Transient Electromagnetic-Thermal FE-Model of a …€¦ · · 2012-01-04Transient Electromagnetic-Thermal FE-Model of a SPICE-Coupled Transformer Including Eddy Currents ... Heat

Transient Electromagnetic-Thermal FE-Model of a SPICE-Coupled Transformer Including Eddy Currents

Holger Neubert*,1, Thomas Bödrich1 and Rolf Disselnkötter2

1 Technische Universität Dresden, Institute of Electromechanical and Electronic Design, Germany, 2 ABB AG, Corporate Research Center Germany, Ladenburg, Germany

* Corresponding author: D-01069 Dresden, Germany, [email protected]

Faculty of Electrical and Computer Engineering Institute of Electromechanical and Electronic Design

H. Neubert, T. Bödrich, R. Disselnkötter: Electromagnetic-Thermal Model of a TransformerCOMSOL Conference Stuttgart 2011, October 26 – 28

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Outline

1. Introduction2. Modelling Approach3. Electromagnetic FE Model4. Thermal Model5. Coupled Time-dependent Simulation6. Summary


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Current Transformers

Used to measure high currents in power grid systemsPrimary winding:- Normally only one turn

(the power line)Secondary windings:- Some hundreds up to

thousands, - Close to short-circuit

condition

1. Introduction

Quelle: Bienzle (Wikimedia)

1i 12

21 i

Ni =


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Bar-typeCurrent Transformer(Low Voltage)

Quelle: ABB Stotz S&J Quelle: ABB

Pole mountedCurrent Transformer(High Voltage)


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2. Modelling Approach

Coupled ModelElectromagnetic FE model of the transformer Network models of the primary and secondary circuitry Thermal FE model of the transformer


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( )rq( )rT

Electromagnetic FE model

Thermal FE model


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3. Electromagnetic FE Model

Parametric geometryAmpère's circuital law and Faraday's law of inductionmf mode (magnetic field only) for time-dependent simulationNon-linear magnetic material behavior in H = f(|B|)eB form


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Modelling of Eddy Currents

Power Line (Primary winding):- Sinusoidal primary current i1(t) is modeled as a total current

density Jz(r,t) inside of the bus bar- Jz(r,t) can not be imposed directly as an external current

density

- A global equation (ge mode) determines Jez inside of the bus bar by

- Jz(r,t) in the primary conductor is calculated from

( ) ( ) ( )tJtJtJ ,,, izezz rrr +=

0prim1 =− Ii

( )∫=prim

zprim

prim1

V

dVJL

I


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Eddy Current in the Power LineSimulated z-component of the total current density with skin effect in the primary conductor (i1peak = 1000 A, RsecExt = 20 Ω, t = 0.23 s)


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Modelling of Eddy Currents

Secondary windings:- Modeled as bulk material (eight prismatic bodies)- Conductivity is set to 10 S/m to supress eddy current effects- Secondary current i2(t) is modeled as an external current

density, derived from the induced voltage


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Eddy Current in the Power LineSimulated z-component of the total current density with skin effect in the primary conductor (i1peak = 1000 A, RsecExt = 20 Ω, t = 0.23 s)


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Non-linear Magnetic Behavior

H = f(|B|)eB form avoids circular variable definitions in constitutiverelationsSeveral approximation approaches with remarkable influence on solution time

no convergenceNearest neighbour

≈ 100Cubic spline interpolation

4.0Linear interpolation

1.3Global rational function

1.0Piecewise cubic interpolation

Relative solution timeApproximation approach


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Non-linear Magnetic Behavior

H = f(|B|)eB form avoids circular variable definitions in constitutiverelationsSeveral approximation approaches with remarkable influence on solution time Cubic spline interpolation may lead to a non-monotonic curves


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Coupling with SPICE Components

cir mode Sinusoidal current source at the primary side External load resistor at the secondary side coupled to the secondary windings (External I vs. U element)Variable

i2coilsec ViRV −=sec

8212i

)(A

VVVNV

+++=

K


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4. Thermal Model

Temperature-dependent electrical conductivity of the conductorsHeat conduction in solids applying the ht mode- Heat sources (mean value over a period of the losses field)- Heat conduction in solids and narrow air gaps,- Thermal contact resistances between solids which are in

mechanical contact- External convection on solid-air interfaces applying empirical

correlationsTime-average of the local power loss density in the time interval [0, ti]

( ) ( )[ ] τστ

dJ

ttq

t

i∫=i

0

2

i,1

,r

r


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Thermal Model


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5. Coupled Time-dependent Simulation

Time-dependent simulationTime scales of the electromagnetic and the thermal model are very different Bi-directionally coupling of the electromagnetic and the thermal modelIterating alternate solutions: - Stationary study steps of the thermal model - Time-dependent study steps of the electromagnetic and circuit

model


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Convergence


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CurrentsSimulated primary current i1 and secondary current i2 ⋅ N2 (i1peak = 1000 A, RsecExt = 25 Ω)Imperfect transformer coupling due to the air gaps in the core


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CurrentsRsecExt = 1 kΩDeformation of the sinusoidal current due to magnetic saturation in the core


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Current densityi1peak = 1000 A, RsecExt = 25 Ω


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Flux densityi1peak = 1000 A, RsecExt = 25 Ω


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6. Summary

Transient Electromagnetic-Thermal FE-Model of a SPICE-Coupled Transformer Including Eddy Currents

Time-dependent simulation of a transformer coupled to an externalcircuitryNon-linear magnetic material properties based on experimental dataEddy current effects are included using a global equationTime-averaged power loss density distributionThe bi-directionally coupled thermal model considers the influenceof the temperature on electrical material propertiesFuture work will focus on- consideration of the transformer core lamination, - the anisotropic material behaviour inside of the coils


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Thank you very muchfor your attention.

Documents

Transient Electromagnetic-Thermal FE-Model of a …€¦ · · 2012-01-04Transient Electromagnetic-Thermal FE-Model of a SPICE-Coupled Transformer Including Eddy Currents ... Heat