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Sensors 2020, 20, 6087 10 of 13 microcoil is as follows: The current density can be simplified to current along the rings. Solving the vector potential, A(r ̄, t) for one ring with the largest radius, R [20], J ̄ r ̄′,t A ̄(r ̄,t)= μ0 coil r d3r ̄′ (A4) 4π |r ̄−r ̄′| After specifying the spherical coordinate system, A ̄ = φˆAφ = φˆμ0I(ω,t)R2rsin(θ)e−jκr (A5) 4(R2 +r2)3/2 ∂A ̄ =φˆjωμ0I(ω,t)R2rsin(θ)e−jκr (A6) ∂t 4 (R2 + r2)3/2 μκ3m 1 j 1 j 1 2 + e−jκr (A7) κr μκ3I(ω,t) 3R2 2R2 R2 B ̄=4π πR2+4+4+4 1 j 1 j 1 (A9) 2 +θˆsinθ where m is the magnetic moment pointing in the direction normal to the surface area made by the coil. B ̄ = rˆ2cosθ 4π (κr) 3 + 3 + m ̄ = IAmˆ (A8) rˆ 2 c o s θ 3 + (κr) 3R2 Allow the definition of a constant, C, ηκ3 3R2 2R2 R2 C=4π π R2+ 4 + 4 + 4 R2 + + 2R2 + φˆsinθ (A10) (A11) (A12) (κr) (κr) (κr) 2 + θˆ s i n θ The general radiated fields from an electrically small coil are represented by the superposition of 3 + four independent, concentric current rings of varying radii. The expression is shown below. e−jκr (κr) (κr) (κr) R2 4π 444 (κr)κr ηκ3I(ω,t) E ̄ = π j 1 2 − 2 + e − j κ r κr j1 E ̄ = φˆCsinθ 2 − e−jκr (κr) κr E ̄ is converted to Cartesian coordinates so that the activating function may be solved at some distance (A13) away from the coil plane. ̄ −1 x 2 + y 2 − y xˆ + x yˆ E=CI(ω,t) tan z x2+y2 √ j 1 −jκ x2+y2+z2 − e κx2+y2+z2 2 κ x2+y2+z2PDF Image | Silver-Nanoparticle Microcoil via Aerosol Jet Printing
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