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 March 4, 2015
1
EE 363M, Dept. of ECE, Univ. of Texas a
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 other connections to the outside world,
the charge on the t
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 variation with respect to the z coordinate, and
furthermore, t
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
additional assumption: the region of space we are in is an
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 z 0
do we recognize a solution to an equation of this f
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. Neikirk 2004, last update March 2, 2015
1
EE 363M, Dept.
EE 363M  Microwave and Radio Frequency
Engineering
Catalog Description:
Design principles in microwave and radio frequency
systems; transmission lines and waveguides; Sparameter representation; impedance matching;
microwave network analysis; microwave de
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
C z
G z
I z z
I z I
V z z
V z V
z
V
Zin(z = l, )
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
one wire is at some time varying voltage relative to t
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
radially outward from the
outer surface of the inner
c
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 j z
I z I o e j z
Vo Z L Z o Z L Z o 1 Z nL 1
L
Vo
Z o
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 z
I z I o e j z
Vo Z L Z o Z L Z o 1 Z nL 1
L
Vo
Z o
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
Gausss law
B 0
plus material properties
D E
r o
B H
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)
input: currents
output: voltage
admittance matrix (Y pa
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
refs
hyperphysics
Fitzpatricks page over in UTPhys
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
r o
ln 1 b
r o
C prll plate
2 b
b
2 b
2

1 b
2
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 z R z
2
2
I
V z
C z
G z
I z z
I z I
V z z
V z V
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 conductors are
closest together
+
+
+
+
+
+
coplanar wa
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
Examples: twin lead
this problem is NOT the same as superp
EE 363M MICROWAVE ENGINEERING
MIDTERM EXAM February 28, 2013
1 hour and 15 minutes
1. A load Z L 50 j100 is connected to a 3 / 8 section of a Z 0 50
transmissionline, which is in turn connected to a Z 0 100 line. Using the
attached Smith chart find (and
E E i ( j n fr,*Sr
I *
i"f I
I V xe =
4
V " V ^ S=  l u f P " $ ?
=
l v  cfw_ _ t ti,^rt e
t
'1q41
Pezerrc
GUKIs cfw_oTI
t
/
lv s =sr
$

g
ULp.
t

k"
V'S + [c'q'=y'
!: f*;
 ?
+
J
3 .r ar=fl
E" = * n iL' + E
2"i* * h'A =/
?*
+iftz4't
= E*cos(uf k
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 Station C0803, Austin, TX 787120240, USA
http:/users.ece.ut