Comparison of Dispersion Formulas for Microstrip Lines
Matthew N. O. Sadiku and Sarhan M. Musa
College of Engineering
Prairie View A&M University
Prairie View, TX 77446
Email: sadiku@ieee.org , sarhan_musa@pvamu.edu
Sudarshan Rao Nelatury
School of Engineering and Engineering Technology
Penn State University, Erie
Erie, PA 16563
Email: srn3@psu.edu
Abstract
This paper presents the comparison of ten dispersion
models for microstrip lines and assesses their accuracy.
Accurate and simple formulas for diverse values of the
microscrip line parameters for microwave circuits have
been compared. This comparison is essential for
computer-aided design (CAD) of microstrip lines that
requires accurate and reliable information on the
dispersion behavior of the microstrip.
1. Introduction
Microstrip lines constitute a vital component of
modern microwave circuits. For example, the microstrip
line is a very attractive transmission line for microwave
integrated-circuit applications involving a large number
of identical units and requiring a high density of
packaging. Microstrip lines do not operate in the TEM
modes when the operating wavelength becomes
comparable with the cross-sectional dimension of the
microstrip lines. Their effective dielectric constant
changes with frequency which makes them dispersive.
The dispersion characteristics of microstrip lines have
been studied over the years. There are several closed-
form dispersion formulas for the microstrip transmission
lines in the open literature. The formulas are either
based on some kind of modeling or on curve fitting of
experimental data. Both analytical and experimental
attempts have been made to describe microstrip line
dispersion. These have led to several closed-form
dispersion models in the literature. The objective of this
paper is to compare ten of such models.
2. Dispersion Models
We now present the ten dispersion models that are
compared in this paper. (They are presented in
chronological order.) In these models,
(0)
re
ε
= zero-
frequency value of
re
ε
r
ε = substrate relative dielectric constant
h = substrate height
w= microstrip line width
o
Z = characteristic impedance
o
μ = permeability of free space
c = velocity of light
f = frequency in Hz unless stated otherwise.
At low frequencies the characteristic parameters of
the microstrip can be found using the following
expressions [1]:
For 1
) 0 (
≤
h
w
eff
+ =
h
w
w
h
Z
eff
eff eff r
o
4
) 0 (
) 0 (
8
ln
) 0 (
60
) 0 (
,
ε
(1)
1/2
,
2
1 12
(0)
1 1
(0)
2 2
(0)
0.04 1
eff
r r
r eff
eff
h
w
w
h
ε ε
ε
-
+
+ -
= +
+ -
(2)
3 0-7803-8367-2/04/$20.00 ©2004 IEEE