plane waves
Statics transverse to z solutions
in STATICS
E 0
now consider a twodimensional statics problem that is
transverse to the z axis, i.e., the symmetry is such that
there is no field variati
plane waves
Lossless uniform plane wave solution to Maxwells equations
assumed
E plane Exo e jt z x
wave
H plane j
Exo e jt z y j j
wave
to understand what this thing looks like, lets make an
add
plane waves
Wave equation form of time harmonic transversetoz Maxwells equations
we could write this in a slightly more compact way:
ETEM to z
2
z 2
H TEM to z
j j ETEM to z 0
2
z 2
j j H TEM to
LC transmission lines
Time domain bounces
review the material form the first few lectures on time domain
bounce behavior for LC t lines.
See the blackboard lectures
Dean P. Neikirk 2004, last update
time varying fields
Summary of electrostatics and
magnetostatics
Maxwells equations for statics
E 0
D E
D v
J E
H J
B 0
r o
B H
r o
but what happens if something changes in time?
Dean P. Ne
EE 363M  Microwave and Radio Frequency
Engineering
Catalog Description:
Design principles in microwave and radio frequency
systems; transmission lines and waveguides; Sparameter representation; imped
transmission lines
Generalized RLCG Tline
note R, L, C, and G are per unit length values
1
1
I z R 2 z L 2 z
Vo
Vo
Z
Zo
Io
Io
Y
Vo Z L Z o
L
Vo
Zo Z L
Z Y
1
1
L z R z
2
2
l L e 2 l
I
V z
transmission lines
Telegraphists equations
lets consider a long piece of something like coax
i.e., a wirepair
one wire carrying a timevarying current out and the other carrying the
return current
capacitance
Static solutions for capacitance: parallel plates
Statics: what happens if we add a slab of dielectric that
completely fills the gap between two charged plates?
assuming there are no othe
capacitance
z
Examples: coax
a
z
inner conducting cylindrical
wire, radius a
z
since its a conductor, all charge
is on the outside
y
all the charge is on the inside
x
by symmetry, the field points
Smith charts
LC transmission line summary
Z()
lossless LC transmission line
Z Y
j L jC j LC j j
j L
L
jC
C
Vo
Z
Zo
Io
Y
V z Vo e j z
V
V(z = l, )
V+
z = l
Le
2 j l
ZL
I z I o e j z
V z Vo e
Matching
LC transmission line summary
Z()
lossless LC transmission line
LC
Vo
Z
Zo
Io
Y
j L
L
jC
C
Vo
Zo
Io
dz
I(z = l, )
V z Vo e j z
V
V(z = l, )
V+
z = l
2 j l
ZL
I z I o e j z
V z Vo e j
time varying fields
Summary of electromagnetics: Maxwells equations
summarizing everything we have so far, valid even if things
are changing in time
Faradays law
Amperes law
B
E
t
D v
D
H J
t
Network representations
for a linear system there should be a set of (possibly
frequency dependent) parameters that relate inputs to
outputs
nport representations
impedance matrix (Z parameters)
in
inductance
BiotSavart (bE'Osuvr) Law
magnetic equivalent of Coulombs Law
a short element of a current carrying line contributes to the Bfield
dB
" source
"
o IdL rsource to observation
4 r 2
dB
capacitance
Example: Twin Lead
pair of wires: twin lead
C z
r o
ln h b
h b
2
1
z
+
z
y
y

x
x
h
wire radius b
Dean P. Neikirk
1
EE 363M, Dept. of ECE, Univ. of Texas at Austin
capacitance

1
capacitance
Loss in Tlines
how do we try to put in loss?
go back to generalized Tline equations, and use the Lair
and Ctotal from the previous equations
Z Y
Z
Zo
Y
I z
1
1
R z L z
2
2
1
1
L
capacitance
Planar wire examples (cross sections)
in general these are hard for C or L calculation
again, charge distribution on surfaces is not uniform, with
higher surface charge density where the
capacitance
Example: Twin Lead
another pair of wires: twin lead
this is like telegraph wire in the air
z
y

+
y
x
x
Dean P. Neikirk
1
EE 363M, Dept. of ECE, Univ. of Texas at Austin
capacitance
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University of Texas at Austin
ENS 116
EE363M Microwave and RadioFrequency Engineering
Prof. Andrea Al
Department of Electrical and Computer Engineering
University of Texas at Austin
1 University Stati