Design Equations (from Gupta) ....................................................................................... 2
Characteristic Impedance and Effective Dielectric Constant ................................... 2
Effect of Strip Thickness .............................................................................................. 4
Effect of Enclosure ........................................................................................................ 5
Effect of Dispersion ....................................................................................................... 7
Losses ............................................................................................................................. 8
Quality Factor-Q ......................................................................................................... 11
Frequency Range of Operation.................................................................................. 14
Design Equations (from Gupta)
Closed-form expressions are necessary for the optimization and computer-aideddesign of microstrip circuits. A complete set design equations for a microstrip is
presented in this section. This includes closed-form expressions for the characteristic
impedance and effective dielectric constant and their variation with metal strip thickness,
enclosure size, and dispersion. Expressions for microstrip loss and quality factor Q are
also described.
Characteristic Impedance and Effective Dielectric Constant
Closed form expressions for Z0m and re have been reported by Wheeler [38],
Schneider [19], and Hammerstand [39]. Wheeler and hammerstand have also given
synthesis expressions for Z0m. The closed–form expressions based on the works of
Wheeler and Schneider are given [39] as
Z0 m 
Z0 m

W
 8h
ln   0.25 
h 
2  re  W
W

 1

 h


  W

W


 1.393  0.667 ln   1.444 

 h

  re h

(A.S1.1.a)
1
W / h  1
(A.S1.1.b)
where   120 
 re 
r  1 r 1

F(W / h)
2
2
1  12h / W 1 / 2  0.041  W / h 2
F( W / h )  
1 / 2
 1  12h / W 
(A.S1.2)
W / h  1
W / h  1
Hammerstand noted [40] that the maximum relative error in re and Z0m is less
than 1 percent .The expressions for W/h in terms of Z0m and re are as follows .For
Z0m  re  89.91 ,that is A>1.52
W/h 
8 exp( A)
exp( 2A)  2
(A.S1.3.a)
for Z0m  re  89.91 , that is, A  1.52
W/h 
r 1 
2
0.61 
ln(
B

1
)

0
.
39

B  1  ln( 2B  1) 



2 r 
 r 
(A.S1.3.b)
where
1/ 2
Z    1
A  0m  r

60  2 
B

r 1 
0.11
0.23 

r  1 
r 
60 2
Z 0m  r
These expressions also provide an accuracy better than one percent.
A more accurate expression for the characteristic impedance Z a0 m of a microstrip
for t = 0 and r = 1 is given by [41]
Z a0 m
2
  f (u )
 2 

ln
 1  
2  u
u 


(A.S1.4)
where
  30.666  0.7528
f (u )  6  2  6 exp  


  u 

(A.S1.5)
and u = W/ h and   120  . The accuracy of this expression is better than 0.01
percent for u  1 and 0.03 percent for u  1000. The effective dielectric constant re may
be expressed as
 re
  1  r  1  10 
 r

1  
2
2 
u
a ( u ) b (r )
(A.S1.6)
  u 3 
1  u 4  (u / 52) 2 
1

a (u )  1  ln  4
ln 1  
 
49  u  0.432  18.7   18.1  
   0.9 

b r   0.564 r
 r  3 
(A.S1.7.a)
0.053
(A.S1.7.b)
The accuracy of this model is better than 0.2 percent for  r  128 and 0.01  u 100.
Finally, the characteristic impedance is
Z0 m 
Za0 m
 re
(A.S1.8)
The results discussed above are based on the assumption that the thickness of the strip
conductor is negligible. But, in practice, the strip has a finite thickness t that affects the
characteristics.
Effect of Strip Thickness
The effect of strip thickness on Z0m and re of microstrip lines has been reported by a
number of investigators [19,31, 38, 41-54]. Simple and accurate formulas for Z0m and re
with finite strip thickness are [31]
Z 0m 

2  re
 8h
W 
ln 
 0.25 e 
h 
 We
W / h  1
(A.S1.9.a)
Z 0m
  We
W


 1.393  0.667 ln  e  1.444 

 re  h
 h

1
W / h  1
(A.S1.9.b)
where
We W 1.25 t 
4W 


1  ln

h
h
 h
t 
We W 1.25 t 
2h 


1  ln

h
h
 h
t 
 re 
r  1 r  1

F( W / h )  C
2
2
( W / h  1 / 2)
( W / h  1 / 2)
(A.S1.10.a)
(A.S1.10.b)
(A.S1.11)
in which
C
r  1 t / h
4.6 W / h
(A.S1.12)
It can be observed that the effect of thickness on Z0m and re is insignificant for small
values of t/h. This agrees with the experimental results reported in [42] for t/h  0.005, 2
 r 10, and W/h  0.1. However, the effect of strip thickness is significant on conductor
loss in the microstrip line.
Effect of Enclosure
Most microstrip circuit applications require a metallic enclosure for hermetic
sealing, mechanical strength, electromagnetic shielding, mounting connectors, and ease
of handling. The effect of the top cover alone [55-57] as well as of the top cover and side
walls [51, 58] has been reported in the literature. Both the top cover and side walls tend
to lower impedance and effective dielectric constant. This is because the fringing flux
lines are prematurely terminated on the enclosure walls. This increases the electric flux in
air. The closed-form equations for a microstrip with top cover (without side walls) are
obtained as [56, 57]
Za0m  Za0  Za0m
(A.S1.13.a)
where Z 0a is the characteristic impedance with infinite shield height and superscript a
designates air as dielectric. The decrease in impedance Z a0m is given by
P
Z a0 m  
P  Q
for W / h  1
for W / h  1
(A.S1.13.b)


h 

P  2701  tanh  0.28  1.2

h



 0.48 W / h  1 

Q  1  tanh 1
2


(
1

h
/
h
)


and h is the height of the top cover above the strip conductor. In this expression, the
distance between the ground plane and the top cover is h + h . The effective dielectric
constant is calculated from the concept of the filling factor q as
 re 
r  1
 1
q r
2
2
(A.S1.14.a)
q  q  q T qc
(A.S1.14.b)
where
 10 
q   q T  1  
u

qT 
a ( u ) b( r )
2 ln 2 t
 W/h h
h
h

q c  tanh 1.043  0.121  1.164 
h
h 

and a(u) and b(r) are obtained from (A.S1.7).
Using the preceding equations, the characteristic impedance of the shielded
microstrip can be calculated from Z 0m  Z a0m /  re .
For the range of parameters 1  r  30, 0.05  W/h  20 , t/h  0.1 and 1< h  /h
<  the maximum error in Z0m and re is found to be less than 1 percent. When h /h  1,
the effect of the top cover on the microstrip characteristics becomes negligible.
Effect of Dispersion
The effect of frequency (dispersion) on re has been described accurately by the
dispersion models given by Getsinger [59], Edwards and Owens [60], Kirschning and
Jansen [61], and Kobayashi [62]. The effect of frequency on Z0m has been described by
several investigators [3, 29, 30, 41, 63]. The accurate expressions of Hammerstad and
Jensen [ 41] for Z0m and Kobayashi [62] for re(f) are
Z 0m f   Z 0m
 re f   1  re
 re  1  re f 
 re f    r 
 r   re
(A.S1.15)
(A.S1.16)
1  f / f 50 m
where
f 50   r 
f k ,T M0

f k ,T M0

0.75  0.75  (0.332 / 1r.73 ) W / h

 re  1
c tan 1   r

 r   re

 r
2h  r   re
m  m0mc




m0  1 


1

 0.32
1 W / h
1  W / h 
1
3

  0.45f 
1.4 



1 
0.15  0.235 exp 
mc   1  W / h 
 f 50 


1

for W / h  0.7
for
W / h  0.7
Z0m ,  re are the quasi-static values obtained earlier, and c is the velocity of light.
Losses
Closed-form expressions for total loss have been reported in the literature [18, 19,
64]. An expression for loss, derived using (A.S1.9) to (A.S1.12), may be written as
T  c  d
(A.S1.17)
The two components  c and d are given by

R s 32  ( We / h ) 2
1
.
38
A

hZ 0 m 32  ( We / h ) 2

c  
6.1  10 5 A R s Z 0 m  re f   W / h  0.667 We / h 
 e


h
We / h  1.444 

dB / unit length
dB / unit length
W / h  1
(A.S1.18)
and
 re (f )  1

4.34
 re (f ) ( r  1)

d  
 27.3  r  re (f )  1 tan 

 r  1  re (f )  0
dB / unit length
(A.S1.19)
dB / unit length
R s  f 0  c ;  c = resistivity of the strip conductor
   0  r tan  = conductivity of the dielectric substrate
W / h  1
and

 h
B
2W

1 

W / h 

2 

1 

W / h 

2 

The dielectric loss is normally very small compared with the conductor loss for dielectric
substrates. The dielectric loss in silicon substrates (used for monolithic MICs), however,
is usually of the same order or even larger than the conductor loss. This is because of the
lower resistivity available in silicon wafers. However, higher resistivity can be obtained
in GaAs, and therefore the dielectric loss is lower for this material. Values of conductor
and dielectric losses per unit length for 50- microstrip lines on various substrates
(dielectric as well as semiconductor) are plotted in Figure A.S1.1 as functions of
frequency. At a given frequency the total loss can be obtained by adding the two values.
Figure A.S1.2 compares the total loss for 50  microstrip lines on RT / duroid,
quartz, and alumina substrates. At the same h/ 0 ratio, high dielectric constant substrates
result in greater loss.
Microstrip loss data presented above require that the conductor thickness be greater than
about four times the skin depth. The microstrip conductor loss calculation for any
metalization thickness has been reported by Lee and ltoh [65] based on the
phenomenological loss equivalence method and by Faraji-Dana and Chow [66, 67] based
on ac resistance. Faraji-Dana and Chow's calculated conductor loss data for thick
conductors agree very well with the calculated results obtained using expressions
reported by Pucel et al. [18]. Aksun and Morkoc [68] have reported dielectric loss results
for microstrip substrates consisting of different GaAs layer thicknesses on an Si substrate.
Figure A.S1.1 Conductor and dielectric losses as a function of frequency for microstrip
lines on various substrates.
Figure A.S1.2 Total loss versus substrate thickness (h) for RT/Duroid (  r  2.32) , Quartz
(  r  3.8) , and alumina (  r  10.0) at various frequencies
Quality Factor-Q
The quality factor, Q, of a microstrip can be related to the total loss in the line by
[69]
QT 

2 T
(A.S1.20)
where QT is the total Q of the resonator ( quarter wavelength) ,  T is the total loss in the
resonator, and   2 /  m . When losses in a resonant line are considered, another loss
factor,  T due to radiation at the open-end discontinuities must also be taken into
account [69, 70] .The corresponding radiation Q-factor is given by [69]
QT 
Z 0m
480h /  0 2 R
(A.S1.21)
where
R
 re (f )  1  re (f )  12 
  re (f )  1


ln


3/ 2
 re (f )
2 re (f )

  re (f )  1

(A.S1.22)
Note that the effect of dispersion is considered, as described by (A.S1.16).
The total Q of the resonator can be expressed by
1
1
1
1



QT Qc Qd Qr
(A.S1.23)
Here, Qc, Qd, and Qr, are the quality factors corresponding to conductor, dielectric, and
radiation losses, respectively. Finally, the circuit quality factor, Q0, is defined as
    d 
1
1
1


 0 c
Q0 Qc Qd
  re (f )
(A.S1.24)
The variation with frequency of Q0, Qr and QT for a quarter-wave resonator on GaAs,
alumina, and quartz substrates is shown in Figure A.S1.3. A quarter-wave 50  resonator
on a 25-mil-thick alumina substrate has a Q0 of about 240 at 2.0 GHz and 550 at 10.0
GHz, whereas QT is 230 at 2.0 GHz and nearly 160 at 10.0 GHz. This is due to the fact
that the radiation losses are higher than conductor and dielectric losses at higher
frequencies. On the other hand, a quarter-wave 50  resonator on a 10-mi1 GaAs
substrate has Q0 of about 82 at 2.0 GHz and 160 at 10.0 GHz, whereas QT is 82 at 2.0
GHz and nearly 145 at 10.0 GHz. This is explained by smaller radiation losses for thin
substrates. Thus, the commonly accepted rule that thick substrates should be used for
high Q circuits does not apply to microstrip lines because of high radiation losses
incurred under this condition.
The variation, with substrate thickness, of the total Q for half-wave resonators on
RT/duroid, quartz, and alumina is shown in Figure A.S1.4 for f = 30 GHz,50 GHz, and
100 GHz. For a given frequency, there is an optimum substrate thickness at which the Q
is maximum. This optimum value of h decreases with increasing frequency and
decreasing dielectric constant value, mainly because of radiation.
Figure A.S1.3 Variation of Q-factors with frequency for quarterwave microstrip
resonators on Quartz, Alumina, and GaAs substrates.
Figure A.S1.4 Total Q for a half-wave resonator on RT/Duroid ( r = 2.32), Quartz (r =
3.8) and Alumina (r = 10.0) versus substrate thickness.
Frequency Range of Operation
The maximum frequency of operation of a microstrip is limited due to several
factors such as excitation of spurious modes, higher losses, pronounced discontinuity
effects, low Q due to radiation from discontinuities, effect of dispersion on pulse
distortion, tight fabrication tolerances, handling fragility, and, of course, technological
processes. The frequency at which significant coupling occurs between the quasi-TEM
mode and the lowest order surface wave spurious mode is given below [72] .
fT 
150
2
tan 1  r 
h  r  1
(A.S1.25)
where fT is in gigahertz and h is in millimeters. Thus the maximum thickness of the
quartz substrate (r  3.8) for microstrip circuits designed at 100 GHz is less than 0.5
mm.
The excitation of higher order modes in a micros trip can be avoided by operating
it below the cut-ff frequency of the first higher order mode, which is given approximately
by [72]
fc 
300
 r (2W  0.8h )
(A.S1.26)
where fc is in gigahertz and W and h are in millimeters. This limitation is mostly
applicable to low impedance lines that have wide microstrip conductors and/or thick
substrates commonly used for microstrip patch antenna elements.
An infinitely long microstrip does not radiate as there is no coupling between the
dominant mode and the higher order radiating modes. However, whenever there is some
discontinuity in the line, namely an open end, slit, step in width, bend, or a gap, higher
order radiating modes are excited, which depend upon the substrate thickness and
frequency of operation. The operating frequency at which the radiation becomes
significant may be calculated from the radiation Q factor of a /2 resonator,
approximately given by
Qr 
3 r Z 0 m 20
32h 2
(A.S1.27)
where   120 . Thus for thicker substrates, where Q  Qr, the variation of Q is
proportional to 1/(fh)2. For example, a 50- resonator on a quartz substrate has a Qr 
0.4252  104/ (fh)2 where f is in gigahertz and h is in millimeters. At 100 GHz, the
substrate thickness is less than 0.065 mm for Qr, greater than 100. A substrate thickness
of this order results in attenuation on the order of 1 dB/cm. Thus not only do thin
substrates give rise to high losses, but they are also difficult to handle and result in
narrow conducting strips. Fabrication tolerances and technological processes such as
photoetching limit the minimum strip width and the spacing between two adjacent strips
in the case of coupled lines. High impedance lines of about 120  require strip widths on
the order of 0.02 mm on a 0.065-mm- thick quartz substrate, thereby also setting a limit
on the frequency of operation of microstrip lines because of low radiation Qr.
On a microstrip, phase and group velocities are frequency dependent. Therefore, a
digital pulse signal is distorted due to different velocities of the signals frequency
components. To minimize distortion, the dispersion effects must be minimized by
keeping the fh product as small as possible.