By William I. Orr, Stuart D. Cowan
This authoritative guide covers the layout, development, deploy, and operation of fifty two vertical antennas.
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It is therefore advantageous to design for a spring constant of 10 N/m or larger for high-reliability applications. 10 SELF-ACTUATION OF MEMS CAPACITIVE SWITCHES The power handling capability of MEMS switches is closely tied to the type of the switch contact (metal-to-metal or capacitive), the geometry of the device, and the circuit conﬁguration (series or shunt), and it is covered in detail in Chapter 7. 24. (a, b) Capacitive switches in shunt and series conﬁgurations. (c) Rectifying e¤ect of the V 2 component in the force equation.
4 2 MECHANICAL MODELING OF MEMS DEVICES: STATIC ANALYSIS SPRING CONSTANT OF CIRCULAR DIAPHRAGMS The spring constant of a circular diaphragm (membrane) can be calculated in a similar fashion to the ﬁxed–ﬁxed beam case. For a uniform load on the membrane and for a deﬂection point deﬁned at the center of the membrane, the spring constant is k ¼ k 0 þ k 00 ¼ 16pEt 3 þ 4pst 3R 2 ð1 À n 2 Þ ð2:24Þ where R and t are the membrane thickness and radius, respectively, and E; n, and s are deﬁned above. The spring constant is very high; and for a gold membrane with R ¼ 150 mm, t ¼ 0:5 mm, and s ¼ 5–20 MPa, we have k 0 ¼ 9 N/m and k 00 ¼ 32–126 N/m.
The expression for the spring constant of a ﬁxed–ﬁxed beam with a concentrated vertical load, P (N) in Fig. 1. Fixed–ﬁxed beam with concentrated vertical load P. load position given by  EI d 2y ¼ MA þ RA x dx 2 y¼ M A x 2 RA x 3 þ 2EI 6EI for x a a for x a a ð2:1Þ Pa MA ¼ À 2 ðl À aÞ 2 l RA ¼ P ðl À aÞ 2 ðl þ 2aÞ l3 where l is the length of the beam, MA (NÁm) is the reaction moment at the left end, and RA (N) is the vertical reaction at the left end. The moment of inertia, I , for a rectangular cross section is given by I ¼ wt 3 =12, where w is the width and t is the thickness of the beam.