Fields and Waves I

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Fields and Waves I

• Lecture 12
• Capacitance
• Laplaces and Poissons Equations
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY

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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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Overview

• HW4 on the handout page. http//hibp.ecse.rpi.edu/
  7Econnor/education/Fields/handoutmain.htm
• Boundary Conditions
• Capacitance from Charge
• Energy
• Capacitance from Energy
• Equations of Laplace and Poisson

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Capacitance and Laplaces / Poissons Equations

• Interface and boundary conditions

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Electrostatics and Maxwell equations

• Integral Form
• A constitutive law for the material

• Differential Form

Fields are continuous in an homogenous media
What happens at the interface between 2
materials ?
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BOUNDARY CONDITIONS - Normal Components

• all derived from Maxwells equations

NORMAL COMPONENT
Take h ltlt a (a thin disc)
a
Material 1
TOP
h
Material 2
BOTTOM
Gaussian Surface
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BOUNDARY CONDITIONS - Normal Components
REGION 2 is a CONDUCTOR, D2 E2 0
Case 1
Material 1
Material 2 conductor
Can only really get rs with conductors
Case 2
REGIONS 1 2 are DIELECTRICS with rs 0
Material 1 dielectric
Material 2 dielectric
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BOUNDARY CONDITIONS - Tangential Components
w
Material 1
h ltlt w
h
Material 2
Note
If region 2 is a conductor E1t 0
Outside conductor E and D are normal to the
surface
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Example 0 - BC
The E field on the air side of a
dielectric-dielectric boundary is E 100 ax
100 ay. What is E on the dielectric side?
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Example 0 - BC
You are almost guaranteed to get a problem like
this on a quiz or final exam.
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Example 0 - BC
Consider the two parallel plate geometries below.
Assume that the plate dimensions are large
compared to the separation d and ignore fringe
effects. For the two figures, the electric field
in the air region, (specified by eo) is given by
E -(V0/d) (2er/(1er)) az
figure on left E -(V0/d) az
figure on right
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Example 0 - BC

• For both cases, find E in the dielectric region.
  Find D in both regions. Within a given region, D
  and E do not vary with position.
• Find the charge density on the plates at all
  locations.

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Example 0 - BC
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Capacitance and Laplaces / Poissons Equations

• Capacitance

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CAPACITANCE - parallel plate capacitor
Conductor
Use Gauss Law,
zd
Note that this can also be found from the BC
z0
very general result
Note
C of Parallel Plate capacitor
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CAPACITANCE - parallel plate capacitor
Note that
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CAPACITANCE - ENERGY METHOD

• The energy stored in capacitors is stored in the
  E-field

Define stored energy
Substitute values of C and V for parallel plate
capacitor
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CAPACITANCE - ENERGY METHOD
In general we can write the total stored energy
as
or
In the class notes, this is derived from first
principles
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CAPACITANCE - ENERGY METHOD
Use the Energy Formulation to compute C for the
Parallel Plate Capacitor
We know that,
(E in terms of V is needed)
Compute TOTAL ENERGY
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Example 1 Coaxial Cable
Consider a coaxial cable with inner radius, a,
and outer radius, b, length l. The outer
conductor was grounded, and inner conductor had
voltage V V0. The relation between surface
charge density on the inner conductor and V0 is
?sa e V0/(a ln(b /a )).

• Find the capacitance using C Q/V.
• Find the stored energy in the system by
  integrating the energy density over the system
  volume.
• Find the capacitance using the stored energy from
  part b.

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Example 1 Coaxial Cable
In a previous class, for coaxial cable
The electric field is given by E V0 / (r ln(b
/a )) ar.
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Example 1 Coaxial Cable
Surface Charge Density
Note the length in this expression. The practical
result is for capacitance per unit length, for
which we drop the length.
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Example 1 Coaxial Cable (Continued)
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Example 2 Two Wire Line
The objective of this problem is to determine the
capacitance between 2 conducting wires. We will
assume there are equal magnitude, opposite sign,
uniform surface charge densities on the two
wires. What is the voltage difference between
the wires? What is the capacitance ?
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Example 2 Two Wire Line

• Electric Field for a Single Wire Located at xxo
  and yyo

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Example 2 Two Wire Line

• This yields an equation for the electric field on
  the x-axis between the two wires (the wires are
  parallel to the z-axis)

and y0
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Example 2 Two Wire Line
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Example 3

• Any 2 conductors have capacitance
• lines on circuit board
• Estimate the capacitance of the two wire
  experiment

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Example 3
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Capacitance and Laplaces / Poissons Equations

• 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

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Boundary Conditions

• Dielectric-Dielectric

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Boundary Conditions

• Conductor-Dielectric

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Electrostatics

• Coulombs Law is already a solution
• All other voltage expressions can be checked with
  one of these equations
• This is the most common way of finding electric
  fields

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Example 4 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 4 Poissons Equation
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Example 5 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 5 Laplaces Equation
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Example 5 Laplaces Equation
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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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Numerical Solution Finite Difference Method
At (x,y) (h/2,0)
Vtop
Vright
At (x,y) (-h/2,0)
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Vtop
Numerical Solution Finite Difference Method
Vright
0
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
V1
V2
V4
V3
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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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 3 Finite Difference
10 Iterations with a spreadsheet
100 100 100 100
80 70.00002 55.00001 20
80 45.00001 30.00001 20
0 0 0 0
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Jacobs Ladder

• Commercial device available from Information
  Unlimited Not a safe company

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555 Timer Applications

• 555 timer is used to produce an oscillating
  signal whose voltage output is increased by the
  transformer to a dangerous level, producing
  sparks. DO NOT DO THIS WITHOUT SUPERVISION OR
  TRAINING

http//freespace.virgin.net/michael.tucknott/jacob
s.htm
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Jacobs Ladder