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

2. 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

3. 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

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

5. 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 = 𝐶 𝑉 𝑚 8.85∙ 10 −12 𝐶 2 𝑁 𝑚 2 = 𝐶 𝐽 𝐶 𝑚 8.85∙ 10 −12 𝐶 2 𝑁 𝑚 2
=4.98 ∙ 10 7 𝑚 <<<Huge

6. 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 𝑚 ∙ 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

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

8. 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

9. 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.

10. Dielectric constants

11. 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

12. 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.

13. Permittivity in Epoxy Resin during Complete Cure Cycle

14. 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

15. 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.

16. 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

17. 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.

18. Energy stored in capacitor
Work to move charge across V
𝑊=𝑄 𝑉 𝑎𝑣𝑔 =𝑄 𝑉 𝑜 +𝑉 = 1 2 𝑄𝑉
Define
𝑃𝐸=𝑒𝑛𝑒𝑟𝑔𝑦= 1 2 𝑄𝑉= 1 2 𝐶 𝑉 2 = 𝑄 2 𝐶
Example 17-11
𝑃𝐸= 1 2 𝐶 𝑉 2 = ∙ 10 −6 𝐶 𝑉 𝑉 2 =36 𝐶∙𝑉=36𝐽
𝑃𝑜𝑤𝑒𝑟= 36 𝐽 10 −3 𝑠 =36 𝑘𝑊 V +

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

20. 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