1. Fields and Waves I Lecture 13 K. A. Connor
Laplace’s and Poisson’s Equations – Numerical Methods
K. A. Connor
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Welcome to Fields and Waves I
Before I start, can those of you with pagers and cell phones please turn them off? Thanks.

2. J. Darryl Michael – GE Global Research Center, Niskayuna, NY
These Slides Were Prepared by Prof. Kenneth A. Connor Using Original Materials Written Mostly by the Following:
Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY
J. Darryl Michael – GE Global Research Center, Niskayuna, NY
Thomas P. Crowley – National Institute of Standards and Technology, Boulder, CO
Sheppard J. Salon – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY
Lale Ergene – ITU Informatics Institute, Istanbul, Turkey
Jeffrey Braunstein – Chung-Ang University, Seoul, Korea
Materials from other sources are referenced where they are used. Those listed as Ulaby are figures from Ulaby’s textbook.
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3. The Equations of Laplace and Poisson
Derivation from Maxwell’s Equations
Laplace’s Equation
Poisson’s Equation
Boundary Conditions
Finite differences
Finite Elements
Theremin
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4. Laplace’s and Poisson’s Equations – Numerical Methods
Laplace’s Equation and Poisson’s Equation
Welcome to Fields and Waves I
Before I start, can those of you with pagers and cell phones please turn them off? Thanks.

5. Maxwell’s Equations Electrostatics Integral Form Differential Form
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6. Electrostatics Integral Form Differential Form 12 November 2018
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7. First, the curl equation
Electrostatics
First, the curl equation
Next, the divergence equation
since
&
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8. Expression of the Laplace operator
Laplacian operator
Expression of the Laplace operator
Cartesian system of coordinates
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9. Electrostatics Laplace’s Equation Poisson’s Equation
Plus Boundary Conditions (Voltage or Charge)
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10. Dielectric-Dielectric
Boundary Conditions
In General
Dielectric-Dielectric
Conductor-Dielectric E
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11. Dielectric-Dielectric
Boundary Conditions
Dielectric-Dielectric
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12. Conductor-Dielectric
Boundary Conditions
Conductor-Dielectric
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13. Example 1 – Poisson’s Equation
A charged region of a semiconductor is sandwiched between two grounded conductors as shown below.
Solve for V(z) directly using Poisson’s Equation
Find E and D
Find the charge density on the conductors
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14. Example 1 – Poisson’s Equation
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15. Example 2 – Laplace’s Equation
A coaxial cable has an inner conductor (at r =a ) held at voltage V0 and an outer conductor (at r =b ) that is grounded. There is no charge other than the surface charge on the conductors.
Solve for V(r) directly using Laplace’s Equation
Solve for E and D
What is the charge density on the two conductors?
What is the capacitance per unit length?
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16. Example 2 – Laplace’s Equation
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17. Example 2 – Laplace’s Equation
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18. Laplace’s and Poisson’s Equations – Numerical Methods
Finite differences

19. Numerical Solution: Finite Difference Method
How does one solve for V(r) when the geometry is not so simple?
We rely on numerical methods
Finite Difference
Finite Elements
Method of Moments
Etc.
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20. Finite difference method
Main principle
Put a grid on the space
Calculate the potential on that grid (nodes) h
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21. Numerical Solution: Finite Difference Method
At (x,y) = (h/2,0)
Vtop h
Vleft
Vcenter
Vright h
Vbottom
At (x,y) = (-h/2,0)
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22. Numerical Solution: Finite Difference Method
Vtop
Vright
Vbottom
Vleft
Vcenter h
Now,
Can get similar expression for
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23. Numerical Solution: Finite Difference Method
Finally we obtain the following expression:
Rearrange the equation to solve for Vcenter :
Poisson Equation Solver
Laplace Equation Solver
Linear System of equations
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24. Numerical Solution: Example
60V
100V
30V V1 V2 V3 V4
Solution Technique - by Iteration
Guess a solution : V=0 everywhere except boundaries
10V
V1= V2 = V3 = V4 = 0
Put new values back
Start:
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25. Example 3 – Finite Difference
Find the voltage at the 4 points
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26. Example 3 – Finite Difference
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27. Example 4 -- Spreadsheet
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28. Example 4 -- Spreadsheet
Use a finite difference calculation to solve for the voltage everywhere in the configuration. Assume this figure extends out of the page for a long distance.
Sketch or plot the equipotentials and electric field lines.
Find the charge density on the conductor at point P (0,1.5).
Find the total charge per unit length on the outer and inner conductors.
Determine the capacitance per unit length between the 2 conductors.
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29. 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
A31
A1 to A31 set boundary voltage = 0Volts
Set these cells to 100
Copy B2 formula to rest of cells
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30. Example 4 -- Spreadsheet
At point P, what is rs ?
Get rs from Boundary Conditions:
Approximate
Use spreadsheet to add columns:
Use C=Q/V
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31. Example 4 -- Spreadsheet
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32. Example 4 -- Spreadsheet
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33. Example 4 -- Spreadsheet
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34. Example 4 -- Spreadsheet
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35. Example 4 -- Spreadsheet
Reversed the voltages
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36. Example 4 -- Spreadsheet
Reversed the voltages
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37. Open Boundary Conditions Dielectric Boundary Conditions Other Methods
What Next?
Open Boundary Conditions
Dielectric Boundary Conditions
Other Methods
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38. Dielectric Boundary Conditions
At a dielectric-dielectric boundary, the voltage is continuous, but the normal derivative is not
Before considering how to deal with this, let us also consider open boundaries or boundaries that represent lines of symmetry.
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39. Open Boundaries V1 V2 X Y E
For a parallel plate capacitor the side boundaries are open and the equipotentials are horizontal.
For such voltage lines, the boundary voltage will equal the value of its immediate inside neighbor.
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40. Open Boundaries X Y E
Note that the condition that two points are on the same equipotential is the same as
This type of BC is called a Uniform Neumann Boundary Condition (from Mathematics).
written as
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41. Closed Boundaries V1 Y X V2
For completeness, we should also note that any part of the boundary that is given a fixed voltage has what is called a Dirichlet Boundary Condition
For a unique solution
At least on Dirichlet needs to be specified
(! not both at the same location)
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42. Symmetry Boundaries Lines of symmetry
We can greatly reduce the work to find numerical solutions by using symmetry.
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43. Symmetry Boundaries Lines of symmetry 12 November 2018
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44. Dielectric Boundary Conditions
At a dielectric-dielectric boundary, the voltage is continuous, but the normal derivative is not
What does this means in terms of potential ?
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45. Dielectric Boundary 2 1
If there are two dielectrics, then the boundary condition at the interface must satisfy the diel-diel BC
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46. Dielectric Boundary
For the lower dielectric constant 10X the upper – 2D voltage plot
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47. Dielectric Boundary Note Change in Slope
For the lower dielectric constant 10X the upper – variation of the voltage along a vertical line
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48. Dielectric Boundary
For the lower dielectric constant 10X the upper – 3D plot
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49. Matlab Analysis
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50. Laplace’s and Poisson’s Equations – Numerical Methods
Finite Elements

51. Other Methods Finite Difference Method of Moments FEMM FLUX2D & FLUX3D
Quickfield
Sonnet
Many Others --
Method of Moments
EZNEC (for antennas)
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52. From Flux
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53. From Flux
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54. From FEMM
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55. From FEMM
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56. Laplace’s and Poisson’s Equations – Numerical Methods
Theremin

57. 12 November 2018 Fields and Waves I
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58. The Theremin Clara Rockmore (1911-1998)
The theremin was invented in 1919 by a Russian physicist named Lev Termen (his name was later changed to Leon Theremin).
Besides looking like no other instrument, the theremin is unique in that it is played without being touched. Two antennas protrude from the theremin – one controlling pitch, and the other controlling volume. As a hand approaches the vertical antenna, the pitch gets higher. Approaching the horizontal antenna makes the volume softer. Because there is no physical contact with the instrument, playing the theremin requires precise skill and perfect pitch.
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59. Clara Rockmore 12 November 2018 Fields and Waves I
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60. The Theremin: Music Lydia Kavina: http://www.lydiakavina.com/
Peter Pringle:
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61. The Manatees Jan Vollaard Pamelia Kurstin
The Theremin: Music
The Manatees Jan Vollaard Pamelia Kurstin
Nosebleeds by CHINESEBURN
You Only Live Twice by Project Pimento
3001 by Rita Lee
Latin Soul by the Manatees
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62. Sources of Theremin Music
(contains a lesson from Pamelia Kurstin)
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63. Difference Frequencies
First Signal
Second Signal
Difference Signal
AM technique can be used to produce a low frequency signal that is the difference between two high frequency signals
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64. Theremin: How It Works
The Player’s Hand Adds an Additional Capacitance to the Antenna Impedance
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65. Theremin: How It Works Pitch & Volume Control Using Oscillators
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66. The difference is in the audio range 0-3kHz. Audio signal is amplified
Theremin: How It Works
Fixed pitch oscillator frequency is compared to the frequency of the variable pitch oscillator.
The difference is in the audio range 0-3kHz.
Audio signal is amplified
Volume control works in a similar manner.
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67. Theremin: How It Works
Zero beat oscillator circuit results is very low frequencies
Volume is also controlled by providing some additional capacitance from the player
Voltage controlled amplifier (VCO) is tool Theremin did not have
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68. Schematic of Etherwave Theremin
Theremin: How It Works
Schematic of Etherwave Theremin
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69. Theremin: How It Works Note, for the tank circuit
The remainder of the circuit reduces this frequency slightly
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70. Are There Some Devices Based on Similar Ideas?
Metal Detector: The coupling between coils is affected by the proximity of metal, either magnetic or just conducting.
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71. What Ever Happened to Leon Theremin?
Theremin: An Electronic Odyssey
Leon Theremin was the secret link between sci-fi films, the Beach Boys, and Carnegie Hall. His self-named electronic musical instrument--the first of its kind--took the world by storm in the 1920s and '30s. Theremin: An Electronic Odyssey , winner of Sundance's Filmmakers Trophy, explores the inventor's strange life and times, including his mysterious 50-year disappearance beginning in the 1940s. Interviews with theremin virtuoso Clara Rockmore, synthesizer pioneer Robert Moog, and Theremin's contemporaries, as well as clips from movies such as The Day the Earth Stood Still, featuring the unworldly sounds of his creation, show an eccentric genius working toward success until his sudden vanishing in the Soviet Union. Footage of Theremin at 94 years old, finally rediscovered and rewarded for his achievements, brings a celebratory ending to what could be a grim or at least uncertain story, but instead is a fascinating documentary.
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72. Theremin Information Moog Music: http://www.moogmusic.com
Theremin Info: (Blocked as an attack site)
ThereminWorld:
Art’s 145 Theremin:
Theremin Sensitivity:
120 Years of Electronic Music:
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73. Theremin Information
The approximate relationship of pitch and hand distance for a 145 theremin.
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74. Theremin Information Design description Tank L Tank C Center F
New F for
dF/dC uH pF
kHz
1 pf increase
"Clara Rockmore"
1165
750
-0.114
SWTP ("TECI") max L
300
410
-0.553
SWTP ("TECI") min L
150
-0.783
Theremax max L
350
120
-3.236
Theremax min L
180
-4.512
Doug Forbes C=150
500
-1.937
Doug Forbes C=100
100
-3.559
"Theoretical" 1
200
-1.989
"Theoretical" 2
400
-0.995
From Theremin Sensitivity
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75. From the Day the Earth Stood Still
Parting Words
From the Day the Earth Stood Still
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