Power (P) of a device is the energy (or work) produced or consumed by it per unit time, i.e., the rate of doing work.
The scaling factor for the electromagnetic force is proportional to the length of the conducting wire, represented as ℓ^2.
Acceleration (a) is scaled as a = λ^3 / λ^2 = λ.
Scaling laws help us to improve our understanding and manipulation of Microsystems behavior.
One must weigh the magnitudes of the possible consequences from the reduction on both the volume and surface of the particular device.
In a third-order scaling system, the acceleration (a) remains unchanged (l^0) when the weight is reduced by a factor of 9.
The Trimmer Force Scaling Vector, defined by William Trimmer in 1989, is represented as F = λ^3 λ^2 λ λ^0.
Counterintuitive features or special effects are often noticed even at macro/micro scales.
In a third-order scaling system, the power density (P/V0) will reduce by a factor of 3 when the weight is reduced by a factor of 9.
The pressure drop (ΔP) in capillary conduits is inversely proportional to the fourth power of the radius (r) of the conduit: ΔP ∝ 1/r^4.
The potential energy stored is given by U = -1/2 * ε * E^2 * V^2, where ε is the permittivity, E is the electric field, and V is the voltage.
An electromagnetic force (F) is generated when a conducting wire with passing electric current (I) subjected to an electromotive force (emf) is exposed to a magnetic field (B) with a flux (dΦ).
Force (F) required to move a mass by a distance (s) is the work done by a device and is given by Fs.
The volume (V) is proportional to the cube of the length (ℓ), expressed as V ∝ ℓ^3.
When size diminishes to extremely small levels, quantum mechanics, rather than Newtonian mechanics, has to be used for studying physical systems.
Thermal inertia is a measure of how fast we can heat or cool a solid.
Scaling of phenomenological behavior involves the scaling of both size and material characterizations, such as thermo fluids in microsystems.
The Trimmer Force Scaling Vector relates force scaling with other pertinent parameters in dynamics, such as force (F), acceleration (a), time (t), and power density (P/V0).
The ratio of power loss to available power is proportional to the square of the length (ℓ): (Power loss/Available power) ∝ ℓ^2.
According to Ampere’s Law, the relationship is given by B × L = I × μ₀, where B is the magnetic field, L is the length of the wire, I is the current, and μ₀ is the magnetic permeability of the material.
Electromagnetic forces are not commonly used in microsystems as a preferred actuation force because reducing the wire length by half results in a reduction of the force by 16 times (2⁴), whereas the reduction of electrostatic force with a similar reduction in size results in a factor of 4 (2²).
Time (t) is scaled as t = λ^1.5 / λ^0.5 = λ.
Scaling laws are proportionality relations of any parameter (e.g., volume) of an object (or system) with its length scale (l).
Volume leads to the mass and weight of device components and relates to both mechanical and thermal inertia.
Surface is related to pressure and the buoyant forces in fluid mechanics. For instance, surface pumping by using piezoelectric means is a practical way for driving fluid flow in capillary conduits.
Scaling in geometry involves the scaling of physical size of objects, including rigid body dynamics, electrostatic, and electromagnetic forces.
Electric resistance (R) is proportional to the resistivity (ρ) and length (L) of the conductor, and inversely proportional to the area (A) of the cross-section: R ∝ (L/A).
Resistive power loss (P) is proportional to the square of the voltage (V) and the length (L) of the conductor: P ∝ (V^2/L).
Electric field energy (U) is proportional to the permittivity (Ɛ) of the dielectric and the square of the electric field strength (E): U ∝ ƐE^2.
The force acting on a wire carrying an electrical current in a magnetic field is given by the expression F = IL × B, where F is the force in Newtons (N), I is the current in Amperes (A), B is the magnetic field in Teslas (T), and L is the length of the wire in meters.
The volumetric flow rate (Q) in capillary conduits is proportional to the fourth power of the radius (r) of the conduit: Q ∝ r^4.
In a third-order scaling system, the power consumption will reduce by a factor of 0.33 (3/9) when the volume is reduced by a factor of 9.
According to scaling laws, the electromagnetic force F is proportional to the fourth power of the length of the wire, F ∝ l⁴.
The resistance R of a material is given by R = ρL/A, where ρ is the resistivity, L is the distance, and A is the cross-sectional area through which the current flows. For a constant ρ, R is proportional to L^2, i.e., R ∝ L^2.
The mass moment of inertia (# $$) of the mirror about the y-axis is given by the formula # $$ = M * c^2, where M is the mass and c is the width of the mirror.
The mass (M) of the mirror is calculated using the formula mass = density * volume, where density is the mass density of the mirror material and volume is the product of the length, width, and height of the mirror.
Paschen's effect illustrates that the voltage V scales linearly with the gap d between electrodes, i.e., V ∝ ℓ.
The inertia of a solid is related to its mass and the acceleration required to initiate or stop the motion of a solid device component.
In a third-order scaling system, the time (t) will reduce by a factor of 3 (l^0.5 = 9^0.5 = 3) to complete the motion when the weight is reduced by a factor of 9.
The scaling law for current is derived from Ohm’s law, I = V / R. Given that V ∝ L and R ∝ L^2, the electrical current I scales as I ∝ L^(-1), meaning electrical current is proportional to L^(-1).
The electric field E is defined as volts (V) per distance (L). For a fixed electric field E, the voltage V is proportional to the distance L, i.e., V ∝ L.
Electromagnetic forces are the principal actuation forces in macroscale, or traditional motors and actuators.
A 50% reduction in the dimensions of the mirror will cause a 32 times reduction in the torque required to turn the mirror.
Energy is the capacity for doing work, i.e., Energy = Work.
The surface area (S) is proportional to the square of the length (ℓ), expressed as S ∝ ℓ^2.
Velocity (v) is proportional to the linear scale (l) divided by time (t), denoted as v ∝ l * t^(-1).
The scaling law for weight, inertia force, or electromagnetic force is proportional to the cube of the linear scale (l), denoted as l^3.
Thermal conductivity (K) of solids is a measure of how conductive a solid becomes when it is scaled down. A 10 times reduction in size causes a 10 times reduction in conductivity of the solid.
The scaling exponent for the second moment of area is 4.
The scaling exponent for electrical resistance is -1.
The scaling exponent for elastic potential energy is 2.
The scaling exponent for electric field energy is -2.
The potential energy U scales as U ∝ ℓ^3, meaning a 10 times reduction in the linear size of electrodes results in a 1000 times reduction in potential energy.
Electrostatic forces F_d, F_w, and F_L scale as F ∝ ℓ^2, meaning a 10 times reduction in electrode linear dimensions results in a 100 times reduction in the magnitude of the electrostatic forces.
Rigid body dynamics is applied in the design of micro actuations and micro sensors, such as micro accelerometers (inertia sensors).
The ratio of surface area to volume (S/V) is proportional to the inverse of the length (ℓ), expressed as S/V ∝ ℓ^-1.
Micro mirrors are essential parts of micro switches used in fiber-optic networks in telecommunication. These mirrors are expected to rotate in a tightly controlled range at high rates, where angular momentum is a dominating factor in both rotation control and the rate of rotation.
From particle kinematics, displacement (s) is given by s = v0 * t + (1/2) * a * t^2, where v0 is the initial velocity.
The scaling law for electromagnetic force with constant current density is proportional to the fourth power of the linear scale (l), denoted as l^4.
The Fourier number is related to how fast heat can be conducted in solids. A 10 times reduction in size causes a 100 times reduction in time to heat the solid.
The scaling exponent for strength is 2.
The scaling exponent for electrical capacitance is 1.
The scaling exponent for strength to weight ratio is -1.
The scaling exponent for available power is 3.
It is important to understand the effect of size reduction on power (P), force (F), or pressure (p), and the time (t) required to deliver the motion in order to design effective micro actuations and micro sensors.
Power density (P/V0) of a device can be expressed as P/V0 = P = (F * s) / V0.
A micro or nano elephant might find it easier to fly because the surface area to volume ratio (S/V) increases as the size decreases, enhancing the relative effect of buoyancy forces.
Displacement (s) is proportional to the linear scale (l), denoted as s ∝ l.
The scaling law for fluid force or electrostatic force is proportional to the square of the linear scale (l), denoted as l^2.
The scaling exponent for mass is 3.
The scaling exponent for natural frequency is -1.
The scaling exponent for surface tension and van der Waals force is 1.
The scaling exponent for thermal time constant is 2.
The scaling exponent for electromagnetic force is 3.
A 10 times reduction in length results in a 1000 times reduction in volume (10^3) and a 100 times reduction in surface area (10^2).
A 50% reduction in the dimension of a micro mirror would significantly reduce the torque required to turn it, due to the scaling laws affecting angular momentum and rotational dynamics.
When the initial velocity (v0) is zero, acceleration (a) is given by a = 2 * s / t^2.
William Trimmer proposed a matrix representation for force scaling, called the 'force scaling vector F'.
The scaling exponent for mass moment of inertia is 5.
The scaling exponent for Reynolds number is 2.
The scaling exponent for fluid force is 2.
The scaling exponent for heat capacity is 3.
The scaling exponent for electrostatic force is 2.
An elephant can't fly as easily as a dragonfly because the volume (related to mass) scales differently compared to the surface area (related to buoyancy force), making it harder for larger animals to achieve flight.
The dynamic force (F) acting on a rigid body in motion with acceleration (a) is given by Newton’s second law as F = M * a.
The scaling law for surface tension force is proportional to the linear scale (l), denoted as l^1.
The scaling exponent for bending stiffness is 1.
The scaling exponent for shear stiffness is 1.
The scaling exponent for inductance is 1.
The scaling exponent for resistance power loss is 1.
The scaling exponent for power loss to power available ratio is -2.
