1. Fields and Waves I Lecture 14 K. A. Connor Y. Maréchal
Currents and Resistance
K. A. Connor
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Y. Maréchal
Power Engineering Department
Institut National Polytechnique de Grenoble, France
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. Resistance and Ohm’s Law Numerical Methods
Overview
Current
Total Current (Flux of Current)
Current Density
Resistance and Ohm’s Law
Numerical Methods
Finite Difference in bio impedance
From Flow Nets to PEEC
Examples
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4. Current sources of magnetic fields i.e. currents We will be looking at
description of resistance
We will consider two parameters:
I - Current (in Amps)
J - Current Density (in Amps/unit area)
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5. Current & Current Density
Definition:
Wire with current, I
Charge passing through cross-section in time Dt
Define:
cross-section
In general,
points in direction of current flow
Flux Integral
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6. Current & Current Density
Example: wire with constant current density
, in cylindrical geometry
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7. Example 1 a - Find I in terms of J0 for Peak Density 12 November 2018
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8. Example 1:Current & Current Density
Note that for this current distribution, there is much more current in the outer shell. This is a combination of the increased density and the larger area of each shell.
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9. Surface Current Density
Surface Current Density Js (in units of A/m)
conductors often have current flowing in thin sheet (eg. High Freq.)
Example: current flow in a wire
Current flow region a
But,
However, define:
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10. Current Density and charges
We now obtain an alternate expression for current density
Wire with current, I
Assume all particles move at same v
In Dt, all particles within Dz = v.Dt will pass through the right face
Cross-section = A
Can we define this speed ??
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Do problem 1c

11. Relation between j and E
, in wire pushed electrons to the right
, force (from E-field) driving electrons
s, collide with lattice ions
, balances
at some
Since
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12. Ohm’s Law = Ohm’s Law Conductivity - units of S/m or 1/ohm-m
varies from 107 to 10-15
Good conductor eg. Cu
Good insulator
, is Fields and Waves version of Ohm’s Law
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13. Resistance of a cylindrical wire
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14. Example 2 – Resistance
Consider a 1 mm radius wire that is 10 meters long.
At DC frequencies, the current density is spread uniformly across the entire cross-section. What is the resistance of the wire?
b. A rough model for the wire in example 1a is to assume that the current density is uniform in the region r > 0.88a and zero elsewhere, J = J0 az for r/a = 0.88-> 1 and 0 for r/a < What resistance is obtained with this model?
c. A more accurate value of the resistance is to calculate the voltage drop along the outside of the wire (at r = a ) from J = σ E and get the resistance from R = V/I. What does this model yield?
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15. Example 2 – Resistance
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16. Example 2 - Resistance
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17. Resistance Valid if only j and A are constant What if they are not?
Compute V and I separately and form V/I
Example: Disk with Radial Current t Vb Va
Look at wedge
(sketch)
I is the same at the inner and outer part
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18. Resistance - alternative calculationVb Va
I is the same at the inner and outer part of disk
Compute potential difference between inner and outer part of disk:
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19. General expression of a resistance
General expression of the resistance t V0 V1
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20. Currents and Resistance
Finite difference for Bioelectrical Impedance Analysis

21. Bioelectrical Impedance Analysis
Bioelectrical Impedance Analysis or BIA is considered one of the most exact and accessible methods of screening body fat. In conventional BIA, a person is weighed, then, height, age and gender are entered in a computer. While the person is lying down, electrodes are attached to various parts of the body and a small electric signal is discharged that measures the impedance or resistance to muscle and fat. The more muscle, the lower the value as the electricity passes easily through lean mass. A formula in the computer converts all of the data to indicate what percentage of the body is fat. -- From Tanita
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22. BIA Potential is directed by Laplace equation
We can use the spreadsheet to find equipotentials.
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23. BIA Legs dominate the resistance. 3D View 12 November 2018
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24. BIA We can use the spreadsheet to find flow lines.
The equations solved in this case are not the same.
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25. Bioimpedance Bioimpedance can also be used to image inside objects.
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26. Bioimpedance
Note how the potential structure depends on the system structure.
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27. Bioimpedance The 3D views shows this even better. 12 November 2018
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28. Electrical Impedance Imaging at RPI
Non-invasive medical imaging devices, Adaptive Current Tomographs (ACT)
For example, the heart is full of blood. Blood conducts electricity well, so the heart has a low impedance. The lungs are filled with air. Air does not conduct electricity well, so the lungs have relatively high impedance. Figure 1 below shows the cross-section of a human chest. As the viewer, you are facing the patient, looking down from above. Blue indicates low impedance, red indicates high impedance.
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29. Currents and Resistance
Other numerical method : from net flows to PEEC

30. Numerical Solutions
The equations of Laplace and Poisson apply in a very wide range of disciplines.
To address resistance, note that we are addressing the flow of electrons (current) across equipotentials.
Thus, we should look to disciplines that focus on flows, such as hydrology (flow net).
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31. Flow Net Theory Equipotentials Current Flow Current flow channel V1 V2
Per Unit Length V2
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32. Calculation of this closed loop resistance
Towards PEEC
Calculation of this closed loop resistance
Current flow
channel
Equipotentials V0 V1 V
First approximation
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33. Current flow simulation
V=10
V=0
Current modulus in color shades
And arrows
Potential in red equi lines
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34. More accurate calculation
Towards PEEC
More accurate calculation
Subdivision according to current flow
Equipotentials V0 V1
3 Current flow
channels V
More accurate approximation
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35. For higher frequencies
Current distribution is not uniform (AC current)
Introduce mutual inductance and self inductance
Conductor 1
Model
R, L, M I1 I2
Subdivision
Conductor 2
Conductor 1
Conductor 2
PEEC = Partial Element Equivalent Circuit
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36. Schneider applications
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37. Equivalent Kirchoff-like
Principle
Equivalent Kirchoff-like
network
Geometry
Meshing
Circuit Solve
Evaluation of
locally: j
globally: P, I …
Pipe oriented description
Preferred current flow direction
Analytical PEEC formulations
Energy is located in conductors
Matrix representation
Elementary parallelepipeds
Compliant with both skin and proximity effects
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38. Real life example (Schneider Electric)
Characteristics: A, low voltage
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39. Current distribution 3 phases simulation 12 November 2018
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40. Other applications of PEEC in power electronics
Calculation of interconnection inductances
Closer view
A 6 pack module IGBT
Model of 1/3 on the module
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41. References on PEEC
“Modeling of printed circuit board loop inductance”, IEEE Trans on Mag, Vol 30, n°5, sept 1994
The INCA software, Magsoft Corp, Ballston Spa,
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