Transmission Lines Classification

TEM Lines (Transverse Electric
Magnetic)
quasi_TEM Lines
•
Coaxial
•Microstrip
•
Stripline
•Suspended Stripline
•Inverted Stripline

non-TEM Lines
•
Rectangular waveguide
•
Circular waveguide
•
Fin-line
•Coplanar Lines
TEM Lines
•
They are constituted by two (at least) independent conductors (a voltage
can be applied among them), embedded in a homogeneous medium
•
Electric and magnetic fields of the propagating wave are orthogonal each
other and do not have components in the direction of propagation
•
A TEM transmission line is characterized by:
– A constant characteristic impedance
– A constant phase velocity lower than or equal to the light velocity c:
v 
c
r
with r relative dielectric constant of the medium.
non-TEM Lines
• Are constituted by an single hollow conductor having a section of arbitrary
shape, generally called waveguide
• The electromagnetic field of a non-TEM wave can assume specific
configurations called modes. The modes are characterized by a minimum
frequency (cutoff frequency), below of which they cannot propagate. The
mode with the smallest cutoff frequency in a waveguide is called dominant
mode and it is the most used in the practice
• The modes in non-TEM lines are generally classified in:
• TE modes: only the electric field has no components in the propagation
direction
• TM modes: only the magnetic field has no components in the propagation
direction
Linee non-TEM (cont.)
• non-TEM modes cannot be associated to uniquely defined voltages and
currents (and then also to impedances/admittances).
The power carried by the wave is however always defined
• non-TEM lines are analyzed (in most practical cases) through the reflection
coefficient (and the associated normalized impedance), which does not
require the unique definition of voltages and currents. The characteristic
impedance is then always assumed unitary
quasi-TEM Lines
• These lines consist of two (or more) conductors, surrounded by a non
homogeneous medium
• As a consequence there is at least one component of E or H field in
the direction of wave propagation. To difference of non-TEM modes,
the dominant mode of quasi-TEM lines has the cutoff frequency equal
to zero
• The rigorous study of this kind of lines is rather complex; has been
then developed an approximate (much easier to compute) model,
based on the concept of equivalent TEM line.
• The equivalent TEM model of a quasi-TEM line differs from a ideal
TEM line in the fact that the characteristic impedance and the phase
velocity are a function of the frequency
Attenuation in transmission lines
Power along a matched line (positive z direction):
1 v z
P( z ) 
2
Zc

2
 2
0
1 v

e 2 z
2 Zc
The power lost per unit length is then:
 v 2

P
1
0


z
2
   2 P  z    Pdiss
  2 
e
 2 Zc

z


The definition of  then results:
 
Pdiss
D issipated pow er p.u.l

2P
C arried pow er
Sources of attenuation in transmission lines
• Attenuation J due to losses in the conductors
• Attenuation D due to losses in the medium (dielectric)
The overall attenuation is the sum of the above contributes:
 J D
Attenuation in the conductors J
• It is caused by the finite conductivity of the conductor material employed. In TEM
lines J varies with frequency as the square root of f (skin effect).
• The actual conductivity of a material depends, other that its physical properties,
also on the surface roughness determined by the fabrication process adopted.
Without a suitable processing of the surfaces (polishing, lapping and plating), the
degradation of conductivity due to the surface roughness may reach the 50 % of
the ideal value
Attenuation in the dielectric medium D
•
In a real dielectric medium under sinusoidal excitation, some energy must be
supplied for aligning the elementary dipoles of the material along the electric
field direction. As a consequence the dielectric constant of the medium
become a complex number
•
A parameter called tan is introduced for characterizing dielectric losses,
which represents the ratio between the imaginary and real part of the
dielectric constant. This parameter is practically independent on the
frequency (at microwave frequencies).
•
Expressions of D as function of tan:

tan
0
 tan
d 
0
f c2
d 
1
f
2
(TEM Lines)
( non  TEM Lines)
Recall on characteristic paramenters of
TEM lines
General definition of the characteristic impedance of a TEM line:
P2
V
Zc 

I
 E  dl
  2  1 cost
 Zw
  2  1 cost
 H t ds
t
P1
 Z w  Fz
Zw 
377
cont .
(x,y) e (x,y) represent the lines at constant electric () and magnetic ()
potential defining the contour imposed by the conductors . Note that Et is
tangent to =cost, while Ht is tangent to =cost.
Moreover the curves =cost must be closed (in the real world magnetic
sources do not exist). As a consequence 2 e 1 are computed in the same
point representing the beginning and the end of the integration line.
Attenuation of a TEM line:
J 
Rs
1 Fz
FJ , FJ 
Fz n
2Z w
NOTE: the direction n represents the normal ENTERING the surface
r

Coaxial Line
 (r ,  )  K  ln(r )
 (r ,  )  K  
 r2 
ln  
r1 
 2  1


Fz 
2
1   2
1 Fz
1  Fz Fz 
2  1 r2  1 r1  1 r2  1 r1
 

FJ 



Fz n Fz  r2 r1 
 r2   2 
 r2 
ln  
ln  
 r1 
 r1 
Coaxial Line dimensioning
Given Zc:
 Zc 
Zc 
r2

 exp  2   exp  2  r

r1
Z
377


0 

Zc=50  for r2/r1  2.3 in air
Monomolad propagation up to fmax:
f max 
r2 r1
v
v
 r2 
  r2  r1 
 f max 1  r2 r1
Minimum attenuation assigned the external radius:
FJ 
1 r2 1 r1
ln  r2 r1 
FJ
1  r2 r1
r

1

 0  2  3.6 ( Z c  76 in air)
  r2 r1  r2   r2 r1   r2 
r1
ln  
 r1 
Minimum attenuation assigned fmax:
FJ
r
2  r2 r1  r1 r2

1

 0  2  4.45 ( Z c  97 in air)
  r2 r1  r1  r2   r2 r1  ln  r2 r1 
r1
Other TEM lines
Parallel Plates
Slabline
Stripline
Bi-wires
q1 (a>>t)
quasi-TEM Lines
A quasi-TEM line is obtained when a inhomogeneous medium is used in a TEM
line. As a consequence the electromagnetic field is no more transverse with
respect the direction of propagation.
Strictly speaking in should be no possible in this case to define uniquely the
voltage and current. In the practice the quasi-TEM approximation is
introduced. This consists in assuming an equivalent homogeneous medium
characterized by an effective dielectric constant r,eff, defined as:
 r ,eff  Cm C0
Cm: Capacitance p.u.l. of non-homogenous line
C0: Capacitance p.u.l. of the line with r=1 (air
everywhere)
Note that r,eff is in general a function of the frequency. Also Zc e vf are
then functions of frequency and can be expressed as:
Zc 
377
 r ,eff
Fz ,
vf 
c
 r ,eff
Microstrip
The most important quasi-TEM line in practical applications is the
microstrip. It belongs to the category of planar structures.
Dielettric
t
Conductor
r
w
h
• h substrate thickness
• w strip width
• r relative dielectric constant of substrate
• t metallic thickness
Formulas for microstrip
Quasi-static (t=0):
60
 r , eff
ln(8 h w  0.25 w h )
w h 1
Zc 
120
 r , eff
con:
1.393  w h  0.667 ln  w h  1.444  
 r , eff 
r  1
2

r  1
2 1  10 h w
,
 
1
300
 eff
w h 1
cm sec
Finite metallic thickness:
For taking into account the finite value of t, an effective strip width
(We) is introduced:
w 1.25  t 
 4 w
1
ln



h
 h 
 t



w h  0.159
we h 
w 1.25  t 
 2h  
1  ln 



h
h 
 t 
w h  0.159
A simple model for introducing the frequency variation in r,eff is given
by:
 r ,eff  f GHz    r 
 r   r ,eff  0 
1   hmm Z c 
1.33
 0.43 f
2
GHz
3
 0.009 f GHz

CAD tool for evaluating Zc