finite element analysis- basic study

FINITE ELEMENTS FOR ELECTRICAL ENGINEERING INTRODUCTION TO ELECTROMAGNETISM

© R. Bargalló. ELECTRICAL ENGINEERING DEPARTAMENT. EUETIB-UPC 8 of 36

Magnetic flux continuity

( )( ) 00

0=⋅∇⇒=⋅⋅∇

⎪⎭

⎪⎬

⎫

⋅⋅∇=⋅

=⋅

∫∫∫∫∫∫∫∫∫∫

BdVBdVBdSB

dSB

VVS

S

Charge conservation

( )

( ) ( ) 00

0

=+⋅∇⇒=⋅+⋅⋅∇

⎪⎪⎭

⎪⎪⎬

⎫

⋅⋅∇=⋅

=⋅+⋅

∫∫∫∫∫∫

∫∫∫∫∫∫∫∫∫∫

dtdJdV

dtddVJ

dVJdSJ

dVdtddSJ

VV

VS

VS

ρρ

ρ

SUMMARY OF MAXWELL’S DIFFERENTIAL LAWS Name Integral law

1 Gauss’s law ( ) ρε =⋅⋅∇ E 2 Ampere’s law

dtdDJH +=×∇

3 Faraday’s law dtdBE −=×∇

4 Magnetic flux continuity 0=⋅∇ B 5 Charge conservation ( ) 0=+⋅∇

dtdJ ρ

Differential forms are better than integral forms. Integrals forms are dependents on volume and the surface of integration. Differential forms are independent of these. Constitutive relations The field vectors D and E and also B and H are related by the properties of the materials at any point in the field region. These are often referred to as the constitutive properties of the material and are given by:

(a) ED ⋅= ε (b) HB ⋅= µ (c) EJ ⋅= σ



ε is the permittivity of the material, µ is the magnetic permeability of the material and σ is the conductivity of the material. In some cases these values can be indicated in relative form:

r

r

µµµεεε⋅=⋅=

0

0

ε0, µ0 are the values for free space and εr, µr are the relative values of the material. Equations (1) to (4) and relations (a) to (c) are the well known Maxwell’s equations. With the addition of the continuity conditions we can solve any electromagnetic problem. Maxwell’s equations do not make a distinction between low and high frequency applications, but for practical applications it is possible to adapt them to these two situations. We will be interested in low-frequency phenomena. When describing low frequency problems the Maxwell’s equations can be divided into two groups:

• Electrostatics and • Magnetostatics.

And, an important point:

These can be treated independently! The following pages will be devoted to this approach.

Maxwell’s equations

Low frequency High frequency (Waves)

Electostatics Magnetics

Magnetostatics Magnetodynamics



Second order operators. Laplace operator. It is possible to combine two vector operators on scalar functions and vector functions. One of these is the Laplace operator (called in short, the Laplacian), this is the div of grad of a scalar function U:

)())(( 2 Ugradz

ky

jx

iUUgraddiv ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=∇=

Following table shows the Laplace operator in some common coordinate systems.

Cartesian coordinates 2

2

2

2

2

2

zU

yU

xU

∂∂

+∂∂

+∂∂

Cylindrical coordinates 2

2

2

2

2

11zUU

rrUr

rr ∂∂

+∂∂⋅+⎟

⎠⎞

⎜⎝⎛

∂∂⋅

∂∂

⋅φ

Spherical coordinates 2

2

2222

2 )(sin1)sin(

)sin(11

zU

rU

rrUr

rr ∂∂⋅

⋅+⎟

⎠⎞

⎜⎝⎛

∂∂⋅

∂∂

⋅⋅

+⎟⎠⎞

⎜⎝⎛

∂∂⋅

∂∂

⋅θθ

θθθ

If A is a vector function, we can demonstrate that:

))(())((2 AcurlcurlAdivgradA −=∇ where A2∇ is called the “vector Laplacian” of A. This is written as (in Cartesian coordinates)

zyx AkAjAiA 2222 ∇⋅+∇⋅+∇⋅=∇ where, for example, the component in the Ox direction is:

2

2

2

2

2

22

zA

yA

xA

A xxxx ∂

∂+

∂∂

+∂∂

=∇



Electrostatic, Magnetostatic and Magnetodynamic Fields In general, there are two classes of electromagnetic fields can de described:

• The time independent static and • Time varying fields.

They can be scalar and vector fields. A typical scalar field for example is the electrostatic potential distribution V(x,y,z) between charged electrodes; and the magnetic field intensity H(x,y,z) surrounding a current carrying conductor is a typical vector field. We have to distinguish between the slow and fast varying electrical current flow field with regard to the geometrical dimensions of the current carrying conductor.

The slow varying fields are understood to be fields not leading to current redistributions. This means that there are no eddy current effects as the dimensions of the current carrying conductor are smaller than the penetration depth of the field. The currents at those frequencies are distributed as in the DC case, uniformly over the whole surface of the conductor. Eddy current effects are considered in the fields with fast varying time dependence, due to the low frequency being treated as quasi-stationary. High frequency fields, as focussed on antenna problems and leading to the electromagnetic waves, are not considered in this course.

Electromagnetic fields

static

Non- static

Electric E

Magnetic B

Current flow JElectric E

Magnetic B

Current flow J

Slow varying (quasi-static)

fast varying Quasi-stationary

Electromagnetic waves

Current flow J



Most of the physical issues in energy engineering can be described by quasi-static phenomena. Slowly varying and periodic fields up to 10 kHz are considered to be quasi-stationary. Electrical energy devices such as motors, actuators, induction furnaces and high voltage transmission lines are operated at low frequency. Typical examples of quasi-static fields are the fields excited by coils in rotating electrical machines and transformers. Inside these conductors the displacement current is negligible and the magnetic field H outside the coil is exclusively excited by the free current density J. For those quasi-static fields, Ampere’s law is applicable:

JH =×∇ Deciding whether the displacement current can be neglected or not, depends on the wavelength λ of the problem considered in the frequency domain. If it is large compared to the physical dimensions of the problem L, the displacement current is negligible. In general if

L)10...5(≈λ the field problem can be considered as quasi-static. For this class of problem, the interesting fields vary slowly and can be periodic. So, three categories of problems are distinguished:

• Static • Slowly varying transient • Time-harmonic eddy current

In time-harmonic problems sinusoidal varying field quantities is assumed. In theory, a time-harmonic solution is only valid for linear systems as a sinusoidal excitation does not yield a single frequency response in the non-linear case. Electrostatic fields The two fundamental laws governing these electrostatic fields are Gauss’s law and Faraday’s law, and the constitutive relation between D and E

Gauss’s law ( ) ρ=⋅∇ D 1 Faraday’s law 0=×∇ E 2

Constitutive relation ED ⋅= ε 3 In terms of the electric (scalar) potential V, E is expressed as

VE −∇= or

∫ ⋅−= dlEV

finite element analysis- basic study

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