Nonlinear transmission lines

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Time-Domain Simulation of Nonlinear Transmission
Lines Interface of SABC Finite Elements to
Circuit Analysis
Dr. Andrew F. Peterson Karim N. Wassef
School of Electrical and Computer
Engineering Georgia Institute of Technology
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Purpose
Purpose
Model Signal Propagation on Transmission Lines
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Self and Mutual Effects
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x and y variation gtgt z variation
Nonlinearly Magnetic Conductive Shell
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Transmission Line Matrix Method
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Assuming Z-Uniformity over a Segment
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Finite Element Analysis w/ SABC
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Problem Statement

• To simulate transient field solution in
  ferromagnetic conductive regions.
• No pre-existing method has incorporated a true
  field solution for such materials with fast
  transients.
• This work achieves this via the integration of a
  Transient Surface Admittance Boundary Condition
  Finite Element Method, and a Transmission Line
  Matrix Method to the Circuit Simulators .

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Example of the need for the new SABC Finite
Element method
Given a sample medium STEEL
Frequency 50 KHz Relative Magnetic Permeability
1000 Conductivity 8.3334 x 106 S/m Wire
Radius 1cm
Triangular element area 3 x 10-12 m2
Cross-sectional area 3 x 10-4 m2 Number of
elements required
108
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Model Capabilities

• The rigorous simulation of the transmission
  line component of a circuit solver.
• The prediction of transient signal propagation
  in conductive ferromagnetic regions demonstrates
  a need for the implementation of this method to
  account for
• The surface effect eddy current due to the
  internal excitation of the source conductive
  wires (copper).
• The proximity effect eddy current due to the
  inductive effect of the magnetic fields on nearby
  conductors (copper).
• The ferromagnetic effect due to the nonlinear
  saturation of a surrounding conduit (steel).
• The transient signal is imposed on a time
  marching basis through the circuit solver. Time
  domain is also the only approach capable of
  retaining the history on the nonlinear regions.

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Full Model Integration into a Circuit Solver

• A Time Domain Transmission Line Matrix method
  can predict propagation down a line given
  specific (time and space dependant) per-unit
  parameter matrices
• R, L, C resistance, inductance, and
  capacitance.

• The nonlinear FE-TD method with SABC is used
  for the extraction of these parameters given the
  proper bias conditions
• I current

• The Circuit Solver interfaces with the TLM-TD
  code that calls the FE-TD routines for efficient
  prediction of signal propagation.

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Full Model Integration into a Circuit Solver
Full Transmission line Model
Other Models
VTB
Circuit Interface
Finite Element Extraction of RLC
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Lumped Element TL
Simplest Case LUMPED TL element in algebraic
companion form integrated over one time increment
h
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Signal Propagation with TLM
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Circuit Interface connecting the TLM to the
Network Solver
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The Finite Element Challenge
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History and Existing FE Formulation

• Simple Finite Element Solution of the Excitation
  Fields

Could not account for material conductivities.

• Expanding the FE Formulation to Include Eddy
  Currents

Required meshing of the entire problem domain (
large problem).
Resulted in very stiff matrices (v. large and v.
small elements).
Was limited to cases of low conductivities and
slow transients.

• Proposed Finite Element Formulation

Employs a Surface Admittance Boundary Condition
that overcomes these drawbacks.
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Transformation of the Computational Domain
Original Problem Domain
New Problem Domain
Sources (J)
No Sources
No internal boundaries
Forced internal boundaries
Not meshed
Fully Meshed Space
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Extension Into the Ferromagnetic Problem
The recursive (Prony Expansion) method employing
a first order Leontovich boundary condition is
employed. It allows for local magnetic edge
elements in the form of
Nonlinear effects are included through this
parameter using a relaxed iterative approach with
a relaxation factor 0 lt (g 0.1) lt 1 and a
convergence criterion of emax .01
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Relaxed Iterative Nonlinear Solution
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Parameter Extraction of the RLC
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Auto-Extraction of the per-unit-length L
Dimension of problem NC ( NC 1 ) / 2
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Model Inter-dependence Number of iterations as a
function of problem size
History of Magnetic Vector Potential and Prony
Expansion Coefficients
Circuit Solver
Segment Current
Transmission Line Matrix Simulator
NONLINEAR Finite Element Method using the SABC
Segment Inductance
History of Segment Currents, Voltages, and
Inductances
Repeat x number of segments Repeat x
NC(NC1)/2 Repeat x iterations to NL convergence
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Conclusion
This method links complicated (nonlinear
properties and detailed geometries) transmission
lines into the circuit network solver. A TLM
method solves for excitation currents modified to
establish time-synchronized interface segments at
the terminal ports. An Admittance Boundary
Finite Element method solves the field equations
using a recursive Pronys method for improved
efficiency (time and memory). A Relaxed iterative
solution is used to model nonlinear ferromagnetic
materials. Number of iterations (.25s /
iteration) NZ ( NL NC ( NC 1 ) / 2
) (1,8,3 gt 3s) (3,5,3 gt 8s) (5,5,5 gt 25s)
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Future Directions
1. Efficiency Enhancements Code is designed for
worst case w/ nonlinear segments having
time- varying L-matrices -gt runs
slow. Software can be made smart to decide on
the fly if certain steps are required -gt speed
up by 1 order of magnitude. 2. Extend Input
Parameter Range Modeling a distributed EM system
as a lumped circuit element -gt some restriction
on range of input parameters. Provide more
options within a smart code -gt alleviate
restrictions. 3. Additional validation studies.