3. Instability Considerations

One issue that has been ignored to this point is the stability of the comb-drive actuator. In the analysis presented in Section 3.2, it was assumed that all forces in the y- direction (perpendicular to the stroke) will cancel. However, that assumption is premised on the moving comb-fingers being exactly У2 way between the stationary fingers. In reality, the comb-fingers are always slightly off-center and the force in the y-direction is:

Where A(Ax) is the effective overlap area of a parallel plate capacitor on each side of the moving comb-finger. While A(Ax) does not explicitly take fringing fields into account, any capacitance value including fringing fields can be represented as an equivalent parallel plate case ignoring fringing fields.

We can now define a virtual spring constant in the y-direction (k_y-virtual) as the derivative of F_y with respect to y, evaluated at the Ay=0 (center) position:

This spring constant essentially represents the amount of instability present due to electrostatic forces. When this virtual spring constant of the electrostatic force exceeds the real mechanical spring constant of the suspension in the y-direction (k_y-real), an instability point is reached and both sets of comb-fingers will likely ‘snap’ together. If

we set k_y._virtual = k_y-real, we can find the maximum stable deflection point (Ax_max). For the case of a traditional, planar comb-drive, we start by re-arranging Equation 30 to be:

2

Substituting this V expression into Equation 39 (set to k_y-real), and using the fact that for the planar case A(x)=h-Ax, yields:

Collecting terms and solving for Ax, we find the maximum stable deflection point for a traditional, planar comb-drive to be:

Thus, the maximum displacement is actually dictated by the ratio of spring constants in the x- and y-directions, rather than their absolute value. It should be noted that the spring constants in Equation 42 are real, instantaneous values. While an approximation, Equation 42 can be used as a reasonable guideline for choosing a suspension design to suit your desired displacement needs. Further discussion on the design and performance of comb-drive suspensions is provided in Section 3.4.

For the case of a gray-scale tailored comb-finger however, Equation 42 is no longer applicable. Since the height is now a function of displacement, we must write A(Ax) as an integral:

Making Equation 39:

2

Similarly, Equation 40 no-longer holds as the V (x) relationship is now a complicated function dependent on h(x), Ax, k_x, N, e₀, and d:

Given a particular h(x) profile of the comb-finger, we can solve Equation 46 numerically for different values of displacement (explicit code is given in Appendix A). Using the comb-finger profiles and assumptions from Figure 3.5, k_y._virtuai was calculated as a function of displacement, as shown in Figure 3.10:

In Figure 3.10 a fictitious line has been added to represent an arbitrary value for k_y-reai. It is clear that a device with gray-scale variable height fingers will reach this limiting threshold earlier than a corresponding planar device would. Such behavior is expected from the gray-scale design because improved resolution was obtained by increasing the voltage required to generate the same displacement. Even though the overlap area of the gray-scale comb-fingers is smaller than the planar case, the fact that force scales with V over-compensates for the reduction in overlap area. Thus, vertically shaped gray-scale comb-fingers can be expected to have a net decrease in stability compared to traditional planar comb-drive designs.