Medan Listrik Statis II Material Kapasitansi Numerik Simulasi

Medan Listrik Statis II (Material, Kapasitansi, Numerik Simulasi) Sukiswo sukiswok@yahoo. com Medan Elektromagnetik. Sukiswo 1

ELECTROSTATICS - MATERIALS Sukiswo, Medan Elektromagnetik

CONDUCTORS and DIELECTRICS Conductors Dielectrics High conductivities; s (for Copper) ~ 5. 8 x 107 S/m Low conductivities; s (for Rubber) ~ 1 x 10 -15 S/m or 1/W-m Semiconductors (mid s’s) Permittivities, e = 1 -100 e 0 Note: e 0 is the permittivity of free space/vacuum = 8. 854 x 10 -12 F/m Medan Elektromagnetik. Sukiswo 3

CONDUCTORS Most electrons are stuck to the nucleus But, 1 or 2 electrons per atom are free to move This means that if you apply an external Efield, the free electrons will move Lattice of Nuclei Medan Elektromagnetik. Sukiswo 4

CONDUCTORS - - electrons + + + + + conductor Apply external E-field, • Force on electrons causes free electrons to move • Charge displacement causes response E-field (opposite to applied external E-field) Medan Elektromagnetik. Sukiswo 5

CONDUCTORS The electrons keep moving until, This means that: , in a conductor Conductor is equipotential Medan Elektromagnetik. Sukiswo 6

DIELECTRICS electron cloud + + nucleus electron cloud centered on nucleus Cloud shifts to setup Medan Elektromagnetik. Sukiswo 7

DIELECTRICS Define: dipole moment Polarization partially cancels applied Field Medan Elektromagnetik. Sukiswo 8

DIELECTRICS subtracts out bound charge Define: Displacement Flux Density ( C/m 2 ) Electric Field (V/m) is due to bound/dielectric charge and free charge is due to bound/dielectric charge only and opposite sign is due to free charge only Medan Elektromagnetik. Sukiswo 9

FREE CHARGES Examples of free charges: rs on conductor electron beam doped region of semi-conductor Gauss’ Law uses just free charge Most general form Medan Elektromagnetik. Sukiswo 10

DIELECTRICS Don’t need to know about bound charges to find Many materials have Define , where Typically, Medan Elektromagnetik. Sukiswo 11

DIELECTRIC BREAKDOWN Example: Arc in Air If E-field is too large, it will pull electrons off from atom These electrons are accelerated by the E-field These accelerated electrons then collide with more atoms that knock off more electrons This is an AVALANCHE PROCESS Damages materials - there is a Voltage limit on components, cables in air : = 30 k. V/cm BREAKDOWN OCCURS if Medan Elektromagnetik. Sukiswo 12

BOUNDARY CONDITIONS - Normal Components • all derived from Maxwell’s equations NORMAL COMPONENT Take h << a a Material 1 Material 2 (a thin disc) TOP BOTTOM h Gaussian Surface Medan Elektromagnetik. Sukiswo 13

BOUNDARY CONDITIONS - Normal Components Case 1: Case 2: REGION 2 is a CONDUCTOR, D 2 = E 2 =0 REGIONS 1 & 2 are DIELECTRICS with rs = 0 Can only really get rs with conductors Medan Elektromagnetik. Sukiswo 14

BOUNDARY CONDITIONS - Tangential Components Material 1 w Material 2 h h << w Note: If region 2 is a conductor E 1 t = 0 Outside conductor E and D are normal to the surface Medan Elektromagnetik. Sukiswo 15

ELECTROSTATICS - CAPACITANCE Sukiswo, Medan Elektromagnetik

CAPACITANCE of Coaxial Cable In previous class, for coaxial cable: a Note: b inner conductor outer conductor Define: very general result charge on 1 conductor DV between conductors Note that: Medan Elektromagnetik. Sukiswo 17

Calculation of CAPACITANCE Problems on calculation of C Find Q 1. Method - Assume rs (use symmetry) Find V(rs) 2. Alternate method - Assume V and find Q Medan Elektromagnetik. Sukiswo 18

CAPACITANCE - parallel plate capacitor z=d Use Gauss’ Law, z=0 C of Parallel Plate capacitor Medan Elektromagnetik. Sukiswo 19

CAPACITANCE - parallel plate capacitor Parallel Plate Capacitance To get large C • increase A • increase e • decrease d This is how electrolytics increase C Do problem 1 a or 2 a & 2 b Medan Elektromagnetik. Sukiswo 20

CAPACITANCE - ENERGY METHOD • energy stored in capacitors is stored in the E-field Define stored energy: Substitute values of C and V for parallel plate capacitor: Energy Density Medan Elektromagnetik. Sukiswo Volume 21

CAPACITANCE - ENERGY METHOD In general we can write the total stored energy as: or Medan Elektromagnetik. Sukiswo 22

CAPACITANCE - ENERGY METHOD Use the Energy Formulation to compute C for the Parallel Plate Capacitor We know that, Compute TOTAL ENERGY: Medan Elektromagnetik. Sukiswo 23

CAPACITANCE Any 2 conductors have capacitance Example: • lines on circuit board • Theremin • wires and cables Medan Elektromagnetik. Sukiswo 24

ELECTROSTATICS - Numerical Simulation Sukiswo, Medan Elektromagnetik

Direct Computation of V If we can express entire problem in terms of V then: • we can solve directly for V • derive all other quantities e. g. E-field, D-field, C and r This approach can be used if conductor defines Outer Boundary • can be SYMMETRIC or NON-SYMMETRIC systems Why is this a useful approach? ? • V is a scalar field - easier to manipulate than E-field • We can control V on conductors • Can apply numerical methods to solve problem Medan Elektromagnetik. Sukiswo 26

Use of Laplace and Poisson’s Equations Start with 2 of MAXWELL’s equations: & In rectangular coordinates: Medan Elektromagnetik. Sukiswo 27

Use of Laplace and Poisson’s Equations Poisson’s equation: Laplace’s equation: (when r = 0) Medan Elektromagnetik. Sukiswo 28

Numerical Solution: Finite Difference Method Use the FINITE DIFFERENCE Technique for solving problems Solve for approximate V on the Grid - for 2 -D Problem Vtop h Vcenter Vleft Vright Vcenter at (x, y) = (0, 0) Vbottom h Medan Elektromagnetik. Sukiswo 29

Numerical Solution: Finite Difference Method At (x, y) = (h/2, 0) h Vtop h Vleft Vcenter Vright Vbottom At (x, y) = (-h/2, 0) Medan Elektromagnetik. Sukiswo 30

Numerical Solution: Finite Difference Method 0 Now, Can get similar expression for Medan Elektromagnetik. Sukiswo 31

Numerical Solution: Finite Difference Method Finally we obtain the following expression: Rearrange the equation to solve for Vcenter : Poisson Equation Solver Laplace Equation Solver Medan Elektromagnetik. Sukiswo 32

Numerical Solution: Example 100 V V 1 Solution Technique - by Iteration V 2 10 V 30 V V 3 V 4 Guess a solution : V=0 everywhere except boundaries V 1 = V 2 = V 3 = V 4 = 0 60 V Start: Put new values back Medan Elektromagnetik. Sukiswo 33

Numerical Solution - use of EXCEL Spreadsheet • To get an accurate solution, need lots of points - one way is to use a SPREADSHEET In spreadsheet, A 1 to A 31 set boundary voltage = 0 Volts Set these cells to 100 Copy B 2 formula to rest of cells A 31 Medan Elektromagnetik. Sukiswo 34

Numerical Solution: Problems 3 c. At point P, what is rs ? Get rs from Boundary Conditions: Approximate 3 d. Use spreadsheet to add columns: 3 e. Use C=Q/V Medan Elektromagnetik. Sukiswo 35