dynamics results. These mechanical equations of motion are easily
integrated with the derived electromagnetic equations, and closed-loop systems can be designed to attain
the desired microactuator performance. These equations guide us to the importance of electromagnetic
features in the modeling, analysis, and design of microactuators.
Some Other Aspects of Microactuator Design and Optimization
In addition to the electromagnetic-mechanical (electromechanical) analysis and design, other design and
optimization problems are involved. As an example, let us focus our attention on the planar windings.
The ideal planar microwindings must produce the maximum electromagnetic field, minimizing the
footprint area, taking into consideration the material characteristics, operating conditions, applications,
power requirements, and many other factors. Many planar winding parameters and characteristics can
be optimized, for example, the dc resistance must be minimized to improve the efficiency, increase the
flux, decrease the losses, etc. To attain good performance, in general, microwindings have the concentric
circular current path and no interconnect resistances. For N-turn winding, the total dc resistance r t is
found to be

N
2pr 1
r t = ---------- ∑ -------------------------
(
r Ok/ r Ik )
t w ln
k=1
where ρ is the winding material resistivity, t w is the winding thickness, r Ok and r Ik are the outer and inner
radii of the k-turn winding, respectively.
To achieve the lowest resistance, the planar winding radii can be optimized by minimizing the resis-
tance, and the minimum resistance is denoted as r tmin . In particular, making use of first- and second-
order necessary conditions for minimization, one has

2
dr t d r t
-------- = 0 and -------- > 0
2
dr w
dr w
where r w is the inner or outer radius of an arbitrary turn of the optimized planar windings from the
standpoint of minimizing the resistance.
Then, the minimum value of the microcoil resistance is given by

r tmin = 2pr N
-------------------------------
(
t w r OR /r IR )
where r OR and r IR are the outer and inner radii of the windings (i.e., r O Nth microcoil and r I 1st microcoil ),
respectively.

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