Silver-Nanoparticle Microcoil via Aerosol Jet Printing

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Silver-Nanoparticle Microcoil via Aerosol Jet Printing ( silver-nanoparticle-microcoil-via-aerosol-jet-printing )

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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) 􏰭 􏰎 􏰩3R􏰪2 􏰩2R􏰪2 􏰩R􏰪2􏰏􏰮 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) 􏰩3R􏰪2 Allow the definition of a constant, C, ηκ3 􏰭 􏰎 􏰩3R􏰪2 􏰩2R􏰪2 􏰩R􏰪2􏰏􏰮 C=4π π R2+ 4 + 4 + 4 R2 + + 􏰩2R􏰪2 + 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) 􏰩R􏰪2􏰏􏰮 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 κ􏰰x2+y2+z2 2 κ x2+y2+z2

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