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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

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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