Electromagnetics
(ENGR 367)
Types of T

lines
and Their Applications
Outline of Lecture
•
Identify Types of T

lines versus Waveguides
•
Detail construction and applications of each type
•
Provide formulas for T

line parameters of each
type that depend on
–
LF or HF operation
–
Lossy or Lossless conditions
•
Show examples of T

line parameter calculation
•
Draw some conclusions
Types of T

lines
•
Coaxial cable
•
Two

wire line
•
Stripline and Microstrip Line
•
Other specialized variations
T

lines versus Waveguides
•
T

lines operate in the TEM mode only
(
T
ransverse
E
lectro
M
agnetic waves)
•
Waveguides carry EM waves that
–
May propagate in the TM or TE modes at
higher frequencies and may perform more
efficiently than T

lines in the microwave range
–
Take a zig

zag path as they reflect off the
conducting boundaries and propagate along
the main axis of the waveguide
Detailed Description of Individual
T

line Types and Their Applications
•
Coaxial Cable: Basic Construction
Radio

grade flexible
RG

59
(Z
0
= 75
)
coaxial cable.
A: outer plastic sheath
B: copper screen
C: inner dielectric insulator
D: copper core
Other standard types have
similar construction.
from http://en.wikipedia.org/wiki/Coax
Coaxial Cable
•
Other aspects of basic construction
–
Stiffness options available on the market
•
Flexible type: has braided sheath
•
Rigid type: has a solid sheath
(Sheath in either case typically made of Cu)
–
Dielectric insulating layer
•
Thickness and permittivity determine
–
Characteristic Impedance (Z
0
)
–
Attenuation (
)
•
May be either solid or perforated
Applications of Coaxial Cable
•
Short runs to connect
–
Home video equipment
–
Ham radio setups (transciever
antenna)
–
Satellite TV (dish Rx
set)
–
Cable modem & VSAT for Internet Access
–
Broadcast radio communication (Tx
ant.)
Applications of Coaxial Cable
•
Long runs to connect
–
Formerly radio and TV networks
(Now replaced by optical and satellite networks)
–
Presently cable TV signals
Specialized Variations of Coax
•
Triaxial Cable (triax): includes a 3
rd
layer
of shielding, insulation and sheathing,
the latter is grounded to further reduce
outside interference
•
Twin

axial cable (twinax): balanced
twisted pair within a cylindrical shield
for shielded and balanced differential
signals
•
Multi

conductor coax
Review Skin Depth
•
A measure of the depth electromagnetic waves
penetrate into a conductor where the amplitude
has decayed by e

1
= 0.368
•
Note that as frequency (f)
increases
, the skin
depth becomes
smaller
and more significant!
1
[m]
f
Coax T

line Geometry
T

line Parameters for Coax
(from Hayt & Buck, 7/e, pp. 483

484)
•
Assuming HF operation such that the skin depth
<< a = radius of inner conductor
–
Lossless approx. (
R
<<
w
L
and
G
<<
w
C
):
–
Modifications for the lossy case:
0
2
ln (magnetostatics) (electrostatic
s)
2 ln(/)
1 60
ln ln
2
ext
ext
r
b
a b a
b b
Z
a a
L C
L
C
ext
0
2
1 1 1
(dielectric); (conductors)
ln(/) 2
( )
( )
diel
c
b a
πδσ a b
j
Z
j
w
w
G R
R L
G C
T

line Parameters for Coax
(from Hayt & Buck, 7/e, p. 485)
•
Modifications for LF operation where the skin
effect is negligible (
>> a): current distributes
uniformly throughout conductor cross

sections
–
Resistance of conductors increases
2 2 2
1 1 1
( )
takes into account the entire crosssect
ions
of the inner and outer conductors
c
a c b
R
T

line Parameters for Coax
(from Hayt & Buck, 7/e, p. 485)
•
Modifications for LF operation (
>> a):
–
Internal inductance of conductors becomes significant
•
For intermediate frequencies (
a):
–
Expressions for parameters become more complicated
–
One can refer to handbook values as needed
,int,int
,int,int 0
(as prev. derived); (,) (more complicate
d)
8 8
(with , as modified for LF)
a b
a b
f b c
j
Z
j
w
w
L L
R L
L L L R L
G C
Example of
Parameter Calculations for Coax
•
Exercise 1 (D14.2a, H&B, 7/e, p. 486)
Given
: a coax T

line with
a
= 4 mm,
b
= 17.5 mm, and
c
= 20 mm. Each conductor has
= 2 x 10
7
S/m, and
the dielectric has
r
= 1,
r
= 3, and
/
w
= 0.025.
Find
:
L
,
C
,
R
,
G
, and Z
0
at 150 MHz.
Solution
: 1
st
find the skin depth; compare to
a
6 7 7
6
1 1
(150 10 ) (4 10 )(2 10 )
9.2 10 m 9.2 m 4 mm
use HF model
f
a
Example of
Parameter Calculations for Coax
•
Exercise 1 (continued)
–
Solution
: next calculate coax T

line parameters
7
ext
12
6 7 3 3
c
d
4 10 17.5
ln ln 295 nH/m
2 2
2 2 (3)(8.85 10 )
113 pF/m
ln(/) ln(17.5/4)
1 1 1 1 1 1
266m/m
2 2 (9.2 10 )(2 10 ) 4 10 17.5 10
2
2 (0.025 )
ln(/) ln(
b
a a
b a
a b
b a b
w
L
C
R
G
6 12
2 (0.025)(2 )(150 10 )(3)(8.85 10 )
2.66 mS/m
/) ln(17.5/4)
a
Example of
Parameter Calculations for Coax
•
Exercise 1 (continued)
–
Solution
: check validity of lossless approx. for Z
0
6 9
ext
ext
6 12
9
ext
0
12
2 (150 10 )(295 10 ) 278 /m
266 m/m
2 (150 10 )(113 10 ) 106 mS/m
2.66 mS/m
295 10
51 (lossless approx.)
113 10
Z
w
w
w
w
L
L R
C
C G
L
C
Two

wire Line
•
Basic Construction
–
Two parallel circular conductors of equal radius and
conductivity enclosed in a plastic insulating material
–
Dielectric insulator
•
Provides mechanical spacing and some rigidity
•
Affects Z
0
and
Applications of Two

wire Line
•
As a lead

in to carry low level signals from
antenna over a short run to a TV or FM Rx
•
Connections in regular telephone networks
•
In the conceptual development of a more
sophisticated waveguide systems
(Fast Neutron Research Facility of CMU
www.fnrf.science.cmu.ac.th/theory/
waveguide/Waveguide%20theory%202.html
)
Parameters of Two

wire Line
(from Hayt & Buck, 7/e, pp. 486

487)
•
For HF operation (
<< a):
–
Lossless approximation
1
ext
1
1
ext
0
1
cosh (/2 );
cosh (/2 )
1
cosh (/2 )
120
cosh (/2 )
r
d a
d a
Z d a
d a
L C
L
C
Parameters of Two

wire Line
(from Hayt & Buck, 7/e, pp. 486

487)
•
For HF operation (
<< a):
–
Modifications for Lossy conditions
•
In LF operation:
must modify above by including
R
and
L
over entire cross

section of conductors as for coax
.
1
.
0
1
and
cosh (/2 )
(with and as above)
diel
cond
a d a
j
Z
j
w
w
R G
R L
R G
G C
Parameters of Two

wire Line
(from Hayt & Buck, 7/e, p. 487)
•
For LF operation (
>> a):
–
Inductance per unit length increases by twice the
internal inductance of a straight round wire
–
Resistance per unit length becomes twice the dc
resistance of a wire of radius
a
and conductivity
c
1
1
cosh
4 2
d
a
L
2
c
2
a
R
Example of Parameter Calculation
for Two

wire Line
•
Exercise 2 (D14.3, H&B, 7/e, p. 487)
Given
: a two

wire T

line with conductors each of radius
0.8 mm and conductivity 3 x 10
7
S/m, separated by
0.8 cm in a dielectric where
r
= 2.5,
r
= 1, and
d
= 4x10

9
S/m
Find
:
,
C
,
G
,
L
&
R
at 60 Hz
Solution
: validate LF model by comparing
to
a
7 7
c
1 1
11.9 mm
(60) (4 10 )(3 10 )
0.8 mm so LF model applies
f
a
Example of Parameter Calculation
for Two

wire Line
•
Exercise 2 (D14.3, H&B, 7/e, p. 487)
Solution
: calculate the LF two

wire T

line circuit param’s
1 1
d
1 1
1 1
2 4 2 7
c
(2.5)(8.85 pF/m)
30 pF/m
cosh (/2 ) cosh (8/1.6)
(4 nS/m)
5.5 nS/m
cosh (/2 ) cosh (8/1.6)
1 (400 nH/m) 1
= cosh ( ) cosh ( ) 1.0 H/m
4 2 4 2
2 2
33
(8 10 ) (3 10 )
d a
d a
d d
a a
a
C
G
L
R
m/m
Striplines and Microstrip Lines
•
Basic Construction: Stripline
–
Single or double track strip of Cu imbedded in
dielectric material sandwiched between
conducting ground planes on top
and
bottom
Striplines and Microstrip Lines
•
Basic Construction: Microstrip
–
Single or double track strip of Cu on top of a
dielectric substrate material above a single
conducting ground plane
•
In practice, superstrate material may be
other than air
Striplines and Microstrip Lines
•
Basic Construction: dielectric material
For HF applications, special substrate and
superstrate (if present) materials must be
used with high uniformity and low loss
tangent (tan
=
’/
w
’) such as Rogers RT
Duroid or FR4 as manufactured for this
application (common PCB will not work!)
Applications of
Stripline & Microstrip Line
•
Trace connections between devices in PCB
microelectronic circuits
•
T

line connections between HF devices easily
integrated with surface mount, distributed
elements and microstrip antennas.
•
HF communication systems and devices with
compact, flat, lightweight, constraints and short
run requirements such as cell phones, portable
PCs and and other wireless mobile devices
Parameters for Microstrip Line
(Single Track)
•
If the strip width is large (w >> d), then
the structure acts like a parallel plate line
where for the low loss case
0
377
r
d d
Z
w w
Parameters for Microstrip Line
(Single Track)
•
If w
d or w < d (as typical for microstrip) then a quasi
TEM mode may be assumed to account for the
propagation of waves through the two different materials
(e.g., air or superstrate and substrate dielectrics)
•
At low frequencies (f < 1.5 GHz) assuming negligible
losses over a short run the propagation velocity is
p0
ext 0 0 0
pd
0
0
p
ext r,eff
1 1
(in air only)
1
(in dielectric only)
1
(in dielectricair medium)
r
v c
c
v
c
v c
L C
C
C
L C
Parameters for Microstrip Line
(Single Track)
•
Definition of Effective Dielectric Constant (
r,eff
):
acts as a weighted average of the air (or superstrate)
and substrate dielectric constants with a proportion
determined by the field filling factor (q)
2
r,eff r,eff
0 p
r,eff
1
where ( 1)
2
or 1 ( 1) where 0.5 1
and as 1;0.5
(parallel plate case; simple average
case)
r r
r
c
v
q q
w w
q q
d d
C
C
Parameters for Microstrip Line
(Single Track)
•
Empirical Formulas for
r,eff
assuming w/d > 1.3
–
in terms of w/d for application to the dimensions of a
pre

fabricated line
–
In terms of
r
and Z
0
for finding the necessary
r,eff
based on a desired Z
0
.555
r r
r,eff
1 1
1 10
2 2
d
w
1
r,eff r r r 10 0
[0.96 (0.109 0.004 )(log (10 ) 1)]
Z
Parameters for Microstrip Line
(Single Track)
•
Characteristic Impedance (Z
0
):
–
Based on the air

filled equivalent microstrip line
–
For w/d < 3.3
air
air
0
0 0
r,eff
where may be found
by curvilinear or other numerical method
s
Z
Z Z
air 2
0
60ln 4(/) 16(/) 2
Z d w d w
Example of Parameter Calculation
for Microstrip Line
•
Exercise 3 (D14.4, H&B, 7/e, p. )
Given
: a 2 mm wide microstrip line is fabricated on a
1 mm thick substrate of lithium niobate (
r
= 4.8).
Find
:
r,eff
, Z
0
and v
p
Solution
:
since w/d = 2 > 1.3:
.555
.555
r r
r,eff
air 2
0
2
air
0
0
r,eff
1 1
5.8 3.8 1
1 10 1 10 3.6
2 2 2 2 2
60ln 4(/) 16(/) 2
60ln 4(1/2) 16(1/2) 2 90
90
Z = 47
3.6
d
w
Z d w d w
Z
Conclusions
•
Types of T

lines such as coax, two

wire,
microstrip lines and other variations have
–
relative advantages in certain applications
–
a unique construction that determines their
circuit parameters
•
Circuit parameters of T

lines depend on LF
or HF operation as indicated by skin depth
versus conductor size; parameters may be
found by calculation or in a handbook
Conclusions
•
While coax has shielding to minimize
interference in low signal applications, it
may be too large and bulky for PCB and
microelectronic circuit applications
•
While microstrip lines are easy to fabricate
and compatible with microelectronic
systems, they are analytically complex
requiring numerical methods for accuracy
Conclusions
•
For higher frequency applications,
waveguides (including rectangular or
cylindrical “pipes,” parallel plates,
dielectric slabs and optical fiber types)
become more efficient than T

lines
•
The analysis of waveguides becomes a
more advanced task with fully EM field
wave equations
References and Other Resources
•
Hayt & Buck,
Engineering Electromagnetics
,
7/e, McGraw Hill: New York, 2006.
•
Kraus & Fleisch,
Electromagnetics with
Applications
, 5/e, McGraw Hill: New York,
1999.
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