Fields%20and%20Waves%20I

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Fields and Waves I

• Lecture 13
• Laplaces and Poissons Equations Numerical
  Methods
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY

2
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 Ulabys textbook.
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The Equations of Laplace and Poisson

• Derivation from Maxwells Equations
• Laplaces Equation
• Poissons Equation
• Boundary Conditions
• Finite differences
• Finite Elements
• Theremin

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Laplaces and Poissons Equations Numerical
Methods

• Laplaces Equation and Poissons Equation

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Maxwells Equations
Electrostatics

• Integral Form

• Differential Form

0
0
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Electrostatics

• Integral Form

• Differential Form

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Electrostatics

• First, the curl equation
• Next, the divergence equation

since

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Laplacian operator

• Expression of the Laplace operator

Cartesian system of coordinates
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Electrostatics

• Laplaces Equation
• Poissons Equation
• Plus Boundary Conditions (Voltage or Charge)

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Boundary Conditions

• In General
• Dielectric-Dielectric
• Conductor-Dielectric

E
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Boundary Conditions

• Dielectric-Dielectric

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Boundary Conditions

• Conductor-Dielectric

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Example 1 Poissons Equation
A charged region of a semiconductor is sandwiched
between two grounded conductors as shown below.
Solve for V(z) directly using Poissons
Equation Find E and D Find the charge density on
the conductors
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Example 1 Poissons Equation
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Example 2 Laplaces 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 Laplaces
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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Example 2 Laplaces Equation
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Example 2 Laplaces Equation
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Laplaces and Poissons Equations Numerical
Methods

• Finite differences

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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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Finite difference method

• Main principle
• Put a grid on the space
• Calculate the potential on that grid (nodes)

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Numerical Solution Finite Difference Method
At (x,y) (h/2,0)
Vtop
Vleft
Vcenter
Vright
h
Vbottom
At (x,y) (-h/2,0)
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Numerical Solution Finite Difference Method
Now,
Can get similar expression for
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Numerical Solution Finite Difference Method
Finally we obtain the following expression
Rearrange the equation to solve for Vcenter
Poisson Equation Solver
Laplace Equation Solver
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Numerical Solution Example
Solution Technique - by Iteration
Guess a solution V0 everywhere except
boundaries
10V
V1 V2 V3 V4 0
Put new values back
Start
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Example 3 Finite Difference
Find the voltage at the 4 points
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Example 3 Finite Difference
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Example 4 -- Spreadsheet
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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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Numerical Solution - use of EXCEL Spreadsheet

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

In spreadsheet,
A31
A1
A1 to A31 set boundary voltage 0Volts
Set these cells to 100
Copy B2 formula to rest of cells
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Example 4 -- Spreadsheet
At point P, what is rs ?
Get rs from Boundary Conditions
Approximate
Use spreadsheet to add columns
Use CQ/V
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Example 4 -- Spreadsheet
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Example 4 -- Spreadsheet
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Example 4 -- Spreadsheet
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Example 4 -- Spreadsheet
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Example 4 -- Spreadsheet
Reversed the voltages
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Example 4 -- Spreadsheet
Reversed the voltages
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What Next?

• Open Boundary Conditions
• Dielectric Boundary Conditions
• Other Methods

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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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Open Boundaries
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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Open Boundaries
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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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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Symmetry Boundaries
Lines of symmetry

• We can greatly reduce the work to find numerical
  solutions by using symmetry.

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Symmetry Boundaries
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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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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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Dielectric Boundary
For the lower dielectric constant 10X the upper
2D voltage plot
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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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Dielectric Boundary
For the lower dielectric constant 10X the upper
3D plot
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Matlab Analysis
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Laplaces and Poissons Equations Numerical
Methods

• Finite Elements

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Other Methods

• Finite Difference
• FEMM
• FLUX2D FLUX3D
• Quickfield
• Sonnet
• Many Others -- http//www.emclab.umr.edu/numer.htm
  l
• Method of Moments
• EZNEC (for antennas)

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From Flux
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From Flux
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From FEMM
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From FEMM
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Laplaces and Poissons Equations Numerical
Methods

• Theremin

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http//www.scificool.com/script-review-the-day-the
-earth-stood-still-remake/
http//www.thefilmtalk.com/misc/Day-the-Earth-Stoo
d-Still.jpg
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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.ht
ml
http//www.musicalmuseum.co.uk/theremin.html
http//www.thereminworld.com/learn.asp
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Clara Rockmore
http//www.thereminworld.com
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The Theremin Music

• Lydia Kavina http//www.lydiakavina.com/
• Peter Pringle http//www.peterpringle.com/pg1.htm
  l
• http//www.peterpringle.com/manysamples/satie.mp3

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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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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)

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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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Theremin How It Works

• The Players Hand Adds an Additional Capacitance
  to the Antenna Impedance

http//www.moogmusic.com/
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Theremin How It Works

• Pitch Volume Control Using Oscillators

http//www.moogmusic.com/
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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/
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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/
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Theremin How It Works

• Schematic of Etherwave Theremin

http//www.moogmusic.com/
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Theremin How It Works

• Note, for the tank circuit
• The remainder of the circuit reduces this
  frequency slightly

http//www.moogmusic.com/
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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.

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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/
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Theremin Information

• Moog Music http//www.moogmusic.com
• Theremin Info http//www.theremin.info/ (Blocked
  as an attack site)
• ThereminWorld http//www.thereminworld.com/defaul
  t.asp
• Arts 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
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Theremin Information
The approximate relationship of pitch and hand
distance for a 145 theremin.
http//home.att.net/theremin1/article5.htm
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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 150 410 641.775 640.994 -0.783
Theremax max L 350 120 776.597 773.381 -3.236
Theremax min L 180 120 1082.913 1078.428 -4.512
Doug Forbes C150 500 150 581.152 579.224 -1.937
Doug Forbes C100 500 100 711.763 708.230 -3.559
"Theoretical" 1 200 200 795.775 793.793 -1.989
"Theoretical" 2 100 400 795.775 794.782 -0.995
From Theremin Sensitivity
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Parting Words

• From the Day the Earth Stood Still