Laplace’s Equation and Field/Potential Estimation

1. Laplace’s Equation and Field/Potential Estimation Alan Murray

2. Estimation? • Often we can “spot” some equipotentials (electrodes) • Can we infer others from these? • Can we create field lines and strengths from the equipotentials? • How accurate is this? • How useful is this?! Alan Murray – University of Edinburgh

3. Metal electrode 5V 1V Metal electrode E-Fieldlines 2V 3V 0V 4V Estimation – the principle Given some equipotentials Can we infer the rest? And then the field lines? Voltage contours Alan Murray – University of Edinburgh

4. e b a c d General idea Let’s say we know the height of b, c, d and e’s bums. Can we estimate the height of a’s bum? The “estimate” is 100% accurate Ha’s bum = ¼(Hb’s bum + Hc’s bum + Hd’s bum + He’s bum) Alan Murray – University of Edinburgh

5. e b a c d General idea Let’s say we know the height of b, c, d and e’s bums. Can we estimate the height of a’s bum? The “estimate” is 100% accurate Ha’s bum = ¼(Hb’s bum + Hc’s bum + Hd’s bum + He’s bum) Alan Murray – University of Edinburgh

6. c General idea Let’s say we know the height of a, c, d and e’s bums. Can we estimate the height of a’s bum? Now the estimate is not 100% accurate How could it be made more accurate? e a b d Ha’s bum = ¼(Hb’s bum + Hc’s bum + Hd’s bum + He’s bum) Alan Murray – University of Edinburgh

7. Now for some Maths … • This is Poisson’s Equation, which describes the variation of V(x,y,z) in space when a charge density ρ(x,y,z) is present. • This is Laplace’s Equation, which describes the variation of V(x,y,z) in space when no charge is present. Alan Murray – University of Edinburgh

8. V1 V2 V4 V3 Method … Divide the region of interest into a grid … finer grid means more accuracy and more calculation! … then approximate Ñ 2V Alan Murray – University of Edinburgh

9. V4 (d) V1 V2 (b) (a) (c) V3 D Look at neighbouring grid points V0? Alan Murray – University of Edinburgh

10. V4 (d) V1 V2 (b) (a) (c) V3 D Look at neighbouring grid points V0? Alan Murray – University of Edinburgh

11. Look at neighbouring grid points Alan Murray – University of Edinburgh

12. Look at neighbouring grid points • V0 is the average of its 4 neighbours in 2D • V0 is the average of its 6 neighbours in 3D • Intuitively “correct” … and supported by theory. • Is it useful? • Often we have some equipotentials • Fixed by conducting plates/electrodes/etc. • Use Laplace (approximately) to calculate the other potentials roughly • Repeat until accuracy is achieved • Or you go blue in the face! Alan Murray – University of Edinburgh

13. 40V 40V 40V 0V -40V 0V Worked Example : Coarse Grid,First Guesstimates 100V 50V 50V -100V Alan Murray – University of Edinburgh

14. Worked Example : Coarse Grid,Iterate … 100V =38V =48V 40V 40V 50V 50V (100+50+40+0)/4 (100+48-40+40)/4 0V -40V 0V -100V Alan Murray – University of Edinburgh

15. Worked Example : Coarse Grid,Iterate … 100V =38V =48V 47V 50V 50V -4V -13V -11V -100V Alan Murray – University of Edinburgh

16. Worked Example : Coarse Grid,Iterate … 100V =38V =48V 47V 50V 50V -4V -13V -11V -100V Alan Murray – University of Edinburgh

17. Iteration : Guidelines • Do not spent ages making initial estimates • All values will converge • Sensible values will converge faster • Always use the most recent values of neighbouring potentials • If the accuracy required is “to the nearest volt”, stop when the last change to every value is <1V Alan Murray – University of Edinburgh

18. =90° 75V equipotential E 48V 45V 48V 45V equipotential 0V equipotential -5V -16V -5V Worked Example : Coarse Grid,Final Values 100V 50V 50V -100V Alan Murray – University of Edinburgh

19. Ey = -[100-(-5)]/2D E 48V 45V 48V Ex = -[45-50]/2D -5V -16V -5V D = 0.05 ®E = (50, -1050)V/m Can we calculate the electric field .. approximate (-¶V/¶x, -¶V/¶y)? 100V 50V 50V -100V Alan Murray – University of Edinburgh

20. Finite Differences and Laplace • This is clearly tedious • Not with a computer it’s not • Can make the grid arbitrarily accurate • Can calculate for ages to achieve high accuracy • Can include not-zero charge density • and non-zero conductivity • and AC signals • This is the technique used when a “closed form” solution is not feasible http://www.see.ed.ac.uk/~afm/teaching/em3/ Examples : MS Excel Alan Murray – University of Edinburgh

21. Finite Differences and Laplace • Examples? • Field lines in a MOSFET • Shape and form of “plates” in a cathode ray tube/Scanning Electron Microscope etc • Fields, resistance and inductance of rails on the London Underground • This was done with help from Brian Flynn and me, by one of our own M.Eng. Students! • Aiming to detect cracks in the line • Practically important or esoteric Electromagnetics? • The student was able to solve Maxwell#3 “by hand” to calculate inductance. Alan Murray – University of Edinburgh

22. Example : Train Tracks Note adaptive mesh – finer where necessary Alan Murray – University of Edinburgh

23. Example : Train Tracks colour indicates electric field strength E Alan Murray – University of Edinburgh

24. Example : Train Tracks colour indicates current density J Alan Murray – University of Edinburgh