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

Sukiswo, Medan Elektromagnetik ELECTROSTATICS - MATERIALS Sukiswo, Medan Elektromagnetik

Medan Elektromagnetik. Sukiswo CONDUCTORS and DIELECTRICS Conductors Dielectrics High conductivities; s (for Copper) ~ 5.8x107 S/m Low conductivities; s (for Rubber) ~ 1x10-15 S/m or 1/W-m Semiconductors (mid s’s) Permittivities, e = 1-100e0 Note: e0 is the permittivity of free space/vacuum = 8.854 x 10-12 F/m Medan Elektromagnetik. Sukiswo

Medan Elektromagnetik. Sukiswo 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 E-field, the free electrons will move Lattice of Nuclei Medan Elektromagnetik. Sukiswo

Medan Elektromagnetik. Sukiswo CONDUCTORS - + conductor electrons 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

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

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

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

is due to bound/dielectric charge and free charge DIELECTRICS subtracts out bound charge Define: Displacement Flux Density ( C/m2 ) 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

Medan Elektromagnetik. Sukiswo 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

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

Medan Elektromagnetik. Sukiswo 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 kV/cm BREAKDOWN OCCURS if Medan Elektromagnetik. Sukiswo

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

Medan Elektromagnetik. Sukiswo BOUNDARY CONDITIONS - Normal Components Case 1: REGION 2 is a CONDUCTOR, D2 = E2 =0 Case 2: REGIONS 1 & 2 are DIELECTRICS with rs = 0 Can only really get rs with conductors Medan Elektromagnetik. Sukiswo

Medan Elektromagnetik. Sukiswo BOUNDARY CONDITIONS - Tangential Components w Material 1 h << w h Material 2 Note: If region 2 is a conductor E1t = 0 Outside conductor E and D are normal to the surface Medan Elektromagnetik. Sukiswo

Sukiswo, Medan Elektromagnetik ELECTROSTATICS - CAPACITANCE Sukiswo, Medan Elektromagnetik

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

Medan Elektromagnetik. Sukiswo 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

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

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

Medan Elektromagnetik. Sukiswo 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 Volume Medan Elektromagnetik. Sukiswo

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

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

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

Sukiswo, Medan Elektromagnetik ELECTROSTATICS - Numerical Simulation Sukiswo, Medan Elektromagnetik

Medan Elektromagnetik. Sukiswo 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

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

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

Medan Elektromagnetik. Sukiswo 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 Vleft Vcenter Vright Vcenter at (x,y) = (0,0) Vbottom h Medan Elektromagnetik. Sukiswo

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

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

Medan Elektromagnetik. Sukiswo 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

Medan Elektromagnetik. Sukiswo Numerical Solution: Example 60V 10V 100V 30V Solution Technique - by Iteration V1 Guess a solution : V=0 everywhere except boundaries V2 V3 V4 V1= V2 = V3 = V4 = 0 Put new values back Start: Medan Elektromagnetik. Sukiswo

Medan Elektromagnetik. Sukiswo Numerical Solution - use of EXCEL Spreadsheet To get an accurate solution, need lots of points - one way is to use a SPREADSHEET In spreadsheet, A1 A1 to A31 set boundary voltage = 0Volts Set these cells to 100 Copy B2 formula to rest of cells A31 Medan Elektromagnetik. Sukiswo

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