This ppt is about planar Transmisson Lines

1. Planar Transmission Lines
1
, ,
   h
w
t

2. 2
Planar Transmission Lines
r
Microstrip
w
h r
Stripline
w
b
w
r
Coplanar Waveguide (CPW)
h
g
g
w
r
Conductor-backed CPW
h
g
g
r
Slotline
h
s
r
Conductor-backed Slotline
h
s

3. 3
Planar Transmission Lines (cont.)
 Stripline is a planar version of coax.
 Coplanar strips (CPS) is a planar version of twin lead.
r
Coplanar Strips (CPS)
h
w
w
s r
Conductor-backed CPS
w
w
h
s

4. Stripline
 Common on circuit boards
 Fabricated with two circuit boards
 Homogenous dielectric
(perfect TEM mode*)
Field structure for TEM mode:
(also TE & TM Modes at high frequency)
TEM mode
4
Electric Field
Magnetic Field
* The mode is a perfect TEM mode if there is no conductor loss.
, ,
  
w b
t

5.  Analysis of stripline is not simple.
 TEM mode fields can be obtained from an electrostatic
analysis (e.g., conformal mapping).
Stripline (cont.)
5
A closed stripline structure is analyzed in the Pozar book by using an
approximate numerical method:
, ,
  
a b

w
b
/ 2
b
a
(to simulate stripline)

6. Conformal Mapping Solution (R. H. T. Bates)
0
30 ( )
( )
K k
Z
K k



sech
2
tanh
2
w
k
b
w
k
b


 
  
 
 
   
 
K = complete elliptic
integral of the first kind
Exact solution (for t = 0):
Stripline (cont.)
6
 
/2
2 2
0
1
1 sin
K k d
k






R. H. T. Bates, “The characteristic impedance of the
shielded slab line,” IEEE Trans. Microwave Theory
and Techniques, vol. 4, pp. 28-33, Jan. 1956.

7. Curve fitting this exact solution:
0
0
ln(4)
4 r
e
b
Z
w b



 
  
 
  
Effective width
2
0 ; 0.35
0.35 ; 0.1 0.35
e
w
b
w w
b b w w
b b




  
 
   
 
 

for
for
Stripline (cont.)
ideal 0
0
1 / 2
2 4 4 r
b
b b
Z
w w w




 
  
 
 
Note: The factor of 1/2 in front is from the parallel
combination of two ideal PPWs.
 
ln 4
0.441


Note:
7
Fringing term

8. Inverting this solution to find w for given Z0:
 
 
0
0
0
0
; 120
0.85 0.6 ; 120
ln(4)
4
r
r
r
X Z
w
b
X Z
X
Z





  

 

   

 
for
for
Stripline (cont.)
8

9. Attenuation
Dielectric Loss:
0
tan tan
2 2
r
d d d
k
k
k

  


  
Stripline (cont.)
(TEM formula)
9
0 c
k k jk   
 
   c j

 

  tan c
d
c






, ,
  
b
w
t
s
R
s
R
s
R

10.    
   
3 0
0
0
0
0
4
(2.7 10 ) ; 120
( )
0.16 ; 120
1 2
1 2 ln
( )
1 1
1 0.414 ln 4
2 2
0.7
2
s r
r
c
s
r
R Z
A Z
b t
R
B Z
Z b
w b t b t
A
b t b t t
b t w
B
w w t
t










  
 

 
 
  
 
  

 
   
      
 
   
 
 
      
 
   
 

 
 
r
wher
fo
for
e
wider strips
narrower strips
2
s
R



Stripline (cont.)
Conductor Loss:
10
Note: We cannot let t  0 when we calculate the conductor loss.
  conductivity of metal

11. 11
Stripline (cont.)
Note about conductor attenuation:
It is necessary to assume a nonzero conductor thickness in order to accurately
calculate the conductor attenuation.
The perturbational method predicts an infinite attenuation if a zero thickness is
assumed.
1
0: 0
sz
t J s
s
  
as
Practical note:
A standard metal thickness for
PCBs is 0.7 [mils] (17.5 [m]),
called “half-ounce copper”.
1 mil = 0.001 inch
0
(0)
2
l
c
P
P
 
1 2
2
0
(0)
2
s
l s
C C z
R
P J d
 
  

 
sz
J onstrip
b
t
w
r

s

12.  Inhomogeneous dielectric
 No TEM mode
TEM mode would require kz = k in each region, but kz must be unique!
 Requires advanced analysis techniques
 Exact fields are hybrid modes (Ez and Hz)
For h /0 << 1, the dominant mode is quasi-TEM.
Microstrip
12
Note: Pozar uses (W, d)
, ,
   h
w
t

13. Microstrip (cont.)
13
Figure from Pozar book
Part of the field lines are in air,
and part of the field lines are inside the substrate.
Note:
The flux lines get more concentrated in the substrate
region as the frequency increases.

14. Equivalent TEM problem:
0
eff
r
k
 

Microstrip (cont.)
14
 The effective permittivity
gives the correct phase
constant.
 The effective strip width
gives the correct Z0.
Equivalent TEM problem
0 0 /
air eff
r
Z Z 

 
0 /
Z L C

since
eff
r

eff
w
0
Z
h
Actual problem
r

0

h
w
2
0
eff
r
k


 
   
 

15. 1 1 1
2 2
1 12
eff r r
r
h
w
 

 
 
 
   
 

 
 
Microstrip (cont.)
15
1
/ 0:
2
eff r
r
w h



 
/ : eff
r r
w h  
  
Effective permittivity:
Limiting cases:
(narrow strip)
(wide strip)
Note:
This formula ignores
“dispersion”, i.e., the fact that
the effective permittivity is
actually a function of
frequency.
r

0

h
w

16. 0 0
60 8
ln ; 1
4
; 1
1.393 0.667ln 1.444
eff
r
eff
r
h w w
w h h
Z w
w w h
h h



  
 
 

 


 


 
 
   
 
 
  
 

for
for
Microstrip (cont.)
16
Characteristic Impedance:
Note:
This formula ignores the fact
that the characteristic
impedance is actually a
function of frequency.

17. Inverting this solution to find w for a given Z0:
 
2
8
; 2
2
2 1 0.61
1 ln(2 1) ln 1 0.39 ; 2
2
A
A
r
r r
e w
e h
w
h w
B B B
h

  


 

   
 

        
 
 
  
 

for
for
0
0
0
1 1 0.11
0.33
60 2 1
2
r r
r r
r
Z
A
B
Z
 
 
 

 
 
  
 
  

where
Microstrip (cont.)
17

18. More accurate formulas for characteristic impedance that account for
dispersion (frequency variation) and conductor thickness:
   
 
 
 
 
0 0
1 0
0
0 1
eff eff
r r
eff eff
r r
f
Z f Z
f
 
 
 

  
 

 
 
     
 
0
0 0
0 / 1.393 0.667ln / 1.444
eff
r
Z
w h w h



 
 
  
 
( / 1)
w h 
2
1 ln
t h
w w
t

 
 
     
 
 
 
18
Microstrip (cont.)
h
t
w
r


19.  
2
1.5
(0)
(0)
1 4
eff
r r
eff eff
r r
f
F
 
  
 

 
 
 

 
 
 
1 1 1 1 /
0
2 2 4.6 /
1 12 /
eff r r r
r
t h
w h
h w
  

 
    
   
 
  
    
 
   
  
 
2
0
4 1 0.5 1 0.868ln 1
r
h w
F
h


 
   
 
    
 
   
 
 
 
 
   
19
Microstrip (cont.)
where
( / 1)
w h 
Note:
   
 
0 :
0
:
eff eff
r r
eff
r r
f
f
f
f
 
 


 

As
As
h
t
w
r


20. Microstrip (cont.)
20
2
0
eff
r
k


 
  
 
“A frequency-dependent
solution for microstrip
transmission lines," E. J.
Denlinger, IEEE Trans.
Microwave Theory and
Techniques, Vol. 19, pp.
30-39, Jan. 1971.
Note:
The flux lines get more
concentrated in the
substrate region as the
frequency increases.
Quasi-TEM region
Frequency variation
(dispersion)
Effective
dielectric
constant
7.0
8.0
9.0
10.0
11.0
12.0
Parameters: r = 11.7, w/h = 0.96, h = 0.317 cm
Frequency (GHz)
r

Note:
The phase velocity is a
function of frequency, which
causes pulse distortion.
p
eff
r
c
v



21. Attenuation
Dielectric loss:
 
 
0
1
tan
2 1
eff
r
r r
d d eff
r r
k 
 
 
 
 

  

 
 
0
s s
c
R R
Z w h


 
“filling factor”
very crude (“parallel-plate”) approximation
Conductor loss:
Microstrip (cont.)
21
(More accurate formulas are given on next slide.)
1: 0
eff
r d
 
 
0
: tan
2
eff r
r r d d
k 
   
 
0
h
Z
w

 

 
 

22. 22
Microstrip (cont.)
More accurate formulas for conductor attenuation:
2
0
1 2
1 1 ln
2 4
s
c
R w h h h t
hZ h w w t h

 
   
    
     
    
 
      
 
 
 
     
 
 
   
 
1
2
2
w
h

 
 
2
0
/
2 2
ln 2 0.94 1 ln
2 0.94
2
s
c
w h
R w w w h h h t
e
w
hZ h h h w w t h
h

 
 
  

   
   
  
   
   
      
     
   
 
   
  
   
   
    

 
 
2
w
h

2
1 ln
t h
w w
t

 
 
     
 
 
 
This is the number e = 2.71828 multiplying the term in parenthesis.
h
t
w
r


23. 23
Microstrip (cont.)
REFERENCES
L. G. Maloratsky, Passive RF and Microwave Integrated Circuits, Elsevier, 2004.
I. Bahl and P. Bhartia, Microwave Solid State Circuit Design, Wiley, 2003.
R. A. Pucel, D. J. Masse, and C. P. Hartwig, “Losses in Microstrip,” IEEE Trans.
Microwave Theory and Techniques, pp. 342-350, June 1968.
R. A. Pucel, D. J. Masse, and C. P. Hartwig, “Corrections to ‘Losses in
Microstrip’,” IEEE Trans. Microwave Theory and Techniques, Dec. 1968, p. 1064.