Capacitance and Dielectrics Capacitance examples Energy stored in capacitor Dielectrics Nat’s research (just fun stuff)

Capacitance Electric potential always proportional to charge Point 𝑉= 𝑘𝑄 𝑟 Sheet 𝑉= 𝑄𝑑 𝐴ε 𝑜 ε 0 = 1 4π𝑘 =8.85∙ 10 −12 𝐶 2 /𝑁 𝑚 2 Wire 𝑉= 𝑄 2π ε 𝑜 𝐿 ln 𝑟 Define capacitance as ratio: 𝐶= 𝑄 𝑉 (𝑢𝑛𝑖𝑡𝑠 𝐶 𝑉 ) 𝐶= ε 𝑜 𝐴 𝑑 (𝑢𝑛𝑖𝑡𝑠 (𝐶 2 /𝑁 𝑚 2 ) 𝑚 2 𝑚 = 𝐶 2 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 𝐸= 𝜎 𝜀 𝑜 = 𝑄 𝜀 𝑜 𝐴 𝜀 𝑜 =8.85∙ 10 −12 𝐶 2 𝑁 𝑚 2 Potential between sheets 𝑉= 𝑄𝑑 𝜀 𝑜 𝐴 Capacitance across sheets 𝐶= 𝑄 𝑉 = 𝑄 𝑄𝑑 𝜀 𝑜 𝐴 = 𝜀 𝑜 𝐴 𝑑 With Dielectric between 𝐶= 𝐾 𝜀 𝑜 𝐴 𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 "𝐾"=𝑑𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Capacitance Typical capacitors Temporarily store charge in circuit Example: AC to DC power supply

Capacitance examples 𝐶= 𝑄 𝑉 = 2500∙ 10 −6 𝐶 850 𝑉 =3.06𝜇𝐹 𝑄=𝐶𝑉= 7∙ 10 −6 𝐶 𝑉 12 𝑉 =84𝜇𝐶 𝐶= 𝜀 𝑜 𝐴 𝑑 𝐴= 𝐶𝑑 𝜀 𝑜 = 0.2 𝐶 𝑉 0.0022 𝑚 8.85∙ 10 −12 𝐶 2 𝑁 𝑚 2 = 0.2 𝐶 𝐽 𝐶 0.0022 𝑚 8.85∙ 10 −12 𝐶 2 𝑁 𝑚 2 =4.98 ∙ 10 7 𝑚 2 <<<Huge

Capacitance examples 𝐸= 𝑄 𝜀 𝑜 𝐴 𝑄= 𝜀 𝑜 𝐴𝐸 𝐸= 𝑄 𝜀 𝑜 𝐴 𝑄= 𝜀 𝑜 𝐴𝐸 = 8.85∙ 10 −12 𝐶 2 𝑁 𝑚 2 .0035 𝑚 2 8.5∙ 10 5 𝑉 𝑚 =26.3 𝑛𝐶 𝑉= 𝑄 𝐶 = 72∙ 10 −6 𝐶 0.8∙ 10 −6 𝐶 𝑉 =90 𝑉 𝐸= 𝑉 𝑑 = 90 𝑉 .002 𝑚 =45,000 𝑉/𝑚

Capacitance examples 𝑄=𝐶𝑉 ∆𝑄=𝐶∆𝑉 18 𝜇𝐶=𝐶 ∙ 24 𝑉 𝐶=0.75 𝜇𝐶

Materials can do 2 things: 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 𝐶= 𝑄 𝑉 Polarizable material increases capacitance Partially canceling electric file between plates (battery not hooked up) Drawing more charge to restore field (battery hooked up) 𝐶= 𝑄 𝑉 Capacitance becomes 𝐶= 𝐾ε 𝑜 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

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.

Dielectric modeling in cement paste 1 Cole-Davidson, 2 Debye relaxations4-7 𝑅𝑒 𝐶 𝑙 1+𝜔 𝜏 𝑙 𝛽 +𝑅𝑒 𝐶 𝑚 1+𝜔 𝜏 𝑚 +𝑅𝑒 𝐶 ℎ 1+𝜔 𝜏 ℎ + 𝐶 𝑝 𝜔 𝛾 −𝐼𝑚 𝐶 𝑙 1+𝜔 𝜏 𝑙 𝛽 𝜀 𝑜 𝜔−𝐼𝑚 𝐶 𝑚 1+𝜔 𝜏 𝑚 𝜀 𝑜 𝜔−𝐼𝑚 𝐶 ℎ 1+𝜔 𝜏 ℎ 𝜀 𝑜 𝜔+ 𝐶 𝑖 2010-12-03 www.msi-sensing.com

Model evolution with cement cure MS&T 07 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 𝑊=𝑄 𝑉 𝑎𝑣𝑔 =𝑄 1 2 𝑉 𝑜 +𝑉 = 1 2 𝑄𝑉 Define 𝑃𝐸=𝑒𝑛𝑒𝑟𝑔𝑦= 1 2 𝑄𝑉= 1 2 𝐶 𝑉 2 = 1 2 𝑄 2 𝐶 Example 17-11 𝑃𝐸= 1 2 𝐶 𝑉 2 = 1 2 660∙ 10 −6 𝐶 𝑉 330 𝑉 2 =36 𝐶∙𝑉=36𝐽 𝑃𝑜𝑤𝑒𝑟= 36 𝐽 10 −3 𝑠 =36 𝑘𝑊 V +

Energy stored in capacitor’s field 𝑃𝐸= 1 2 𝐶 𝑉 2 = 1 2 𝜀 𝑜 𝐴 𝑑 𝐸𝑑 2 = 1 2 𝜀 𝑜 𝐸 2 (𝐴𝑑) Energy density 𝑃𝐸 𝑣𝑜𝑙𝑢𝑚𝑒 = 1 2 𝜀 𝑜 𝐶 𝐸 2 𝐴𝑑 𝑣𝑜𝑙𝑢𝑚𝑒 = 1 2 𝜀 𝑜 𝐸 2 Energy Density proportional to field squared! V +

TDR Dielectric Spectroscopy Sensor admittance from incident and reflected Laplace Transforms. Sample complex permittivity from sensor admittance. Differential methods Bilinear calibration methods.1 Non-uniform sampling.2