1. Medan Listrik Statis II (Material, Kapasitansi, Numerik Simulasi)
Medan Elektromagnetik. Sukiswo
Medan Listrik Statis II (Material, Kapasitansi, Numerik Simulasi)
Sukiswo
Medan Elektromagnetik. Sukiswo

2. Sukiswo, Medan Elektromagnetik
ELECTROSTATICS - MATERIALS
Sukiswo, Medan Elektromagnetik

3. 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 = x F/m
Medan Elektromagnetik. Sukiswo

4. 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

5. 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

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

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

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

9. 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

10. 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

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

12. 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

13. 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

14. 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

15. 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

16. Sukiswo, Medan Elektromagnetik
ELECTROSTATICS - CAPACITANCE
Sukiswo, Medan Elektromagnetik

17. 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

18. 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

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

20. 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

21. 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

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

23. 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

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

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

26. 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

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

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

29. 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

30. 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

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

32. 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

33. 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

34. 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

35. 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