Transcript Heat Flow
Topics • • • • Unidentified Flying Objects Alien Abductions Flying Saucers End of the World April 1, 2014 Capacitance and Dielectrics • • • • • Capacitance Capacitance examples Energy stored in capacitor Dielectrics Nat’s research (just fun stuff) Capacitance • Electric potential always proportional to charge – Point 𝑉= – Sheet 𝑉= 𝑘𝑄 𝑟2 𝑄𝑑 𝐴ε𝑜 1 ε0 = 4π𝑘 = 8.85 ∙ 10−12 𝐶 2 /𝑁𝑚2 𝑄 𝑉 = 2πε 𝐿 ln 𝑟 – Wire 𝑜 • Define capacitance as ratio: 𝑄 – 𝐶=𝑉 – 𝐶= ε𝑜 𝐴 𝑑 (𝑢𝑛𝑖𝑡𝑠 (𝑢𝑛𝑖𝑡𝑠 𝐶 𝑉 ) (𝐶 2 /𝑁𝑚2 )𝑚2 𝑚 = 𝐶2 𝑁𝑚 𝐶 = 𝑉) • Measure of geometry’s ability to store charge • Charge create a voltage, but voltage requires charge Capacitance of Parallel Plate Constant electric field between two conducting sheets 𝜎 𝑄 𝑜𝐴 𝜀𝑜 = 8.85 ∙ 10−12 𝐶 2 𝑁𝑚2 𝐸=𝜀 =𝜀 𝑜 Potential between sheets 𝑄𝑑 𝑉= 𝜀𝑜 𝐴 Capacitance across sheets 𝑄 𝑄 𝜀𝑜 𝐴 𝐶= = = 𝑄𝑑 𝑉 𝑑 𝜀𝑜 𝐴 With Dielectric between 𝐶= 𝐾𝜀𝑜 𝐴 𝑑 𝐾 = 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Capacitance • Typical capacitors Capacitance examples 𝐶= 𝑄 𝑉 = 2500∙10−6 𝐶 850 𝑉 = 3.06𝜇𝐹 𝑄 = 𝐶𝑉 = 7 ∙ 10−6 𝐶 𝑉 12 𝑉 = 84𝜇𝐶 𝐶= 𝜀𝑜 𝐴 𝑑 𝐴= 𝐶𝑑 𝜀𝑜 = 0.2 𝐶 𝑉 0.0022 𝑚 8.85∙10−12 𝐶 2 𝑁 𝑚2 = 4.98 ∙ 107 𝑚2 <<<Huge = 0.2 𝐶 𝐽 𝐶 0.0022 𝑚 8.85∙10−12 𝐶 2 𝑁 𝑚2 Capacitance examples 𝑄 𝑜𝐴 𝐸=𝜀 𝑄 = 𝜀𝑜 𝐴𝐸 = 8.85 ∙ 10−12 𝐶 2 𝑁 𝑚2 .0035 𝑚2 8.5 ∙ 105 𝑉 𝑚 = 26.3 𝑛𝐶 𝑉= 𝑄 𝐶 = 72∙10−6 𝐶 0.8∙10−6 𝐶 𝑉 = 90 𝑉 𝑉 90 𝑉 𝐸 = 𝑑 = .002 𝑚 = 45,000 𝑉/𝑚 Capacitance examples 𝑄 = 𝐶𝑉 ∆𝑄 = 𝐶∆𝑉 18 𝜇𝐶 = 𝐶 ∙ 24 𝑉 𝐶 = 0.75 𝜇𝐶 Electrical Properties of Materials • Materials can do 2 things: • Polarize – Initial alignment of charge with applied voltage – Charge proportional to voltage – Temporary short-range alignment • Conduct – Continuous flow of charge with applied voltage – Current proportional to voltage – Continuous long-range movement Dielectrics • Polarizable material increases capacitance • Ability to “cancel” charge on plates draws more charge for given voltage, increases capacitance • Capacitance becomes 𝐶= 𝑘ε𝑜 𝐴 𝑑 (𝑘 𝑖𝑠 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) • Actually k isn’t a “constant”. Can vary with frequency, temperature, orientation, etc. Dielectric constants Dielectric Spectroscopy (Nat’s Research) • Most insulators contain polar molecules and free ions • • • These can align as a function of frequency (up to a point) Where they fail to align is called “relaxation frequency” Characteristic spectrum 2010-12-03 www.msi-sensing.com TDR Dielectric Spectroscopy • Sensor admittance from incident and reflected Laplace Transforms. Y ( ) G • v( ) r ( ) v( ) r ( ) Sample complex permittivity from sensor admittance. ( ) Y ( ) iCo • Differential methods • Bilinear calibration methods.1 • Non-uniform sampling.2 13 Dielectric Permittivity in Epoxy Resin 1 MHz -1 GHz • Aerospace resin Hexcel 8552. • High frequency range 1 MHz – 1 GHz. • Temperature constant 125°C, transition decreases with cure. • TDR measurement method. www.msi-sensing.com Permittivity in Epoxy Resin during Complete Cure Cycle www.msi-sensing.com Application to cement hydration • Cement Conductivity - Variation with Cure • Imaginary counterpart of real permittivity (’’). • Multiply by to remove power law (o’’). • Decrease in ion conductivity, growth of intermediate feature with cure • Frequency of intermediate feature does not match permittivity www.msi-sensing.com Basic signal evolution in cement paste3 • Permittivity (ε’ ) and conductivity (εoωε’’) from 10 kHz to 3 GHz. • Initial behavior at zero cure time. • Evolution with cure time. • Low, medium, and high (free) relaxations. 17 Dielectric modeling in cement paste • 1 Cole-Davidson, 2 Debye relaxations4-7 𝑅𝑒 −𝐼𝑚 2010-12-03 𝐶𝑙 1 + 𝜔𝜏𝑙 𝐶𝑙 1 + 𝜔𝜏𝑙 𝛽 𝛽 + 𝑅𝑒 𝜀𝑜 𝜔 − 𝐼𝑚 𝐶𝑚 𝐶ℎ + 𝑅𝑒 + 𝐶𝑝 𝜔 𝛾 1 + 𝜔𝜏𝑚 1 + 𝜔𝜏ℎ 𝐶𝑚 𝐶ℎ 𝜀𝑜 𝜔 − 𝐼𝑚 𝜀 𝜔 + 𝐶𝑖 1 + 𝜔𝜏𝑚 1 + 𝜔𝜏ℎ 𝑜 www.msi-sensing.com Model evolution with cement cure • Free-relaxation decreases as water consumed in reaction. • Bound-water8, grain polarization9 forms with developing microstructure. • Variations in frequency and distribution factor. • Conductivity decrease does not match free-water decrease. 2010-12-03 www.msi-sensing.com Energy stored in capacitor • Work to move charge across V 1 2 𝑊 = 𝑄𝑉𝑎𝑣𝑔 = 𝑄 𝑉𝑜 + 𝑉 = 𝑄𝑉 + • Define 𝑃𝐸 = 𝑒𝑛𝑒𝑟𝑔𝑦 = 1 𝑄𝑉 2 = 1 𝐶𝑉 2 2 = 1 𝑄2 2 𝐶 • Example 17-11 • Energy Density 1 2 𝑃𝐸 = 𝐶𝑉 2 = 1 𝜀𝑜 𝐴 2 2 𝐸 𝑑 2 𝑑 1 2 = 𝜀𝑜 𝐸 2 (𝐴𝑑) • Energy Density proportional to field squared! V 0