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1. Introduction
.
2/20/20131 Electromagnetic Field Theory by R. S. Kshetrimayum
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1.1 Electromagnetic theory in a nutshell Electromagnetic field theory is the study of fields produced
by electric charges at
rest or
in motion
Electromagnetic theory can be divided into three sub-
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum2
divisions electrostatics,
magnetostatic and
time-varying fields as depicted in Fig. 1.1
depending on whether the charge which is the source ofelectromagnetic field is at rest or motion
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1.1 Electromagnetic theory in a nutshell Electrostatic fields are produced by static electric charges
Magnetostatic fields are produced by electric charges movingwith uniform velocity also known as direct current
Time-varying fields are produced by accelerated or
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum4
-
currents
An accelerated or decelerated charge also produces radiation
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1.2 Computational electromagnetics
Computational electromagnetics (CEM) is an
interdisciplinary field where
we apply numerical methods and
use computers
to solve
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum5
practical
and real-life electromagnetic problems
which usually do not have simple analytical solutions
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1.2 Computational electromagnetics
1.2.1 Why do we need Computational electromagnetics?
Maxwell’s equations along with the electromagnetic boundary conditions
describe any kind of electromagnetic phenomenon in nature
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum6
exc u ng quan um mec an cs Due to the linearity of the four Maxwell’s equations in the
differential forms,
it may appear rather easy to solve them analytically
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1.2 Computational electromagnetics
But the boundary and interface conditions make them hard to
solve analytically for many practical electromagnetic
engineering problems
Hence one has to resort to use
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum7
,
approximate or
computational methods
to solve them
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1.2 Computational electromagnetics
An advantage of this is that it is possible to simulate a
device/experiment/phenomenon
any number of times
as per our requirements
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum8
,
we can try to achieve the best or optimal result
before actually doing the experiments
Sometime experiments are dangerous to perform
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1.2 Computational electromagnetics
1.2.2 Computational electromagnetics in a nutshell
For any computational solution in Computationalelectromagnetics,
it is necessary to develop the required equations and
solve them using a computer also known as equation solvers
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum9
There are two types of equations: integral or
differential equations and
correspondingly two solvers: integral or
differential equation solvers
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1.2 Computational electromagnetics
Integral equations are equations in which the unknown is
under an integral sign just like
in differential equation your unknown function is under a
differential sign
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum10
,
unknown line charge density λ
It is an integral equation since the unknown λ is under an
integral sign
0
( ') '( )
4 ( , ')
x dxV x
r x x
λ
πε = ∫
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1.2 Computational electromagnetics
Similarly,
is a differential equation because the unknown function f(x) isunder a differential sign
Sometimes com lex e uations can constitute both inte ral
2
2
2
( )d f x x
dx− =
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum11
as well as differential equations also known as integro-differential equation
In general, all the available Computational electromagnetics
methods may be classified broadly into two categories: a) differential equation solvers and
b) integral equation solvers
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1.2 Computational electromagnetics
Computationalelectromagnetics
Integral
equation solver
Differential
equation solver
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum12
equation solver
Frequency domainintegral equation solver
Fig. 1.2 Computational electromagnetics in a nutshell
Time domain differentialequation solver
Frequency domain
differential equation solver
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1.2 Computational electromagnetics
Time Domain Integral Equation (TDIE) solver: solves
complex electromagnetic engineering problems in the form
of integral equations in time domain
Frequency Domain Integral Equation (FDIE) solver: solves
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum13
of integral equations in frequency domain A suitable example for this is Method of Moments (MoM)
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1.2 Computational electromagnetics
Time Domain Differential Equation (TDDE) solver: solves
complex electromagnetic engineering problems in the form
of differential equations in time domain
A possible example for this is Finite Difference Time Domain
(FDTD)
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum14
Frequency Domain Differential Equation (FDDE) solver:solves complex electromagnetic engineering problems in the
form of differential equations in frequency domain
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1.3 General curvilinear coordinate system
1.3.1 Coordinate systems
Note that it is possible to develop one general expressions
also known as general curvilinear coordinate system for
divergence,
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other vector operations
of the three orthogonal coordinate systems viz.
Rectangular,
Cylindrical and
Spherical coordinate systems
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1.3 General curvilinear coordinate system
A point in space represented by a1, a2 and a3 in the
general curvilinear coordinate system
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1.3 General curvilinear coordinate system
Differential elements can be expressed as dl1=s1da1,
dl2
=s2
da2
, dl3
=s3
da3
where s1
, s2
and s3
are the scale
factors
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1.3 General curvilinear coordinate system
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1.3 General curvilinear coordinate system
θ
r̂ ˆ z
φ̂
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum19
φ
θ
ˆθ
ˆ x
ˆ y
φ θ
ˆ
(a)(b)
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1.3 General curvilinear coordinate system
φ̂
ˆ yˆ zθ
r̂ ˆ z
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ˆ ρ φ
ˆ xθ̂
φ̂ ˆ
(c) (d)
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1.3 General curvilinear coordinate system
Fig. 1.4
(a) Coordinate systems and their variables
(b) Geometry relationship between the Rectangular and
Spherical coordinate systems
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum21
c eometry re at ons p etween t e ectangu ar an
Cylindrical coordinate systems and
(d) Geometry relationship between the Spherical and
Cylindrical coordinate systems
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1.3 General curvilinear coordinate system
1.3.2 Direction cosines
Direction cosines of a vector are the cosines of the angles
between the vector and three coordinate axes
For instance, the direction cosines of a vectorr
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum22
with the x-, y- and z- axes are:
( , , ) x y z
x y z x y z= + +
2 2 2
ˆ( , , )cos
( , , )
x
x y z
A A x y z x
A x y z A A Aα
•= =
+ +
r
r
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1.3 General curvilinear coordinate system
2 2 2
ˆ( , , )cos
( , , )
y
x y z
A A x y z y
A x y z A A A
β •
= =
+ +
r
r
2 2 2
ˆ( , , )cos z
A A x y z z
A xγ
•= =
r
r
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum23
where α, β and γ are respectively the angles vector makes
with the x-, y- and z- axes
x y z
r
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1.3 General curvilinear coordinate system
In a more general sense, direction cosine refers to the cosine
of the angle between any two vectors
They are quite useful for converting one coordinate system
to another (or coordinate transformation)
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum24
ˆ ˆˆ ˆ ˆ ˆsin cos cos cos sin
ˆ ˆˆ ˆ ˆ ˆsin sin cos sin cos
ˆ ˆˆˆ ˆ ˆcos sin 0
x r x x
y r y y
z r z z
θ φ θ θ φ φ φ
θ φ θ θ φ φ φ
θ θ θ φ
• = • = • = −
• = • = • =
• = • = − • =
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1.3 General curvilinear coordinate system
(b) Cylindrical and Rectangular
ˆˆˆ ˆ ˆ ˆcos sin 0
ˆˆˆ ˆ ˆ ˆsin cos 0
ˆˆˆ ˆ ˆ ˆ0 0 1
x x x z y y y z
z z z z
ρ φ φ φ ρ φ φ φ
ρ φ
• = • = − • =
• = • = • =
• = • = • =
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum25
(c) Spherical and Cylindricalˆ ˆˆ ˆ ˆˆ sin cos 0
ˆ ˆ ˆ ˆ ˆˆ 0 0 1
ˆ ˆˆˆ ˆ ˆcos sin 0
r
r
z r z z
ρ θ ρ θ θ ρ φ
φ φ θ φ φ
θ θ θ φ
• = • = • =
• = • = • =
• = • = − • =
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1.3 General curvilinear coordinate system
1.3.3 Coordinate transformations
(a) Spherical to Rectangular and vice versa
( ) [ ] ( )
sin cos cos cos sin
sin sin cos sin cos , , , ,
x r
y
A A
A A A x y z T A r θ
θ φ θ φ φ
θ φ θ φ φ θ φ
−
= ⇒ =r r
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cos sin 0 z A A φ θ θ
−
sin cos sin sin cos
cos cos cos sin sin
sin cos 0
xr
y
z
A A
A A
A A
θ
φ
θ φ θ φ θ
θ φ θ φ θ
φ φ
= −
−
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1.3 General curvilinear coordinate system
(c) Spherical to Cylindrical and vice versa
( ) [ ] ( )sin cos 0
0 0 1 , , , ,
cos sin 0
r
sc
z
A A A A A z T A r
A A
ρ
φ θ
φ
θ θ φ θ φ
θ θ
= ⇒ =
−
r r
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sin 0 cos
cos 0 sin
0 1 0
r
z
A
A
A
ρ
θ φ
φ
θ θ
θ θ
= −
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Vector calculus
Vector differential
calculusVector integral
calculus
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum29
Gradient
Divergence
Fig. 1.3 Vector calculus
Curl Laplacian
Divergence
theoremStoke’s
theorem
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1.4 Vector differential calculus1.4.1 Gradient of a scalar function:
The gradient of any scalar functionΨ
is a vector whosecomponents in any direction are given by the spatial rate
change ofΨ along that direction
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1 2 3
1 1 2 2 3 3
a a as a s a s a
ψ ψ ψ ψ
∂ ∂ ∂∇ = + +
∂ ∂ ∂
) ) )
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1.4 Vector differential calculusHow to memorize this formula?
Note that in each of the three terms in the gradient of scalar
function above,
we have a unit vector,
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum31
corresponding variable and divide by the corresponding scale factor
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1.4 Vector differential calculus
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1.4 Vector differential calculus1.4.2 Divergence of a vector:
( ) ( ) ( )2 3 1 1 3 2 1 2 3
1 2 3 1 2 3
1 A s s A s s A s s A
s s s a a a
∂ ∂ ∂∇ • = + +
∂ ∂ ∂
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It is a measure of how much the vector spreads out(diverge) from the point in question
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1.4 Vector differential calculusHow to memorize this formula?
Note that in the expression of divergence of a vector above,
outside the third bracket, we have division by product of all
scale factors, and
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum34
Each term contains a
partial differential w.r.t. one of the variable
to the product of corresponding vector component and
scale factors of the remaining two axes
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1.4 Vector differential calculus
Divergence
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1.4 Vector differential calculus1.4.3 Curl of a vector:
1 1 2 2 3 3
1 2 3 1 2 3
1
s a s a s a
As s s a a a
∂ ∂ ∂∇ × =
∂ ∂ ∂
) ) )
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How much that vector curls around the point in
question?
1 1 2 2 3 3
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1.4 Vector differential calculusHow to memorize this formula?
Note that in the expression of curl of a vector above,
outside the determinant, we have division by product of all scale
factors
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum37
, ,
have multiplied by corresponding scale factors to unit vectorsand vector components respectively
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1.4 Vector differential calculus
Curl
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1.4 Vector differential calculus
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum40
Fig. 1.5 (b) Positive divergence and curl around z-axis
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1.4 Vector differential calculus Scalar triple product
( ) ( ) ( ) x y z
x y z
x y z
A A A
A B C B B B B C A C A B
C C C
• × = = • × = • ×r r r r r rr r r
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Note that the above three vector scalar triple productsare the same from the definition of scalar triple product
Vector triple product (“bac-cab” rule)
( ) ( ) ( ) A B C B A C C A B× × = • − •r r r r r rr r r
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1.4 Vector differential calculusSome useful vector identities:
This means curl of a gradient of scalar function is always
zero
( ) 0ψ ∇ × ∇ =
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum42
This means divergence of a curl of vector is always zero
( ) 0 A∇ • ∇ × =r
( ) ( ) ( ) A B B A A B∇ • × = • ∇ × − • ∇ ×r r rr r r
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1.4 Vector differential calculus This means that divergence of cross product of two vectors is
equal to
the dot product of second vector and curl of first vector
minus dot product of first vector and curl of second vector
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum43
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1.4 Vector differential calculus1.4.4 Laplacian of a scalar or vector function:
Laplacian is an operator which can operate on a scalar or
vector
Laplacian of a scalar function:
2/20/2013Electromagnetic Field Theory by R. S. Kshetrimayum44
Laplacian of a vector function:
2ψ ψ ∇ = ∇ • ∇ 2 3 1 3 1 2
1 2 3 1 1 1 2 2 2 3 3 3
1 s s s s s ss s s a s a a s a a s a
ψ ψ ψ ∂ ∂ ∂ ∂ ∂ ∂
= + + ∂ ∂ ∂ ∂ ∂ ∂
( ) ( ) A A A∇×∇× =∇ ∇• − ∇•∇Q ( ) 2 A A= ∇ ∇• −∇
( )2 A A A∇ = ∇ ∇ • − ∇ × ∇ ×
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1.5 Vector integral calculus
1.5.1 Scalar line integral of a scalar function
where is the scalar function
( ) ( )( )1 2 3 1 2 3 1 1 1 2 2 2 3 3 3, , , ,a a a dl a a a s da a s da a s da aψ ψ = + +∫ ∫
r) ) )
ψ
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and is the vector line element
1.5.2 Scalar line integral of a vector field
d l
( )
( ) ( ) ( ){ } ( )
1 2 3
1 1 2 3 1 2 1 2 3 2 3 1 2 3 3 1 1 1 2 2 2 3 3 3
, ,
, , , , , ,
A a a a dl
A a a a a A a a a a A a a a a s da a s da a s da a
•
= + + • + +
∫
∫
rr
) ) ) ) ) )
1 5 Vector integral calculus
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1.5 Vector integral calculus
where is the vector field and
and is the vector line element
1.5.3 Scalar surface integral of a vector field
A ds A nds• = •r rr )
Ar
d lr
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where is the vector field and
is the normal to surface element ds
r
n̂
1 5 Vector integral calculus
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1.5 Vector integral calculus
1.5.4 Divergence Theorem
It is also known as Green’s or Gauss’s theorem
Consider a closed surface S in presence of a vector field as
shown in Fig. 1.8 (a)
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et t e vo ume enc ose y t s c ose sur ace e g ven y
Then according to the Divergence theorem
dv Asd A
S V
∫ ∫∫∫ •∇=•rrr
1 5 Vector integral calculus
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1.5 Vector integral calculus
Fig. 1.8 (a) Divergence theorem (Converts closed surface
integrals to the volume integrals)
Ar
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V
n̂dsr
da
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1.5 Vector integral calculus
1.5.5 Stokes theorem
Consider a closed curve C enclosing an area S in presence of
a vector field
Then, Stokes theorem can be written as
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∫ ∫∫ •×∇=•C S
sd Ald A rrrr
1 6 SVector calculus
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1.6 Summary
Vector differential calculus
Gradient
Vector integral calculus
CurlStoke’s theorem
)∫ ∫∫ •×∇=•
C S
sd Ald A rrrr
∂ ∂ ∂ s a s a s a) ) )
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Divergence
Fig. 1.9 Vector calculus in a nutshell
Laplacian
Divergence theorem( )dv Asd A
S V
∫ ∫∫∫ •∇=•rrr
1 2 3
1 1 2 2 3 3
a a as a s a s a
ψ ∇ = + +∂ ∂ ∂
) ) )
( ) ( ) ( )2 3 1 1 3 2 1 2 3
1 2 3 1 2 3
1 A s s A s s A s s A
s s s a a a
∂ ∂ ∂
∇ • = + + ∂ ∂ ∂
1 2 3 1 2 3
1 1 2 2 3 3
1 A
s s s a a as A s A s A
∂ ∂ ∂∇ × =
∂ ∂ ∂
2 3 1 3 1 2
1 2 3 1 1 1 2 2 2 3 3 3
1 s s s s s s
s s s a s a a s a a s a
ψ ψ ψ ∂ ∂ ∂ ∂ ∂ ∂
= + + ∂ ∂ ∂ ∂ ∂ ∂
2
ψ ψ ∇ = ∇ • ∇