NONDESTRUCTIVE TESTING HANDBOOK -
Electromagnetic Testing
Manual de Ensaio Não Destrutivo - Ensaio Eletromagnético
- Parte 1. Modêlo do Fenômeno do Ensaio Eletromagnético
- Introdução
- Equações Diferenciais Básicas para Campos Eletromagnéticos
- Modêlos Analítico e Numérico
- Parte 2. Modêlo do Meio Condutor Homogêneo
- Fundamentos
- Modêlos Analíticos
- Modêlos de Dodd e Deeds
- Extensões dos Modêlos de Dodd e Deeds
- Modêlos Tridimensionais
- Perturbação e Expansão da Função de Engen
- Conclusões
- Parte 3. Modêlos Analíticos e Integral para Simular Trincas
- Introdução
- Elementos da Teoria de Trincas
- Plano de Impedâncias
- Dipolo da Corrente
- Mono polo da Corrente Estática
- Dipolo de Campo Estático
- Pequena Inclusão Esférica
- Dipolo Dinâmico da Corrente
- Resposta da Sonda
- Pequenas Descontinuidades
- Trincas Longas
- Técnicas Avançadas
- Campo Elétrico na Abertura da Trinca
- Trinca Impenetrável
- Distribuição do Dipolo da Corrente na Superfície
- Formulação Integral
- Resultados dos Elementos de Contorno
- Teoria da Trinca de Pouca Penetração
- Formulações Alternativas
- Regime de Pouca Penetração
- Parte 4. Modêlo Computacional do Campo de Correntes Parasitas
- Bases Matemáticas do Modelo
- Tipos de Modêlo
- Visão Geral da Modelagem Analítica e Numérica
- Modêlo Analítico
- Técnica de Solução Integral
- Modêlo Numérico
- Técnica das Diferenças Finitas
- Representação das Diferenças Finitas
- Formulação das Diferenças Finitas para Problemas de Campo Bidimensional e Axissimétricos
- Contornos e Condições de Contorno
- Malhas Não Uniformes e Não Retangulares
- Solução do Sistema de Equações
- Solução Interativa
- Solução por Matriz de Inversão
- Técnica de Elementos Finitos
- Formulação de Elementos Finitos para Geometrias Bidimensionais e Axissimétricas
- Energia Funcional para Problemas de Correntes Parasitas
- Discretização de Elementos Finitos
- Formulação de Elementos Finitos
- Elementos Isoparamêtricos Quadrilaterais
- Minimização Funcional
- Condições de Contorno
- Cálculos com Vetor Magnético Potencial
- Modelagem da Física do Ensaio de Correntes Parasitas
- Modelagem para Projeto de Sondas
- Projeto por Elementos Finitos de Sondas de Correntes Parasitas Absoluta e Diferencial
- Modelagem para Simulação
- Conclusões
1 MODÊLO DO FENÔMENO DO ENSAIO ELETROMAGNÉTICO
1.1 INTRODUÇÃO
Modelos
matemáticos são usados para simular o fenômeno das correntes
parasitas e suas aplicações em ensaios não destrutivos. Os modelos
tipicamente simulam um ensaio de correntes parasitas e predizem o sinal
da sonda associado a uma descontinuidade específica (uma região onde a
condutividade ou permeabilidade muda abruptamente) sob diferentes
condições experimentais. Os resultados desses estudos paramétricos são
úteis no projeto de sondas, na visualização da interação do campo com
as descontinuidades, na otimização da configuração do ensaio e na
geração de assinaturas de descontinuidade que podem ser usadas para
desenvolver algoritmos de interpretação de sinais. Os modelos de
simulação são relativamente baratos em comparação com os dados
adquiridos experimentalmente a partir de descontinuidades artificiais.
Todos
os fenômenos eletromagnéticos, incluindo aqueles relacionados ao
vazamento de fluxo magnético e aos ensaios de correntes parasitas, são
governados por equações diferenciais. (R01)
1.2 EQUAÇÕES DIFERENCIAIS BÁSICAS PARA CAMPOS ELETROMAGNÉTICOS (R02)
As
equações diferenciais que governam campos eletromagnéticos gerais,
variáveis no tempo, em baixas frequências, em regiões que incluem
materiais magnéticos e condutores e densidades de corrente aplicadas,
são derivadas das equações de Maxwell: (R01)
onde B é a densidade de fluxo magnético (tesla), D é a densidade de fluxo elétrico (coulomb por metro quadrado), E é a intensidade do campo elétrico (volt por metro), H é a intensidade do campo magnético (ampère por metro), J é a densidade de corrente (ampère por metro quadrado), t é o tempo (segundo) e ρ é a densidade de carga (coulomb por metro cúbico).
A
Equação 2 depende da aproximação quase-estática, que negligencia a
corrente de deslocamento. A técnica de micro-ondas necessita da
corrente de deslocamento, mas sua omissão é justificável na técnica de
correntes parasitas, pois as frequências mais altas encontradas são da
ordem de alguns megahertz. Nessas frequências, a corrente de condução
em metais é tipicamente muitas ordens de magnitude maior que a corrente
de deslocamento. A carga pode se acumular em limites de descontinuidade
e na superfície de condutores, causando um salto na componente normal
do campo elétrico. No entanto, a Eq. 2 implica que \/.J = 0, o que
significa, por exemplo, que a corrente normal a uma superfície que
adquire carga é desprezível. Embora a corrente de carga possa ser
desprezada, o efeito da carga no campo elétrico não pode ser ignorado.
Se o limite não for abrupto, a carga incidente se distribui por um
volume.
Observe
que, ao igualar todas as derivadas temporais a zero, essas equações
podem ser usadas para descrever fenômenos de fuga de fluxo magnético. O
mesmo modelo numérico usado para ensaios de correntes parasitas pode
ser aplicado a ensaios de fuga de fluxo magnético, igualando-se a
frequência da corrente da fonte a zero.
Além das equações de Maxwell, as seguintes relações descrevem meios lineares e isotrópicos:
A permissividade ou constante dielétrica ε (farad por metro), a permeabilidade magnética μ (henry por metro) e a condutividade elétrica σ
(siemens por metro) são tratadas aqui como constantes escalares. Em
meios anisotrópicos, cada uma se torna um tensor 3 x 3. O comportamento
não linear de qualquer uma das três propriedades pode existir em uma
determinada situação. Embora a não linearidade na condutividade e na
permissividade seja raramente encontrada em problemas de correntes
parasitas, a não linearidade de materiais magnéticos é comum e se
expressa como a dependência da permeabilidade em relação ao campo. Para
aplicações práticas de correntes parasitas, os níveis de excitação
geralmente são baixos o suficiente para justificar a suposição de
linearidade para materiais magnéticos.
Usando essa suposição e substituindo a Eq. 5, a Eq. 2 se torna:
Isso,
no entanto, não é suficiente para especificar completamente os campos
dentro da região da solução, pois a densidade de corrente J contém duas
fontes diferentes. A primeira e mais óbvia é a densidade de corrente
aplicada Js. Uma segunda componente é a densidade de corrente parasita induzida Je. Assim, a Eq. 8 torna-se:
Neste ponto, é útil introduzir o potencial vetor magnético A, que é definido como segue:
Substituindo isso na Eq. 8 e na Eq. 1, obtemos as Eqs. 11 e 12 para uma região livre de fontes:
O campo elétrico na Eq. 12 é:
A
Eq. 13 mostra que o campo elétrico pode ser dividido em um termo de
potencial vetor magnético e uma contribuição escrita como o gradiente
de um potencial escalar. O gradiente do potencial é incluído para
expressar o campo elétrico como uma forma geral que satisfaz a Eq. 12.
O potencial escalar é eliminado quando a Eq. 13 é substituída na Eq. 1,
porque o rotacional do gradiente é identicamente zero.
Portanto, o campo eletromagnético está definido para qualquer problema físico específico, mas A e Φ
ainda não estão definidos. Por exemplo, um gradiente de potencial
diferente poderia ser adicionado ao termo do potencial vetor em vez do \/Φ original, e A
poderia ser ajustado para fornecer o campo elétrico correto. A
expressão resultante satisfaria a Eq. 12 e produziria o mesmo fluxo
magnético da Eq. 1. Portanto, há flexibilidade na escolha de A e Φ.
Para garantir que os potenciais sejam definidos de forma única, a
partição do campo deve ser fixada de alguma forma. Isso geralmente é
feito completando a definição de A.
Um
campo vetorial pode ser definido, além de uma constante arbitrária,
especificando seu rotacional e sua divergência. No caso do potencial
vetor magnético, o rotacional é dado pela Eq. 10. É necessário apenas
decidir sobre a divergência para que ela esteja totalmente
especificada. A especificação da divergência é chamada de condição de
calibre.
Substituindo a Eq. 13 na Eq. 12 fornece:
Expandindo o lado esquerdo com a identidade vetorial \/ x \/ x = \/\/-\/, obtemos:
A divergência de A
é comumente definida como zero (condição de calibre de Coulomb), mas
isso, em geral, não separaria os potenciais escalar e vetorial. Em vez
disso, a condição de calibre é escolhida:
o que elimina os dois últimos termos da equação 15, resultando em:
A equação 17 se assemelha à equação de difusão para fluxo de calor e possui soluções semelhantes no domínio do tempo.
A
maioria dos ensaios de correntes parasitas, no entanto, é realizada com
corrente alternada, cuja dependência temporal é simplesmente uma
oscilação harmônica no tempo. A oscilação harmônica é caracterizada por
uma amplitude e uma fase, que podem ser convenientemente representadas
na forma fasorial: A(r,t) = R{A(r) ejwt}, onde A(r)
é um vetor complexo que representa a amplitude e a fase das componentes
do potencial vetor magnético e onde j = \/(-1), R denota a operação de
extrair a parte real e ω
é a frequência angular (radianos por segundo). Observe que o mesmo
símbolo é usado aqui para representar tanto a quantidade real
dependente do tempo A(r,t) quanto a quantidade complexa A(r),
mas elas são distinguidas por seus argumentos. Em outros lugares, os
argumentos não serão fornecidos e a distinção entre as duas deve ser
reconhecida pelo contexto. A derivada temporal fornece:
Portanto, para a teoria harmônica temporal, jω é substituído por δ.(δt)-1 na Eq. 17 e o potencial vetor pode ser visto como um fasor complexo.
Dessa forma, a Eq. A equação 17 torna-se a equação 19:
1.3 MODÊLOS ANALÍTICO E NUMÉRICO
Existem
diferentes tipos de modelos. Alguns são analíticos e outros numéricos.
Os modelos analíticos são computacionalmente mais eficientes do que os
modelos numéricos. No entanto, os modelos numéricos são muito mais
flexíveis e podem ser usados para modelar geometrias complexas de
descontinuidades, não linearidade do material e outras complexidades
associadas a cenários de ensaiosreais.
A
seguir, são descritos modelos analíticos que caracterizam o
comportamento de correntes parasitas em meios condutores homogêneos
livres de descontinuidades, particularmente o modelo proposto por Dodd
e Deeds (R10) e
suas extensões. Soluções analíticas e integrais, técnicas numéricas que
abrangem descontinuidades em materiais, também são descritas a seguir,
assim como técnicas numéricas baseadas em análise de diferenças finitas
e elementos finitos.
2. MODÊLO DO MEIO CONDUTOR HOMOGÊNEO
2.1 FUNDAMENTOS
Os ensaios quantitativos de correntes parasitas baseados em modelos
evoluíram de forma constante com as melhorias na capacidade
computacional. O foco na modelagem precisa levou a uma compreensão
completa dos ensaios de correntes parasitas e à automação total dos ensaios de campo. (R02) (R07) A
modelagem é realizada resolvendo as equações de Maxwell, e as soluções
podem ser expressas analiticamente ou numericamente. As soluções
analíticas fornecem expressões de forma fechada para os parâmetros de
interesse nos ensaios de correntes parasitas e são o tema da presente
discussão.
Os
modelos de ensaios de correntes parasitas podem ser usados para o
projeto da bobina, seleção da frequência de ensaio e interpretação dos
dados de ensaio. Grandezas importantes a serem calculadas são a
distribuição de correntes parasitas induzidas no espécime submetido ao
ensaio, bem como a mudança de impedância resultante da bobina. O
cálculo
e a visualização do padrão de fluxo de correntes parasitas podem ser
usados para avaliar a profundidade real de penetração no material e a
interação com descontinuidades específicas. Dessa forma, a configuração
da bobina pode ser otimizada para garantir a máxima interação com
determinados tipos de descontinuidade, levando em consideração
adequadamente a frequência e os parâmetros do material. O cálculo e a
visualização dos planos de impedância podem ser usados para
comparação com sinais de ensaios reais. Essa comparação proporciona uma
melhor compreensão das variações de impedância decorrentes de
descontinuidades conhecidas de tamanho e orientação específicos, bem
como de características materiais e espaciais particulares do objeto de
ensaio.
Problemas
relacionados à indução de correntes parasitas são formulados por meio
de equações diferenciais, que determinam o campo magnético e grandezas
correlatas em um determinado ponto em função da densidade de corrente
de uma fonte existente. O fluxo de correntes parasitas é calculado
utilizando-se a equação diferencial de difusão, que é convenientemente
expressa em termos do potencial vetor magnético. Existem duas maneiras
de resolver essa equação diferencial: técnicas analíticas e numéricas.
Analiticamente,
a equação é resolvida pela técnica de separação de variáveis dentro
de uma região da geometria. A influência de fontes externas à região é
considerada pela imposição de condições de contorno apropriadas.
Soluções analíticas podem lidar com problemas bidimensionais, problemas
axisimétricos e, em certos casos, tridimensionais, desde que as
equações correspondentes sejam lineares e a geometria das fronteiras e
fontes seja relativamente simples. Como a classe de geometrias que
podem ser tratadas geralmente se restringe a problemas com fronteiras
canônicas (regiões planas, cilíndricas e esféricas), essas técnicas
permitem apenas uma aproximação para problemas com fronteiras não
canônicas ou descontinuidades. As soluções obtidas por técnicas
analíticas são gerais e exatas, proporcionando uma compreensão mais
profunda do problema. Elas são normalmente obtidas na forma de uma
relação matemática, que pode então ser usada para análise, estudos
paramétricos e calibração de sistemas de ensaio. Um aspecto importante
dos modelos analíticos é que as expressões de forma fechada são
facilmente codificadas, seja com linguagens de programação de alto
nível ou com pacotes matemáticos comerciais, exigindo, portanto, um
esforço mínimo do desenvolvedor. Quando as soluções são codificadas,
elas são muito mais rápidas do que as técnicas numéricas, que exigem
tempos de computação significativamente maiores.
Soluções
analíticas também são usadas para validação de soluções obtidas por
técnicas numéricas mais complexas. Estas últimas produzem resultados
numéricos em vez de expressões de forma fechada, e sua precisão pode
ser confirmada independentemente por modelos analíticos, que fornecem
uma alternativa de baixo custo à verificação experimental de resultados
numéricos.
Modelos
para problemas com fronteiras canônicas são descritos abaixo, começando
com os modelos bem estabelecidos desenvolvidos por Dodd e Deeds.
Extensões desses modelos, bem como modelos tridimensionais e modelos
semianalíticos para problemas envolvendo fronteiras canônicas, são
apresentados. Soluções aproximadas com aplicação à modelagem de
descontinuidades são apresentadas em outro local, abaixo.
2.2 MODÊLOS ANALÍTICOS
No
caso de uma geometria axisimétrica bidimensional com simetria
rotacional em torno do eixo Z, a Eq. 17 em uma região livre de fontes
torna-se:
A
equação acima é resolvida adotando-se a técnica de separação de
variáveis. Embora muitas aplicações possam ser modeladas com uma
geometria axisimétrica, muitas aplicações são descritas por uma
geometria tridimensional que apresenta dificuldades específicas. Essas
dificuldades surgem ao usar coordenadas curvilíneas para a descrição do
problema, porque os componentes de A (Eq. 17) estão acoplados nas
equações diferenciais escalares resultantes. Nesse caso, a técnica de
separação de variáveis não pode ser aplicada. O inconveniente é
evitado usando o potencial vetorial de segunda ordem W, que foi
introduzido por Smythe (R08). Para o caso de um solenoidal A, com
divergência zero como na Eq. 17, W é definido como:
onde μ é um vetor unitário fixo e Wa e Wb são duas funções escalares ortogonais que satisfazem a equação escalar:
Como
a equação acima é separável em vários sistemas de coordenadas,
formulações baseadas em W podem ser usadas efetivamente para a
separação da equação diferencial vetorial da Eq. 17.
Modelos
analíticos adequados para ensaios de correntes parasitas foram
desenvolvidos ao longo dos anos por pesquisadores em ensaios não
destrutivos e em geofísica, bem como por projetistas de ímãs, motores e
aceleradores. Inicialmente, o problema básico estudado era o de uma
fonte de corrente filamentar próxima a um objeto condutor de ensaio. Uma
revisão e uma lista de soluções são apresentadas por Tegopoulos e
Kriezis (R09) para
uma variedade de configurações em relação à forma das fontes e à
geometria dos meios condutores. Os problemas bidimensionais são
estudados usando o potencial vetor magnético A, enquanto os problemas
tridimensionais são tratados usando o potencial vetor de segunda ordem
W.
2.3 MODÊLOS DE DODD E DEEFS
Na teoria do ensaios por correntes parasitas, o trabalho de Dodd e Deeds (R10) forneceu
a base para um dos modelos mais populares. Com base em uma série de
trabalhos anteriores, eles apresentaram soluções para distribuições de
correntes parasitas, na forma de integrais de Fourier-Bessel, para
diversas configurações de bobinas axissimétricas frequentemente
encontradas em aplicações de ensaios por correntes parasitas. Essas
soluções foram aplicadas ao cálculo de correntes parasitas produzidas
por bobinas cilíndricas em condutores planos e cilíndricos, na análise
de mudanças de impedância da bobina causadas pela presença de tais
condutores e na previsão de mudanças de impedância causadas por
descontinuidades no seu interior. (R11) (R12) Uma
característica essencial da análise de Dodd e Deeds é que, em
frequências típicas de correntes parasitas, uma bobina de múltiplas
espiras enrolada com fio isolado circular pode ser aproximada por uma
lâmina de corrente, obtendo-se o campo eletromagnético por superposição.
A
equação diferencial resolvida foi a Eq. 20 e a impedância da bobina foi
calculada a partir da seguinte expressão para simetria axial:
onde Acs é
a área da seção transversal (metros quadrados) e N é o número de
espiras na bobina. O princípio da superposição é aplicado integrando o
potencial vetor magnético sobre a área da seção transversal da bobina.
Expressões
analíticas para o campo eletromagnético e a impedância da bobina foram
obtidas para uma variedade de geometrias comuns de objetos ensaiado
(Fig. 1): para uma bobina cilíndrica de seção transversal retangular
acima de um plano estratificado, circundando uma haste estratificada ou
dentro de um furo cilíndrico estratificado. A configuração esférica da
Fig. 1c também foi considerada, mas o caso particular de uma bobina de
seção transversal retangular foi analisado por Nikitin. (R13) (R14) Uma
vez realizados os cálculos usando uma única bobina, a análise pode ser
estendida para configurações de múltiplas bobinas simplesmente pela
superposição das soluções. (R11) (R15) Os
modelos de Dodd também foram estendidos para um número arbitrário de
camadas, utilizando a técnica matricial proposta por Cheng, Dodd e
Deeds. (R16) (R18)
 
Figura 1. Geometrias de objetos de teste para modelos de Dodd e Deeds:
(a) semi-espaço estratificado;
(b) furo estratificado;
(c) esfera estratificada.
Para
o caso de uma bobina sobre um semi-espaço condutor homogêneo (Fig. 2a),
a expressão analítica para a impedância da bobina é dada por:
onde:
e:
onde a é a variável de integração, J1(x) é a função de Bessel de primeira espécie e primeira ordem, l é a largura da bobina (metro), lo é a distância (metro), r1 é o raio interno da bobina (metro), r2 é o raio externo da bobina (metro), μ é a permeabilidade magnética relativa (adimensional), μo é a permeabilidade magnética (henry por metro) do vácuo e σ é a condutividade (siemens por metro).
 
Legenda:
r 1 = raio interno da bobina = 2 mm (0,08 pol.)
r 2 = raio externo da bobina = 4 mm (0,16 pol.)
l = largura da bobina = 1 mm (0,04 pol.)
μr = permeabilidade magnética relativa do semi-espaço (razão) = 1
σ = = 35,4 MS-m-1 (61% International Annealed Copper Standard)
Figura 2. Bobina acima da placa de metal:
(a) configuração geométrica;
(b) exibição do plano de impedância normalizado.
A densidade de corrente parasita é calculada a partir do potencial vetor magnético:
No caso de uma bobina normal acima de um condutor de semi-espaço (Fig. 2a), a densidade de corrente induzida é dada por:
onde J é a raiz quadrada média da corrente na bobina.
As
equações 24 e 28 envolvem o cálculo numérico de uma integral infinita.
Técnicas de integração numérica disponíveis na maioria dos softwares de
análise numérica podem ser usadas para calcular as integrais.
A
Figura 2b é uma representação de impedância gerada por computador para
uma bobina de superfície. A impedância é representada normalizada,
usando a reatância indutiva da bobina no ar como fator de normalização.
(Essa grandeza também pode ser calculada a partir da Eq. 24, definindo
a condutividade como zero, a₁ = a). Essas representações de impedância demonstram a frequência ideal para um ensaio específico.
Essa
frequência geralmente é aquela que produz a melhor diferença de fase
entre os lugares geométricos de dois parâmetros. O material condutor do
semi-espaço é o alumínio e a curva sólida representa o lugar geométrico
produzido pela variação da frequência de excitação. Como a
condutividade e a frequência sempre aparecem como um produto na Eq. 22,
a mesma curva teria sido produzida para uma frequência de excitação
constante e uma condutividade variável. As linhas tracejadas são as
curvas de afastamento e representam a variação da impedância com o
afastamento da bobina. As curvas pontilhadas mostram a variação da
impedância com a frequência para diferentes permeabilidades magnéticas
do material do semi-espaço.
A
Figura 3 é um exemplo de uma representação gerada por computador dos
contornos das correntes parasitas induzidas por uma bobina de
superfície em várias frequências. Como esperado, as frequências mais
altas resultam em uma menor penetração das correntes parasitas no
objeto condutor. Usando a Equação 28 para uma variedade de bobinas,
observa-se que as densidades de pico das correntes parasitas associadas
a bobinas maiores diminuem mais lentamente com a profundidade do que
aquelas produzidas por bobinas menores. Uma investigação semelhante
conduzida por Mottl (R19) mostrou
que a profundidade de penetração padrão e o atraso de fase linear com a
profundidade, obtidos como soluções para o caso de onda plana,
raramente se aproximam da distribuição de correntes parasitas em
amostras condutoras sob uma bobina real. A profundidade de penetração
padrão permanece um parâmetro do material, e não uma medida real de
penetração.
Figura 3. Contornos das correntes parasitas induzidas por bobina de superfície em várias frequências:
(a) 1 kHz;
(b) 10 kHz;
(c) 100 kHz.
Os
modelos de Dodd e Deeds provaram ser muito úteis, pois foram
bem-sucedidos na previsão de dados experimentais de medições de
correntes parasitas. Desde a década de 1970, eles têm sido amplamente
utilizados pela comunidade de ensaios não destrutivos no projeto de
ensaios por correntes parasitas. Mais especificamente, têm sido usados
para otimizar tipos gerais de ensaios por correntes parasitas, como
medições de espessura e condutividade, para otimizar testes específicos
para problemas específicos e para auxiliar no projeto de instrumentação
de indução geral para controle de processos.
2.4 EXTENSÕES DOS MODÊLOS DE DODD E DEEDS
Os
modelos de Dodd e Deeds assumem uma variação harmônica no tempo para a
solução da equação de difusão. Técnicas de modelagem semelhantes podem
ser usadas no caso de excitações transientes de bobinas, como funções
de tempo em degrau ou pulsos retangulares. Essas excitações de corrente
são usadas na técnica de correntes parasitas pulsadas, que é aplicada à
detecção de perdas de metal ou trincas em maiores profundidades.
Além
da superposição de bobinas, diferentes frequências também podem ser
sobrepostas para obter a resposta de um sistema de correntes parasitas
transientes. Uma técnica simples para avaliar campos transientes é
obter, por meio de uma transformada de Fourier, o espectro de
frequência do pulso de corrente de excitação e calcular a resposta de
tensão em cada frequência, adquirindo assim o espectro de
tensão-frequência. A resposta de tensão transiente é então obtida por
uma transformada inversa de Fourier. Uma vantagem distinta dessa
técnica é que ela pode ser aproximada numericamente usando a
transformada rápida de Fourier. Bowler (R20) usa
essa abordagem para uma excitação pulsada com a forma de uma função
degrau com a bobina localizada acima de um sistema estratificado.
consistindo em duas placas. A configuração imita geometrias encontradas
na detecção e identificação de metal em juntas sobrepostas de aeronaves.
Outra
técnica para avaliar campos transientes é calcular a transformada de
Laplace das equações de campo, resolver as equações transformadas e
recuperar o comportamento no domínio do tempo por meio de uma
transformada inversa de Laplace. Essa abordagem é seguida por Waidelich (R21) , Ludwig (R22) , Sapunov (R23) e Bowler (R24) para
obter a resposta de tensão de uma bobina situada acima de um plano
condutor estratificado. No caso de um semi-espaço condutor homogêneo ou
para sistemas de placas finas simples (R25) ,
a transformada inversa de Laplace pode ser obtida analiticamente, mas
no caso de um semi-espaço estratificado isso não é possível e técnicas
numéricas são necessárias para obter a resposta em função do tempo.
Nessa situação, uma rotina numérica robusta deve ser usada para
calcular a transformada inversa de Laplace. Em outras situações, é
preferível trabalhar com a solução no domínio da frequência, como já
descrito, usando a transformada de Fourier.
As
Figuras 4 a 6 mostram as respostas de tensão obtidas para o caso
descrito por Bowler.(R20) A resposta de tensão é calculada avaliando-se
numericamente a transformada inversa de Laplace. Observa-se que certas
características do pulso, como a amplitude, o tempo de chegada do
máximo e o ponto de cruzamento, são sensíveis a diferentes
características geométricas, possibilitando assim a estimativa da perda
no metal.
Legenda:
Figura 4. Perda metálica na placa superior em um sistema de duas placas:
(a) configuração;
(b) potencial elétrico transiente.
O sinal apresentado é a tensão da bobina subtraída da resposta da mesma bobina devido ao semi-espaço condutor.
A porcentagem de variação do parâmetro é em função da espessura de uma placa.
Legenda:
Figura 5. Separação entre placas em um sistema de duas placas:
(a) configuração;
(b) potencial elétrico transiente.
O sinal apresentado é a tensão da bobina subtraída da resposta da mesma bobina devido ao semi-espaço condutor.
A porcentagem de variação do parâmetro é em função da espessura de uma placa.
Legenda:
Figura 6. Perda metálica na placa inferior acima do sistema de duas placas:
(a) configuração;
(b) potencial elétrico transiente.
O sinal apresentado é a tensão da bobina subtraída da resposta da mesma bobina devido ao semi-espaço condutor.
A porcentagem de variação do parâmetro é em função da espessura de uma placa.
Outras
extensões da técnica de modelagem de Dodd dizem respeito aos perfis de
condutividade e permeabilidade dos objetos ensaiado. As aplicações
incluem têmpera superficial, tratamento térmico, bombardeio iônico ou
processos químicos, que produzem perfis de condutividade e
permeabilidade próximos à superfície com variação suave. Nesses casos,
onde, por exemplo, a condutividade σ(z)
na Eq. 20 é uma função contínua da profundidade, o campo
eletromagnético e a impedância da bobina podem ser avaliados de duas
maneiras.
A
primeira é resolver a Eq. 20 analiticamente para formas especiais de
variações de condutividade. Tais soluções, que resultam em expressões
de forma fechada envolvendo funções transcendentais de ordem superior,
foram derivadas por muitos pesquisadores para funções específicas não
apenas de condutividade, mas também de perfis de permeabilidade
magnética. (R26) (R29) Essa abordagem é muito mais rápida do que a abordagem por partes mais geral descrita a seguir.
Como discutido acima, Cheng (R17) estendeu
os modelos de Dodd e Deed para regiões estratificadas com um número
arbitrário de camadas. Se os perfis contínuos de condutividade e
permeabilidade forem substituídos por perfis constantes por partes,
então é possível aproximar numericamente a impedância da bobina
implementando a técnica acima. Quanto maior o número de camadas, melhor
a aproximação. Usando esta técnica, Uzal (R26) estudou
o problema de um condutor revestido cuja condutividade do revestimento
variava continuamente com a profundidade e a permeabilidade. Embora
esta técnica seja mais lenta do que a baseada na solução analítica para
cada perfil específico, ela é mais geral e particularmente útil quando
se deseja resolver o problema inverso, ou seja, avaliar o perfil a
partir de medições de frequência variável. A abordagem por partes
também foi estendida a objetos de teste cilíndricos e esféricos por
Uzal e Theodoulidis, respectivamente. (R30) (R31)
2.5 MODÊLOS TRIDIMENSIONAIS
Os
modelos descritos até agora são bidimensionais e axissimétricos. Sua
simplicidade reside no fato de o potencial vetor magnético ter apenas
uma componente e a técnica de separação de variáveis ser aplicável.
Uma quantidade significativa de trabalhos aborda modelos de bobinas com
formatos diferentes da bobina cilíndrica clássica ou posições que
destroem a axissimetria. Um problema de grande interesse é a avaliação
do campo eletromagnético tridimensional para uma bobina com formato e
orientação arbitrários sobre um semi-espaço condutor.
The models described so far are
two-dimensional and axisymmetric. Their
simplicity lies in the fact that the
magnetic vector potential has only one
component and the technique of
separation of variables is applicable.
A significant amount of work concerns
models of coils that have shapes other
than the classical cylindrical coil or
positions that destroy the axisymmetry. A
problem of great interest is the evaluation
of the three-dimensional electromagnetic
field for a coil with an arbitrary shape and
orientation above a conducting half space.
Weaver (R32) apresentou
uma teoria geral da indução eletromagnética em um semi-espaço condutor
por uma fonte magnética externa usando os vetores de Hertz elétrico e
magnético, enquanto Hannakam (R33) forneceu
soluções para uma bobina filamentar usando a formulação similar do
potencial vetor de segunda ordem. Com base nesta última formulação,
Kriezis (R34) avaliou
a densidade de corrente parasita induzida em um semi-espaço condutor
por uma bobina filamentar cujo eixo é paralelo à superfície.
Weaver (R32) presented a general theory of
electromagnetic induction in a
conducting half space by an external
magnetic source using the electric and
magnetic hertz vectors whereas
Hannakam (R33) provided solutions for a
filamentary coil using the similar second
order vector potential formulation. Based
on the latter formulation, Kriezis (R34)
evaluated the eddy current density
induced in a conducting half space by a
filamentary coil whose axis is parallel to
the surface.
Outros pesquisadores, como Beissner e Bowler (R35), têm
preferido as funções diádicas de Green na resolução do problema. Bowler
conseguiu apresentar expressões analíticas para a densidade de
correntes parasitas de uma bobina cilíndrica orientada verticalmente
sobre um semi-espaço condutor, estendendo assim os resultados de
Kriezis para uma bobina de sonda de correntes parasitas de seção
transversal finita. Beissner (R37) e Tsaknakis (R38) apresentaram
fórmulas para a distribuição de correntes parasitas provenientes de
fontes cilindricamente simétricas inclinadas em um ângulo arbitrário em
relação à normal da superfície. A solução geral para uma fonte não
simétrica assume a forma de uma integral de Fourier bidimensional.
Other researchers like Beissner?S and
Bowler (R35) have favored Green’s dyadic
functions in solving the problem. Bowler
was able to present analytical expressions
for the eddy current density of a vertically
oriented cylindrical coil over a conducting
half space, thus extending the results of
Kriezis to an eddy current probe coil of
finite cross section. Beissner (R37) and
Tsaknakis (R38) presented formulas for the
eddy current distribution from.
cylindrically symmetric sources inclined
at an arbitrary angle with respect to the
surface normal. The general solution for a
nonsymmetric source is in the form of a
two-dimensional fourier integral.
Os
cálculos numéricos para o caso não simétrico são, portanto, mais
exigentes do que aqueles necessários para avaliar os campos a partir
das fórmulas de Dodd e Deeds, onde as integrais são unidimensionais. Um
modelo semianalítico também foi apresentado por Juillard (R39) para
o mesmo problema, onde a bobina é dividida em vários elementos
denominados fontes de corrente pontuais. O problema é resolvido para
cada fonte de corrente pontual e a superposição é aplicada para
calcular o campo eletromagnético de toda a bobina. Outra técnica para
calcular o campo magnético, baseada na transformada de Fourier, foi
apresentada por Panas (R40) e Sadeghi (R41), que resolveram o problema de uma bobina elíptica e de uma bobina retangular em posição inclinada, respectivamente.
Numerical computations for the
nonsymmetric case are therefore more
demanding than those needed to evaluate
fields from Dodd and Deeds formulas,
where the integrals are one-dimensional.
A semianalytical model was also presented
by Juillard (R39) for the same problem where
the coil is divided in a number of
elements called point current sources. The
problem is solved for each point current
source and superposition is applied to
compute the electromagnetic field from
the whole coil. Another technique for
computing the magnetic field, based on
the fourier transform, was presented by
Panas (R40) and Sadeghi, (R41) who solved the
problem of an elliptical coil and a
rectangular coil in an inclined position,
respectively.
Uma
conclusão importante de todos esses estudos é que as correntes
parasitas induzidas no condutor fluem paralelamente à superfície do
condutor, independentemente da forma da bobina indutora. As Figuras 7 e
8 mostram as correntes parasitas induzidas na superfície de um
semi-espaço metálico condutor por uma bobina retangular quando a bobina
está paralela e perpendicular ao metal.
An important conclusion of all these
studies is that the eddy currents induced
in the conductor flow parallel to the
surface of the conductor, irrespective of
the shape of the inducing coil. Figures 7
and 8 show the eddy currents induced on
the surface of a conducting metal half
space from a rectangular coil when the
coil is parallel and perpendicular to the
metal.
 
Ficure 7. Eddy current testing with rectangular coil parallel
to test object:
(a) setup;
(b) eddy current pattern.
Figura 7. Teste de correntes parasitas com bobina retangular paralela ao objeto de teste:
(a) configuração;
(b) padrão de correntes parasitas.
 
Ficure 8. Eddy current testing with rectangular coil
perpendicular to test object:
(a) setup;
(b) eddy current
pattern.
Figura 8. Teste de correntes parasitas com bobina retangular perpendicular ao objeto de teste:
(a) configuração;
(b) padrão de correntes parasitas.
O problema de uma bobina de formato arbitrário adjacente a um sistema condutor cilíndrico foi estudado por Hannakam (R42) com o potencial vetorial de segunda ordem e por Grimberg (R43) (R44) com funções de Green diádicas. Hannakam (R45) , Theodoulidis (R46) e Mrozynski (R47) estenderam
a formulação do potencial vetorial de segunda ordem no sistema de
coordenadas esféricas para resolver o problema de uma bobina de formato
arbitrário adjacente a uma esfera condutora. Uma conclusão importante
foi que as correntes parasitas fluem em superfícies esféricas
concêntricas à superfície do condutor.
The problem of an arbitrarily shaped
coil beside a cylindrical conducting
system was studied by Hannakam (R42) with
the second order vector potential and by
Grimberg (R43)(R44) with dyadic Green’s
functions. Hannakam, (R45) Theodoulidis (R46)
and Mrozynski (R47) extended the second
order vector potential formulation in the
spherical coordinate system to solve for
an arbitrarily shaped coil beside a
conducting sphere. An important
conclusion was that the eddy currents
flow in spherical surfaces concentric with
the conductor’s surface.
Todas
as soluções analíticas acima mencionadas referem-se ao campo
eletromagnético, com ênfase na densidade de correntes parasitas
induzidas. A variação da impedância da bobina, por outro lado, é
calculada em duas etapas: (1) primeiro, o problema tridimensional da
avaliação do campo eletromagnético é resolvido analiticamente e (2) em
seguida, aplica-se a expressão geral da variação da impedância de uma
bobina. Uma expressão para a variação da impedância foi derivada por
Auld (R48) .
Demonstrou-se, por meio do teorema da reciprocidade de Lorenz, que a
variação da impedância de uma sonda de correntes parasitas na presença
de uma descontinuidade é expressa em termos de uma integral avaliada
sobre qualquer superfície fechada S que contenha a descontinuidade.
All of the above analytical solutions
concern the electromagnetic field with
emphasis on the induced eddy current
density. The impedance change of the
coil, on the other hand, is calculated in
two steps: (1) first the three-dimensional
problem of evaluating the electromagnetic
field is solved analytically and (2) then
the general expression of the impedance
change of a coil is applied. An impedance
change expression was derived by Auld.(R48)
It was shown, through the lorenz
reciprocity theorem, that the change in
the impedance of an eddy current probe
in the presence of a discontinuity is
expressed in terms of an integral
evaluated over any closed surface S
containing the discontinuity.
where n is the unit vector normal to the surface and where E and H are the electric
and magnetic field intensities; the primed
symbols denote the fields in the presence
of the discontinuity and the unprimed
symbols denote the fields in the absence
of the discontinuity. The ΔZ formula is
well suited to derivation of general
expressions and can also be used
effectively to compute the impedance
change of a coil in canonical problems.5
This development is significant because
the coil geometry does not appear
explicitly (no integrals appear over the
volume of the coil) and allows the choice
of planar, cylindrical and spherical
boundaries in keeping with the symmetry
of the problem.
onde n é o vetor unitário normal à superfície e onde E e H são
as intensidades dos campos elétrico e magnético; os símbolos com
apóstrofo denotam os campos na presença da descontinuidade e os
símbolos sem apóstrofo denotam os campos na ausência da
descontinuidade. A fórmula ΔZ é
adequada para a derivação de expressões gerais e também pode ser usada
efetivamente para calcular a variação de impedância de uma bobina em
problemas canônicos.⁵ Este desenvolvimento é significativo porque a
geometria da bobina não aparece explicitamente (nenhuma integral
aparece sobre o volume da bobina) e permite a escolha de condições de
contorno planas, cilíndricas e esféricas, em consonância com a simetria
do problema.
No
caso particular de uma bobina com forma e orientação arbitrárias, sobre
um semi-espaço condutor, a superfície de integração coincide com a
superfície do semi-espaço, fechada por uma superfície no infinito, que
não contribui. Seguindo essa abordagem e resolvendo analiticamente o
campo eletromagnético tridimensional, Burke* apresentou a seguinte
expressão geral para a impedância de qualquer bobina sobre um
semi-espaço condutor:
In the particular case of a coil with
arbitrary shape and orientation, above a
conducting half space, the surface of
integration coincides with the surface of
the half space, closed by a surface at
infinity, which makes no contribution.
Following this approach and solving
analytically for the three-dimensional
electromagnetic field, Burke*?>° presented
the following general expression for the
impedance of any coil over a conducting
half space:
where uw
andv are integration variables,
onde μ e v são variáveis de integração,
e:
O termo B^5z(μ,v) denota
a transformada dupla de Fourier da componente normal do campo magnético
da fonte na superfície do plano metálico. Para formas de bobina
simples, possui uma expressão analítica em termos de μ e
v. Para formas mais complexas, deve ser calculado numericamente usando
a lei de Biot-Savart. A mesma abordagem foi seguida por Theodoulidiss (R51) (R52) para
avaliar a impedância de uma bobina retangular sobre um semi-espaço
condutor e foi posteriormente estendida a coordenadas cilíndricas para
avaliar a impedância de uma bobina em posição deslocada em relação a um
tubo, simulando assim o sinal de oscilação presente durante os testes
de tubo.
The term B^5z(μ,v) denotes the double
fourier transform of the normal
component of the source magnetic field
on the surface of the metal plane. For
simple coil shapes, it has an analytical
expression in terms of μ and v. For more
complex shapes, it has to be calculated
numerically using the Biot-Savart law. The
same approach was followed by
Theodoulidiss (R51)(R52) for evaluating the
impedance of a rectangular coil over a
conducting half space and was further
extended to cylindrical coordinates for
evaluating the impedance of a bobbin coil
in an offset position to a tube, thus
simulating the wobble signal present
during tube tests.
2.6 PERTURBAÇÃO E EXPANSÃO DA FUNÇÃO DE ENGEN
The class of problems that can be solved
analytically can be extended with the aid
of perturbation techniques, which are
often used to provide solutions to
physical problems that would otherwise
be difficult or time consuming to treat.
Perturbation techniques are inherently
approximate and their main applicability
is in the modeling of discontinuities. Such
techniques can be used by assuming that
the conductivities of the discontinuity
and the surrounding medium do not
differ very much or by considering
limiting cases such as a high frequency
limit. (R53)
A
classe de problemas que podem ser resolvidos analiticamente pode ser
estendida com o auxílio de técnicas de perturbação, frequentemente
utilizadas para fornecer soluções a problemas físicos que, de outra
forma, seriam difíceis ou demorados de tratar. As técnicas de
perturbação são inerentemente aproximadas e sua principal
aplicabilidade reside na modelagem de descontinuidades. Tais técnicas
podem ser utilizadas assumindo-se que as condutividades da
descontinuidade e do meio circundante não diferem muito ou
considerando-se casos limite, como um limite de alta frequência. (R53)
Nevertheless, perturbation techniques
have also been applied to models of
canonical problems. A technique called
the layer approximation, based on the
analytic transfer matrix solution for the
electric field in a layered metal, was used
by Satveli (R54) to calculate the impedance
change in a number of canonical
problems. Burkes (R55) also has presented a
perturbation technique, which enables
the impedance computation in the high
frequency limit when the conducting
region is canonical. The technique was
applied to the cases of a two-dimensional
conducting wedge anda slot in a
conducting half space.
Não
obstante, as técnicas de perturbação também têm sido aplicadas a
modelos de problemas canônicos. Uma técnica denominada aproximação de
camadas, baseada na solução analítica da matriz de transferência para o
campo elétrico em um metal estratificado, foi utilizada por Satveli (R54) para calcular a variação de impedância em diversos problemas canônicos. Burkes (R55) também
apresentou uma técnica de perturbação que permite o cálculo da
impedância no limite de alta frequência quando a região condutora é
canônica. A técnica foi aplicada aos casos de uma cunha condutora
bidimensional e uma fenda em um semi-espaço condutor.
Expansões
das funções de Eigen também podem ser usadas para ampliar ainda mais a classe
de problemas que podem ser resolvidos analiticamente. (R56) (R58) O
problema é resolvido novamente usando separação de variáveis; como a
região de interesse é finita, condições de contorno adicionais limitam
o domínio da solução. Como resultado, a solução envolve séries em vez
de integrais. Os coeficientes da série são calculados resolvendo-se um
sistema matricial, formado pela imposição das condições de interface e
de contorno do problema.
O cálculo numérico dos coeficientes classifica a técnica como semianalítica. A técnica foi efetivamente usada por Theodoulidis (R59) para
derivar uma expressão para a impedância de uma bobina de sonda com
núcleo de ferrite sobre um semi-espaço condutor em camadas.
2.7 CONCLUSÕES
As
soluções analíticas em ensaios por correntes parasitas, embora
restritas a certas geometrias em comparação com as soluções numéricas
mais gerais, possuem uma forma explícita e fechada. Os modelos não são
computacionalmente intensivos e oferecem soluções precisas. Eles têm
escopo limitado, mas não valor limitado.
Sempre
que plausível, as soluções analíticas são preferíveis às numéricas
porque são mais fáceis de aplicar, menos dispendiosas
computacionalmente, mais precisas e, finalmente, permitem estudos
paramétricos fáceis da geometria do ensaio.
3. MODÊLOS ANALÍTICOS E INTEGRAL PARA SIMULAR TRINCAS
3.1 INTRODUÇÃO
Eddy current nondestructive testing uses
inductive probes to excite currents in
electrical conductors. The simple fact that
the coil carrying an alternating current
can sense a discontinuity in a metal is
intuitively easy to understand but
evaluating the signal for a given
configuration of coil and discontinuity is
not always easy. The present discussion
describes calculations of probe signals
from cracks, starting with a review of the
basic theoretical concepts and moving on
to a number of related techniques for
evaluating probe response.
Early investigators applied concepts
from other fields of electromagnetism to
problems in eddy current testing. The
researcher in relatively unexplored areas
of electromagnetic theory inevitably
brings concepts from the parent discipline
and adapts them for the new field of
investigation. As advances in the new area
begin to mature, the new discipline
adopts distinct themes and approaches
that are successful and rewarding. At the
end of the twentieth century, eddy
current nondestructive testing was at a
point of early maturity. Basic problems
had been solved satisfactorily yet many
problems remained open and relatively
underdeveloped.
This discussion of crack theory briefly
reviews a few significant early
developments relevant to the treatment of
crack problems in eddy current testing,
including the analysis of the spherical
inclusion and the penny shaped crack.
Recent advanced developments in the
evaluation of crack signals are then briefly
outlined. Two approaches are described:
(1) integral techniques that represent the
effect of a discontinuity in terms of dipole
distribution and (2) approaches valid at
high frequencies that use small
approximations of standard depth of
penetration.
3.2 ELEMENTOS DA TEORIA DE TRINCAS
The pioneering achievements of Friedrich
Forster and his colleagues in eddy current
nondestructive testing resulted from
extensive theoretical and experimental
investigations,© laying the foundations
on which others have built over the
intervening half century. Early uses of
eddy current testing investigated by
Forster are metal sorting, hardness
measurement and the evaluation of heat
treatments through the effects of electrical
resistivity variations. In developing
instruments for these measurements,
Forster recorded the impedance change of
a solenoid when it was near an electrically
conducting material. In the initial
investigations, the solenoid impedance
changes due to the cylindrical rods acting
as cores were measured using an
inductance bridge. It soon became
apparent that the measurements yielded
results dependent on the dimensions of
the rod. Consequently, much effort would
be devoted to the problem of separating
the effects of variations in the sample
dimensions and the variation in
resistivity. Forster’s ultimate success was
made possible by his willingness and
ability to analyze the problem
theoretically. (R61)
Forster used analytical expressions for
the impedance of an infinite solenoid in
the presence of a conducting rod to
account for the effects of variations in rod
diameter and material properties. Later
Dodd and Deeds derived closed form
integral expressions for the field and
impedance of an axial coil of finite length
encircling an infinitely long rod.!° In
addition, they derived integral expressions
for the impedance and field of a
normal
coil above a layered half-space conductor,
a normal coil being one whose axis is
normal to the surface of the conductor.
Although ferrite cored probes may be
preferred for discontinuity detection
because of their enhanced sensitivity, air
cored coils have been widely used in
calculations because of the ease of
evaluating the field using the formulas of
Dodd and Deeds. Usually numerical
techniques are needed to calculate the
fields of probes with ferrite cores.!,2
However, Theodoulidis has shown that
solutions satisfying Maxwell’s equations
for axially symmetric ferrite cored probes
can be found.5?3 Other significant and
interesting results using the analytical
solutions of Maxwell’s equations are
described elsewhere in this chapter.
More thana decade after Férster’s work
became widely known, an embryonic
discontinuity theory was given in the
dissertation of Michael Burrows.** Central
to the thesis is the idea that a small
discontinuity in a conductor, such as a
tiny spherical cavity, produces a perturbed
field that is the same as that of a suitably
chosen dipole. Because the discontinuity
is small compared with the standard
depth of penetration and small on the
scale of other spatial variations of the
unperturbed field, the field can be
approximated as locally uniform and the
polarization of a spheroidal discontinuity
can be found by using standard textbook
theory.®> Having determined the dipole
intensity, Burrows found the induced
electromotive force in a pickup coil due to
the discontinuity by using an expression
derived from reciprocity principles.
(R66)
Because key elements of this approach
arise in more advanced treatments of
discontinuities, the dipole analysis will be
summarized later.
The small discontinuity analysis is
itself of limited practical application but
the principle of representing the effect of
a discontinuity by an equivalent
electromagnetic source distribution can be
applied to arbitrary discontinuities using
either multipole expansions or a dipole
distribution. Multipole techniques for
representing the field have not been
pursued®’ extensively in nondestructive
testing although they may be fruitful.
However, numerous approaches have
been developed based on the
representation of a discontinuity in terms
of a current dipole distribution. Early
developments in which a volume dipole
density was expanded in terms of volume
elements were made by the geophysics
community,°*-7° followed by an adaption
of the technique by McKirdy’! and by
Bowler, Jenkins, Sabbagh and Sabbagh”*73
to the solution of problems in eddy
current testing. An account of the volume
element technique is given in this
handbook and elsewhere.
Although the equivalent source
representation is a common feature of a
number of crack response calculations, a
seminal article by Kahn, Spal and
Feldman”! can be seen as a significant
initial step for developments that have
taken a different path. In the essentially
two-dimensional problem, the field is
uniform along the length of a crack of
constant depth and negligible opening. If
the standard depth of penetration is small
compared with the crack depth, the
current flow follows stream lines parallel
to the crack faces except at the corners
where the crack meets the surface of the
conductor and in the region of the crack
edge. Kahn gives local solutions for the
corner, face and edge field, each of which
contribute to the impedance change. An
interesting feature of the edge field is that
it has the same mathematical form as that
given by Sommerfeld’ for the diffraction
of a plane wave bya half-plane barrier.
The diffraction of an electromagnetic
wave at a thin conducting barrier and the
flow of eddy currents around the edge of
a crack are physically distinct phenomena
but both are governed by Maxwell’s
equations and are subject to comparable
boundary conditions. For a time
harmonic field, the physical difference
between the two cases is manifest in the
wave number, a number that is real in a
lossless medium but complex in a
conductor. Hence the solutions are
essentially the same, differing only in the
nature of the wave number.
Before describing in more detail the
implications of the equivalent source
approaches, typical examples of the
outcomes of such calculations in the form
of probe signals due to cracks are
reviewed.
3.2.1 PLANO DE IMPEDÂNCIAS
The impedance of an eddy current probe
varies with frequency and with its
proximity to the conductor as measured
bya liftoff parameter, defined here as the
distance from the surface of the
conductor to the base of the coil. Energy
dissipated by induced current is related to
an increase in the resistive part of the
driving point impedance whereas the
reactive component of impedance is
reduced by the induced current as a
consequence of Lenz’s law. Following
Forster, the probe impedance
Zn = Ry + jXn, Normalized with respect to
the magnitude of the free space coil
reactance Xo, varies with frequency as
shown on the impedance plane diagram
(Fig. 9),”6 where the normalized reactance,
X, =X-Xo" is plotted against the
normalized resistance Ry= (R —
Ro)-(Xo)"!,
Ro being the free space coil resistance. In
the low frequency limit, X, = 1 and
Rn = 0, as represented by a point at the
top of the main curve. In the high
frequency limit, the curve intersects the
reactance axis at a value of X,, about 0.68
in this case, which depends on the coil
geometry. For flat pancake coils with a
small liftoff, the limiting value of the
normalized reactance has a lower value
than for a longer solenoidal coil with
larger liftoff. Thus, the high frequency
intersection point is a measure of the
coupling between the probe and the work
piece, having a low normalized reactance
for greater coupling.
The data for the main curve in Fig. 9
were calculated from a Dodd and Deeds
formula!® and hence represent results of
an idealization that neglects interwinding
capacitance and the effects of a finite
penetration depth in the windings. The
parameters of a coil are taken from a
benchmark experiment on simulated
cracks in aluminum.’° Superimposed on
the diagram are two signals calculated
using the lowest and highest frequencies
of the experiments, 250 Hz and 50 kHz
respectively, for the same simulated
planar crack.
Ficure 9. Calculated normalized impedance
variation with frequency of normal coil. Two
discontinuity signals from semielliptical
simulated crack are also shown.
Discontinuity responses were calculated for
excitation frequencies of 250 Hz, upper
trace, and 50 kHz, lower trace, for same
simulated crack. Details of coil parameters
and simulated crack are given by Harrison
and Burke.76
In Fig. 10, the calculated discontinuity
signals are displayed with the background
coil impedance removed. The response is
for a normal coil whose axis is in the
plane of the crack. Taking the crack plane
to be the x = 0 plane, then the impedance
variation shown occurs as the coil is
moved in the horizontal Y direction from
one end of the crack to the other. The
numerical techniques used for calculating
these impedance variations are described
elsewhere in this chapter. First some
general comments are in order,
concerning the nature of numerical
schemes.
Ficure 10. Normalized impedance due to semielliptical
simulated crack shown as impedance plane locus. Trace is
obtained from impedance variation as coil position is varied
along crack. Note that impedance of discontinuity has been
normalized by dividing by free space coil reactance at
designated frequency. Details of coil parameters and
simulated crack are given by Harrison and Burke.76
The discontinuity impedance is
calculated from the electromagnetic field
in the presence of the discontinuity.
Simple cases that can be dealt with
analytically are discussed first. For more
complicated geometries, numerical
techniques are needed. Numerical
techniques for solving electromagnetic
field problems are traditionally
categorized as differential or integral
techniques. Finite element and finite
difference techniques are the most
common in the differential category
whereas the integral techniques can be
classified as boundary element and
volume element techniques.
Most numerical schemes introduce a
set of localized functions defined with
respect to a grid or mesh. Often these
functions are low order polynomials,
which interpolate between nodal points
or the edges of cells. Typically, they do
not satisfy Maxwell’s equations (or the
integral equivalent) nor does a linear
superposition of them forma solution.
Nevertheless, it is postulated that a
superposition of such functions gives a
reasonably accurate numerical
approximation of a solution.
The numerical results rarely come with
a guarantee of accuracy. Because of the
way in whicha solution is constructed,
the results are dependent on a mesh or
grid. In the absence of error estimates,
and these are rarely given, it is important
that code is validated because, even if it is
bug free, the onus is on the author to
demonstrate that the results are reliable.
Elementary techniques, on the other
hand, provide a means of predicting
limited results. A
number of simple
formulas for evaluating discontinuity
signals are given below, preceded by a
summary of the basic expressions for a
current dipole field. The dipole theory is
presented in a way that anticipates the
more advanced numerical techniques for
homogeneous conducting media, in
which integral formulations are used.
3.3 DIPOLO DA CORRENTE
3.3.1 MONO POLO DA CORRENTE ESTATÍSTICA
The current dipole is formed from two
monopoles of opposite polarity adjacent
to one another. A current monopole is a
point source of current with intensity I. In
an unbounded homogeneous region, the
current spreads uniformly in all directions
from the source. Hence, the current
density obeys an inverse square law and is
directed radially from the source. Suppose
a current monopole located at a position
represented by the vector r’ gives rise to a
current density J at some other point
whose coordinate is r. Then:
where R = Ir-r’l and R is a radial unit
vector. Expressing the electric field as
E =-V®, then the current monopole
potential is:
where 6p is the electrical conductivity of
the medium. The potential satisfies the
laplace equation, V7® = 0, except at the
singular point where the point source is
located. The (4nR)-! dependence ofa static
potential due to a point source is
identified as a scalar Green’s function for
a laplacian problem in three dimensions.
3.3.2 DIPOLO DE CAMPO ESTÁTICO
Let two current monopoles of opposite
polarity approach one another while
keeping constant the product of their
source intensity and their separation.
With initial separation dr, the dipole
potential is:
In general, the limit of f(r’ + 5r) — f(r’)
as the separation dr’ tends to zero can be
written as 6r’-V f(r’). Hence the limit
above can be related to the gradient of R!
with respect to the primed source
coordinates. The gradient may be written
in terms of the unprimed field
coordinates with a reversal of sign. Also
expressing the dipole moment as the
(finite) limit of p = dr’ gives the static
current dipole potential:
where p is the dipole moment (ampere
meter).
By taking the negative gradient to find
the electric field and multiplying by the
conductivity, the current density can be
written:
Although the scalar product here can
be seen as producinga scalar function on
which the first gradient acts, the above
expression can also be interpreted as a
dyadic operator, VV(4nR)"!, acting on the
vector p. The final result is the same but
the second viewpoint prompts the idea
that the dyad may be detached from the
vector on which it acts and given a
separate mathematical life. Studying the
properties of dyadic Green’s functions’”
leads to distinct ways of finding solutions
of Maxwell's equations as outlined below.
Before returning to the role of the
dyadic Green’s functions, a simple
illustration of the fundamental utility of
the current dipole is given. The dipole
field is used to express the solution of a
problem in which a uniform current in an
otherwise homogeneous conductor of
electrical conductivity 69 encounters a
spherical inclusion of uniform
conductivity σ.
3.3.3 PEQUENA INCLUSÃO ESFÉRICA
The spherical inclusion problem, usually
found in textbooks as a problem in
electrostatics involving a dielectric sphere,
has a solution that satisfies the laplace
equation inside and outside the sphere.
Interface conditions on its surface ensure
that the normal current and tangential
electric field are continuous. Given a
uniform field Ep in the Z direction, which
is also the polar direction of a spherical
coordinate system (z = R cos 8) and
defining the parameter s as the
conductivity ratio s = o-o9', the internal
potential (volt) is:65
whereas outside the sphere the potential
is:
where Θ is the polar angle (radian). The
external potential can also be written:
where the dipole intensity and direction
are given by:
Perhaps of greater interest here is the
fact that the external electric field can be
written:
where Eo = £o2. This goes beyond the basic
textbook account by expressing the field
of the dipole in terms of a dyadic Green’s
function, 69 !VV(42R)"1.
Equation 42 can apply to a dipole of
arbitrary orientation. Figure 11 shows the
current associated with the perturbed field
that when added to the unperturbed
current 6oEo2 gives the total current
density.
Ficure 11. Perturbed current at small spherical inclusion in
metal.
An additional point of interest is that
the dipole intensity can be related to a
uniform current dipole density P
distributed in the spherical region. By
puttingp = 4.3"! x na8P, it is found that:
where E is the electric field in the sphere
given by taking the negative gradient of
Eq. 38.
3.3.4 DIPOLO DINÂMICO DA CORRENTE
In eddy current testing, the fields are
dynamic rather than static. Therefore, the
dynamic current dipole has a more
significant elemental discontinuity field
than does the field of static current
dipole. The dynamic current dipole for a
time harmonic field is described by
essentially the same equations as those
used for the textbook treatment of the
hertzian dipole.’® The difference arises
from the fact that in eddy current
applications the host medium is a
conductor, not air. In a good conductor
such as a metal, the charge current is
much larger than the displacement
current. Consequently, the latter can be
neglected. This means that Ampére’s law
(Eq. 2), V x
H =J, is adequate and
Maxwell’s addition of the displacement
current j@D to the right hand side of this
relationship is not needed. Here, the field
is expressed in terms of complex phasors,
which means, for example, that the
magnetic field varies in time as the real
part of He/, m being the angular
frequency (radian per second) of the
excitation. The neglect of displacement
current means that solutions are sought in
the quasistatic limit. As a short cut from
the description of waves in air to fields in
a conductor, the displacement current
j@eE, which appears in standard hertzian
dipole theory,”* can usually be replaced
with the charge current ooE.
It is convenient to express the dynamic
field in terms of a magnetic vector
potential A, related to the magnetic flux
density:
and having a gage condition:
replacing the usual lorenz condition. For a
current dipole in an unbounded
conductor of conductivity op and
permeability of vacuum, the magnetic
vector potential is:
where k = V(-j@oL09), taking the root with
a positive real part and I-p = p. The
parameterk is related to the standard
depth of penetration 6 (meter):
where:
The identity dyad I in Eq. 46 has been
inserted to express the magnetic vector
potential as a dot product of a dyadic
operator acting on a vector source, this
being the appropriate general form for the
relationship between a vector source and
a vector field A. The magnetic field due to
the current dipole is found from:
The electric field is found from Ampére’s
law in the form:
Combining Eqs. 46, 49 and 50 gives:
Equation 51 has been derived from the
identity V x V x = VV--V? and from the
fact that the vector potential satisfies
Helmholtz’s equation.”? A discussion of
the dyadic form between the braces has
been given by Tai.’78°
Clearly, the dynamic dipole field
reduces to the static case, Eq. 42 in the
limit, as angular frequency @ goes to zero.
It also reduces to the static case in the
near field, where the first term of the
dyadic operator is negligible. This is a
reminder of the fact that a local field on a
scale small compared with the standard
depth of penetration 6 can often be
analyzed using electrostatic or
magnetostatic theory.
Equation 51 may be generalized to give
the perturbed field due to a volumetric
discontinuity by representing the effect of
such a discontinuity as a general dipole
distribution P(r’). Then the perturbed
electric field is found by replacing the
point dipole p in Eq. 51 by P(r’) and
integrating with respect to the source
coordinate r’ over the region of the dipole
density. This field representation is used
in volume integral formulations and is a
preliminary step toward a volume element
calculation of the dipole density.”2
Similarly, the effects of a thin crack can be
represented by a surface dipole layer and
form the basis of a boundary element
formulation.®4 In either case, the dipole
density is determined by an integral
equation. Having founda solution, the
probe signal due to the discontinuity can
be calculated from the probe response
formulas below.
3.3.5 RESPOSTA DA SONDA
An eddy current probe senses
discontinuities through changes of
impedance. There are a
number of
techniques for calculating the
discontinuity response depending on the
details of the approach used. For example,
Kahn and others used the integration of
the poynting vector over a surface.”4 Auld
uses a reciprocity relationship attributed
to Lorenz*® whereas others use a
reciprocal relationship associated with
Rumsey.°°8! Rumsey’s relationship is used
next.
The coil current density can be
represented by a function J. With E®
defined as the perturbed field due to the
discontinuity, the probe impedance
change due to the discontinuity is:
where the integration is over the coil
region denoted by Q,.
The coil current can be used as a phase
reference and taken to be real. Although
the coil current is confined to the coil
windings, these are usually on such a
small scale that the current density can be
approximated as a smooth function,
usually a constant, over the coil cross
section. In a calculation in which the
effects of the discontinuity are
represented by a dipole volume
distribution P, Rumsey’s reciprocal
relations may be invoked to write the
impedance change in terms of the
unperturbed electric field at the
discontinuity E:72
where the integration is now over the
region Qp where the discontinuity
conductivity differs from that of the host.
Equation 53 is advantageous because P is
usually calculated directly by an integral
equation technique whereas the
evaluation of E) for Eq. 52 requires an
additional step once the dipole density
has been found. In general:
which defines P(f) for an arbitrary
discontinuity whose conductivity o(r)
differs from that of the host conductivity
Go. For the special case of the small
spherical region with constant
conductivity, a similar relationship
(Eq. 43) is used.
3.3.6 PEQUENAS DISCONTINUIDADES
For a small spherical discontinuity, such
as a gas bubble or spherical inclusion in a
conductor, the impedance change sensed
by a probe is given by:
An explicit expression for the response
can be found using a suitable unperturbed
field, for example the normal coil field in
a half-space conductor.!° A simpler case is
one where the field at the surface of the
conductor is uniform. This approximation
may in practice be reasonable if the probe
dimensions are larger than the standard
depth of penetration. With Ho as the
tangential magnetic field in the
(horizontal) Y direction and the Z
direction normal to the surface of the
conductor, the unperturbed
electromagnetic field in a conductor
below the plane z =0 is given by:
e:
By substituting the expression for the
dipole density of a small spherical
inclusion given in Eq. 41 into the relation
Eq. 55 with E® given by Eq. 57, it is
found that, for a small spherical cavity
(o = 0), centered at r = ro and of radius a
(meter), the impedance is:
The impedance change is proportional
to a? simply because the dipole intensity
varies in proportion to the volume of the
sphere. Note that the ratio Ho-I"' is real for
a magnetic field uniform at the surface.
However, it may be useful to estimate the
small sphere response for a
nonuniform
field, for which Eq. 55 applies if the
unperturbed field is known. Note that the
maximum value of the ratio Hy-I-! can be
regarded as a figure of merit for the probe
because the signal intensity depends on
its square. Note also that the factor 2jkz
in the exponential of Eq. 58 indicates that
the signal is attenuated over a path of
length 2z, representing the round trip
distance from the surface to the
discontinuity and back.
Another small discontinuity result that
can be found by elementary means is the
response due to a semicircular surface
crack of negligible opening whose radius
is smaller than the standard depth of
penetration. The assumption of a
relatively large standard depth of
penetration means that the local field can
be treated as static in the sense that it
may be described by a potential satisfying
the laplace equation. The surface of the
conductor acts as a plane of reflection,
allowing a conversion of the semicircular
crack problem to a circular crack problem
by appealing to the technique of images.
The problem can then be solved as if the
crack were a thin disk in a uniform stream
of incompressible fluid. With a uniform
applied electromagnetic field given by
Eqs. 56 and 57, the impedance change
due to such a crack is:82:83
Of practical importance is the question
of what limits the detection of small
cracks. Equation 59 yields insight and
significant basic information in this
regard. First, note that the response
depends on the third power of the crack
radius. Second, the impedance change
increases in proportion to the frequency
because k? = — jmpbtoo:
Thirdly note that for a strictly uniform
field, the change in impedance is purely
inductive (imaginary), HoT"! being real.
Even if the assumptions that went into
the derivation of the simple relation given
by Eq. 59 are not precisely satisfied, the
equation can provide an approximate
answer. If the accuracy is inadequate,
improvements may be made by extending
the results to higher order terms by using
perturbation theory or by taking into
account nonuniformities in the field by
an extension of the basic analytical
technique.*?
3.3.7 TRINCAS LONGAS
A long crack of constant depth d (meter)
may be treated as a two-dimensional
problem provided that the unperturbed
field does not vary along its length. Such
a configuration does not relate directly to
most practical problems but its solution
has had an impact on the understanding
of crack fields. The problem can be solved
analytically in the low and high
frequency regimes that correspond to
small and large standard depths of
penetration compared with the crack
depth.
According to the thin penetration
approach of Kahn, assuming the crack is
in the plane x = 0, the field on the crack
faces has the form:
e:
where n is the characteristic impedance of
the medium:
This field can be used to evaluate the
complex time average poynting vector P
(not to be confused with dipole density P)
at the crack faces from:
where the asterisk (*) denotes the complex
conjugate. The uniform face field means
that:
where the characteristic impedance of the
medium is 1 = jk-(oo)-!. The upper and
lower signs on the right side refer to the
positive and negative sides of the crack,
respectively. Integrating the poynting
vector over the crack surface and equating
the result to the energy transferred at the
drive point of the probe gives an
impedance:
per unit length of the crack. To Eq. 65
must be added the corner and edge effects
that together with Zs give rise to a
combined impedance:
The three contributions to the
impedance per unit length include the
field at the edge (represented by the 1 in
parentheses) and the corner field (the
8-1! term). A complete analysis of the
above expression is given elsewhere.*4 The
impedance in this problem therefore
contains a dominant face term that varies
as the square root of frequency, is
proportional to the crack depth and has a
phase angle of m-4-! with respect to the
drive current. The additional terms due to
the edge and corner are resistive.
Complementary to the kahn thin
penetration result is a formula valid in the
low frequency regime that can be found
froma solution valid in the static, direct
current limit. In this regime, ikd is a small
parameter; this fact can be exploited to
find a field solution in the form of an
ordered series using Rayleigh-Ritz
perturbation theory. Likewise, the
impedance can be expressed as an ordered
series:83
For a uniform excitation field, the
leading term at low frequency is purely
inductive and increases linearly with
frequency and as the square of the crack
depth.
The long crack theory is readily
extended in range from the high
frequency limit to lower frequencies by
accounting for the interaction between
the edge and corner fields through the
Weiner-Hopf technique®s and by applying
the perturbation technique to extend the
range of validity of the low frequency
approximation.®’ The impedance results
of these extensions, shown in Fig. 12,
have been compared with numerical
results of a boundary element code.*? In
these figures, the impedance is
normalized by writing:
Legenda:
- - - = teoria de alta frequência
___ = teoria de baixa frequência
o = elemento limite
Ficure 12. Analytical and numerical results of change in
normalized impedance Z, due to long surface breaking
crack: (a) for inductive, or imaginary, component;
(b) for resistive, or real, component.
Hence, the kahn impedance (Eq. 66) is
written in terms of the normalized
impedance:
The main benefit of the study of the two
dimension problem is that it provides a
simple test bed for new techniques,
including an adaption of the geometrical
theory of diffraction,*® to problems in
eddy current crack interaction.
3.4 TÉCNICAS AVANÇADAS
Two types of advanced techniques for
evaluating probe signals due to cracks are
considered next. First, equivalent source
techniques are discussed, of which the
Burrows small discontinuity theory® is an
elementary precursor. Second, the thin
penetration approaches, prototyped by
Kahn and others” and applicable to both
ferromagnetic and nonferromagnetic
materials, are described.
The equivalent source techniques cover
all frequencies and are closely linked with
field formulations based on integral
equations. They can be used to evaluate
fields at cracks in ferromagnetic material
but here the description will be limited to
materials with the permeability of a
vacuum.
Finding a numerical solution from
integral equations can be more
demanding in the thin penetration regime
because a large number of volumetric cells
or boundary elements may be needed to
give an accurate result. Usually, a grid
containing several cells per standard
depth of penetration is required, so the
number of unknowns and the
computational cost are usually high in the
thin penetration regime. Because this cost
is avoided in approaches that explicitly
take advantage of small penetration depth
approximations, the techniques described
here are complementary. To understand
dipole and thin skin techniques, it is
helpful to consider the behavior of the
electric field near the crack mouth and
the properties of the field at the crack
face.
3.4.1 CAMPO ELÈTRICO NA ABERTURA DA TRINCA
The crack opening is typically much
smaller than the standard depth of
penetration. Therefore, a local field theory
for this region can be based on Maxwell’s
equations in the static limit. Because the
electric field varies relatively slowly along
the crack mouth away from the ends, a
two-dimensional solution in a plane
perpendicular to the mouth direction
adequately captures the significant
features. This approach implies that the
solution of the laplace equation in two
dimensions is suitable for the task.
The geometry of the problem (Fig. 13)
lends itself to the Schwarz-Christoffel
theory,8788 which yields a conformal
transformation to map the domain of the
crack and the adjoining half plane above
it into a half plane. An elementary
solution for the half plane will lead to a
fixed potential difference across the crack.
Then, an inverse transform can be applied
to produce a representation of the electric
field at the crack mouth. In this case, a
suitable analytic inverse transform is
apparently lacking and the mapping must
be done numerically by using, possibly,
the newton-raphson iterative technique or
the brent algorithm.89
Forster?° and others?! have used
conformal mapping to determine the
magnetic flux leakage at the crack mouth.
In fact, the mapping is used widely to
find the magnetic field at the gap between
two pole pieces such as the field at the
gap between the poles of a magnetic
recording head.% In eddy current
problems, the electric field is needed
rather than the magnetic field but the
solution is essentially the same (Fig. 13).
Ficure 13. Electric field at crack opening.
At the corners, the electric field is
singular, varying in magnitude in air close
to the corner as (feorner)!/3, Where (Feorner)
is the radial distance from the apex of the
corner. This behavior is characteristic of
the field in the vicinity of a right angled
wedge.°? Between the crack faces, the field
tends to become more uniform deeper
into the crack. The magnitude of the field
between the faces depends on how deep
and wide the crack is. If the crack is made
narrower while the potential across the
crack remains the same, then the
magnitude of the electric field increases.
In the limit of closure without contact,
the electric field forms a singular layer,
infinitely strong, of infinitesimal
thickness. It is this limiting case that will
be explored here because the singular
layer has a simple mathematical
representation.
3.4.2 TRINCA IMPENETRÁVEL
In calculations of the field perturbation
due to a crack, it is usual and convenient
to apply a boundary condition that states
that the normal component of the current
density in the conductor at the crack face
is zero. Although the surface of the crack
supports a distribution of electrical charge
and the charge must get there somehow,
in the quasistatic approximation the
charging current is neglected. In a
conductor, the displacement current jweoE
is neglected because it is very much
smaller than the charge current o9£. Even
at high eddy current test frequencies,
~10 MHz, where the magnitude of
displacement current is greater than at
lower frequencies, the ratio g&@-09! is on
the order of 10~° for a low conductivity
metal, 0.58 MS-m-! (1 percent of the
International Annealed Copper Standard).
However, the accuracy of a boundary
condition that neglects the charging
current at the crack face is dependent on
crack width. Therefore, it is necessary to
seek a justification for the quasistatic
approximation in this context.
The normal component of the true
current, to use Maxwell’s term for the sum.
of the displacement and charge current, is
continuous across an interface. Therefore,
the displacement current between the
faces and directed across the crack is equal
to the charging current at the conducting
side of the crack face. Hence, the
boundary condition is justified if the
displacement current jwgoE, across the
crack is negligible compared with the
tangential charge current ooE; at the crack
face. In the following argument, these
currents are estimated and compared.
Applying Stokes’ theorem to Faraday’s
induction law in differential form gives an
integral form of the induction law in
which the line integral of the electric field
arounda closed path is equated to the
rate of change of magnetic flux through
the surface S bounded by the path. If it
happens that the rate of change of
magnetic flux through S can be neglected.
Then the line integral is approximately
zero:
whereC is the path bounding S and ds is
an incremental displacement along the
path.
For this case, the path links points
ABCD (Fig. 14) in the limit as the points
approach the crack surface. By
considering an exponential field at the
crack face, it can be shown that the
magnetic flux throughS is less than the
path integral of E over a crack face by a
factor on the order w-8!, wherew is crack
width (meter) and6 is standard depth of
penetration (meter). Hence, if w is small
compared with the standard depth of
penetration, as it usually is, then Eq. 70 is
a reasonable approximation. This
equation indicates that the following are
of roughly comparable order of
magnitude: 2Eod = E,w, where E, is the
normal component (volt per meter) of the
electric field in the crack and Ep is the
tangential field (volt per meter) at the
outer surface. That being the case, the
ratio of the displacement current across
the crack jwegE,, to the tangential face
current O9£p is small if:
Ficure 14. Integration path C, crossing crack.
This condition for the validity of the
quasistatic approximation at the crack is
usually satisfied. For example, if
d-w-! = 104, then meod-(ogw)! = 10-5 at
10 MHz in a conductor with a low
conductivity, 0.5 MS-m-! (1 percent
International Annealed Copper Standard).
Assuming the quasistatic approximation
for a
nonconducting crack, the zero
normal current at the crack face is
written:
where the + sign denotes points on one or
the other crack face approached from the
interior of the conductor.
3.4.3 DISTRIBUIÇÃO DO DIPOLO DA CORRENTE NA SUPERFÍCIE
A basic problem to be considered initially
is the probe response to an ideal crack,
defined as having negligible opening
compared with the standard depth of
penetration but satisfying the requirement
in Eq. 72 for the validity of the quasistatic
approximation. The crack is therefore
impenetrable to the flow of electric
current. The ideal crack is defined, for
example, with respect to an open surface
So bounded by the crack edge and by the
intersection of the crack with the surface
of the conductor (Fig. 15).
Ficure 15. Side view of coil and crack, showing crack in Y,Z
plane of coordinate system. Surface So is part of the Y,Z
plane occupied by the crack.
Eddy currents flow around the buried
crack edge such that the current density is
different at points adjacent to one
another on opposite faces. The fact that
the crack opening is neglected means that
the ideal crack gives rise to a discontinuity
in the tangential current density at So
and, consequently, a discontinuity in the
tangential electric field. The solution of
the ideal crack problem can be found by
evaluating the discontinuity in the field
directly or by expressing the jump in the
field in terms of an equivalent dipole
source distribution, either electric °4 or
magnetic.°> The relationship between the
field and the equivalent current dipole
source is described next.
For an open crack, the volume dipole
density P is defined by Eq. 54 and, like
the electric field in the crack, is larger for
cracks of narrower opening. However, the
integral of P along a path C,, across the
crack is expected to tend to a finite value
in the limit as the crack opening becomes
infinitesimal. With w as the width (meter)
of the crack opening, the limit is written:
where p is the surface dipole density
having the vector representation
p = fp.
For a crack whose interior has zero
conductivity, it can be seen from the
definition (Eq. 54) that P = —oE.
Therefore:
This relation can be used in formulas
for the line integral (Eq. 70) along a path
Co around a segment of the surface of an
ideal crack (Fig. 16) to give:
Ficure 16. Integration path Co crosses crack at points A and
B and is formed in limit as At and B+ approach surface So.
where the subscript t denotes components
tangential to Sp and where $s is a
displacement vector between points A and
B on the surface So (Fig. 16):
Because 4g is arbitrary, it can be seen that:
A similar relationship between the
jump in the electric field at a surface and
the gradient of the surface dipole density
exists for the electrostatic charge dipole
layer.® Here it relates the discontinuity in
the dynamic tangential electric field at an
ideal crack surface Sp to the surface
distribution of dynamic current dipoles
whose orientation is normal to So.
Two properties of the dipole density
are worthy of note at this point. Firstly, it
tends to zero at the buried crack edge.
Secondly, the derivative dP-(dz)"! is zero at
the crack mouth, z being the coordinate
whose axis is normal to the surface of the
conductor (Fig. 15). These properties are
written as:
e:
where re is the coordinate of an edge
point and r,, is the coordinate of a point
at the crack mouth. For example, Eq. 80
gives the dipole density for a long straight
crack of depth d in a uniform unperturbed
field Eo:83
Note that p(z) vanishes at z =-d and
that the derivative with respect to z
vanishes at z = 0 in keeping with the
general properties in Eqs. 78 and 79. In
addition, it is important to be aware that
the electric field has a half-power
singularity at the edge of an ideal crack
varying locally as: 96
wherep is the perpendicular distance
(meter) of a point from the edge and9 is
an angle (radian) measured from the
surface Sp in a plane perpendicular to the
edge. This means that, in general, the
dipole density varies as:
near the edge.
The solution of the eddy current ideal
crack problem has been reduced to one of
finding the surface dipole density p. Thus
a scalar replaces a two-component vector,
the jump in the tangential electric field.
Consequently fewer unknowns are needed
for a
numerical solution. To calculate p, it
is necessary to know the continuity
conditions that apply to the magnetic
field at the crack surface Sy because these
conditions will be needed in the
derivation of an equation from which the
dipole density can be calculated.
Although the details of these derivations
will not be given here, it is useful to
understand the continuity conditions that
apply to the magnetic field at the ideal
crack surface.
The jump in the tangential electric
field at the ideal crack is inseparable from
the singular property of the electric field
between the crack faces, as expressed here
in terms of a current dipole layer.
However, no such singular behavior
occurs in the magnetic field. The truth of
this can be demonstrated by following an
argument like the one for the electric field
but applying Stokes’ theorem to Ampére’s
law rather than to the induction law,
thereby forming the line integral of H
around the path Co. Following this
parallel argument, it can be deduced that
the line integral
of H vanishes as the
closed path A_A,B,B_ (Fig. 16) collapses
onto the crack but no singular behavior of
the magnetic field in the crack could lead
to a discontinuity in the tangential
magnetic field. It is concluded that:
at So. In addition, it may be recalled that
the normal magnetic flux B (tesla) is
continuous at an interface.®> At a crack,
which is in fact a double interface, the
same relationship holds:
To confirm the consistency of the
continuity conditions at So, note that
Faraday’s induction law implies that the
normal magnetic flux density at So is:
By using this relationship to express
the difference B,, — B,_ in terms of the
jump in the tangential electric field and
substituting for the jump in the transverse
electric field using Eq. 77, the transverse
curl acts on the transverse gradient of the
dipole density to give zero. Thus, the
continuity of the normal flux density is
ensured by the fact that the jump in the
tangential electric field is expressed as the
tangential gradient of a scalar function.
Having now defined the continuity
conditions at the surface So, one is
equipped for the task of finding a
governing equation for the dipole
density p.
3.4.4 FORMULAÇÃO INTEGRAL
The most common approach to the
solution of electromagnetic field problems
at low frequencies, such as the modeling
of electrical machines, electromagnets and
eddy current discontinuity detection, is to
use a differential formulation as the basis
of a finite element solution. However, in
the area of antennas and electromagnetic
wave propagation, integral techniques are
used more commonly than the finite
element scheme. In the approaches
described here, the aim is to compute
solutions for simple but realistic
geometries using relatively few unknowns
and adapt the forward problem solver for
the task of iterative inversion. Integral
equation techniques are better suited to
this strategy, particularly if the region of
the required solution can be confined to
the discontinuity. The implication is that
the number of unknowns is small and the
forward solver is fast.
In antenna theory, the hertzian dipole
is used as a fundamental solution from
which the field of a wire antenna is found
by integration over the wire structure, a
step that is justified by the principle of
superposition. The elementary current
dipole field (Eq. 51) like the hertzian
dipole, plays the role of a fundamental
solution in a conductor. It allows the field
of an extensive discontinuity in a
conductor to be expressed as an integral
over a discontinuity region. The
fundamental solution is written here as:
where G(rir’) is a dyadic Green’s function
transforming the current dipole source p
into the electric field. For a dipole
embedded in an unbounded domain, the
dyadic Green’s function is given in the
braces of Eq. 51. A more representative
configuration in eddy current testing is
one in which a probe in air interacts with
a discontinuity in a conducting plate. If
the standard depth of penetration is
smaller than the plate thickness, the
conductor can be considered as occupying
a half space (Fig. 15). The dyadic Green's
function for a half space, like the
fundamental solution, is known in
explicit analytical form.°*°4 Hence the
discontinuity field can be written as an
integral over the discontinuity in the
knowledge that the integral kernel will
ensure that the correct continuity
conditions will be satisfied automatically
at the interface of air and conductor.
For a crack in a half-space conductor
(z = 0), the electric field is written as the
sum of the unperturbed probe field E©
and the discontinuity field:
Here the field due to the crack is
expressed in terms of its equivalent
sourcep as superposition of dipole fields
written as an integral over the crack
surface So. It should be noted that, rather
than simply invoking the principle of
superposition, the formal techniques of
deriving integral equations for the field
are based on Green’s second theorem.’”
Equation 87 is multiplied by the
conductivity o) and the condition (Eq. 72)
is applied so that the normal component
of the current density at a point at the
crack surface is zero:
onde:
e:
It is to be understood that the field point
whose coordinate is r approaches a point
r* on the crack and that this limiting
process takes place after the integration
has been performed. Equation 88
determines the current dipole density on
the surface So.
Rather than seeking a solution of the
integral equation itself, an approximation
is constructed by expanding the unknown
p(r) as a linear superposition of a set of N
basis functions and the expansion
coefficients determined by using the
moment technique.” By this
approximation procedure, a matrix
equation replaces the integral equation as
the means of finding the field. The
solution of the matrix equation can then
be found by standard numerical
techniques.*° The classic text on the
moment technique in electromagnetism is
by Harrington’ and a more recent
volume on the subject, which includes
the treatment of dyadic Green’s functions,
is by Wang.98
Having calculated a discrete estimate of
the dipole density p(ro), ro € So, the coil
impedance change due to the
discontinuity is determined from a variant
of Eq. 53:
where the integration is over the surface
So. In applying the moment technique to
the ideal crack problem,% the discrete
approximation of the dipole density
converts the impedance integral to a
summation.
3.4.5 RESULTADOS DOS ELEMENTOS DE CONTORNO
Results have been calculated using a
version of the moment technique in
which the dipole density is approximated
as a piecewise constant with respect to a
regular grid of rectangular boundary
elements. For a piecewise constant
solution, it is necessary to find the value
of the constant coefficient for each of, say,
N cells. This value is obtained by
expressing the dipole density as a linear
superposition of N rectangular pulse
functions, substituting the expansion into
Eq. 88 and demanding that the resulting
equation is satisfied at the center of each
and every rectangular cell, a step known
as point matching or collocation. The
procedure leads to an N x N matrix
equation for the coefficients of the
piecewise constant approximation.
In general, the moment technique
proceeds by expanding the unknown
function in terms of suitable set basis
functions defined with respect to a grid or
a set of nodal points subdividing the
domain of the solution. Therefore, the
dipole density can be approximated by
using a set of basis functions that lead to
a smoother representation of the solution
than does the piecewise constant
approximation. This approximation.
certainly leads to improved results.°?
However, despite the relatively crude
approximation of the piecewise constant
solution, the results (Fig. 17) agree
reasonably well with experiment on a
semielliptical artificial crack. 76
Legenda:
__ - = plotagem teórica fara 16 x 8 células
- - - = plotagem teórica fara 32 x 16 células
___ = plotagem teórica fara 40 x 20 células
o = observações
Ficure 17. Variation with probe position for coil whose axis is
in plane of semielliptical simulated crack in aluminum:
(a) resistance change; (b) reactance change.76
Incidentally, note that the theoretical
predictions computed with a grid of
16 x 8 elements are also used to generate
the 250 Hz impedance plane plot in
Fig. 10. The computed results in Fig. 17
are plotted for three different rectangular
cell sizes showing the dependence of the
results on the number of unknowns. A
reasonably accurate result can be achieved
with only 128 unknowns and the finer
grid results are consistent with each other.
Figure 18 shows similar low frequency
(250 Hz) results for a simulated crack
whose shape is shown in Fig. 15. At
intermediate frequencies, the crack
opening must be taken into account!
and at high frequencies, the number of
boundary elements must be increased.
However, in the thin penetration regime,
boundary elements can be avoided
altogether as discussed in the following
section.
Legenda:
___ = plotagem teórica fara 32 x 16 células
o = observações
3.4.6 TEORIA DA TRINCA DE POUCA PENETRAÇÃO
A number of approaches have been used
to determine the electromagnetic field at
a crack for the thin penetration regime. In
this regime, in which the standard depth
of penetration is very much smaller than
the length and depth of the crack, eddy
currents are confined to a region close to
the conductor and to the crack surface. It
is found that their distribution over the
crack is governed by the solution of the
laplace equation in the domain of the
crack face. The reduction to a
two-dimensional laplace problem is
theoretically attractive because a number
of standard techniques can be adopted to
solve such problems. From the practical
point of view, it is often desirable to carry
out eddy current testing and experiments
in the thin penetration regime because
the sensitivity to discontinuities is greater
at high frequencies. In testing
ferromagnetic materials for cracks, the
standard depth of penetration is usually
much smaller than the overall
discontinuity dimensions. Hence, the
high frequency limit has important
practical significance.
The main theoretical question to be
faced in seeking a solution of the
two-dimensional laplace problem is,
“What are the boundary conditions?”
Beginning in the early 1980s, a research
group at University College London in the
United Kingdom produceda series of
articles on the alternating current
potential drop technique for measuring
cracks. A
number of these articles were
based on the unfolding model.'0!,102 This
model was successfully applied to the
problem of finding the depth of cracks in
ferromagnetic steel in the thin
penetration regime. The problem domain
can be divided into two equal parts, each
consisting of a half plane at the surface of
the conductor anda crack face at right
angles to it. The line adjoining the half
plane and the crack face is called the fold
line. By unfolding the crack face into the
surface plane of the conductor, a modified
problem domain is formed. A scalar
potential representing the electromagnetic
field in the plane was deemed to be
continuous and have continuous normal
gradient at the fold line. At the crack
edge, a boundary condition on the
potential was deduced from the fact that
the electric field tangential to the tip is
zero. These constraints are sufficient to
form a well posed, two-dimensional
laplace problem that was solved to give
results in agreement with experiment.
Estimates of crack depth in steel
components using alternating current
potential drop were improved as a result
of this work.
The unfolding model is not valid for
nonferrous material but an alternative
thin penetration theory was developed for
eddy current testing in such materials by
Auld and others, who considered cracks in
aluminum alloys.48!93 Auld’s boundary
condition assumes that the external
magnetic field tangential to the conductor
surface is not perturbed by the crack. The
assumption may have been inspired by
Kahn’s two-dimensional long crack
problem’? because it is exact when the
magnetic field is uniform along the length
of a crack of uniform depth but, for a
nonuniform probe field at a finite crack, it
is approximate. The approximation is
reasonable provided the coil diameter is
large compared with the crack size but
this limitation leaves room for
improvement in the predictions.
It became evident in the late 1980s
that the differences between the London
group’s model and Auld’s approach ought
to be reconcilable in a unified theory that
would be valid for arbitrary permeability.
In seeking the unified approach, the
perturbation in the magnetic field at the
crack mouth was taken into account by
Lewis, Michael, Lugg and Collins, 104105
who derived a boundary condition using
a flux conservation argument applied to a
region around the opening. The resulting
theory is applicable to materials of
arbitrary relative permeability and
corroborates the unfolding model in the
high permeability limit.
3.4.7 FORMULAÇÕES ALTERNATIVAS
A more formal approach to obtaining the
unified theory is to start with a technique
valid at an arbitrary frequency and
specialize it systematically for the thin
penetration regime. A suitable
formulation for this strategy is one where
the electromagnetic field in the conductor
is expressed in terms of transverse electric
and transverse magnetic hertz
potentials,!°° y and y’ respectively. Then,
the electric and magnetic fields take the
forms:
e:
where z <(0 and where the preferred
direction 2 is normal to the crack plane.
In a half-space problem formulated
using hertz potentials, it is usual to
choose the preferred direction as the
normal to the interface between the air
and the conductor. In this way the two
potentials are decoupled at the interface.
Although the present choice of preferred
direction leads to coupled interface
conditions, the chosen modes are
decoupled at the crack surface. In fact, the
transverse electric mode does not interact
directly with an ideal crack at all. Instead,
it is perturbed indirectly through its
coupling with the transverse magnetic
mode at the surface of the conductor.
Because direct transverse electric
interaction with the crack is absent, the
transverse electric potential and its
gradients are continuous at the ideal crack
plane. In contrast, the transverse
magnetic potential is subject to a direct
interaction of the crack with the field and
therefore exhibits a discontinuity at the
crack.
‘To examine the discontinuity of the
transverse magnetic hertz potential, it is
necessary to reconsider the properties of
the electromagnetic field at the crack.
First, the fact that the normal
component of the electric field at the
surface of the crack is zero means that:
Second, in the absence of direct
transverse electric interaction with the
crack, the continuity of the tangential
magnetic field (Eq. 83) implies that:
as can be deduced from Eqs. 93 and 83.
Thus the transverse magnetic potential
itself is continuous at the crack surface So.
Third, noting that the jump in the
electric field is due solely to the transverse
magnetic mode, it can be seen from the
form of the transverse magnetic
contribution in Eq. 92 combined with
Eq. 77 that:
It is concluded that the transverse
magnetic potential has a discontinuity in
its normal gradient at the crack surface So.
It can be shown that the transverse
electric hertz potential y, expressed as the
sum of unperturbed and perturbed
components, is given by:
for arbitrary frequency and standard
depth of penetration. The Green’s
function G(r,r’) accounts for the cross
coupling between transverse electric and
transverse magnetic modes. 107
Several approaches for finding a
solution to the ideal crack problem follow
immediately from Eq. 97, both at an
arbitrary frequency and for the thin
penetration regime. For example, without
restricting the frequency, one can use a
symmetry argument to write the jump in
the derivative of the potential at the crack
as 2(0y)-(0x)"!. Differentiating Eq. 97 with
respect to x and assigning the field
coordinate to a point at the crack face
denoted by r* will give an equation for
the normal derivative of y. From the
solution, p can be found from Eq. 96 and
the probe impedance due to the
discontinuity found from Eq. 91. The
following approach has appeared in the
literature.
Setting the field coordinate in Eq. 97 to
r* and using Eq. 96 gives:
Essentially the same equation as Eq. 98 is
found using a magnetic vector potential
formulation.!°8 Findinga solution relies
on the fact that the unknown potential
w(r*) satisfies the laplace equation on So
(Eq. 94) and must be determined
simultaneously with p(f). These two
unknown functions can indeed be found
from the same equation simultaneously
by imposing further constraints. The
additional constraints are not the
boundary conditions on the laplace
problem for y at the crack face, because
these are not defined. Instead, the
boundary conditions at an arbitrary
frequency (Eqs. 78 and 79) are imposed
on p.
In finding a solution using the
moment technique using N equations for
N unknowns, a reduction in the
unknowns needed to approximate p can
be made because the prior knowledge
derived from Eqs. 78 and 79 restricts its
behavior at the perimeter of the crack.
This technique releases some degrees of
freedom that can be used to represent
w(r*) as a solution of the laplace equation
on the crack face. By management of the
unknown coefficients in this way, a
solution can be found that agrees with
experiment. 108
Legenda:
___ = plotagem teórica fara 32 x 16 células
o = observações
Ficure 19. Inductance and resistance variation with probe position for coil whose axis is in
plane of semielliptical artificial crack in aluminum: (a) inductance plot; (b) resistance plot.
Theory (solid line) is compared with experimental results (points) acquired at 50 kHz. See
Harrison and Burke for details of coil parameters and simulated crack. 76
3.4.8 REGIME DE POUCA PENETRAÇÃO
As Auld has shown, a suitable boundary
condition for formulating a well defined
laplace problem on Sp in the thin
penetration regime can be derived from
the magnetic field at the crack mouth.
The transverse magnetic component of
the magnetic field in the Y direction can
be written:
where wy is the perturbed potential (volt)
due to the crack. As it stands, Eq. 99
cannot be used immediately as a
boundary condition because the
perturbed field at the mouth is not known
in advance. Auld got around this problem
by neglecting the perturbation of the
magnetic field at the crack mouth, a
reasonable approximation because it can
be small for nonferromagnetic materials.
Taking the field perturbation into account
increases the complexity of the
problem!°” but improves the accuracy of
the results for nonferrous alloys and gives
results valid for ferromagnetic
materials.
109
Results of impedance predictions!!°
and measurements for a semielliptical
artificial crack are shown in Fig. 19. The
experimental data are taken froma series
of measurements made at
16 frequencies.”° For a comparison with
thin penetration theory, results at the
highest frequency (SO kHz) are shown.
Calculations were performed with
conformal mapping.!!° At this frequency,
the depth of the simulated crack,
8.61 mm (0.339 in.), is more than
18 times the standard depth of
penetration, 0.47 mm (0.019 in.). Note
that the theory underpredicts the resistive
component by about 10 percent.
However, this component, is small
compared with the inductive reactance,
which has a
maximum value over 600 Q.
The overall accuracy of the predictions is
good.
4. MODÊLO COMPUTACIONAL DO CAMPO DE CORRENTES PARASITAS
4.1 BASES MATEMÁTICAS DO MODELO
Computer modeling is used to simulate
reality. In the case of eddy current testing,
the computer can be programmed to
replicate (1) the physics of the testing
media, (2) the characteristics of the test
object and (3) the geometry of the test —
and to then display a dynamic visual
image of all three during testing.
Modeling may be used for computer
aided design of eddy current test
components, as well as research on
specific or general test applications. Both
of these potential uses reduce the need for
trial and error manufacturing of sample
components and provide an alternative to
actual tests that are difficult, hazardous or
costly.
By definition, modeling is the design of
a mathematical system that obeys certain
fact based conditions. The behavior of the
model is then used to understand an
analogous physical system. The value of
the model relies directly on its ability
accurately to duplicate the behavior of the
system for which it is the analog.
All electromagnetic phenomena,
including those related to eddy current
testing, are described by Maxwell’s
equations (Eqs. 1 to 4). In performing an
eddy current test, these relations are used
even if this is not explicitly known to the
inspector. The designs of eddy current
tests and equipment are based on
Maxwell’s equations, regardless of the
actual technique used in the design
process. In modeling electromagnetic
phenomena, it is natural to rely on the
solution of Maxwell’s equations. The more
accurately these equations can be
modeled, the better the resulting general
model. The usefulness of eddy current
tests and the information available from
interpretation of their test signals can
only be as good as the understanding of
the underlying principles. Authoritative
decisions regarding a test signal cannot be
made if all aspects of the magnetic field’s
interaction with materials and material
discontinuities are not fully and uniquely
understood.
The solution of Maxwell’s equations is
at the heart of any eddy current
discontinuity characterization scheme.
The ability to interpret signals, design
tests and equipment and ultimately to
solve the inverse problem in eddy current
testing is directly related to the ability to
solve these equations within realistic
testing geometries.
Although the solution of field
problems has preoccupied scientists and
engineers since the publication of
Maxwell's Treatise! in 1873, the
complexity of field relations and
interactions has limited such attempts to
well behaved, simple problems. Such an
extensive effort has yielded few specific
results and not one general model capable
of describing all electromagnetic field
phenomena. Models that could be applied
generally could be developed only after
the introduction of digital computers.
This is not surprising considering the
complexity of the interactions involved.
Eddy current techniques of
nondestructive testing rely on alternating
current excitation that induces secondary
currents and fields in the test material.
Discontinuities in the test object cause
changes in the induced fields, which are
usually monitored by measurable changes
in coil impedances. Thus, the technique
requires indirect measurement (through
impedance changes) of secondary
(induced) fields and currents.
The nature of field problems,
nondestructive testing applications in
particular, leads to three-dimensional,
nonlinear, partial differential equations
within awkward boundary conditions.
The solution domain often includes
complicated discontinuity shapes. For the
case of moving probe applications, the
solution is a function of time and
position. In addition, the solution domain
is unbounded: the field only decays to
zero at infinity. These very general
requirements encompass the whole
spectrum of possible difficulties (with the
exception of high frequency problems).
These complications have led to a
definite reliance on experimental
techniques!!!-113 and on analytical models
wherever such models could be found.
4.1.1 TIPOS DE MODELO
Experimental Modeling. Experimental
techniques are derived from
measurements in eddy current tests, either
actual or simulated. The value of such
models is limited because of their
empirical nature and the fact that their
extension to other geometries is more or
less speculative. This limitation does not
mean that accurate, controlled
experimental data are not valuable in
modeling. Both analytical and especially
numerical modeling rely on such data for
confirmation.
Experimental techniques have many
limitations and, although very useful at
times, are not always reliable for
modeling. The generality required for
modeling does not exist with
experimental techniques.
Analytical Modeling. Analytical models
are derived from elementary field and
circuit theory relations. At the very basis
of this approach is the fact that some
simplifying assumptions must be made
with regard to the test environment.
These assumptions include (1) those that
are satisfied with little or no errors, such
as linearity in eddy current calculations,
and (2) those that imply large errors or
even a modification of the geometry,
relying on the hope that, by doing so, the
solution is still an approximation of the
actual problem modeled. In this category
are symmetry considerations, boundary
conditions and discontinuity shape
approximations. In spite of extensive
simplification, analytical models are
extremely complicated and the results
tend to be limited to a single geometry or
class of problems.
Numerical Modeling. Numerical modeling
in nondestructive testing is an outgrowth
of the failure of analytical models reliably
to predict the necessary field interactions
with any degree of generality. A
numerical
model uses a digital computer to solve the
governing equations directly, with few
simplifying assumptions. This in itself is
enough to explain the value of such
models. Numerical modeling allows the
solution of very complex problems and, at
the same time, does not require the user
to know the intricacies of electromagnetic
theory or differential calculus. All that the
user is required to do is input the problem
variables and, if necessary, verify the
results experimentally.
4.1.2 VISÃO GERAL DA MODELAGEM ANALÍTICA E NUMÉRICA
The existence of models for eddy current
testing phenomena depends entirely on
the ability to solve Maxwell’s equations
with or without approximations. The
value of such models in solving the
inversion problem satisfactorily is beyond
dispute. Solution of this important
problem is possible only with the
development of good theoretical models,
capable of predicting the complex
interactions of a multitude of factors in
the test object. A good, reliable theoretical
model for nondestructive testing should
be able to satisfy the following conditions.
- The model should describe the physics
of interaction between the applied
alternating current field, induced
currents and discontinuities in the test
object.
- The model should serve as a
theoretical test bed for situations
difficult or impossible to replicate
experimentally.
- The model should generate eddy
current output signals for a wide
variety of discontinuities and
specimen shapes, avoiding costly
sample preparation and helping to
determine discontinuity
characterization parameters.
- The model should provide training
data for automated discontinuity
characterization systems and
equipment.
- The model should aid in the design of
eddy current probes for specific
applications.
For the purpose of deriving such a
model, two main avenues are available:
the analytical approach and the numerical
approach.
4.2 MODELO ANALÍTICO
Analytical models are derived from basic
field and circuit theory considerations. In
effect, an attempt is made to solve
Maxwell’s equations directly. These
equations are generally three-dimensional,
nonlinear, partial differential equations
within complex boundaries and
discontinuity shapes. In addition, for
moving probe problems, the solution is
both time and position dependent. It is
therefore not surprising to find that such
solutions are only possible for the most
elementary of test geometries, with
simplifying assumptions in terms of
geometry, dimensionality, discontinuity
shapes and sources. This
oversimplification accounts for the fact
that analytical models are limited in
scope, applicable only to selected
problems and not easily extended to other
geometries. On the other hand, the
solution to problems for which an
analytical model applies is relatively
simple and allows parameter change
studies.
Analytical modeling has its roots in the
pioneering work of Ampére, Orsted,
Faraday, Lenz, Gauss, Helmholtz, Henry
and Foucault, culminating in Maxwell’s
Treatise,! in which many practical
problems are addressed and solved,
including many that are directly related to
nondestructive testing. In view of this
background and the work of Hughes,!!4 it
is surprising that the first serious attempt
at modeling was undertaken only after
Steinmetz introduced the complex
notation for field quantities,!!> thus
paving the way for the early modeling
work of Forster and the introduction of
impedance plane diagrams as an accepted
technique of presenting eddy current test
data.
The phenomena associated with eddy
current testing can be examined only after
Maxwell’s equations have been
manipulated and simplified into a form
suitable for solution by one of the
techniques for solving partial differential
equations. Such techniques as separation
of variables, bessel functions, power series
and the various transform techniques,
especially the fourier transform, are used.
Forster and Stambke!! used bessel
functions to find the complex effective
permeability of a metal rod encircled by a
secondary search coil and an alternating
excitation coil. The effective permeability
concept is used to directly connect the
dissipative and inductive quantities in the
specimen with the resistive and inductive
terms in the impedance plane plot.
Hochschild!!! solves a fairly simple
problem (a cylindrical sample and a
concentric coil), using bessel functions
directly for the magnetic flux density B
(tesla) within the conductor, in terms of
the flux density at the conductor’s surface,
using the following expression:
wherer is radial distance (meter) and y is a
function of the standard depth of
penetration.
This leads to an expression for the flux
linkage with the encircling coil from
which the induced voltage and the real
and imaginary parts of the coil impedance
are found. The well known comma
shaped curves can be derived by plotting
the impedance for various frequencies and
conductivities.
The same geometry is solved by
Libby!!® by using the definition of
magnetic vector potential and substituting
it for the flux density. To obtain the coil’s
impedance, the magnetic vector potential
is related to the induced voltage Ag (volt)
in a current loop and then the impedance
of the coil is obtained as a closed form
expression in a source free region:
Waidelich and Renken!" used the
image coil concept to calculate the
impedance of a coil in the vicinity of
conducting media and compared the
results with experimental data. Vine!!8
shows this to be the limiting case of a
single current loop above a conducting
plate of finite thickness. Cheng"!
examines the same situation and, using
the magnetic vector potential in
cylindrical coordinates, obtains an
integral expression for the coil
impedance.
Expanding on this idea, Dodd!“
obtains expressions for the coil impedance
of a rectangular cross section coil above a
two-conductor plane and encircling a
two-conductor rod.
To avoid the complexities of Maxwell’s
integral equations, Graneau and Swann!?!
and Graneau!”? replace conducting media
with an infinite number of filamentary
circuits corresponding to streamlines of
current flow. This substitution leads to a
coupled circuit model and a power series
representation for the induced currents.
Burrows® introduces magnetic and
electric dipoles to represent small
discontinuities. Dodd and others!! predict
the induced voltage in a circular coil due
to small discontinuities.
Many other theoretical and
experimental concepts’*!23-!27 have been
used to approximate the solution to
nondestructive testing problems with
varying degrees of simplification and
success. The underlying assumptions
made in deriving these models are of such
a restrictive nature that their application
to other problems, in more realistic
geometries, is all but impossible.
4.2.1 TÉCNICA DE SOLUÇÃO INTEGRAL
To obtain a closed form solution for
simple geometries, analytical techniques
require solution of integral equations and,
if the integration can be carried out, a
solution may be obtained for a particular
geometry. Beyond the simplification of
the actual geometry and a lack of
generality, a new consideration should be
introduced: integration processes that
may or may not be done analytically.
In such cases, numerical integration is
used. Because of this need to numerically
integrate analytical expressions, the
technique is considered a hybrid that
bridges the gap between the two
techniques. Inherent in it are all the
initial approximations of an analytical
technique combined with the flexibility
of numerical techniques.
The names integral solution technique
and boundary value solution should not be
confused with similar or identical terms
used for finite element solution of field
problems in integral form. This later usage
refers to the formulation of Maxwell’s
equations (or any other physical system)
as a boundary integral problem, usually
solved by surface discretization of a
volumetric region. The term boundary
value solution is here much more restricted
and reflects the fact that an integral
expression is obtained and its solution is
in terms of orthogonal functions along
the boundaries of the solution
region. 123,128
Dodd 15 128 129 obtained such
expressions for a variety of testing
geometries that fall into two major
categories: (1) multilayered conductors
and (2) multiple coaxial cylindrical layers.
These two categories include many of the
more practical and useful testing
geometries.
The solutions are in the following form: 15
where A(r,z) is the magnetic vector
potential in the nth plane for a coil above
a number of planar conductors, d
indicates a differential, J is the current per
turn, J is a bessel function of the first kind
and first order, o is the separation
constant of the differential equation, 1, is
the turn density of the coil, u is the
magnetic permeability (henry per meter),
V22,Vi2(n,1) and V29(n,1) are
transformation matrices and where:
where R, and R, are the inner and outer
coil radii (meter).
Although these expressions are
complicated and very difficult to evaluate
on paper, they are easily integrated
numerically on computers.
Once the magnetic vector potential has
been evaluated, any magnetic or related
quantity may be calculated. These are
derivable either directly from Eq. 102 or
through other known relations.
For example, if the driving coil is
coaxial with the pickup coil in an
axisymmetric geometry, the induced
voltage in the pickup coil can be written:
where a is the cross sectional area (square
meter) of the coil winding and n is the
number of turns in the coil.
The integration is performed over the
coil cross section and is particularly
convenient for coils with rectangular cross
sections (as is usually the case with eddy
current coils).
The impedance Z of a coil may be
found from Eq. 104 by dividing by I:
Similarly, eddy current densities, flux
densities, stored and dissipated energies or
forces due to eddy currents may be
calculated.
The technique described above has
been programmed for minicomputers as
well as mainframes and the computer
programs are available in the open
literature, 128-130 accounting for the
technique’s success.
The integral solution technique has
several advantages over purely analytical
techniques, the most important being the
wider range of application afforded by
numerical integration. Because of this, the
integral solution technique has been
applied to eddy current problems ranging
from single-coil, single-conductor
situations to situations using multiple
coils and multilayered materials for
simulation of such important problems as
discontinuity detection and measurement
of cladding thickness, conductivity,
permeability and liftoff.
The technique still suffers from the
problems associated with analytical
techniques: (1) it is not general and (2) it
requires the evaluation of an integral
expression for each class of problems. In
addition, because of its reliance on
orthogonal functions on the boundaries,
it can only take into account relatively
simple geometries and discontinuity
shapes.
Lastly, the technique assumes linear
material properties and superposition of
solutions, a difficulty found in all
boundary integral techniques and not
easily extendable to nonlinear problems.
This assumption is acceptable for eddy
current testing applications but is not
general enough to form the basis of an
all-purpose solution technique, as is the
case with the more general numerical
techniques.
4.3 MODÊLO NUMÉRICO
Numerical modeling is different from
analytical modeling. For the purposes of
nondestructive testing, the most
important aspect of numerical modeling
is the fact that none of the simplifying
assumptions made in the analytical
approach are necessary to reach a
satisfactory solution.
Starting with Maxwell’s equations,
there are many different ways to proceed
and each requires the formulation of the
original equations in some particular
form.!3!-133 Assumptions are made for the
sake of simplicity and economics of the
solution and not to render the equations
solvable. Thus, a two-dimensional or
axisymmetric solution is assumed if the
geometry is approximately
two-dimensional or has axial symmetry. A
two-dimensional solution is likewise a
special case of the more general
three-dimensional solution. Similarly, an
axisymmetric solution is the solution of a
three-dimensional problem in cylindrical
coordinates. Other assumptions help to
shorten the solution process or to obtain
insight into the problem before a more
general solution is attempted. A linear
formulation may be used asa first
approximation or in cases where the
problem is indeed linear.
The application of a numerical process
for the purpose of obtaining a model is, to
a large extent, a matter of choosing the
numerical technique to be used, a matter
of making correct assumptions for
geometry and the nature of the solution
and, perhaps most important, a matter of
economics. Within these constraints, any
accuracy can be obtained regardless of
geometry, linearity, nonlinearity or
dimensionality of the problem.
Numerical techniques in general are far
more powerful than analytical techniques.
At the same time, the solution is obtained
as numerical data rather than a closed
form solution. As such, the solution to a
particular problem may not be usable for
the analysis of a different, perhaps similar
problem. Thus, parameter study requires
repetitive solution and it is not always
possible to deduce parameters from the
solution, as is the case with analytical
solutions. This disadvantage is hardly
significant: the same repetitive process
allows study of parameters for which the
analytical approach cannot be applied
(such as changes in the geometry of an
arbitrarily shaped discontinuity).
4.3.1 TÉCNICA DAS DIFERENÇAS FINITAS
The finite difference technique has been
used as a general means to solve partial
differential equations. The reasons for its
widespread application are many. The
technique is relatively easy to apply, as
well as being general. It is equally
applicable to direct current fields, to
quasistatic or transient fields and to linear
and nonlinear problems. In its simplest
form, the formulation of the field
equations consists of simply replacing the
partial derivatives by appropriate
difference formulas. A solution can then
be obtained for the dependent variable at
discrete points within the solution region
either by an iterative process or by the
solution of a system of algebraic
equations, depending on which finite
difference formula is applied.
The application of the finite difference
technique is complicated by problems of
convergence and stability of the solution
as well as by restrictions on the
discretization process. Although regular
sets of discretization points (grids) are
easy to handle, irregular grids are not.
Discretization of complex geometries into
regular grids is not practical and irregular
grids may in some cases make the
solution nonconvergent. In field
problems, the inability to properly
discretize small areas (such as air gaps or
discontinuities) is detrimental to the finite
difference technique. In addition, the
technique is a nodal technique and
cannot take into account distributed
parameters such as current densities,
conductivities and permeability. These
have to be described as equivalent nodal
quantities with all the associated errors.
The obtained solution is valid only at the
nodal points.
4.3.2 REPRESENTAÇÃO DAS DIFERENÇAS FINITAS
If an attempt is made to solve a partial
differential equation such as Eq. 100 or
101, it should be possible either to
integrate the equation or to represent the
partial derivatives in terms of the
unknowns themselves at discrete points in
space. The finite difference algorithm is
an implementation of the second
approach.
Considering Fig. 20, where a general
function is described, the true derivative
dy-(dx) at a point x; is the tangent to the
curve at this point. An approximation to
the derivative can be found by taking two
points, (one point on each side of x;) and
passing a straight line through them. If
the two points are chosen to be equally
spaced about point x; (as in Fig. 20), the
following expression for the slope of the
line can be obtained:
Ficure 20. General function and finite difference
approximation to true derivative.
By denoting in short form y() as Yi, Yix+ay)
aS Yi) ANA VyAS Yi, a Simpler
expression linking the approximation y’,)
to the function value at xj_1) and x(.1) can
be written:
The same result can be obtained
formally by using a taylor series
expansion. By expanding the function
y = fl) about the point x; for x = x;- Ax
and x =
x;
+ Ax, the following results are
obtained:
By subtracting Eq. 109 from Eq. 108 and
rearranging the terms, the first derivative
can be written as:
Then, neglecting terms with (Ax)? or
higher powers of Ax, the expression in
Eq. 107 is obtained. This technique is less
intuitive than the one used to derive
Eq. 107 but shows two important points.
- The error produced by the
approximation is on the order of (Ax)?.
Thus, there is a simple way of
estimating the error and, at the same
time, the solution can be improved by
reducing the size of Ax.
- The finite difference formula in
Eq. 110 was obtained by choosing an
appropriate expansion to cancel
specific terms of the expansions. This
indicates that higher order derivatives
and different approximations to the
same derivatives can be obtained.
Indeed, many useful difference
formulas have been derived.134
An approximation for the second
derivative is obtained by adding the two
expansions in Eqs. 108 and 109 and
rearranging the terms:
This particular approximation also
introduces an error on the order of (Ax)2.
The finite difference expressions
derived here use points on both sides of
the point at which the derivatives are
calculated. They are therefore called
central difference expressions. Backward and
forward difference formulas may also be
used. 134
4.3.3 REPRESENTAÇÃO DAS DIFERENÇAS FINITAS PARA PROBLEMAS DE CAMPO BIDIMENSIONAL E ASSIMÉTRICOS
The first step in the field formulation
consists of replacing the partial derivatives
by a difference equation. Referring to the
grid in Fig. 21, Eq. 19 reduces in the case
of an axisymmetric geometry in
cylindrical coordinates to:
Ficure 21. Simple finite difference grid.
Here the current density J, was replaced by
an equivalent nodal current J;;. A similar
expression may be written for the
two-dimensional field equation.
If the grid is equally spaced in both
directions (Wr = Wz = h) and a constant
permeability can be assumed for all
points, Eq. 112 is further simplified as:
Equation 112 forms the basis for
solution of eddy current problems in
axisymmetric geometries. In its present
form, the equation is of little use because,
besides applying only to regular grids, it
can be used to describe only nonmagnetic
media. This limitation can be seen by
testing where the interface between two
materials is shown (Fig. 22). When
calculating the value of the magnetic
vector potential for points on the
interface, some points lie in areas of
different permeability. Equation 113
requires a single value for all points
forming the expression. Similar problems
are encountered with description of
conductivities and current densities
because all three properties are volumetric
properties rather than point values.
Ficure 22. Section of finite difference grid
showing domain of node i. Five nodes
shown are needed to approximate second
derivatives of magnetic vector potential with
respect to x and y or to rand zin
axisymmetric geometries.
To circumvent these problems, some
special techniques were derived by several
researchers. One such technique came
from Erdelyi and Fuchs!35.136 and was
later refined by Demerdash!37.138 for
problems in electrical machines. If the
five nodes associated with the calculation
of the magnetic vector potential are
considered (as in Fig. 22), four distinct
regions are observed. Assuming that nodes
must coincide with material boundaries,
each such domain may have different
material properties and current densities
associated with it.
Under the present assumptions, an
equivalent material property that is a
weighted average of the properties of the
four domains is produced. The current at
the central node then becomes:
The weights S; to Sy are functions of
the four domains around node i. These
areas need not be equal or rectangular.
Similarly, average values for
conductivity o and reluctivity v are
defined as:
and:
The net effect of these approximations
is to change the material properties at the
edges of material discontinuities,
assuming that the errors introduced by
doing so are small. This approximation
may or may not be good, depending on
the grid. The same technique is used to
calculate the equivalent material
properties in nonregular and
nonrectangular grids but the weighting is
more complicated. 137
4.3.4 CONTORNOS E CONDIÇÕES DE CONTORNO
The first aspect to be considered is the
structure of the grid itself. If the grid is
kept uniform, the boundaries can be
located only along lines connecting
nodes. A curved boundary becomes a
jagged line, as shown in Fig. 23. This
approximation may be good for very fine
grids where outer boundaries are
involved. Curved material boundaries
where the field gradients are high require
better fitting to reduce errors.
Ficure 23. Boundary representation in
regular grid.
This fitting can be done as shown in
Fig. 24. The technique consists of
calculating scale factors that effectively
move nodes from their original location
to the boundary. By doing so, the
regularity of the grid is lost and, in the
solution process, Eq. 113 will have to be
modified to account for the different grid
spacings used.
Ficure 24. More accurate representation of
boundaries by using irregular spacings at
boundary.
The boundary conditions most often
encountered in eddy current field
problems are dirichlet boundary
conditions (prescribed values of the
magnetic vector potential) and, in most
cases relevant to nondestructive test
applications, the values are zero. These are
taken into account automatically by
Eq. 113 when the five-node stencil of
Fig. 22 hits the boundary. Other types of
boundary conditions may be used with
the finite difference technique.
4.3.5 MALHAS NÃO UNIFORMES E NÃO RETANGULARES
The idea of scaling the distance between
nodes to fit the grid to the boundary of
the solution domain or material interfaces
may be used with nonuniform grids.
There is no requirement that the grid be
rectangular. If nonuniform grids are used,
Eq. 111 will have to be modified because
the definition of the second derivative is
based on a uniform grid. One approach is
to rewrite the difference expression in
Eq. 111 in the following form:
The four weights are then calculated as
weighted averages!3” that depend on the
grid structure and material permeabilities.
The calculation of the current at nodal
points follows a technique similar to the
one described above.
unknown at the node being assembled
and four neighboring nodes, according to
Eq. 113. For each node in the solution
region, a linear algebraic equation is
assembled.
To find the unknown values of the
magnetic vector potential, there are two
basic techniques available: (1) the iterative
solution and (2) the matrix inversion
solution.
4.3.6 SOLUÇÃO DO SISTEMA DE EQUAÇÕES
4.3.7 SOLUÇÃO INTERATIVA
The iterative solution is the simplest way
to reach a solution and is often used with
the finite difference technique. It consists
of assuming an initial solution
throughout the solution region (either an
approximation or, if this is not possible,
zero). The correct current distribution
together with the boundary conditions
are next applied. Then, the finite
difference equation is applied to each
interior node and the value at each node
is updated in turn until a new solution is
obtained. The process is repeated until the
change in the solution is smaller than a
predetermined value. The number of
iterations required to obtain a good
solution may be quite large. In some cases
the solution may not converge to the
final solution. This technique is called the
explicit technique and because of its
simplicity is often used in nonlinear and
time dependent problems.!#? There are
ways to refine the basic technique to
make it more stable!#° or to accelerate
convergence.!4! It is also possible to
estimate beforehand if the solution is
convergent, based on the grid spacing and
the type of finite difference formulas
used.
4.3.8 SOLUÇÂO POR MATRIZ DE INVERSÃO
Instead of assuming an initial solution,
the unknown values may be entered in
the finite difference equation (Eq. 113)
and a system of N equations with N
unknowns is assembled (N is the number
of interior grid points excluding boundary
nodes). The final result is a system of
complex algebraic equations:
where [G] = the coefficient matrix
resulting from the finite difference
description of the partial derivatives,
{I} = a vector of equivalent applied
currents at the nodes in the solution
region and [R] = a matrix resulting from
the eddy current distribution.
The imaginary part of the matrix is due
to eddy currents alone and will disappear
for direct current applications. The
technique outlined above is an implicit
technique, absolutely stable, and therefore
does not require iteration. The matrix in
Eq. 118 can be solved using any standard
technique such as the gauss elimination
or the conjugate gradient technique.142
The derivation presented refers to the
axisymmetric field equation. An identical
procedure can be used for
two-dimensional geometries, replacing the
necessary partial derivatives by
appropriate finite difference expressions.
4.4 TÉCNICA DE ELEMENTOS FINITOS
The finite element technique has a briefer
history than the finite difference
technique. It evolved in the late 1950s as
a numerical technique in structural
analysis 143 but has spread quickly to
become a major analysis tool in diverse
areas of engineering 144-147 as well as in the
physical sciences 148,149 and medical
research. 150
Because of its success in modeling
intricate geometries efficiently and
accurately, the potential for its application
to electrical and magnetic fields was
recognized in the early 1970s and has
been applied with great success to the
study of direct current and low frequency
electromagnetic fields in electrical
machines,151-153 Jarge magnet structures 154
and permanent magnet design.155
The finite element technique has
considerable advantages over the finite
difference technique, including the ease
of handling boundary conditions and the
ability to follow awkwardly shaped
boundaries.!5° The technique is by
definition a volumetric technique where
various parameters are associated with the
volume (or the surface, in the case of
two-dimensional and axisymmetric
formulations). Therefore, it is naturally
suited to the modeling of continuum
problems.
The finite element technique is also
quite flexible in terms of the
discretization process. Being a discrete
technique, it requires discretization of the
solution region but no restrictions are
imposed on the shape, size and number of
finite elements.!*3 The solution process as
well as the formulation is not affected by
the size and shape of the elements used.
Furthermore, although the finite
difference technique assumes linear
relations between the unknowns, the
finite element technique can handle
higher order relations as well.!43,157
Problems with convergence have no
meaning in the context of finite elements.
These factors are of particular
importance for the simulation of
electromagnetic test techniques and the
technique has received considerable
attention. Numerical models based on the
finite element technique have been
developed for two-dimensional 158,159 and
three-dimensional 160,161 eddy current
applications.
When compared to finite difference
techniques, problems solved by the finite
element technique generally require larger
computer resources, especially for
nonlinear and time dependent problems.
The technique does not lend itself well to
the solution of transient problems because
it cannot efficiently handle time
discretization.
The two techniques are
complementary, each being suited to the
solution of different situations. Time
integration in finite element computer
codes is usually handled by various forms
of finite difference schemes. 162
In the following discussions, the finite
element formulation of the
electromagnetic field equations is outlined
with reference to a particular element
shape. The technique of formulation is
completely general, however, and any
other element shape can be used with
relatively minor changes in the
formulation.
4.4.1 FORMULAÇÃO DE ELEMETOS FINITOS PARA GEOMETRIS BIDIMENSIONAIS E AXISSIMÉTRICAS
The finite element technique does not
provide a direct solution to
electromagnetic field equations. Rather,
the solution is obtained by first
formulating these equations into a
suitable form for finite element solution
and then solving the resulting set of
simultaneous algebraic equations for the
magnetic vector potential at discrete
points in the solution region.
The formulation of the
two-dimensional and axisymmetric field
equations is presented here using the
magnetic vector potential and an energy
functional equivalent to the original
equations. The following assumptions are
made throughout this derivation.
- The source current density J, and the
magnetic vector potential A vary
sinusoidally with time. Harmonics in
both the source and induced fields are
absent.
- The source medium is assumed to be
infinitely conducting, thus effectively
neglecting eddy currents in the source.
In the case of eddy current probes, this
is equivalent to subtracting the coil’s
direct current resistance from the
resulting impedance, a
common
practice in interpreting eddy current
signals.
- Electrical conductivity o and magnetic
permeability 1:
of materials in the
solution region are single-valued
within each element. Each element is
therefore a linear region but spatial
variations between neighboring
elements is allowed both in wt
and o.
Also, these values can be different in
each direction within the element,
thus allowing modeling of anisotropic
media. The linearity assumption
provides satisfactory results because
the applied and induced current
densities in practical testing
applications are very low.
- The model is based on
two-dimensional or axisymmetric
assumptions. Thus, only one
component of the magnetic vector
potential is present, either in the Z
direction (two-dimensional) or in the
8 direction (axisymmetric) and is
perpendicular to the cross section of
the geometry modeled.
Among the many techniques available
for finite element formulation of general
problems, the weighted residual technique
(Galerkin’s technique) and the use of an
energy functional stemming from a global
energy balance concept,143 in conjunction
with variational techniques, are the most
commonly used. Both procedures allow a
direct formulation based on the original
equations and both are satisfactory in
terms of the resulting solution. Galerkin’s
technique is more convenient when only
the differential equations and their
boundary conditions are available.157
When physical interpretation of the
problem is important, the energy balance
formulation introduced by Oden 163,164 in
1969 offers an attractive alternative.
A general energy functional for
electromagnetic field problems can be
written as:
where E, is the stored energy (joule) due
to the magnetic field, EF; is the input
energy (joule) derived from impressed
current densities, E, is the energy (joule)
dissipated through eddy current densities
in the conducting parts of the geometry,
excluding sources, and v is volume (cubic
meter).
The derivation that follows is limited
to two-dimensional and axisymmetric
geometries. It uses a particular type of
isoparametric element (four-node
quadrilateral element) but it is completely
general and applies to other types of
elements as well. A complete derivation
for two-dimensional and axisymmetric
problems in terms of triangular elements
has been published.158 Researchers
describe a three-dimensional derivation in
terms of tetrahedral elements 165 and a
three-dimensional derivation in terms of
eight-node and twenty-node hexahedral
isoparametric elements.160
4.4.2 ENERGIA FUNCIONAL PARA PROBLEMAS DE CORRENTES PARASITAS
The general energy functional in Eq. 119
applies to any field situation regardless of
dimensionality, because it is a statement
of energy balance in the system. In terms
of the magnetic vector potential A, the
functional for two-dimensional or
axisymmetric geometries can be written:
where J, is source current density (ampere
per square meter) and where, for the
axisymmetric case, y is replaced by z and
[0A-(0x)-!] is replaced by [0A-(0r)-! + A-r].
In the two-dimensional case, dv = dxdy
and in the axisymmetric case dv =
2nrdrdz.
From the principles of variational
calculus, it can be shown that a correct
solution to the governing partial
differential equation is obtained by
minimizing the energy functional
throughout the solution region. Although
proof of this statement!® is not included
here, it is a very important step because it
also defines the natural boundary
conditions that are implicit in the
formulation and that need not be
explicitly applied. The most important of
these conditions are on the interfaces
between different materials.
4.4.3 DISCRETIZAÇÃO DE ELEMENTOS FINITOS
The solution of the variational
formulation of the eddy current equations
presented above is performed by finding a
set of functions that minimize the
functional F(A). Because it is not possible
to minimize the functional everywhere, it
is minimized at discrete points (nodes) in
a bounded region (solution region). The
discretization of the solution region is
therefore a very important step in the
finite element technique because the
number of nodes as well as their location
in the solution region has an impact on
the solution. A discretization (mesh) with
few nodes in regions of high gradients in
the solution will introduce errors whereas
too many nodes will unnecessarily
complicate and lengthen the solution. It
is at this stage in the finite element
process where efficient discretization
techniques combined with correct
judgment can havea significant impact
on the efficiency and accuracy of the
solution.
A large variety of volumetric finite
elements can be used to discretize the
region but, depending on the equations to
be solved, some are more useful than
others. The most common finite elements
used for two-dimensional field
calculations are the triangular and
quadrilateral elements.!5’ Linear or first
order elements are defined as having a
node at each vertex of the element as in
Fig. 25a. Parabolic or second order
elements are defined by including a node
between each pair of adjacent vertices:
 
Ficure 25. Quadrilateral finite elements and
shape functions: (a) element in local
coordinate system; (b) mapped, curvilinear
element in global coordinate system.
where € and 1 are finite element
coordinates. Higher order elements can be
defined similarly.
Automatic mesh generators for a
variety of element shapes are available
and offer a simple and efficient way to
define the necessary input data.!°6167
The details of the discretization process
are discussed in the literature.!5’ The
following steps and assumptions are
general and applicable to the
discretization of a region into finite
elements.
- The solution region is subdivided into
finite elements. The number and
shape of the elements are not
restricted in any way. The element
density must be chosen for the
geometry of the region and expected
gradients in the solution. Small, dense
elements must be used in regions of
high curvature or high gradients.
- Material interfaces within the solution
region and on the boundaries must
coincide with element boundaries. An
element cannot cover more than one
material.
- The current density and each
conductivity and permeability
component are assumed to be
constant within the element.
Calculated quantities are either nodal
values, as in the case for the magnetic
vector potential, or quantities
associated with the element such as
flux densities or energy. In this case,
the calculated value is associated with
the volume (energy) or with the
centroid of the element (flux density).
- At the outer boundaries of the
solution region, the magnetic vector
potential is either zero (by ensuring
that the discretized region extends far
enough to have negligible flux density
on the boundary) or is otherwise
prescribed from known or calculated
conditions.
- The discretizations for
two-dimensional and axisymmetric
geometries are identical. The
difference between the two
formulations manifests itself in the
integration over the element volume
and in the form of the functional
itself.
4.4.4 FORMULAÇÃO DE ELEMENTOS FINITOS
The discretization of the solution region is
a geometrical procedure and by itself is
not sufficient to ensure that the chosen
elements can be used for finite element
solution. A set N; of special functions,
called interpolating or shape functions, must
be chosen for the element: 143
where 4A; is the nodal vector potential
(weber per meter) and n is the number of
nodes in the element.
These must meet two conditions to
ensure convergence of the solution as the
size of the elements decreases: (1) at
element interfaces C,, continuity must be
maintained (compatibility requirement);
and (2) within an element C,,,, continuity
must be met (completeness requirement).
In these conditions, r is the necessary
continuity order of the function at the
element boundaries and C, continuity
means that the function and its first r
derivatives are continuous. For field
problems formulated in terms of the
magnetic vector potential, only Co
continuity is necessary, meaning that only
the function is continuous. Its first
derivatives define the flux density
components, which are not necessarily
continuous.
The shape functions for a given
element can be obtained bya variety of
techniques, each leading to a separate set
of functions. One common technique is
to define polynomials to fit the sides of
the element based ona natural system of
coordinates.!43,157 The technique used
here is to define the shape functions in a
convenient, local system of coordinates
and then to map the functions into the
cartesian or cylindrical system
(isoparametric elements) in which the
solution is required.
4.4.5 ELEMENTOS ISOPARAMÉTRICOS QUADRILATEROS
The interpolation functions for
quadrilateral elements are those of the
serendipity family.143,157 The elements are
created in a local system &,n where the
functions are found by testing.
Considering Fig. 25, the shape functions
for the four nodes are written in the
following form:
where &;, n; = +1 or-1 and where K is a
constant.
The shape functions for quadrilateral
elements are shown in Eq. 121 fora
particular choice of local coordinates.
These shape functions are mapped into
the global coordinate system using the
shape functions themselves for the
mapping (isoparametric mapping). The
elements then become curvilinear as
shown schematically in Fig. 25b.
The variation of the magnetic vector
potential within the element can be
written in terms of the shape functions
and the nodal unknowns as:
Thus, a complete description of the
magnetic vector potential within the
finite element has been obtained in terms
of (1) the shape functions and (2) the
unknown values of the magnetic vector
potential at the nodes of the element. The
energy functional in Eq. 120 requires the
definition of the first derivatives of A with
respect to x and y. These can be written
as:
and:
The shape functions are derived in local
coordinates. Their derivatives are obtained
in the local system using the chain rule of
differentiation:
and:
Rewriting these equations in matrix form
and inverting the system to obtain the
derivatives of the shape functions in the
global system, the following is
obtained: 143
where [J] is the jacobian matrix calculated
for Eqs. 127 and 128. Similarly, the
following expression may be used for
two-dimensional formulation:
and Eq. 131 applies to the axisymmetric
formulation:
To evaluate the elemental contribution
for the two-dimensional formulation,
Eqs. 124 to 126 must be integrated over
the element:
or for the axisymmetric formulation:
where /(§,n) is the transformed f(x,y). The
equations above can only be integrated
numerically using a technique like the
gaussian quadrature.
4.4.6 MINIMIZAÇÃO FUNCIONAL
Energy balance in the solution region is
achieved by minimizing the energy
functional in Eq. 120 at every node of the
region. The first partial derivative of F(A)
with respect to each nodal value A is set
to zero:
Instead of performing this operation
over the entire region, it is convenient to
do it element by element and then to sum
the contribution of individual elements in
order to obtain N simultaneous linear
algebraic equations in N unknown values
of the magnetic vector potential for the
entire solution region. In this way, a
repeatable process is performed on each
element, a process well adapted for
automatic assembly on a computer. The
size of the elemental matrix is 4 x 4 (fora
four-node element) and the individual
contributions to the elemental matrix are
summarized below:
where the indices i,j vary from 1 to 4.
Each coefficient is numerically
integrated over the volume of the element
using gaussian quadrature!5’ and then
summed into a complex elemental matrix
of the form:
where {a} is the 4 x 1 vector of unknowns,
{q} is the source vector, [r] is the
imaginary part and [s] is the real part of
the elemental matrix.
The elemental matrices of all the
elements in the solution region are
summed into a global system of the form:
where there are a total of N equations in
N unknowns, N being the total number of
nodes in the solution region. This matrix
is symmetric and banded. The bandwidth
depends on the number of elements, the
number of nodes per element and
especially on the way the nodes are
numbered.
The system in Eq. 139 can be solved by
any standard solution technique (such as
gauss elimination or the conjugate
gradient technique) to yield the nodal
values of the magnetic vector potential.
4.4.7 CONDIÇÕES DE CONTORNO
The field equations (formulated in terms
of finite elements in Eq. 139 for eddy
current problems) can only be solved
provided a correct set of boundary
conditions is specified. Either dirichlet
boundary conditions (for which the
function A is known on the boundary) or
neumann boundary conditions (for which
the first derivative of A is known) can be
specified. In the finite element analysis of
magnetic field problems, it is more
convenient to specify dirichlet boundary
conditions because the global matrix in
Eq. 139 can accommodate the function
value A but not its derivative. Moreover,
the neumann boundary conditions are
implicit in the formulation in Eq. 139 and
need not be specified.
4.4.8 CÁLCULOS COM VETOR MAGNÉTICO POTENCIAL
Although the developed formulation
yields a correct solution to the problem at
hand, this solution is in terms of the
magnetic vector potential. Being an
auxiliary function, it is not measurable by
itself and therefore is of little value for
comparison with measurements and other
calculations. From the definition of the
magnetic vector potential as B = V x A,
the flux density is immediately defined.
Other quantities are calculable, including
coil impedances, stored and dissipated
energy and eddy current density. The
derivation of quantities calculated for the
magnetic vector potential can be found in
a number of sources. 15.159,161
The solution (either by finite
differences or finite elements) is correct in
terms of the magnetic vector potential.
The magnetic vector potential is an
auxiliary function used to simplify the
solution and is not by itself a
measurable
quantity. It is necessary to calculate other
quantities such as flux densities, eddy
current densities and coil impedances.
Because coil impedance is of greater
importance in eddy current testing, its
derivation is outlined below. The
literature may be consulted for the
calculation of other quantities. 158-161
In axisymmetric geometries, the
impedance of a coil is calculated from the
value of the magnetic vector potential in
the coil’s cross section, starting with the
general formula for the impedance of a
loop of wire carrying an alternating
current Is:
Because the magnetic vector potential
has only one component in the direction
of the current for one turn of a coil
havinga radius r, the impedance is:
where A; is the magnetic vector potential
(volt) at r; and J, is the root mean square
value of the current (ampere).
By integrating this over an elemental
area with N, turns per unit area (square
meter), the expression becomes:
where Aci is the magnetic vector potential
at the same point, rci is the average
distance (meter) of the area chosen from
the axis of symmetry and 4i is the
elemental area (square meter).
In a finite element calculation, r.;, Ac;
and A; are the distance from the axis to
the centroid of the element, the magnetic
vector potential at the centroid of the
element and the area of the element,
respectively. In the case of finite
differences, the same quantities can be
used by imagining the area between four
nodes as being an elemental area. The
centroidal value of the magnetic vector
potential can be calculated as the average
of the four nodal values of the element.
When this value is summed over the
elements in the coil’s cross section and it
is noted that
N,I,, the coil impedance
becomes:
For differential probes, the impedance is
calculated separately for each coil and the
two impedances are added:
In three-dimensional geometries, this
technique cannot be used because it
assumes that the magnetic vector
potential is constant along the
circumference of the coils. The impedance
can be determined by calculating the
stored and dissipated energies, then
finding the resistance of the coils from
the dissipated energy and the inductance
from the stored energy.168,169
4.5 MODELAGEM DA FÍSICA DO ENSAIO DE CORRENTES PARASITAS
The interaction of electromagnetic fields
with material discontinuities, the basis of
all electromagnetic testing methods, is a
complicated phenomenon. Attempts at
solving Maxwell’s equations are in effect
attempts to describe these interactions in
some detail.
If the basic transformer equivalent
model associated with an eddy current
test were to be used, the various
equivalent circuit parameters can be
determined and such questions as the coil
impedance for a certain, simplified
geometry may be answered. It is quite a
different task to analyze with any degree
of generality the details of magnetic field
interaction with the material. To do this, a
continuous impedance plane trajectory is
needed and the details of field
distribution in materials and
discontinuities must be calculable.
The numerical approach to this
problem provides these data as an integral
part of the calculation. Field distributions,
eddy current densities and impedance
values are calculated for any or all probe
positions. The user may choose to test all
or part of these data, either personally or
in a computerized testing procedure.
To demonstrate the value of a
numerical model for understanding field
interactions with materials, the geometry
in Fig. 26 can be analyzed. It comprises a
conducting nonmagnetic tube inside a
carbon steel plate with a gap 0.4 mm
(0.015 in.) between the two. The tube is
1.3 mm (0.05 in.) thick, 22 mm (0.88 in.)
in diameter and the plate is 19 mm
(0.75 in.) thick. The response of the
differential eddy current probe is obtained
by moving the coils from a point where,
because of the distance from the steel
plate, no eddy currents are induced. The
probe is moved a very short distance
toward the plate and the impedance as
well as the field distribution are
recalculated. Repeating this process, a
large number of probe positions are
calculated, resulting in a smooth,
continuous impedance plane trajectory.
Ficure 26. Steam generator geometry.
To perform this calculation, the
geometry in Fig. 26 is discretized into
some 3000 quadrilateral elements
(6000 triangular elements could be used
to produce an identical mesh with
identical results in half the geometry
because of symmetry). This produces a
mesh with 3146 nodes and a system of
equations with 3146 unknowns and a
semibandwidth of 27. The mesh allows
movement of the probe in 140 probe
positions to produce a curve composed of
140 impedance points. In general, 30 to
50 probe positions are sufficient but more
positions may be needed to model
complex or composite discontinuities.
Figure 27 showsa series of five probe
positions, the respective impedance plane
trajectory in the upper left corner and the
flux distribution around the coils. In this
sequence, a small inside diameter
axisymmetric slot is simulated in the tube
at the center of the support plate.
    
Ficure 27. Finite element modeling of circumferential inside
diameter slot under center of steel support plate: (a) probe
far from plate; (b) probe approaches support plate and
leading coil experiences large change in field and produces
impedance plot; (c) coils are centered with support plate
and closed contour has been described; (d) probe leaves
support plate and trailing coil experiences large change;
(e) probe is far from support plate and complete impedance
plane trajectory has been described.
In Fig. 27a, the coils are both well away
from the steel plate, so both have
essentially the same field distribution. The
single point to the left represents the
impedance of the probe at this particular
probe position (zero impedance point).
In Fig. 27b, the leading coil is close to
the edge of the plate and a dramatic
change in the flux distribution has taken
place compared with Fig. 27a. The trailing
coil, however, has a distribution that has
changed very little. The change in the
impedance is clearly noticeable. It should
be remembered, however, that there are
some 35 probe positions between those
shown in Figs. 27a and 27b.
Figure 27c represents the situation
where the coils are centered with the
middle of the support plate. Because the
two coils are differentially connected, this
situation is identical to the one in Fig. 27a
in terms of impedance and the impedance
trajectory now describes a close curve. The
effect of the plate (large lobe) and the
discontinuity are shown.
As the leading coil leaves the support
plate region, the process is repeated and
Figs. 27d and 27e are in effect, a reflection
of Figs. 27b and 27a. Thus, a full
symmetric impedance plane trajectory has
been described. The signal in Fig. 27 was
obtained as a combined signal of the steel
plate and the slot. The two signals were
quite distinct because the discontinuity is
relatively far from the edge of the steel
plate.
In a sequence of this type, not only is
it possible to obtaina full description of
the impedance plane trajectory but also to
test the details of subtle changes in flux
and eddy current distribution, the effects
of conductivity and permeability on flux
lines and the progression of impedance
changes.
As a second example of a similar
situation, consider the sequence in
Fig. 28. It repeats the sequence of Fig. 27
but this time the discontinuity is directly
under the edge of the steel plate.
Starting with Fig. 28a, far from the
plate and discontinuity, the situation is
identical to that in Fig. 27a. As the probe
approaches the plate, the difference
between the two geometries becomes
apparent. The flux lines are affected by
the plate and the discontinuity, producing
a pronounced composite signal. This first
lobe of the composite signal is completed
with Fig. 28c. The rest of the curve repeats
Figs. 27d and 27e and the second part of
the curve is identical to that in Fig. 27
because the discontinuity has no
influence when the probe leaves the plate
region.
By calculating the impedance at a large
number of probe positions in the
sequence and by photographing these as
an animation sequence, a new dimension
in field modeling is achieved. Not only
does the sequence reveal all the necessary
details on the measurement but also
produces a startling and unique view of
the changes in the magnetic field as they
occur in real time. The figures presented
for Fig. 28 are in fact still frames from an
animated movie that describes this and
other simple testing geometries.170
    
Ficure 28. Finite element modeling of circumferential inside
diameter slot under edge of steel support plate: (a) probe is
far from plate; (b) probe approaches support plate and
leading coil experiences large change in field and produces
impedance plot; (c) coils are centered with support plate
and closed contour has been described; (d) probe leaves
support plate and trailing coil experiences large change;
(e) probe is far from support plate and complete impedance
plane trajectory has been described and, because of
asymmetry in geometry, signal is asymmetric.
4.5.1 MODELAGEM PARA PROJETO DE SONDAS
Another very important aspect of eddy
current modeling is the probe itself. It is
safe to assume that the quality of probes
used in eddy current testing has more
effect on the results than all other factors,
yet their design has been based in the past
on empirical considerations and on
experience. This implies a trial and error
procedure by whicha probe is designed
and built, then tested. The process is
repeated until a satisfactory result is
obtained. A numerical approach allows a
more detailed design and a potentially
better product can be obtained at much
lower expense and in shorter design
times. Both general purpose probes 171 and
specialized probes. 172 can be designed.
Detailed studies can be performed about
the various parameters of the probe — its
frequency response and its response to
various discontinuities and material
properties. These data may be used either
to evaluate existing probes or to design
new ones. In addition, a
numerical model
evaluates parameters impossible to
evaluate in any other way. A simple
example is the calculation of flux
densities in any part of the tested
material. Knowledge of flux densities can
reveal saturation effects and possible
nonlinear behavior of the tested material.
4.5.2 PROJETO POR ELEMENTOS FINITOS DE SONDAS DE CORRENTES PARASITAS ABSOLUTA E DIFERENCIAL
The geometry being modeled is shown in
Fig. 29a, where a differential eddy current
probe is placed inside a tube of heat
resistant nickel chromium alloy (Unified
Numbering System N06600) with a
simulated discontinuity. The geometry is
discretized by using triangular finite
elements.171 In this situation, both the
spacing and the width of the coils can be
varied.
Legenda:
r = coodenada radial
z = coordenada axial
Ficure 29. Differential eddy current probe inside tube with
outside diameter axisymmetric slot: (a) geometry; (b) finite
element mesh (half region).
The geometry in Fig. 29a is discretized
into a large number of triangular elements
as shown in Fig. 29b. The finite element
technique is applied to solve for the
magnetic vector potential at each node of
the mesh in Fig. 29b. From these values,
the impedance of the coil is calculated at
discrete probe positions to form the
impedance plane trajectory caused by the
discontinuity.
Symmetry exists about the Z axis and
only half of the geometry is analyzed
using the axisymmetric formulation. Also,
because symmetry exists about the center
of the discontinuity, the probe is allowed
to move up to the point where it is
centered with the discontinuity and the
calculated impedance values are reflected
to form a full impedance plane trajectory.
Figure 30 compares the experimental
results (Fig. 30a) and finite element results
(Fig. 30b) from a differential probe with
coils 2
mm (0.08 in.) wide at spacings
from 1 to 8.5 mm (0.04 to 0.34 in.). The
indication is from a slot (shown in
Fig. 29a) measuring 1 mm (0.04 in.) wide
and 0.4 mm (0.015 in.) deep on the outer
surface of a 22 mm (0.87 in.) tube made
of heat resistant nickel chromium alloy
(Unified Numbering System N06600).
 
Ficure 30. Impedance plane trajectories for outside diameter
axisymmetric slot and distance d between two coils of probe:
(a) experimental element; (b) finite element.
The experimental results were obtained
using a specially designed eddy current
probe with interchangeable coils and
variable spacing between the coils. These
results show clearly that as the spacing of
the coils increases the resulting
impedance plane trajectory loses its
differential nature and the probe behaves
increasingly as two distinct absolute
probes. On the other hand, decreasing the
spacing widens the loops but also reduces
the amplitude of the trajectories.
Figure 31 compares different sized coils
at a constant spacing for the same
discontinuity as in Fig. 30. The spacing is
2.5 mm (0.1 in.) and the coil width varies
from 0.5 to 7.5 mm (0.02 to 0.3 in.). In
this case, as the coil becomes wider, the
amplitude increases and the shape
becomes narrower. From these
calculations and experiments, it is clear
that a good compromise is achieved by
choosing a probe whose coil width and
spacing is comparable to the width of the
discontinuity. Further finite element
predictions were made to investigate this
model as a design tool.
 
Ficure 31. Impedance plane trajectories for different coil sizes
at 100 kHz and constant spacing between coils for slot in
Fig. 30: (a) experimental; (b) finite element.
Impedance plane trajectories were
calculated and plotted by varying the
following parameters in a given probe.
- For frequency, impedance plane
trajectories were calculated at 50, 100
and 150 kHz (Fig. 32).
- For discontinuity geometry, two
different discontinuities were
simulated: a 1 mm (0.04 in.) wide,
0.4 mm (0.016 in.) deep outside
diameter discontinuity and a 1
mm
(0.04 in.) wide, 0.76 mm (0.030 in.)
deep outside diameter discontinuity.
- For coil spacing, calculations were
performed at coil spacings of 1.0, 2.5,
4.0, 5.6, 7.0 and 8.6 mm (0.04, 0.10,
0.16, 0.22, 0.28 and 0.34 in.). The
plots in Fig. 30 show the relation in
the amplitude for smaller and larger
discontinuities and the importance of
choosing the correct spacing for the
probe if meaningful signals are to be
obtained.
- For signal rotation with frequency due
to discontinuities, rotation of the
signal from two probes at 1 mm
(0.4 in.) and 6.3 mm (0.25 in.)
spacings were modeled at various
frequencies for a 10 mm (0.4 in.) wide,
0.76 mm (0.03 in.) deep slot and for a
19 mm (0.75 in.) carbon steel support
plate.
 
Ficure 32. Finite element predicted impedance plane
trajectories for different discontinuities, frequencies and coil
spacings. Small signals are for outside diameter axisymmetric
slot; larger signals are for a deeper slot (d = coil spacing).
Figure 33 indicates that the rotation is
more or less linear at higher frequencies
(the optimal frequency for this particular
probe is about 125 kHz) but is nonlinear
at lower frequencies. The same
phenomena are observed experimentally
and must be taken into account when
relating measured parameters to signal
rotation.
Legend:
A. 1mm (0.04 in.) spacing for probe 1 and axisymmetric slot.
B. 6.4 mm (0.25 in.) spacing for probe 1 and axisymmetric slot.
C. 4.5 mm (0.18 in.) spacing for probe 1 and support plate
Ficure 33. Signal rotation versus frequency for one probe
with different coil spacings.
As a second example of the application
of the finite element model to probe
design, the geometry in Fig. 34 was
studied. The component is a section of a
steam generator tube of heat resistant
nickel chromium alloy (Unified
Numbering System N08800) inside the
tube sheet region. The steam generator
contains rolled tubes where the rolling
region can be at varying distances from
the tube sheet inner surface. The absolute
coil is 1
mm (0.04 in.) thick and has a
length of 1 mm (0.04 in.), which needs to
be optimized for the particular
application.
Ficure 34. Geometry of steam generator section showing
absolute coil, tube sheet and corrosion resistant nickel alloy
tube.
In addition, the signal from the rolling
region is to be modeled for identification
of the tube condition.173
To determine the probe length needed
to obtain the best signal for different
locations of the rolling region relative to
the tube sheet surface, three coil lengths
— 1,3 and 9 mm (0.04, 0.12 and 0.36 in.)
— were modeled for the same three
distances. The finite element results for
these nine situations are plotted in
Fig. 35.
        
Ficure 35. Impedance plane trajectories for coils of length a and for spacing ¢,
6
where/ is distance between tube sheet and rolling distance to inner surface: (a) a= 1 mm
(0.04 in.),
(0.04 in.),
(0.12 in.),
= 1 mm (0.04 in.); (b) a=1 mm (0.04 in.),
¢
= 3 mm (0.12 in,); (©)
9 mm (0.36 in.); (d) a= 3 mm (0.12 in.), = 1 mm (0.04 in.); (e)
3 mm (0.12 in.); (f) a= 3 mm (0.12 in.), £ =
9 mm (0.36 in.); (g) a
mm
3mm
mm
(0.36 in.), 6 = 1 mm (0.04 in.); (h) a= 9 mm (0.36 in.), £= 3 mm (0.12 in); (i) a= 9 mm
(0.36 in.), ¢ = 9 mm (0.36 in.).
The longer the coil, in comparison
with the distance between the two factors
that cause the signal change (tube sheet
and rolling region), the less distinct the
phenomena are in the signal. Thus a coil
9 mm (0.36 in.) long, testing for the
rolling region that is only 1 mm (0.04 in.)
away from the tube sheet surface,
produces a flat composite signal in which
the rolling and the tube sheet cannot be
distinguished as in Fig. 35g. The other
extreme is when the coil is much smaller
than the distance as in Fig. 35c. Here the
two signals are simply superimposed and
one signal does not affect the other.
The curves in Fig. 35 are generated at
100 kHz and are, in general, a composite
signal. The lower, comma shaped parts of
the curves are due to the effect of the tube
sheet. The upper part is due to the rolling
region. These curves compare very well
with experimental results, such as the
curve in Fig. 36 taken at 100 kHz. The
choice of coil size and shape might be
complicated by additional factors, such as
the minimum number of required turns,
but as can be seen from these results the
coil should be of the same general length
as the effect it is measuring.
Ficure 36. Experimental impedance plane
trajectory at 100 kHz from 3 mm (0.12 in.)
long coil at nominal spacing of tube sheet
and rolling.
These results show that the numerical
model is a powerful tool in probe design.
It is less expensive and more accurate
than empirical techniques and can be
used beyond the restricting
approximations of an analytical model.
This versatility is more important for
complex, multicoil probes where the
interaction between the coils, and
possibly ferrite cores and shields,
complicates the design. The responses due
to three-dimensional and subsurface
discontinuities as well as changes in
material properties can also be modeled
and used as valuable input in the design
process.
4.5.3 MODELAGEM PARA SIMULAÇÂO
A third application of the numerical
models is in the area of simulation of test
geometries that are difficult, expensive or
impossible to simulate experimentally. In
this case, a
numerical model is not an
option but rather a necessity. Examples of
these difficult geometries include
subsurface discontinuities, arbitrarily
shaped geometries and discontinuities in
nuclear power plant structures, where
confirmation of the discontinuity shape
cannot be obtained by visual testing.
Information regarding these
discontinuities is of great importance but
cannot be obtained reliably by any other
means. Even if the cost of producing
reference standards to calibrate testing
equipment could be justified, these would
be approximations of real discontinuities
and would be limited to the particular
discontinuity prepared.
The numerical model, once its
versatility and accuracy have been
demonstrated, can handle the task of
producing training data in an economical
and convenient way. It can produce the
data necessary for many testing
configurations (two-dimensional,
axisymmetric and three-dimensional) and
produce the output in a form that is
directly compatible with computer
analysis of raw data.
One situation where measurements are
not generally possible is the testing for
buildup of corrosion products such as
magnetite. Producing sample test objects
by packing such small gaps with
magnetite is very difficult and the
numerical model presents the only
reasonable alternative.
The model described above was applied
to a detailed numerical study of magnetite
buildup in the crevice gap of pressurized
water reactor steam generators.!® There is
uncertainty about how the magnetite
accumulates in the crevices. Similarly, in
the chemical flushing process, the
magnetite is removed but the signal
obtained while monitoring the process
depends on the way the magnetite is
flushed. Several possibilities have been
modeled numerically: (1) radial or axial
buildup (or flushing), (2) axial buildup
from one side of the support plate and
(3) flushing in the presence of tube
denting. Because of the large number of
impedance plane trajectories obtained,
only representative data are presented
here.
The geometry is presented in Fig. 37. It
consists of a tube of high temperature
nickel chromium alloy (Unified
Numbering System N06600), 22 mm
(0.87 in.) in diameter and 1.3 mm
(0.05 in.) in wall thickness inside a
19 mm (0.75 in.) carbon steel support
plate. The crevice gap between the tube
and support plate is nominally 0.4 mm
(0.015 in.) and the signals were calculated
for a differential eddy current probe at
100 kHz. The dashed lines in Fig. 37
schematically represent the area in which
denting of the tubes was modeled.
Ficure 37. Geometry used to study effect of
magnetite accumulation in crevice gap
between tube and support plate. Dashed
line represents area in which denting is
modeled.
The first part of the study dealt with
radial buildup of magnetite from the
support plate toward the tube, or flushing
of magnetite from the tube toward the
support plate. Figure 38 shows the
geometry involved and the impedance
plane trajectories for magnetite buildup in
layers of 0.08 mm (0.003 in.). The change
in signal from a clean gap to one full of
magnetite is quite dramatic, both in shape
and amplitude.
Ficure 38. Modeling of radial buildup of magnetite of
thickness @.
The second part of the study assumed
an axial buildup of magnetite from the
center of the support plate outward.
Although this direction is not very likely
during buildup, it is representative of
chemical flushing, where the chemical
agents attack the magnetite from both
sides. Figure 39 shows this situation: the
changes betweena full gap and partially
filled gap are dramatic.
Ficure 39. Modeling of axial buildup of magnetite
(or extent) outward from center.
Another aspect of testing steam
generator tubing is that of denting due to
magnetite buildup. Similarly, monitoring
the flushing of magnetite in this situation
is more important to ensure complete
cleaning of the magnetite in the gap.
Figure 40 shows the impedance plane
trajectories of axial magnetite buildup (or
flushing) in various amounts from a clean
gap to a gap full of magnetite. From these
plots, it is evident that even small
amounts of magnetite in the gap affect
the impedance plane trajectory.
Legenda:
d = 0.1. mm (0.004 in.)
wabilu = magnetite
Ficure 40. Modeling of magnetite buildup in presence of
denting.
In addition to the data presented
above, studies were carried out
(1) for axial buildup from one side of the
support plate (Fig. 41), (2) for axial
buildup from one side of the support
plate with a portion of the gap clean on
both sides of the magnetite band (Fig. 42)
and (3) for denting of the tube without
the presence of magnetite (Fig. 43). For
comparison, the clean crevice gap
trajectory is given again in Fig. 43.
Ficure 41. Modeling of buildup of magnetite of extent
from one side of support plate
Legenda:
d = 0.1. mm (0.004 in.)
wabilu = magnetite
Figure 42. Modeling of buildup from one side of support
plate with portion of gap clean.
Ficure 43. Modeling of denting in tube without presence of
magnetite, representative of flushed gap d.
The geometries modeled above indicate
the extent and versatility of the numerical
model. There still remains the question of
comparison with real, known data. The
experimental measurement of a support
plate signal in the presence of a clean
support plate is only part of the answer.
Most of the trajectories in Figs. 38 to 43
cannot be reproduced experimentally
because of the difficulty of building
experimental setups that reflect test
conditions.
The geometries modeled above indicate
the extent and versatility of the numerical
model. There still remains the question of
comparison with real, known data. The
experimental measurement of a support
plate signal in the presence of a clean
support plate is only part of the answer.
Most of the trajectories in Figs. 38 to 43
cannot be reproduced experimentally
because of the difficulty of building
experimental setups that reflect test
conditions.
To partly answer the question of
comparisons with known data, the
experiment in Fig. 44 was carried out.
Here, a 25 mm (1 in.) thick support plate
was drilled to provide a 1.5 mm (0.06 in.)
gap, which was then packed axially with
magnetite powder at about 30 percent
(Fig. 44a), 60 percent (Fig. 44b) and
90 percent (Fig. 44c) of the gap length.
The resemblance to corresponding
numerical trajectories is immediately
evident.
  
Ficure 44. Experimental impedance plane trajectories from crevice gaps of 1.5 mm (0.06 in.) width in 25 mm (1 in.)
support plate hole packed with magnetite in various
amounts: (a) 30 percent full; (b) 60 percent full;
(c) 90 percent full.
More convincing evidence of the
ability, accuracy and usefulness of
numerical modeling is provided in Fig. 45.
4.5.1 MODELAGEM PARA PROJETO DE SONDAS
4.5.2 PROJETO POR ELEMENTOS FINITOS DE SONDAS DE CORRENTES PARASITAS ABSOLUTA E DIFERENCIAL
4.5.3 MODELAGEM PARA SIMULAÇÃO
Ficure 45. Experimental data taken during flushing process
in model boiler: (a) impedance plane trajectory of support
plate before flushing; (b) part of gap cleaned, indicating
unequal flushing from both sides; (c) clean gap after
flushing.
These impedance plane trajectories were
obtained during the chemical flushing of
a model boiler. The data are from a tube
with a 0.1 mm (0.004 in.) radial dent at
various stages of flushing. Figure 45a was
taken before flushing began and shows an
identifiable dent filled with magnetite. As
the flushing progresses, the gap shows
various stages of cleaning. Thus, for
example, in Fig. 45b, one side of the gap
is almost completely clean, as indicated
by the large lower lobe. The rest is still
packed with magnetite, indicating an
uneven flushing from both sides of the
support plate. Figure 45c shows a clean
gap while the dent in the tube is visible.
This particular experiment provides a
convincing experimental confirmation of
the numerical model and demonstrates its
value in interpreting data and in
monitoring the flushing process.
4.5.4 CONCLUSÕES
testing phenomena has evolved from an
experimental state through more
sophisticated analytical techniques into
general numerical models. The three
techniques of modeling each have
advantages and shortcomings but the
numerical technique possesses the
generality needed to model the intricacies
of interactions between field and
discontinuity.
As with almost any computer
application, the numerical solution of
eddy current problems is based on known
physical relations. Analysis is part of the
process of acquiring knowledge.
Autores:
- Lalita S. Upda, Michigan State University, East Lansing, Michigan
- Nathan Ida, University of Akron, Akron, Ohio (Parts 1 and 4)
- John R. Bowler, Iowa State University, Ames, Iowa (Part 3)
- Theodoros Theodoulidis, Aristotle University of Thessaloniki, Thessaloniki, Greece (Part 2)
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