1
EE256 Numerical Electromagnetics
H. O. #4
Marshall
24 June 2008
Summer 2008
LECTURE 2
2.1
PARTIAL DIFFERENTIAL EQUATIONS AND PHYSICAL SYSTEMS
Most physical systems are described by one or more partial differential equations, which can be derived by
direct application of the governing physical laws. For electromagnetic phenomena, the governing physical
laws are experimentally based and are Faraday’s law, Ampere’s law, Coulomb’s law, and the conversation
of electric charge. Application of these laws lead to Maxwell’s equations. For voltage and current waves
on transmission lines, application of Kirchoff’s voltage and current laws (which are contained in Maxwell’s
equations) leads to the
Telegrapher’s equations
:
∂
V
∂z
=

R
I 
L
∂
I
∂t
[2.1
a
]
∂
I
∂z
=

G
V 
C
∂
V
∂t
[2.1
b
]
where
V
(
z, t
)
and
I
(
z, t
)
are respectively the line voltage and current, while
R
,
L
,
G
, and
C
are re
spectively the distributed resistance, inductance, conductance, and capacitance of the line. Manipulation of
[2.1
a
] and [2.1
b
] leads to the general
wave equation
for a transmission line:
∂
2
V
∂z
2
=
RG
V
+ (
RC
+
LG
)
∂
V
∂t
+
LC
∂
2
V
∂t
2
[2.2]
Note that making the substitution
V ↔I
in [2.2] gives the wave equation in terms of current
I
(
z, t
)
.
A similar wave equation can be derived in terms of the electric field
E
by manipulating [1.1] and [1.3],
for simple media for which
,
μ
,
σ
, and
σ
m
are simple constants:
∇
2
E
=
σσ
m
E
+ (
μσ
+
σ
m
)
∂
E
∂t
+
μ
∂
2
E
∂t
2
[2.3]
The corresponding equation for the magnetic field
H
can be obtained by making the substitution
E ↔
H
.
General solutions of equations [2.2] and [2.3] are in general quite complex, except in special cases.
Solutions for [2.2] take a simple and analytically tractable form when we assume that the variations in time
are sinusoidal (timeharmonic), or when we assume that the loss terms (i.e., those involving
R
,
G
in [2.2])
are zero. Solutions for [2.3] also become tractable for the timeharmonic and lossless (assuming that terms
involving
σ
,
σ
m
in [2.3] are zero) cases, if we make additional simplifying assumptions about the direction
and spatial dependency of
E
(
x, y, z, t
)
.
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Partial differential equations arise in a variety of ways and in the context of a wide range of different
physical problems, remarkably having forms quite similar to equations [2.1] and [1.1] through [1.5]. The
basic reason for the commonality of the form of governing equations for different physical systems is the fact
that most of classical physics is based on principles of conservation, whether it be principle of conservation
of mass, energy, momentum, or electric charge.
For example, application of the principle of conservation of momentum to an incremental volume of a
compressible fluid (e.g., a gas such as air) gives the ‘telegrapher’s equations’ for acoustic waves:
Transmission

line Analog
Acoustic waves
∂
V
∂z
=

L
∂
I
∂t
∂u
x
∂x
=

1
γ
g
p
a
∂p
∂t
∂
I
∂z
=

C
∂
V
∂t
∂p
∂x
=

ρ
v
∂u
x
∂t
[2.4]
where
u
x
is the velocity in the
x
direction,
p
is the variation in pressure (above an ambient level),
ρ
v
is the mass per unit volume,
γ
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 '09
 Electromagnet, Tn, Partial differential equation, wave equation, finite difference

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