Fields and Waves I Lecture 13 Laplaces and
- Slides: 75
Fields and Waves I Lecture 13 Laplace’s and Poisson’s Equations – Numerical Methods K. A. Connor Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute, Troy, NY Fields and Waves I
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. 11/9/2020 Fields and Waves I 2
The Equations of Laplace and Poisson § § § § Derivation from Maxwell’s Equations Laplace’s Equation Poisson’s Equation Boundary Conditions Finite differences Finite Elements Theremin 11/9/2020 Fields and Waves I 3
Laplace’s and Poisson’s Equations – Numerical Methods Laplace’s Equation and Poisson’s Equation Fields and Waves I
Maxwell’s Equations Electrostatics § Integral Form § Differential Form 0 11/9/2020 0 Fields and Waves I 5
Electrostatics § Integral Form 11/9/2020 § Differential Form Fields and Waves I 6
Electrostatics § First, the curl equation since § Next, the divergence equation & 11/9/2020 Fields and Waves I 7
Laplacian operator § Expression of the Laplace operator Cartesian system of coordinates 11/9/2020 Fields and Waves I 8
Electrostatics § Laplace’s Equation § Poisson’s Equation § Plus Boundary Conditions (Voltage or Charge) 11/9/2020 Fields and Waves I 9
Boundary Conditions § In General E § Dielectric-Dielectric § Conductor-Dielectric 11/9/2020 Fields and Waves I 10
Boundary Conditions § Dielectric-Dielectric 11/9/2020 Fields and Waves I 11
Boundary Conditions § Conductor-Dielectric 11/9/2020 Fields and Waves I 12
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 11/9/2020 Fields and Waves I 13
Example 1 – Poisson’s Equation 11/9/2020 Fields and Waves I 14
Example 2 – Laplace’s Equation A coaxial cable has an inner conductor (at r =a ) held at voltage V 0 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? 11/9/2020 Fields and Waves I 15
Example 2 – Laplace’s Equation 11/9/2020 Fields and Waves I 16
Example 2 – Laplace’s Equation 11/9/2020 Fields and Waves I 17
Laplace’s and Poisson’s Equations – Numerical Methods Finite differences Fields and Waves I
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. 11/9/2020 Fields and Waves I 19
Finite difference method § Main principle • • Put a grid on the space Calculate the potential on that grid (nodes) h h 11/9/2020 Fields and Waves I 20
Numerical Solution: Finite Difference Method At (x, y) = (h/2, 0) h Vleft Vtop Vcenter Vright h Vbottom At (x, y) = (-h/2, 0) 11/9/2020 Fields and Waves I 21
Numerical Solution: Finite Difference Method h Vtop 0 h Now, Vleft Vcenter Vright Vbottom Can get similar expression for 11/9/2020 Fields and Waves I 22
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 11/9/2020 Fields and Waves I 23
Numerical Solution: Example 100 V 10 V V 1 V 2 V 3 V 4 60 V Start: 11/9/2020 Solution Technique - by Iteration Guess a solution : V=0 everywhere 30 V except boundaries V 1 = V 2 = V 3 = V 4 = 0 Put new values back Fields and Waves I 24
Example 3 – Finite Difference Find the voltage at the 4 points 11/9/2020 Fields and Waves I 25
Example 3 – Finite Difference 11/9/2020 Fields and Waves I 26
Example 4 -- Spreadsheet 11/9/2020 Fields and Waves I 27
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. 11/9/2020 Fields and Waves I 28
Numerical Solution - use of EXCEL Spreadsheet • To get an accurate solution, need lots of points - one way is to use a SPREADSHEET In spreadsheet, A 1 A 31 A 1 to A 31 set boundary voltage = 0 Volts Set these cells to 100 Copy B 2 formula to rest of cells 11/9/2020 Fields and Waves I 29
Example 4 -- Spreadsheet At point P, what is rs ? Get rs from Boundary Conditions: Approximate Use spreadsheet to add columns: Use C=Q/V 11/9/2020 Fields and Waves I 30
Example 4 -- Spreadsheet 11/9/2020 Fields and Waves I 31
Example 4 -- Spreadsheet 11/9/2020 Fields and Waves I 32
Example 4 -- Spreadsheet 11/9/2020 Fields and Waves I 33
Example 4 -- Spreadsheet 11/9/2020 Fields and Waves I 34
Example 4 -- Spreadsheet Reversed the voltages 11/9/2020 Fields and Waves I 35
Example 4 -- Spreadsheet Reversed the voltages 11/9/2020 Fields and Waves I 36
What Next? § Open Boundary Conditions § Dielectric Boundary Conditions § Other Methods 11/9/2020 Fields and Waves I 37
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. 11/9/2020 Fields and Waves I 38
V 1 Open Boundaries V 2 Y X 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. 11/9/2020 Fields and Waves I 39
Open Boundaries Y X E § Note that the condition that two points are on the same equipotential is the same as written as § This type of BC is called a Uniform Neumann Boundary Condition (from Mathematics). 11/9/2020 Fields and Waves I 40
Closed Boundaries V 1 Y X V 2 § 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) 11/9/2020 Fields and Waves I 41
Lines of symmetry Symmetry Boundaries § We can greatly reduce the work to find numerical solutions by using symmetry. 11/9/2020 Fields and Waves I 42
Lines of symmetry Symmetry Boundaries 11/9/2020 Fields and Waves I 43
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 ? 11/9/2020 Fields and Waves I 44
Dielectric Boundary 2 1 If there are two dielectrics, then the boundary condition at the interface must satisfy the diel-diel BC 11/9/2020 Fields and Waves I 45
Dielectric Boundary For the lower dielectric constant 10 X the upper – 2 D voltage plot 11/9/2020 Fields and Waves I 46
Dielectric Boundary Note Change in Slope For the lower dielectric constant 10 X the upper – variation of the voltage along a vertical line 11/9/2020 Fields and Waves I 47
Dielectric Boundary For the lower dielectric constant 10 X the upper – 3 D plot 11/9/2020 Fields and Waves I 48
Matlab Analysis 11/9/2020 Fields and Waves I 49
Laplace’s and Poisson’s Equations – Numerical Methods Finite Elements Fields and Waves I
Other Methods § Finite Difference • • • FEMM FLUX 2 D & FLUX 3 D Quickfield Sonnet Many Others -- http: //www. emclab. umr. edu/numer. html § Method of Moments • EZNEC (for antennas) 11/9/2020 Fields and Waves I 51
From Flux 11/9/2020 Fields and Waves I 52
From Flux 11/9/2020 Fields and Waves I 53
From FEMM 11/9/2020 Fields and Waves I 54
From FEMM 11/9/2020 Fields and Waves I 55
Laplace’s and Poisson’s Equations – Numerical Methods Theremin 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 11/9/2020 Fields and Waves I 57
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, theremin is unique in that it is played without being touched. Two antennas protrude from 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 theremin requires precise skill and perfect pitch. http: //www. musicalmuseum. co. uk/theremin. html 11/9/2020 http: //www. synthtopia. com/artists/ Clara. Rockmore. html Fields and Waves I http: //www. thereminwo rld. com/learn. asp 58
Clara Rockmore http: //www. thereminworld. com 11/9/2020 Fields and Waves I 59
The Theremin: Music § Lydia Kavina: http: //www. lydiakavina. com/ § Peter Pringle: http: //www. peterpringle. com/pg 1. html • 11/9/2020 http: //www. peterpringle. com/manysamples/satie. mp 3 Fields and Waves I 60
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 11/9/2020 Fields and Waves I 61
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) 11/9/2020 Fields and Waves I 62
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 11/9/2020 Fields and Waves I 63
Theremin: How It Works § The Player’s Hand Adds an Additional Capacitance to the Antenna Impedance http: //www. moogmusic. com/ 11/9/2020 Fields and Waves I 64
Theremin: How It Works § Pitch & Volume Control Using Oscillators http: //www. moogmusic. com/ 11/9/2020 Fields and Waves I 65
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 -3 k. Hz. § Audio signal is amplified § Volume control works in a similar manner. http: //www. moogmusic. com/ 11/9/2020 Fields and Waves I 66
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/ 11/9/2020 Fields and Waves I 67
Theremin: How It Works § Schematic of Etherwave Theremin 11/9/2020 Fields and Waves I http: //www. moogmusic. com/ 68
Theremin: How It Works § Note, for the tank circuit § The remainder of the circuit reduces this frequency slightly http: //www. moogmusic. com/ 11/9/2020 Fields and Waves I 69
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. 11/9/2020 Fields and Waves I 70
What Ever Happened to Leon Theremin? Theremin: An Electronic Odyssey http: //imdb. com/title/tt 0108323/ 11/9/2020 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 1920 s and '30 s. 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 1940 s. 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. Fields and Waves I 71
Theremin Information § Moog Music: http: //www. moogmusic. com § Theremin Info: http: //www. theremin. info/ (Blocked as an attack § § site) Theremin. World: http: //www. thereminworld. com/default. asp Art’s 145 Theremin: http: //home. att. net/~theremin 1/145. 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 11/9/2020 Fields and Waves I 72
Theremin Information The approximate relationship of pitch and hand distance for a 145 theremin. http: //home. att. net/~theremin 1/article 5. htm 11/9/2020 Fields and Waves I 73
Theremin Information Design description Tank L Tank C u. H "Clara Rockmore" p. F Center F New F for k. Hz d. F/d. C 1 pf increase 1165 750 170. 266 170. 152 -0. 114 SWTP ("TECI") max L 300 410 453. 803 453. 251 -0. 553 SWTP ("TECI") min L 150 410 641. 775 640. 994 -0. 783 Theremax L 350 120 776. 597 773. 381 -3. 236 Theremax min L 180 120 1082. 913 1078. 428 -4. 512 Doug Forbes C=150 500 150 581. 152 579. 224 -1. 937 Doug Forbes C=100 500 100 711. 763 708. 230 -3. 559 "Theoretical" 1 200 795. 775 793 -1. 989 "Theoretical" 2 100 400 795. 775 794. 782 -0. 995 From Theremin Sensitivity 11/9/2020 Fields and Waves I 74
Parting Words § From the Day the Earth Stood Still 11/9/2020 Fields and Waves I 75
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