Darryl Michael/GE CRD

Fields and Waves

Lesson 3.6

ELECTROSTATICS - Numerical Simulation

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

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

Use of Laplace and Poisson’s Equations

Start with 2 of MAXWELL’s equations:

D

0 E

& ED

E

VE

VVE )( V2

In rectangular coordinates:2

2

2

2

2

22

z

V

y

V

x

VV

Use of Laplace and Poisson’s Equations

Poisson’s equation:

V2

Laplace’s equation: (when = 0)

02 V

Do Problem 1

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

Vbottom

VrightVleft Vcenter Vcenter at (x,y) = (0,0)

h

h

h

VV

x

V

x

VE centerrightx

)(

Numerical Solution: Finite Difference Method

Vtop

Vright

At (x,y) = (h/2,0)

Vbottom

VleftVcenter

h

h

At (x,y) = (-h/2,0) h

VV

x

V

x

VE leftcenterx

)(

Numerical Solution: Finite Difference Method

z

E

y

E

x

EEVV zyx

2

0

2

20,2

0,2

h

VVV

h

hE

hE

x

E

x

E leftrightcenterxx

xx

Now,

Can get similar expression fory

E y

Numerical Solution: Finite Difference Method

Finally we obtain the following expression:

22 4

h

VVVVVV centerbottomtopleftright

Rearrange the equation to solve for Vcenter :

2

4

1 hVV neighborscenter

Poisson Equation Solver

neighborscenter VV4

1 Laplace Equation Solver

Numerical Solution: Example

60V

10V

100V

30V

V1 V2

V3 V4

Solution Technique - by Iteration

Guess a solution : V=0 everywhere except boundaries

V1= V2 = V3 = V4 = 0

Start:

321 1010041 VVV

412 3010041 VVV

413 601041 VVV

324 603041 VVV

5.27)0110(41

1 V

375.39)05.27130(41

2 V

375.24)05.2770(41

3 V

4375.38)375.24375.3990(41

4 V

Put new values back

Numerical Solution - use of EXCEL Spreadsheet

Do Problem 2

• To get an accurate solution, need lots of points - one way is to use a SPREADSHEET

In spreadsheet, 22314

12 CABBB

A1

A31

A1 to A31 set boundary voltage = 0Volts

Set these cells to 100

Copy B2 formula to rest of cells

Numerical Solution: Problems

Do Problem 3a and 3b

Helpful Hints for Problems 3c - 3e

3c. At point P, what is s ?

Get s from Boundary Conditions: snn ED

Approximate x

VEn

3d. Use spreadsheet to add columns: AreasmallQ s _

AreaFullQ s _

3e. Use C=Q/V