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SCALING LAWS ยท PART 3 OF 3

Power density decides the winner.

Power is work per time, P = F d / t. For an actuator, what matters is how much power each bit of volume delivers: P / V.

P / V โˆ ( Sf ร— S1 รท S(4 โˆ’ f)/2 ) รท S3 = S(3f โˆ’ 8)/2
A negative exponent means the power per volume rises as the device shrinks. Try each force law in the simulator.

FOUR CASES

Two force laws win, two lose.

S1 force โ†’ P/V โˆ Sโˆ’2.5 Rises fast as the device shrinks
S2 force โ†’ P/V โˆ Sโˆ’1 Rises as the device shrinks
S3 force โ†’ P/V โˆ S0.5 Falls as the device shrinks
S4 force โ†’ P/V โˆ S2 Falls fast as the device shrinks
The dividing line is 3f โˆ’ 8 = 0, about f = 2.7: force laws that scale faster than S2 lose power density when they are made smaller.

ELECTROSTATIC vs MAGNETIC

Why MEMS run on electrostatics.

Electrostatic, [S2]

Shrink the electrodes 10ร—: the force drops 102 = 100ร—, but the power density rises 10ร—.

Magnetic, [S4]

A current-carrying wire in a field: field โˆ S, current โˆ S2, wire length โˆ S, so force โˆ S4. Shrink 10ร—: the force drops 104 = 10,000ร— and the power density falls 100ร—.

These numbers assume the electric field and the current density stay the same as the device shrinks. Real designs push both harder at small sizes, but the trend holds.

Comb drives and micromirrors are electrostatic; the motors in a car are magnetic. Each wins at its own size.

THE SCALING TABLE

How other properties scale.

Friction and adhesion Capillary [S1]; van der Waals, hydrogen bonding, electrostatic [S2]
Strength รท weight [Sโˆ’1]: small beams carry far more than their own weight
Stiffness, k = EA / L [S1]
Resonant frequency, โˆš(k / m) โˆš(S1 รท S3) = [Sโˆ’1]
Resistance, L / ฯƒA [Sโˆ’1]
Current, J ร— A [S2]
Capacitance, ฮตA / d [S1]
Reynolds number, ฯvd / ฮผ [S2] when the speed also scales with size
Diffusion time, about L2 / D [S2]: diffusing 10 ยตm is a million times faster than diffusing 1 cm

PUTTING IT TO WORK

What the table means for a design.

Resonators ring high Frequency โˆ Sโˆ’1, so a proof mass a few hundred microns across resonates in the kHz range, far above any shaking it measures.
Flow is smooth Reynolds number โˆ S2 is tiny in a microchannel, so flow is laminar and liquids mix only by diffusion, which is fast at these sizes.
Parts stick Adhesion [S1, S2] beats weight and spring forces, so released structures need dimples, careful drying and release holes.
Actuate electrostatically Power density โˆ Sโˆ’1: comb drives and parallel plates are the natural choice.
Before you design anything small, ask how each force scales. The answer usually picks the design for you. Try it in the Virtual Fab Lab.

UNIT 4 STUDY COMPLETE

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Ready for the Fab Challenge?

You've covered Power Density & Scaling in Real MEMS.

Based on Prof. Sam Kassegne's ME 585 lecture on scaling laws (San Diego State University) and William Trimmer's work on the scaling of micromechanical devices.