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EEE 431Computational Methods in
Electrodynamics
Lecture 3
By
Dr. Rasime Uyguroglu
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Energy and Power
We would like to derive equations governing EM energy and power.
Starting with Maxwell’s equation’s:
(1)
(2)imp c
BXE
tD
XH J Jt
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Energy and Power (Cont.)
Apply H. to the first equation and E. to the second:
.( )
.( )imp c
BH XE
tD
E XH J Jt
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Energy and Power (Cont.)
Subtracting:
Since,
.( ) .( ) .( ) .( )imp c
B DH XE E XH H E J J
t t
.( ) .( ) .( )
.( ) .( ) .( )imp c
EXH H XE E XH
B DEXH H E J J
t t
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Energy and Power (Cont.)
Integration over the volume of interest:
.( ) [ .( ) .( ) ]imp c
v v
B DEXH dv H E J J dv
t t
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Energy and Power (Cont.)
Applying the divergence theorem:
ˆ. . .( )imp c
s v v
B DEXH nds H dv E J J dv
t t
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Energy and Power (Cont.)
Explanation of different terms: Poynting Vector in
The power flowing out of the surface S (W).
2( / )W m
P EXH
0 ˆ.s
P P nds
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Energy and Power (Cont.)
Dissipated Power (W)
Supplied Power (W)
2( . ) .d c
v v v
P E J dv E Edv E dv
( . )s impvP E J dv
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Energy and Power
Magnetic power (W)
Magnetic Energy.
2
. .
1
2
m
v v
m
v
B HP H dv H dv
t t
H dv Wt t
,mW
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Energy and Power (Cont.)
Electric power (W)
electric energy.
2
. .
1
2
e
v v
ev
D EP E E dv
t t
E dv Wt t
,eW
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Energy and Power (Cont.)
Conservation of EM Energy
0 ( )s d e mP P P W Wt
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Classification of EM Problems
1) The solution region of the problem, 2) The nature of the equation describing
the problem, 3) The associated boundary conditions.
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1) Classification of Solution Regions:
Closed region, bounded, or open region, unbounded. i.e Wave propagation in a waveguide is a closed region problem where radiation from a dipole antenna is an open region problem.
A problem also is classified in terms of the electrical, constitutive properties. We shall be concerned with simple materials here.
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2)Classification of differential Equations
Most EM problems can be written as:
L: Operator (integral, differential, integrodifferential)
: Excitation or source : Unknown function.
L g
g
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Classification of Differential Equations (Cont.)
Example: Poisson’s Equation in differential form .
2
2
v
v
V
L
g
V
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Classification of Differential Equations (Cont.):
In integral form, the Poisson’s equation is of the form:
2
2
4
4
v
v
v
v
V dvr
dvL
r
g V
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Classification of Differential Equations (Cont.):
EM problems satisfy second order partial differential equations (PDE).
i.e. Wave equation, Laplace’s equation.
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Classification of Differential Equations (Cont.):
In general, a two dimensional second order PDE:
If PDE is homogeneous. If PDE is inhomogeneous.
2 2 2
2 2a b c d e f g
x x y y x y
( , ) 0g x y
( , ) 0g x y
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Classification of Differential Equations (Cont.):
A PDE in general can have both: 1) Initial values (Transient Equations) 2) Boundary Values (Steady state
equations)
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Classification of Differential Equations (Cont.):
The L operator is now:2 2 2
2 2L a b c d e f
x x y y x y
2
2
2
, 4 0
, 4 0
, 4 0
If b ac Elliptic
If b ac Parabolic
If b ac Hyperbolic
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Classification of Differential Equations (Cont.):
Examples: Elliptic PDE, Poisson’s and Laplace’s
Equations:2 2
2 2
2 2
2 2
( , ) ' .
0 '
g x y Poisson s Eqnx y
Laplace s Eqnx y
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Classification of Differential Equations (Cont.):
For both cases a=c=1,b=0.
An elliptic PDE usually models the closed region problems.
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Classification of Differential Equations (Cont.):
Hyperbolic PDE’s, the Wave Equation in one dimension:
Propagation Problems (Open region problems)
2 2
2 2 2
10
x u t
2 , 0, 1a u b c
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Parabolic PDE, Heat Equation in one dimension.
Open region problem.
Classification of Differential Equations (Cont.):
2
2
1, 0
kx t
a b c
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Classification of Differential Equations (Cont.):
The type of problem represented by:
Such problems are called deterministic. Nondeterministic (eigenvalue) problem is
represented by:
Eigenproblems: Waveguide problems, where eigenvalues corresponds to cutoff frequencies.
L g
L
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3) Classification of Boundary Conditions:
What is the problem? Find which satisfies
within a solution region R. must satisfy certain conditions on
Surface S, the boundary of R. These boundary conditions are Dirichlet
and Neumann types.
L g
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Classification of Boundary Conditions (Cont.):
1) Dirichlet B.C.: vanishes on S.
2) Neumann B.C.: i.e. the normal derivative of vanishes on S.
Mixed B.C. exits.
( ) 0,r r on S
( )0, .
rr on S
n