scaling laws in mems
TRANSCRIPT
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Scaling Laws
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Micro Intuition
• Linear extrapolation is easy but we are at a loss when considering the implications that shrinking of length has on surface area to volume ratios and on the relative strength of external forces
• Micro intuition can be misleading.
• Our aim is to develop a systematic approach about the likely behavior of downsized systems so we do not need to rely on micro intuition alone.
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Scaling Laws
• They allow us to determine whether physical phenomena will scale more favorably or will scale poorly.
• Generally, smaller things are less effected by volume dependent phenomena such as mass and inertia, and are more effected by surface area dependent phenomena such as contact forces or heat transfer.
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As you decrease the size
• Friction > inertia
• Heat dissipation > Heat storage
• Electrostatic force > Magnetic Force
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Surface Area to Volume Ratio
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Surface Area : Volume Ratio
For Example
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What are the implications of this?
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• Volume relates, for example, to both mechanical and thermal inertia. Thermal inertia is a measure on how fast we can heat or cool a solid. It is an important parameter in the design of a thermally actuated devices.
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Thermal Scaling
• Small object will loose heat rapidly, the dissipation of waste heat is not problematic in many cases.
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Mathematical Approach
• Mathematically, a scaling law is a law that describes the variations of physical quantities with the size of the system.
• Use of dimensional analysis.
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Scaling effects on spring constant (k)
• Consider a beam, length L, width w, Thickness t, and Youngs Modulus E.
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Stress in a rod connected to a mass experiencing a
constant acceleration
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Resistance
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Resistance
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Capacitance
Given a parallel plate capacitor with plate area wL=A and plate separation d, the capacitance is given as
C = ε A / d
where ε = permittivity of gap insulator material
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Electrostatic Forces
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Electrostatic Forces
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Electromagnetism
• Faraday’s law governs the induced force (or a motion) in the wire under the influence of a magnetic field.
• The scaling of electromagnetic force follows: F ∝ S4.
• For electromagnets, as S decreases, these forces decrease because it is difficult to generate large magnetic fields with small coils of wire.
• However permanent magnets maintain their strength as they are scaled down in size, and it is often advantageous to design magnetic systems that use the interaction between an electromagnet and a permanent magnet.
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Magnet Scaling
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Fluid Mechanics
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Use of Matrix Formalization
• To design micromechanical actuators, it is helpful to understand how forces scale. Use of a matrix formalism notation is very handy to describe how different forces scale into the small (and large) domain. It is called the Trimmer’s vertical bracket approach.
• Formulated by William Trimmer in 1986.
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Use of Matrix Formalization
• The top element in this notation refers to the case where the force scales as S1. The next one down refers to a case where the force scales as S2, etc.
• If the system becomes one-tenth its original size, all the dimensions decrease by a tenth. The mass of a system, m, scales as (S3) and, as systems become smaller, the scaling of the force also determines the acceleration (a), transit time (t), and the amount of power per unit volume (PV-1)
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Scaling of forces
• The force due to surface tension scales as S1
• The force due to electrostatics with constant field scales as S2
• The force due to certain magnetic forces scales as S3
• Gravitational forces scale as S4
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• Summarizing The Trimmer Notation
Order Force Scale, F Acceleration, a Time, t Power Density, P/V
1 1 -2 1.5 -2.5
2 2 -1 1 -1
3 3 0 0.5 0.5
4 4 1 0 2
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• List of Physical Phenomena and their scaling
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Benefits Of Scaling
• Speed (Frequency increase, Thermal Time constraints reduce)
• Power Consumption (actuation energy reduce, heating power reduces)
• Robustness (g-force resilience increases)
• Economy (batch fabrication)