Abstract
This chapter begins with a detailed analysis on the design and modeling varies micro-relays. Analytical formulations for the switching voltages, spring design and modeling, and the dynamic behavior of micro-relays are established. These delay and energy models are then used for relay energy-delay optimization and scaling in Chap. 5.
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Notes
- 1.
The second solution to the force-balance equation is a mono-stable position (marked by the square symbol in Fig. 2.3). This is because, at such position, a small increase in the displacement will result in the electrostatic force to be greater than the spring force, causing the gap to close abruptly. Similarly, a small reduction in the displacement will result in the spring force to dominate, which will force the mechanical structure to move to the stable position (as marked by the circle symbol).
- 2.
To evaluate the integral \( {\displaystyle \underset{0}{\overset{x}{\int }}}{\left\langle x-{x}_0\right\rangle}^ndx \), we note that \( {\displaystyle \underset{0}{\overset{x}{\int }}}{\left\langle x-{x}_0\right\rangle}^ndx\equiv {\displaystyle \underset{0}{\overset{x}{\int }}}{\left(x-{x}_0\right)}^nu\left(x-{x}_o\right)dx \);
For x < xo, \( {\displaystyle \underset{0}{\overset{x}{\int }}}{\left(x-{x}_0\right)}^nu\left(x-{x}_o\right)dx=0 \);
For x ≥ xo, \( {\displaystyle \underset{0}{\overset{x}{\int }}}{\left(x-{x}_0\right)}^nu\left(x-{x}_o\right)dx={\displaystyle \underset{x_o}{\overset{x}{\int }}}{\left(x-{x}_0\right)}^ndx=\frac{1}{n+1}{\left(x-{x}_0\right)}^{n+1} \)
Therefore \( {\displaystyle \underset{0}{\overset{x}{\int }}}{\left(x-{x}_0\right)}^nu\left(x-{x}_o\right)dx=\frac{1}{n+1}{\left(\mathrm{x}-{x}_0\right)}^{n+1}u\left(x-{x}_o\right)=\frac{1}{n+1}{\left\langle x-{x}_0\right\rangle}^{n+1} \)
- 3.
Note that the capacitance associated with the wires used to interconnect a relay-based circuit would likely scale with \( \sqrt{A} \) rather than A. Furthermore, parasitic interconnect capacitance would likely to increase with beam length L. However, using the linear dependence on A greatly simplifies the calculations, and the overall findings are relatively unaffected by these simplifications.
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Appendix: Spring Constant of a Pinned-Pinned Beam
Appendix: Spring Constant of a Pinned-Pinned Beam
To calculate the spring constant of a pinned-pinned beam, we first take advantage of the symmetry (as depicted in Fig. 2.26) to obtain the reaction force at the pinned-end to be F/2:
Deflection of a pinned-pinned beam
For a load uniformly distributed throughout the span of the beam, the bending moment at x can be expressed by
At the pinned end, bending moment is zero; therefore Eq. (2.98) becomes
Substituting Eq. (2.99) into the beam equation, we obtain
where E/(1 − v2) is used instead of Young Modulus E to account for the plate effect. Integrating both sides by x, the slope of the beam can be obtained:
By symmetry, \( {\left.\frac{dz}{dx}\right|}_{x=L/2}=0 \); therefore
And the beam deflection is
Hence, the maximum beam deflection at x = L/2 is
Finally, the spring constant can readily be obtained:
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Kam, H., Chen, F. (2015). Design and Modeling of Micro-relay. In: Micro-Relay Technology for Energy-Efficient Integrated Circuits. Microsystems and Nanosystems, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2128-7_2
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