Electromagnetic Fields & Waves

murmerlastUrban and Civil

Nov 16, 2013 (3 years and 8 months ago)

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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 cross-sect
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 dielectric-air 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.