ECE 5317-6351
Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 10
Waveguides Part 7:
Planar Transmission Lines
1
Stripline
 Common on circuit boards
 Fabricated with two circuit boards
 Homogenous dielectric
(perfect TEM mode)
 , 0 , 
W
b
TEM mode
(also TE & TM Modes)
Field structure for TEM mode:
Electric Field
Magnetic Field
2
Stripline (cont.)
 Analysis of stripline is not simple
 TEM mode fields can be obtained from static analysis
 Closed stripline structure is analyzed in the Pozar book
b
W
 , 0 , 
b/2
a
a >> b
3
Stripline (cont.)
Inductance / unit length
For lossless TEM mode:
Capacitance / unit length
1
vp 



vp
Z0 

1
LC
     LC
L

C
LC
1

C
v pC
We can find Z0 if C is known
4
Stripline (cont.)
From static, conformal mapping solution (S. Cohn)
Exact solution:
30 K (k )
Z0 
K (k ')
K  elliptical integral
 W 
k  sech 

 2b 
 W 
k '  tanh 

 2b 
5
Stripline (cont.)
Curve fitting this exact solution:
 0
Z0  
4 
r


b


ln(4)
 We 
b
Note :

Effective width
Z0
as W

 0.441
Fringing term

0

We W 


2
b
b 
W
 0.35 


b 

Note: Z0PPW  1   b / 2   b 
2  W 
ln  4 
4W
0b
4W  r
; for
W
 0.35
b
; for 0.1 
W
 0.35
b
The factor of 1/2 in front is from the
parallel combination of two PPWs.
6
Stripline (cont.)
Inverting this solution to find W for given Z0:
 X ; for  r Z 0  120  
W 

b 
0.85  0.6  X ; for  r Z 0  120   
X
30 ln(4)

 r Z0 
7
Stripline (cont.)
Attenuation
Dielectric Loss:
 d  k  
k 
k 
k
tan   0 r tan   0 r
2
2
2
 c   c  j c    j
  c 
 
  c 
(TEM formula)


Rs
t
W
b
 , 0 , 
8
Stripline (cont.)
Conductor Loss:
Rs r Z 0

3
(2.7

10
)
A;

30

(
b

t
)

c  
 R 

0.16  s  B

 Z 0b 
for
 r Z 0  120   
;
 r Z 0  120  
Rs 
where
A  1 2
B  1
for

2
W
1  b  t   2b  t 
 
 ln 

(b  t )   b  t   t 
1
t
1  W
   0.414 
ln  4
W
W 2 
t

 2
  0.7t 
2

b



9
Microstrip
 Inhomogeneous dielectric
 ,  ,
W
d
 No TEM mode
Cannot phase match across dielectric interface
 Requires advanced analysis techniques
 Exact fields are hybrid modes (Ez and Hz)
For d /0 << 1  dominate mode is quasi-TEM
10
Microstrip (cont.)
11
Microstrip (cont.)
For the equivalent TEM problem:
vp 
c
 reff
  k0 
Actual problem
eff
r
W
0
r
W
d
d
 reff
Equivalent TEM problem
12
Microstrip (cont.)
Effective permittivity:
 reff

 r 1  r 1 
1



2
2 
d
1

12

W







Limiting cases:
r 1
2
W / d  0:
 reff 
W / d :
 reff   r
(narrow strip)
(wide strip)
13
Microstrip (cont.)
Characteristic Impedance:
Z0 
 60
W
 8d W 
ln

;
for
1



eff
d
 W 4d 
  r

120
;

W
W


 reff   1.393  0.667 ln   1.444  

d

d
for
W
1
d
14
Microstrip (cont.)
Inverting this solution to find W gives Z0:

8e A
W
;
for
2

2A
e

2
d
W 
 
d
 2  B  1  ln(2 B  1)   r  1  ln  B  1  0.39  0.61   ;
 
2 r 
 r 

for
W
2
d
where
A
B
Z0  r  1  r 1 
0.11 

0.33



60
2
r 1
r 
377
2Z 0  r
15
Microstrip (cont.)
Attenuation
Dielectric loss:
d
“filling” factor
   reff  1 
k0  r

tan   effr
2
  r  r  1 
Conductor loss:
c 
Rs
Z 0W
very crude (“parallel-plate”) approximation
(More accurate formulas are given later.)
16
Microstrip (cont.)
More accurate formulas for characteristic impedance
that account for dispersion and conductor thickness:
(W / d  1)
  reff  f   1   reff  0 
Z 0  f   Z 0  0   eff
 eff

0

1


 r
 r  f 
Z0  0 
120
 reff  0  W  / d   1.393  0.667 ln  W  / d   1.444  
W W 
W
r
t 
 2d  
1

ln
 
 
 t 
t
d
17
Microstrip (cont.)
where
 reff

eff
r
 0 

 r   reff (0) 
eff

 f     r (0) 
1.5

1  4F


 r  1   r  1  
2
(W / d  1)
   1   t / d 
 r




2
 2   1  12  d / W    4.6   W / d 
1

d 

 W
F  4    r  1  0.5  1  0.868ln 1 

d


 0 




2




Note:
As
W
r
f :
 reff  f    r
t
d
18
Microstrip (cont.)

r
Frequency variation
r
eff
r

 
 k0 
2
"A frequency-dependent
solution for microstrip
transmission lines," E. J.
Denlinger, IEEE Trans.
Microwave Theory and
Techniques, Vol. 19, pp.
30-39, Jan. 1971.
W
19
Microstrip (cont.)
More accurate formulas for conductor attenuation:
1 W

2
2 d
W
2
d
 R  1
c   s  
 hZ0   2
 W 
 1  

   4d 
2

d
d   2d  t  
1



  
 ln 


  W W   t  d  


 R  W  2 
d
d   2d  t 
W
    W  W  /  d   
 c   s    ln  2 e   0.94    
1



 ln    

W


hZ
d

2
d
h
W

W




  t  d 
 0 
 
 0.94  
2d


2
W W 
W
r
t
t 
 2d  
1  ln   

 
 t 
d
20
Microstrip (cont.)
Note:
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
J sz 
s
t  0:
as
s0
Practical note: A standard thickness
for PCBs is 0.7 [mils] (17.5 [m]),
called “half-ounce copper”.
J sz
s
W
r
1 mil = 0.001 inch
t
d
21
TXLINE
This is a public-domain software for calculating the properties of
some common planar transmission lines.
http://web.awrcorp.com/Usa/Products/Optional-Products/TX-Line/
22
TXLINE (cont.)
TX-Line™ Transmission Line Calculator
TX-Line is a FREE, easy-to-use, Windows-based interactive transmission line
calculator for the analysis and synthesis of transmission line structures. TX-Line
enables users enter either physical characteristics or electrical characteristic for
common transmission medium such as:
Microstrip
Stripline
Coplanar waveguide
Grounded coplanar WG
Slotline
TX-Line runs on Microsoft® Windows® 2000-SP4, XP-SP2, Vista-SP1, Windows® 7
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.
24