BASIC OF ELECTROMAGNETISM:
MAXWELL EQUATIONS
AND
NUMERICAL TECHNIQUES
Giuseppe Mazzarella
D.I.E.E. – Universitá di Cagliari
Pula, 11 June 2012
OUTLINE
i) A hierarchy of Maxwell equations
ii) Fundamental theorems of EM theory
iii) Full–wave numerical techniques
A Hierarchy of Maxwell Equations
Fundamental forms of Maxwell equations:
∂b
∇×e=−
∂t
∂e
1
c2 ∇ × b =
+
j
∂t
ε0 tot
1
ρtot
∇·e=
ε0
d
e · dl = −
dt
Ic
I
Z
b · dS
SZ
d
h · dl = I +
d · dS
dt
S
Ic
d · dS = Q
∇·b=0
IS
F eynman′ s Lectures on P hysics
Franceschetti, Campi Elettromagnetici
S
b · dS = 0
Fig. 1: Geometry for the integral form of M.E.
1st digression: Microscopic vs. Macroscopic
Macroscopic equations are averages of microscopic ones:
Space average
over a small volume
Simple but naive. Difficult to evaluate
the volume size. Difficult to use for the fields
Statistical average
over an ensemble
Microscopic fluctuations
can have a macroscopic effect
Low–pass filtering
in wavenumber space
More effective
(Robinson: Macroscopic Electromagnetism)
A Hierarchy of Maxwell Equations
Local (differential) form of macroscopic equations:
∂b
∂t
∂d
∇×h=
+ jtot
∂t
∇×e=−
in regular points, but also
(d2 − d1 ) · in = ρs
in × (h2 − h1 ) = js
d(t+ ) = d(t+ )
(b2 − b1 ) · in = 0
in × (e2 − e1 ) = 0
b(t+ ) = b(t+ )
(Continuity conditions)
2nd digression: Materials
Macroscopic Maxwell equations are largely underdetermined since the same sources give
different fields in different environments, and all those field are solution of Maxwell equations.
Three vector relations are needed, describing the E.M. behavior of the involved materials. Such
constitutive relations are usually expressed as i–o relations between field vectors. The media
are therefore described as 4D dynamical systems.
Though fundamental fields are e and b, the constitutive relations assume as inputs e and
h, since the latter are the main vectors used in engineering applications. This because e
and h depends directly on controllable sources (voltage differences between conductors, and
conduction currents), while b depends on the total currents, including also polarization ones,
which are neither easily osservable, nor controllable.
The material response in (r0 , t0 ) depends on e(r, t) and h(r, t), for ∀r and ∀t ≤ t0 . The response
is, therefore, dispersive.
2nd digression: Materials
Almost all materials are time–invariants and (except for very high fields) linear, so that a
typical constitutive equations reads
d(r, t) = ε0 e(r, t)+
Z Z
G(r, rs , ts −t)·e(rs , ts ) dVs dts +
Z Z
GH (r, rs , ts −t)·h(rs , ts ) dVs dts
where G and GH are the Green functions (impulse response ) of the material. GH 6= 0 for chiral
media.
In this case the steady–state solution to sinusoidal sources is sinusoidal. Such solution can be
obtained by the Maxwell Equations in the Frequency Domain (FD), namely
∇ × E = −jω B − M
∇ × H = jω D + J
where the constitutive equations becomes t–independent
Z
Z
Ĝ(r, rs ) · E(rs ) dVs +
ĜH (r, rs ) · H(rs ) dVs
D(r) = ε0 E(r) +
A Hierarchy of Maxwell Equations
. . . and a hierarchy of
Analytical
solution approaches
By far the best
Available only in few cases
but nonetheless very useful
Test of numerical procedures
For extrapolation
To develop physical insight
Semi–analytical
Model based
Full wave
Asymptotic or variational solutions
Approximate a problem
in terms of simpler ones
Direct solution of Maxwell equations
Often General Purpose
OUTLINE
i) A hierarchy of Maxwell equations
ii) Fundamental theorems of EM theory
iii) Full–wave numerical techniques
Fundamental theorems of EM theory
Poynting theorem: energy balance in EM fields
Starting from the density of Lorentz force in time domain f = ρ e + ρ v0 × b, the power received
by the EM field in volume V is (if averaged over a period)
Z
1
Pa (sources → field) = Re[PG ]
where
PG = −
E · J∗0 dV
V 2
Inserting Maxwell equations (for local, isotropic media) in PG we get
I
Z
Z 1
1
1
S · in dS + ω
ε2 |E|2 dV = −j 2ω
µ|H|2 − ε1 |E|2 dV + PG
2
4
S
V
V 4
where S = 12 E × H∗ is the Poynting vector (in FD) and ε1 − jε2 is the effective permittivity
of the medium.
Fundamental theorems of EM theory
The real part is the energy balance for EM field
I
Z
1
Sr · in dS +
ω
ε2 |E|2 dV = Re [PG ]
2
S
V
active power
power dissipated
active power
+
=
from sources
leaving V
inside V
Moreover, from the imaginary part
I
Z 1
1
µ|H|2 − ε1 |E|2 dV = Im [PG ]
Si · in dS
+2ω
4
4
V
S
reactive power
unbalanced EM energy
reactive power
+
= from sources
leavingV
inside V
Fundamental theorems of EM theory
Reciprocity theorem
Two different EM fields (E1 , H1 ) and (E2 , H2 ) can be connected together in various forms.
Assuming the same (local and isotropic) medium for both we have
I
∂V
[E1 × H2 − E2 × H1 ] · in dS =
Z
V
[E2 · J1 − H2 · M1 ] dV −
Z
V
[E1 · J2 − H1 · M2 ] dV
The r.h.s integrals are known as reactions of fields over currents.
The reactions are equal if
• ∂V is a PEC, PMC or an impedance surface (including good conductors)
• ∂V is a spherical surface at infinity (where the Sommerfeld conditions holds)
• V contains all sources (or none).
Fundamental theorems of EM theory
Equivalence theorem
States the equivalence of two different EM configuration. These are equivalent in the sense that
the EM field in a given region is the same for both configurations.
The same field outside S can be obtained by volume, or by equivalent, surface currents:
Js = in × H
Outside S depends on the choice of in
Ms = −in × E
Fundamental theorems of EM theory
Equivalence theorem
Main applications:
• E and H on S are known, or have been
measured
• E and H on S can easily be approximated
• The whole field outside S depends only on E and H on S (and can be computed from them)
Fundamental theorems of EM theory
Main corollaries of equivalence theorem
After application of the equivalence theorem, the medium inside S can be modified, without
affecting the field.
Replacing the medium inside S by a PEC makes the electric surface currents useless, since they
do not produce any field. We can measure only the Etan on S to compute the whole field
outside S (Near field–Far field transformation).
Fundamental theorems of EM theory
Main corollaries of equivalence theorem
In reflector, and scattering, analysis, we can replace the metallic structure by the currents
induced on it.
If the Js were known, the radiated field could be computed by a simple integration. However,
for structures with a large radius of curvature (respect to λ), the physical optics (PO)
approximation can be used. PO assumes that Js can be bf locally computed, as the conductor
surface were the infinite tangent plane.
PO current errors can be considered randomply distributed. So PO is accurate only close to
the broadside direction, since there the errors adds in phase.
OUTLINE
i) A hierarchy of Maxwell equations
ii) Fundamental theorems of EM theory
iii) Full–wave numerical techniques
Full–wave numerical techniques
There have been devised huge number of (more or less) different full–wave
numerical techniques.
=⇒
There have been proposed many taxonomies to classify them.
but the top–level classification is
Synthetic
Complete solution is obtained
form a limited amount of information
Analytical
Look for solution values
in the whole solution domain
Synthetic Full–wave numerical techniques
Modal solutions
The field is expressed in different regions as a generalized Fourier series or integral, using the
modes of that region as basis functions.
In EM theory, a mode is a field configuration which can exists alone in a structure, i.e., which
fulfills both the Maxwell quations and all the boundary conditions in that structure.
The Fourier coefficients (discrete, or with a lower dimensionality than the EM field) are then
the unknown of the problem. Enforcing the boundary conditions between regions gives the set
of resolving equations.
The complete Fourier representation of the fields must be truncated in order to solve the
problem. The explicit truncation order must be different in each region, since a stable solution
requires that the quickest variation of the field in each region must be the same (relative
convergence phenomenon).
Example: Mode matching in a waveguide problem
The field in each region is expressed as a modal series. To define the truncation level, we must
define a cut–off frequency level F0 (unrelated from the problem frequency), and retain only
modes with cut–off frequency smaller than F0 .
Synthetic Full–wave numerical techniques
Method of Moments (MoM)
Equivalence theorem states that the 3D field can be computed once (equivalent or induced)
currents are known. The latter are 2D, or even 1D, so a far less complex problem can be solved.
The actual field in each region i–th is expressed as an integral over the unknown currents, using
a suitable Green function Gi , depending on the problem at hand:
Z
Hi (r) =
Gi (r, rs ) · M(rs ) dSs = Li [M]
S
Then the continuity conditions on the same surface S are enforced, to get an integral equation
n
o
n
o
in × L1 [M] + Hinc = in × L2 [−M]
P
The MoM requires then that the unknown is discretized M =
cp fp , and the integral equation
is enforced in a weak form
Z
n
o
gm · in × L1 [M] + Hinc − L2 [−M] dSs
S
to get the resolving linear system
Z
Z
n
o
n
o
X
cp
gm · in × L1 [fp ] + L2 [fp ] dSs =
gm · in × − Hinc dSs
p
S
S
Synthetic Full–wave numerical techniques
Method of Moments (MoM)
Discretization fp and weighting gm functions are quite arbitrary, but usually fall into two
categories
P
• Entire domain: all functions are defined in the whole S, so M =
cp fp is a sort of Fourier
expansion;
• Subsectional: all functions are defined on a small subset of S (so M is approximated by piecewise
constant, or piecewise linear funtions, or so);
Matrix filling time require the solution of a field problem for a source fp , so that this phase is
easier, and can be more time–effective, using subsectional basis functions. Using entire domain
basis functions gives a far smaller system, and usually better conditioned.
Synthetic Full–wave numerical techniques
Method of Moments (MoM)
The MoM matrix is ill–conditioned (it comes out from a First Kind Fredholm integral equation),
so the discretization must be chosen in order to constraint the spatial bandwidth of the solution.
In some problems, the MoM equation is approximate (e.g., current on a wire antenna, see
Collin: Antennas and Radiowave propagation). The discretization bandwidth must then be
chosen also to remain into the validity interval of the approximation.
It is well–known that the frequency behavior of the currents is more or less resonant. This prevent a robust interpolation of them. But the system matrix, which contains reaction integrals,
is a smooth function of the frequency. Therefore it allows a simple and robust interpolation.
Since the solution time (for entire domain basis function) is small, this matrix interpolation is
very effective to evaluate a frequency response in many problems.
Analytic Full–wave numerical techniques
Solve more or less directly the Maxwell equation in the whole solution domain, using a suitable
dicretization. The problem is therefore very easily formulated.
They allow also to device general purpose solution softwares. But have a computational complexity orders of magnitude larger than synthetic procedures.
Many hybrid techniques have been proposed, trying to merge the advantages of the parent
techniques.
A typical example derives from the MoM requirement of the field of the basis functions. This
field can, sometimes, computed using an analytic technique.
Analytic Full–wave numerical techniques
Finite Differences in Time Domain (FDTD) – Yee, 1966
Maxwell equations are differential equations, so a direct discretization of them is possible, replacing derivatives with finite differences. This is usually one in time domain, and the frequency
response is obtained with Fourier transform of the computed time evolution.
A rectangular grid of sampling points is required (and the inclusion of curved boundary is still
an open problem), and the standard leap–frog approach requires two grids, one for E and the
other for H, spaced half a step apart.
Its main drawback is the computational complexity. Space discretization ∆x smaller than λ/10
is mandatory (and in many problems even smaller than √λ/20 is required). The time step is
further constrained by the Yee stability condition c∆t ≤ 3∆x.
Analytic Full–wave numerical techniques
Finite Integration Technique (FIT) – Weiland, 1977
In FDTD, the relation between the EM field are approximated by finite differences. In FIT
the approach is reversed: FIT equations are exact equations for averaged (thus approximate)
field values. Solution domain is divided into small rectangular cells, and the integral form of
Maxwell equations are enforced on those cells. Unknown are therefore line integral of electric
field and fluxes of magnetic field (or vice–versa).
Numerical solution is similiar to FDTD, though FIT can be implemented in frequency domain
also. However, since the equations are exact, some formal manipulations are possible in FIT.
It is worth noting that averages of field values are the results of field measurements, so its use
could not be considered a limitation.
Analytic Full–wave numerical techniques
Finite Element Method (FEM) – Structural Analysis, circa 1940
In this case, the whole solution domain is divided into small cells, whose shape is unrestricted
(rectangular, tetrahedral or even more irregular). Then a weak form of the problem is enforced,
typically using the energy (which is extremal for the correct solution), or even looking for a Least
Square solution. In each cell, the solution is approximated by small–order polynomial, and the
energy is expressed in terms of the coefficients of these polynomial. Finally, minimization gives
a linear (sparse) system in the unknown coefficients.
FEM equations are therefore more complex to devise than FDTD, but the density of unknown
can be smaller. As a matter of fact, non–uniform discretization is the rule with FEM.
Thank you for your attention
Giuseppe Mazzarella
D.I.E.E. – Universitá di Cagliari
Piazza D’Armi – I–09123 Cagliari
mazzarella@diee.unica.it