PROBLEM DESCRIPTION
The purpose of this project is to compare signal distortion on a microstrip line and a stripline.
The microstrip and stripline geometries are shown below. The lines extends to infinity in the z
direction. A signal generator is attached at z = 0 with the polarity shown. The signal generator
applies a voltage signal that is
( ) 0 1, 0
0, s
V tt
v t
< < =
otherwise
where 10
0t 1.0 10− = × [s].
FORMULATION AND CALCULATION
Formulate the transmission line voltage v(z,t) at a point z > 0, giving enough details to make the
derivation complete. However, you do not need to re-derive anything in your write-up that is
2
already derived in the class notes. Note that the Fourier transform of the input voltage function
can be obtained in closed form, and you should do this as part of your analysis. Then implement
the numerical calculation of the voltage v(z,t) using any software package that you prefer (e.g.,
Matlab). Please use the CAD formulas given below for all of the microstrip line calculations.
RESULTS
A) Frequency-Domain
1) Plot the characteristic impedance vs. frequency for both the microstrip and the stripline
structures. Ignore loss for this calculation (the characteristic impedance will therefore be a
real number).
2) Plot the conductor attenuation αc, the dielectric attenuation αd, and the total attenuation α
(all in np/m) versus frequency, from 0 to 50 GHz, for both the microstrip and the stripline
structures.
B) Time-Domain
1) Plot the transmission-line voltage v(z,t) versus t for z = 1 cm, 5 cm, 10 cm, 20 cm, and 40
cm. Account for all loss (dielectric and conductor).
2) Repeat the above plots assuming there is no dielectric or conductor loss.
Note: To remove the dielectric loss, set the loss tangent of the dielectric to be zero. To remove
the conductor loss, set the conductivity of the metal to be high enough so that the metal behaves
essentially as a perfect conductor.
Validation
In part (2) of the Time-Domain results, the voltage pulse should propagate at the phase velocity
/ p r v c = ε without changing shape for the stripline. Hence, you know what the correct results
should be for this case. Make sure that your numerical solution gives you the correct results!
(Your results will exhibit the Gibbs phenomenon, however; please see the note below in the
3
discussion of numerical issues.) For extra validation, you can try getting results for the case of a
pulse propagating on a microstrip line as shown in the class notes.
FORMAT GUIDELINES
Your project should consist of a write-up that contains an Abstract, an Introduction section, a
Formulation section, a Results section, and a Conclusions Section. You may also include a
References section and an Appendix, if appropriate. The project should be done on a word
processor, with the equations done in the word processor. You may use any word processor that
you wish. However, it is recommended that you write the report using Microsoft Word along
with MathType to do the equations. (This is how this project document was written.) For a free
30-day trial version of MathType, please visit www.mathtype.com.
The Results section should provide the results that are required, and also provide a thorough
discussion of the results. All plots should be done using software such as Matlab, etc., and should
look professional.
A very significant part of your grade will depend on the accuracy of your results, so you are
encouraged to do as much numerical checking as possible to have confidence in your results.
Also important is the thoroughness of your discussion and your interpretation of the results.
You will also be graded on the neatness and quality of your write-up (including grammar,
format, and appearance), and the quality of your results. Please use a good choice of scales and
professional-looking axis labeling when you plot your results, and make sure that your plots are
easy to read and look nice.
NUMERICAL ISSUES
Numerical experimentation will probably be required to make sure that you have a sufficient
limit of integration ωmax for the integration variable ω in the numerical calculation of the inverse
Fourier transform integral. You may also need to experiment to make sure that you have a
sufficient sample density (number of integration points) when you compute the integral. You
may wish to plot the integrand to help you with this. Note that the numerical integration may
4
require a finer sample density as the distance z increases, since the integrand will oscillate faster
as a function of ω as z increases. It might be helpful to break up the integral from zero to ωmax
into several sub-intervals.
Also, be aware that you will encounter what is known as the “Gibbs phenomenon” since the
input pulse has sharp edges. This means that there will be “ringing” near the edges of the pulse,
even in the ideal case with no loss and no dispersion, because your numerical calculation has a
finite integration limit. To read more about the Gibbs phenomenon, please visit
https://en.wikipedia.org/wiki/Gibbs_phenomenon.
PARAMETERS OF MICROSTRIP LINE
εr = 2.2
tanδ = 0.0009 (loss tangent of substrate)
h = 1.575 [mm] (62 [mils])
w = 4.83 [mm]
t = 0.0175 [mm] (corresponding to 0.5 oz copper /ft2 for the copper cladding)
σ = 3.0 × 107 [S/m] (for both of the copper conductors, the strip and the ground plane)
Note: This substrate corresponds to Rogers 5880 material
(https://rogerscorp.com/en/advanced-connectivity-solutions/rt-duroid-laminates).
Note: The characteristic impedance should be 50 [Ω] at 1 GHz (according to TXLINE).
PARAMETERS OF STRIPLINE
εr = 2.2
tanδ = 0.0009 (loss tangent of substrate)
b = 3.15 [mm] (124 [mils])
5
w = 2.55 [mm]
t = 0.0175 [mm] (corresponding to 0.5 oz copper /ft2 for the copper cladding)
σ = 3.0 × 107 [S/m] (for both of the copper conductors, the strip and the ground plane)
Note: This substrate corresponds to Rogers 5880 material
(https://rogerscorp.com/en/advanced-connectivity-solutions/rt-duroid-laminates).
Note: The characteristic impedance should be 50 [Ω] at 1 GHz (according to TXLINE).
6
MICROSTRIP GEOMETRY
r ε
SIDE VIEW
t
h
z
y
v t s ( ) +
−
Coax
Probe
TOP VIEW
z
w
x
Probe
w r ε
END VIEW
x
y
w t
h
Coax
Probe
7
STRIPLINE GEOMETRY
TOP VIEW
z
w
x
Probe
w r ε
END VIEW
x
y
t w
b
Coax
Probe
SIDE VIEW
r ε
t
h
z
y
v t s ( ) +
−
Coax
Probe
b
8
APPROXIMATE CAD DESIGN FORMULAS FOR MICROSTRIP
Note: In these formulas, εr is the real part of the complex relative permittivity.
Phase Constant
0 ( ) eff
r β ε = k f
where the “effective relative permittivity” is
( )
2
1.5
(0) (0) 1 4
eff
eff eff r r
r r f F
ε ε
ε ε −
− = +
-
( ) ( )
1 1 1 / 1 0
2 2 1 12 / 4.6 /
eff r r r
r
t h
h w w h
ε ε ε
ε
+ − − = + − +
( / 1) w h ≥
2
0
4 1 0.5 1 0.868ln 1 r
h w F
h
ε
λ
= − ++ +
Characteristic Impedance
( ) ( ) ( )
( )
( )
( ) 0 0
1 0
0
0 1
eff eff
r r
eff eff
r r
f Zf Z f
ε ε
ε ε
− = −
where
( ) ( ) ( ) (( ) ) 0
120 0
0 / 1.393 0.667 ln / 1.444 eff
r
Z
w h w h
π
ε = ′ ′ ++ +
( / 1) w h ≥
2 1 ln t h
w w
π t
′ =+ +
9
Conductor Attenuation
Note: In these formulas, Z0 means Z f 0 ( ).
1
2
w
h π
≤ :
2
0
1 4 1 1 ln
2 4
s
c
R w hh wt
hZ h w w t w
π
α
π π
′ = − ++ + ′ ′
1 2
2
w
π h
< ≤ :
2
0
1 2 1 1 ln
2 4
s
c
R w h h ht
hZ h w w t h
α
π π
′ = − ++ − ′ ′
2 w
h
≥ :
( ) 2
0
2 2 / ln 0.94 2 1 ln
2 0.94
2
s
c
R w w w w h hh ht
e
hZ h h h w w w t h
h
π
α π
π π
− ′′ ′ ′ = + + + ++ − ′ ′ ′ +
where
1 Rs σδ =
0
2 δ
ωµ σ = (This is the skin depth, assuming that the metal is nonmagnetic).
2 1 ln t h
w w
π t
′ =+ +
Note: In the term 2πe in the last αc formula, e is the number e (sometimes called “Euler’s
number”), e = 2.718281828.
10
Dielectric Attenuation
0
tan
2 d r q k δ
α ε
= ,
where the “filling factor” q is
( )
( ) 1
1
eff
r r
eff
r r
f
q f
ε ε
ε ε
− = −
.
11
APPROXIMATE CAD DESIGN FORMULAS FOR STRIPLINE
Note: In these formulas, εr is the real part of the complex relative permittivity.
Phase Constant
0 r β ε = k
Characteristic Impedance
0
0
4 ln(4) r e
b Z
w b
η
ε
π
= +
Dielectric Attenuation
0 tan tan
2 2
r
d dd
k k k ε α δδ ′ =≈ ≈ ′′
Conductor Attenuation
[ ] ( )
[ ] ( )
3 0
0
0
0
0
4 (2.7 10 ) ; 120
( )
0.16 ; 120
s r
r
c
s
r
R Z A Z
b t
R B Z
Z b
ε
ε
η
α
ε
− × ≤ Ω − ≈ ≥ Ω
for
for
wider strips
narrower strips
where
2
0 ; 0.35
0.35 ; 0.1 0.35
e
w
w w b
b b w w
b b
≥
= −
− ≤ ≤
for
for
12
1 2 1 2 ln
( )
1 1 1 0.414 ln 4
2 2 0.7
2
w bt bt A
bt bt t
b tw B
w w t t
π
π
π
+ −
=+ + − −
=+ + + + +
and
1 Rs σδ =
0
2 δ
ωµ σ = (This is the skin depth, assuming that the metal is nonmagnetic).
13
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.