Dielectrics capacitance 2017mk

Nguyễn Công Phương Engineering Electromagnetics Dielectrics & Capacitance Contents I II III IV V VI VII VIII IX X XI XII XIII XIV XV Introduction Vector Analysis Coulomb’s Law & Electric Field Intensity Electric Flux Density, Gauss’ Law & Divergence Energy & Potential Current & Conductors Dielectrics & Capacitance Poisson’s & Laplace’s Equations The Steady Magnetic Field Magnetic Forces & Inductance Time – Varying Fields & Maxwell’s Equations Transmission Lines The Uniform Plane Wave Plane Wave Reflection & Dispersion Guided Waves & Radiation Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectrics & Capacitance Dielectric Materials Boundary Conditions for Perfect Dielectric Materials Capacitance Using Field Sketches to Estimate Capacitance Current Density & Flux Density Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectric Materials (1) + – – d E Q + E • Dipole moment: p = Qd • Q: the positive one of the bound charges • d: the vector from the negative to the positive charge Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectric Materials (2) • Dipole moment: p = Qd • If there are n dipoles per unit volume, then the total dipole moment in Δv: ptotal = n∆v ∑ pi i =1 • The polarization: n∆v P = lim pi ∑ ∆v→0 ∆v i =1 • Unit: C/m2 Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectric Materials (3)ΔS E Density: n molecules/m3 ∆v = d cos θ∆S + + – – ΔS – ∆Qb = nQ∆v + + + – + + + – + θ – – → ∆Qb = nQd cos θ∆S – – = nQd.∆S → ∆Qb = P.∆S p = Qd → P = nQd → Qb = − ∫ P.dS S Gauss’s law: QT = QT = Qb + Q ∫S ε 0E.dS → Q = QT – Qb + →Q= – d cos θ d cos θ d ∫ S (ε 0E + P ) dS (Q: the total free charge) Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectric Materials (4) Q= ∫ S ( ε 0E + P ) dS Gauss’s law: Q = Divergence theorem: ∫ S ∫ S D.dS → D = ε 0E + P D.dS = ∫ ∇.Ddv v Q = ∫ ρ v dv → ∇.D = ρv V Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectric Materials (5) • D = ε0E + P • In an isotropic material, E & P are always parallel, regardless of the orientation of the field • P = χeε0E • χe : the electric susceptibility • → D = ε0E + P = ε0E + χeε0E = (χe + 1)ε0E • εr = χe + 1: the relative permitivity • → D = ε0εrE = εE • ε = ε0εr : the permitivity Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectric Materials (6) N Ida Engineering Electromagnetics Springer, 2015, pp 175 Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectrics & Capacitance Dielectric Materials Boundary Conditions for Perfect Dielectric Materials Capacitance Using Field Sketches to Estimate Capacitance Current Density & Flux Density Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 10 Capacitance (7) C= C= d1 d2 + ε1S ε S = ε1S1 + ε S d Area, S Conducting plates 1 + C1 C2 ε2 d2 ε1 d1 d Conducting plates = C1 + C2 S1 S2 ε1 ε2 Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn d 24 ρ L R01 V1 = ln 2πε R1 − ρ L R02 V2 = ln 2πε R2 y Capacitance (8) P(x, y, 0) R2 R1 (– a, 0, 0) (a, 0, 0) ρ L  R01 R02  → V = V1 + V2 = − ln  ln  2πε  R1 R2  ρL R01R2 = ln 2πε R02 R1 R01 = R02 R1 = ( x − a) + y R2 = ( x + a )2 + y – ρL z +ρL ρL ( x + a )2 + y →V = ln 2πε ( x − a )2 + y ρ L ( x + a )2 + y = ln 4πε ( x − a )2 + y Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 25 x y Capacitance (9) ρ L ( x + a) + y V = ln 4πε ( x − a )2 + y P(x, y, 0) R2 R1 (– a, 0, 0) (a, 0, 0) Choosing an equipotential surface V1, we define: – ρL z +ρL K1 = e4πεV1/ρL K1 + 2 → K1 = → x − ax + y + a =0 2 K1 − (x − a) + y ( x + a) + y 2  a K1   K1 +  → x−a   + y =   K1 −  K −    Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 26 x y Capacitance (10) K1 = e P(x, y, 0) R2 4πεV1/ ρL R1  a K1   K1 +  →x−a   + y =   K1 −  K − 1    (– a, 0, 0) (a, 0, 0) – ρL +ρL z • The V = V1 surface is independent of z → it is a cylinder • It intersects the xy plane in a circle of radius: b= 2a K1 K1 − K1 + & this circle is centered at (x = h, y = 0) where h = a K1 − Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 27 x y Capacitance (11) V0 = h V1 The V1 surface intersects the xy plane in a circle of radius K1 + & centered at (x = h, y = 0) where h = a K1 −  a = h − b2  → 2 h + h − b  K1 = b  x b 2a K1 b= K1 − 4πε V1 → ρL = ln K1 z If h, b & V1 are given then a, ρL & K1 can be found K1 = e 4πεV1 /ρ L 2πε L 2πε L ρ L L 4πε L = → C plane, cylinder = = = V1 ln K1 ln[( h + h − b )/b] cosh −1 (h/b) Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 28 Capacitance (12) Ex Given the system, find the location & the magnitude of the equivalent line charge, & the location of the 50V equipotential surface a= y V0 = The equivalent line charge h − b = 132 − = 12 m K1 = h+ h −b b 2 = 13 + 12 =5 x V1 = 100 V h = 13 m b=5m → K1 = 25 4π × 8.854 × 10−12 × 100 4πε V1 → ρ L = = 3.46 nC /m ρL = ln 25 ln K1 2πε 2π × 8.854 × 10−12 C plane, cylinder = = = 34.6 pF/m −1 −1 cosh (h/b) cosh (13/ 5) Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 29 Capacitance (13) Ex Given the system, find the location & the magnitude of the equivalent line charge, & the location of the 50V equipotential surface y V0 = The equivalent line charge K = e 4πε V2 / ρ L =e π ×8.854 ×10 −12 × 50 / 3.46 ×10 −9 = 5.00 x V1 = 100 V h = 13 m 2a K 2 × 12 → b2 = = = 13.42 m K2 − 5−1 K2 + +1 h2 = a = 12 = 18 m K2 −1 −1 b=5m V3 = 25 V → b3 = 29.06 m, h3 = 31.44 m Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 30 y Capacitance (14) C plane, cylinder = V0 = h V1 2πε L ln[( h + h − b2 )/b ] b≪h → C plane, cylinder = C plane, wire → Cwire, wire = x b z 2πε L = 2h ln b y h πε L 2h ln b x z Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 31 Dielectrics & Capacitance Dielectric Materials Boundary Conditions for Perfect Dielectric Materials Capacitance Using Field Sketches to Estimate Capacitance Current Density & Flux Density Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 32 Using Field Sketches to Estimate Capacitance (1) • A conductor boundary is an equipotential surface • The electric field intensity E & the electric flux D are both perpendicular to the equipotential surfaces • E & D are perpendicular to the conductor boundaries & posses zero tangential values • The lines of electric flux, or streamlines, begin & terminate on charge & therefore, in a charge-free, homogeneous dielectric, begin & terminate only on the conductor boundaries Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 33 Using Field Sketches to Estimate Capacitance (2) ∆Ltan =1 ∆LN E & D are both perpendicular to the equipotential surfaces ∆ψ E= ε ∆Ltan ∆V E= ∆LN ∆ψ ∆V → = ε ∆Ltan ∆LN ΔLtan ∆Ltan ∆ψ → = const = ∆LN ε ∆V B A ΔLN Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn B’ A’ 34 Using Field Sketches to Estimate Capacitance (3) Q C= V0 Q = N Q ∆Q = N Q ∆ψ V0 = NV ∆V →C = ∆Ltan ∆LN NQ ∆ψ NV ∆V ∆ψ = const = =1 ε ∆V N Q ∆Ltan NQ →C= ε =ε NV ∆LN NV Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 35 Using Field Sketches to Estimate Capacitance (4) 15 30 46 100 V 80 62 Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 36 Dielectrics & Capacitance Dielectric Materials Boundary Conditions for Perfect Dielectric Materials Capacitance Using Field Sketches to Estimate Capacitance Current Density & Flux Density Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 37 Current Density & Flux Density J = σ Eσ D = ε Eε Eσ = −∇Vσ Eε = −∇ Vε J.dS = σ ∫ Eσ dS ∫S Vσ = − ∫ Eσ dL Q = ε ∫ Eε dS S Vε = − ∫ Eε dL I= S  − ∫ Eσ dL Vσ R = = I  σ ∫ Eσ dS  S →  Q ε ∫ S Eε dS = C = Vε − ∫ Eε dL  ε → RC = σ Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 38 ... Waves & Radiation Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn Dielectrics & Capacitance Dielectric Materials Boundary Conditions for Perfect Dielectric Materials Capacitance Using... sites.google.com/site/ncpdhbkhn Dielectrics & Capacitance Dielectric Materials Boundary Conditions for Perfect Dielectric Materials Capacitance Using Field Sketches to Estimate Capacitance Current Density & Flux Density Dielectrics. .. 3.2  ε1   Dielectrics & Capacitance - sites.google.com/site/ncpdhbkhn 16 Dielectrics & Capacitance Dielectric Materials Boundary Conditions for Perfect Dielectric Materials Capacitance Using