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.

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. 12 November 2018 Fields and Waves I

The Equations of Laplace and Poisson Derivation from Maxwell’s Equations Laplace’s Equation Poisson’s Equation Boundary Conditions Finite differences Finite Elements Theremin 12 November 2018 Fields and Waves I

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.

Maxwell’s Equations Electrostatics Integral Form Differential Form 12 November 2018 Fields and Waves I

Electrostatics Integral Form Differential Form 12 November 2018 Fields and Waves I

First, the curl equation Electrostatics First, the curl equation Next, the divergence equation since & 12 November 2018 Fields and Waves I

Expression of the Laplace operator Laplacian operator Expression of the Laplace operator Cartesian system of coordinates 12 November 2018 Fields and Waves I

Electrostatics Laplace’s Equation Poisson’s Equation Plus Boundary Conditions (Voltage or Charge) 12 November 2018 Fields and Waves I

Dielectric-Dielectric Boundary Conditions In General Dielectric-Dielectric Conductor-Dielectric E 12 November 2018 Fields and Waves I

Dielectric-Dielectric Boundary Conditions Dielectric-Dielectric 12 November 2018 Fields and Waves I

Conductor-Dielectric Boundary Conditions Conductor-Dielectric 12 November 2018 Fields and Waves I

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 12 November 2018 Fields and Waves I

Example 1 – Poisson’s Equation 12 November 2018 Fields and Waves I

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? 12 November 2018 Fields and Waves I

Example 2 – Laplace’s Equation 12 November 2018 Fields and Waves I

Example 2 – Laplace’s Equation 12 November 2018 Fields and Waves I

Laplace’s and Poisson’s Equations – Numerical Methods Finite differences

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. 12 November 2018 Fields and Waves I

Finite difference method Main principle Put a grid on the space Calculate the potential on that grid (nodes) h 12 November 2018 Fields and Waves I

Numerical Solution: Finite Difference Method At (x,y) = (h/2,0) Vtop h Vleft Vcenter Vright h Vbottom At (x,y) = (-h/2,0) 12 November 2018 Fields and Waves I

Numerical Solution: Finite Difference Method Vtop Vright Vbottom Vleft Vcenter h Now, Can get similar expression for 12 November 2018 Fields and Waves I

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 12 November 2018 Fields and Waves I

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: 12 November 2018 Fields and Waves I

Example 3 – Finite Difference Find the voltage at the 4 points 12 November 2018 Fields and Waves I

Example 3 – Finite Difference 12 November 2018 Fields and Waves I

Example 4 -- Spreadsheet 12 November 2018 Fields and Waves I

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. 12 November 2018 Fields and Waves I

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 12 November 2018 Fields and Waves I

Example 4 -- Spreadsheet At point P, what is rs ? Get rs from Boundary Conditions: Approximate Use spreadsheet to add columns: Use C=Q/V 12 November 2018 Fields and Waves I

Example 4 -- Spreadsheet 12 November 2018 Fields and Waves I

Example 4 -- Spreadsheet 12 November 2018 Fields and Waves I

Example 4 -- Spreadsheet 12 November 2018 Fields and Waves I

Example 4 -- Spreadsheet 12 November 2018 Fields and Waves I

Example 4 -- Spreadsheet Reversed the voltages 12 November 2018 Fields and Waves I

Example 4 -- Spreadsheet Reversed the voltages 12 November 2018 Fields and Waves I

Open Boundary Conditions Dielectric Boundary Conditions Other Methods What Next? Open Boundary Conditions Dielectric Boundary Conditions Other Methods 12 November 2018 Fields and Waves I

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. 12 November 2018 Fields and Waves I

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. 12 November 2018 Fields and Waves I

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 12 November 2018 Fields and Waves I

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) 12 November 2018 Fields and Waves I

Symmetry Boundaries Lines of symmetry We can greatly reduce the work to find numerical solutions by using symmetry. 12 November 2018 Fields and Waves I

Symmetry Boundaries Lines of symmetry 12 November 2018 Fields and Waves I

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 ? 12 November 2018 Fields and Waves I

Dielectric Boundary 2 1 If there are two dielectrics, then the boundary condition at the interface must satisfy the diel-diel BC 12 November 2018 Fields and Waves I

Dielectric Boundary For the lower dielectric constant 10X the upper – 2D voltage plot 12 November 2018 Fields and Waves I

Dielectric Boundary Note Change in Slope For the lower dielectric constant 10X the upper – variation of the voltage along a vertical line 12 November 2018 Fields and Waves I

Dielectric Boundary For the lower dielectric constant 10X the upper – 3D plot 12 November 2018 Fields and Waves I

Matlab Analysis 12 November 2018 Fields and Waves I

Laplace’s and Poisson’s Equations – Numerical Methods Finite Elements

Other Methods Finite Difference Method of Moments FEMM FLUX2D & FLUX3D Quickfield Sonnet Many Others -- http://www.emclab.umr.edu/numer.html Method of Moments EZNEC (for antennas) 12 November 2018 Fields and Waves I

From Flux 12 November 2018 Fields and Waves I

From Flux 12 November 2018 Fields and Waves I

From FEMM 12 November 2018 Fields and Waves I

From FEMM 12 November 2018 Fields and Waves I

Laplace’s and Poisson’s Equations – Numerical Methods Theremin

12 November 2018 Fields and Waves I http://www.scificool.com/script-review-the-day-the-earth-stood-still-remake/ http://www.thefilmtalk.com/misc/Day-the-Earth-Stood-Still.jpg 12 November 2018 Fields and Waves I

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. http://www.synthtopia.com/artists/ClaraRockmore.html http://www.musicalmuseum.co.uk/theremin.html http://www.thereminworld.com/learn.asp 12 November 2018 Fields and Waves I

Clara Rockmore 12 November 2018 Fields and Waves I http://www.thereminworld.com 12 November 2018 Fields and Waves I

The Theremin: Music Lydia Kavina: http://www.lydiakavina.com/ Peter Pringle: http://www.peterpringle.com/pg1.html http://www.peterpringle.com/manysamples/satie.mp3 12 November 2018 Fields and Waves I

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 12 November 2018 Fields and Waves I

Sources of Theremin Music http://www.electrotheremin.com/sounds.html http://www.thereminworld.com/bands.asp http://www.thereminvox.com/directory/66/ http://www.teleura.com/shop/productview/7/3/ http://woub.org/sync/feature-theremin.html (contains a lesson from Pamelia Kurstin) 12 November 2018 Fields and Waves I

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 12 November 2018 Fields and Waves I

Theremin: How It Works The Player’s Hand Adds an Additional Capacitance to the Antenna Impedance http://www.moogmusic.com/ 12 November 2018 Fields and Waves I

Theremin: How It Works Pitch & Volume Control Using Oscillators http://www.moogmusic.com/ 12 November 2018 Fields and Waves I

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. http://www.moogmusic.com/ 12 November 2018 Fields and Waves I

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 http://www.moogmusic.com/ 12 November 2018 Fields and Waves I

Schematic of Etherwave Theremin Theremin: How It Works Schematic of Etherwave Theremin 12 November 2018 Fields and Waves I http://www.moogmusic.com/

Theremin: How It Works Note, for the tank circuit The remainder of the circuit reduces this frequency slightly http://www.moogmusic.com/ 12 November 2018 Fields and Waves I

Are There Some Devices Based on Similar Ideas? http://www.ionet.net/~tgomez/coil.html Metal Detector: The coupling between coils is affected by the proximity of metal, either magnetic or just conducting. 12 November 2018 Fields and Waves I

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. http://imdb.com/title/tt0108323/ 12 November 2018 Fields and Waves I

Theremin Information Moog Music: http://www.moogmusic.com Theremin Info: http://www.theremin.info/ (Blocked as an attack site) ThereminWorld: http://www.thereminworld.com/default.asp Art’s 145 Theremin: http://home.att.net/~theremin1/145/145.html Theremin Sensitivity: http://www.dogstar.dantimax.dk/theremin/thersens.htm 120 Years of Electronic Music: http://www.obsolete.com/120_years/ http://thereminworld.com/darestore.asp 12 November 2018 Fields and Waves I

Theremin Information The approximate relationship of pitch and hand distance for a 145 theremin. http://home.att.net/~theremin1/article5.htm 12 November 2018 Fields and Waves I

Theremin Information Design description Tank L Tank C Center F New F for dF/dC uH pF kHz 1 pf increase "Clara Rockmore" 1165 750 170.266 170.152 -0.114 SWTP ("TECI") max L 300 410 453.803 453.251 -0.553 SWTP ("TECI") min L 150 641.775 640.994 -0.783 Theremax max L 350 120 776.597 773.381 -3.236 Theremax min L 180 1082.913 1078.428 -4.512 Doug Forbes C=150 500 581.152 579.224 -1.937 Doug Forbes C=100 100 711.763 708.230 -3.559 "Theoretical" 1 200 795.775 793.793 -1.989 "Theoretical" 2 400 794.782 -0.995 From Theremin Sensitivity 12 November 2018 Fields and Waves I

From the Day the Earth Stood Still Parting Words From the Day the Earth Stood Still 12 November 2018 Fields and Waves I