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Comparison of Approximate Formulas for the Capacitance of Microstrip Line 
 
Matthew N. O. Sadiku, Sarhan M. Musa 
College of Engineering 
Prairie View A&M University 
Prairie View, TX 77446 
Email: sadiku@ieee.org, smmusa@pvamu.edu 
and 
Sudarshan R. Nelatury 
School of Engineering 
Penn State University, Erie 
Erie, PA 16563 
Email: srn3@psu.edu 
 
 
 
Abstract 
 
In the development of CAD routines at rf or microwave 
frequencies, closed form models for the capacitance per 
unit length of microstrip interconnects are easily 
incorporated. Numerous formulas have been proposed 
based on analytical, numerical, and empirical approaches. 
When there are a host of formulas found in the literature, 
knowledge of their accuracy against measured values and 
a case study of comparison is all the more vital in the 
design practice. This paper considers 12 such closed form 
models published over the past four decades and makes a 
critical comparison and seeks a way of improving the 
accuracy. While new formulas are proposed that give rise 
to less % error compared to few others taken as reference, 
it is found that composite formulas inherit combined 
goodness from which they are derived. Our case study 
provides some interesting results. 
 
1. Introduction 
 
Printed transmission lines such as microstrip lines are 
widely used because they provide circuits that are compact 
and light in weight. Also, its simple structure and 
ruggedness make it a good substitute for conventional 
coaxial and waveguides in several applications. However, 
the interconnect capacitance between junctions on a chip 
plays a major role in digital circuit performance. At 
frequencies above a few megahertz, the interconnect 
capacitance is heavily responsible for determining the 
characteristic impedance of the traces. 
 
Several attempts have been made to calculate the 
capacitance of microstrip line by rigorous numerical 
analysis. This type of computation is time-consuming 
when implemented in computer-aided design (CAD) 
programs. To overcome this situation, simple formulas 
which are accurate enough for engineering analysis have 
been proposed. Some of these formulas were compared by 
Bogatin [1] in 1988. Since then other models have been 
proposed. The purpose of this work is to improve and 
extend the work in [1]. 
 
This paper compares a variety of published models for 
calculating the capacitance per unit length for microstrip 
geometry shown in Fig. 1. For each model, we define the 
effective relative permittivity 
 
 
eff
o
C
C
ε = (1) 
and the line impedance 
eff
o
o
Z
u C
ε= (2) 
 
where Co = capacitance per length when 1rε = , C = 
capacitance per unit length when filled with the material, 
and uo = speed of light in free space = 3 x 108 m/sec. 
 
 
 
 
 
 
Fig. 1 Microstrip geometry 
 
w
t 
o rε ε h
1-4244-1029-0/07/$25.00 ©2007 IEEE. 427
In what follows, these models are written down in 
chronological order. Comparison is made with the bench 
mark of measurements reported in [1]. 
 
2. Models 
 
We now present the twelve models for calculating the 
capacitance per unit length of the microstrip. They are 
presented in chronological order. The models presented 
here may readily be handled using a pocket calculator. 
They can be incorporated easily in CAD routines. 
 
 
Model 1: Parallel plate [2] 
o r
wC
h
ε ε= (3) 
 
Model 2: Kaupp [3] 
 
 
60 ln 5.98
eff
o
eff
C
hu
w
ε= ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
 (4) 
0.475 0.67
0.8
eff r
eff
e
w w t
ε= +
= + (5) 
 
 
Model 3: Schneider [4,2] 
 
6
860 ln
4 , / 1
2.42 0.44 1 , / 1
eff
o
o eff
h wu
w hC w h
w h h w h
h w w
ε
ε ε
⎧⎪ ⎛ ⎞⎪ +⎜ ⎟⎪ ⎝ ⎠= ≤⎨⎪ ⎡ ⎤⎛ ⎞⎪ + − + − ≥⎢ ⎥⎜ ⎟⎪ ⎝ ⎠⎢ ⎥⎣ ⎦⎩
 
 
 (6) 
where 
 
0.51 1 101
2 2
r r
eff
h
w
ε εε
−+ − ⎛ ⎞= + +⎜ ⎟⎝ ⎠ (7) 
 
 
 
Model 4: Kumar et al. [5] 
 
2
4ln
o eff
wC
hh
t
πε ε
⎡ ⎤⎢ ⎥⎢ ⎥= + ⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
 (8) 
where 
 
11 1 1
2 2 10
r r
eff
w
t
ε εε
−+ − ⎛ ⎞= + +⎜ ⎟⎝ ⎠ (9) 
 
 
 
 
Model 5: Wheeler [6,10] 
 
2
2
4
1 8 8 8ln 1
2
o eff
eff eff eff
C
h h h
w w w
ε ε
π
= ⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪⎢ ⎥+ + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦⎩ ⎭
 
 
 (10) 
 
where 
 
2
2
4ln
1
1.10
eff
t ew w
t
wh
t
π
π
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪= + ⎨ ⎬⎪ ⎪⎡ ⎤⎪ ⎪⎢ ⎥⎛ ⎞⎪ ⎪⎢ ⎥+⎜ ⎟ ⎛ ⎞⎪ ⎪⎝ ⎠ ⎢ ⎥+⎜ ⎟⎪ ⎪⎢ ⎥⎝ ⎠⎣ ⎦⎩ ⎭
 
 (11) 
 
 
0.5
1 1 101
2 2
r r
eff
eff
h
w
ε εε
−⎛ ⎞+ −= + +⎜ ⎟⎜ ⎟⎝ ⎠
 
 (12) 
 
Model 6: Poh et al. [7] 
 
For w/h ≤ 0.6, 
2 2
(1 )
18 1ln 0.041 0.454
16(1 )
o r
r
r
r r
hC
h w ww
w h h
ε ε π
εε ε ε
+= ⎧ ⎫⎡ ⎤−⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + −⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
 
 (13) 
 
and for w/h > 0.6, 
 
( ) ( )
1/ 2
2 21 1 2.230 4.554 4.464 3.89o r r r
r
w h h hC In
h w w w
ε ε ε επε
⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞= − + − − − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎩ ⎭
 (14) 
428
 
Model 7: Sakurai and Tamaru [8] 
 
 
0.222
1.15 2.80o r
w tC
h h
ε ε ⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
 
 (15) 
 
 
Model 8: T. C. Edwards [9] 
 
For w/h < 3.3 
 
1
2
11
0
4 1 1 1 42.78 10 2( 1) 16 2
2 1 2
r
r
r r
h hC In In In
w w
ε πε ε ε ε π
−
−
⎡ ⎤⎧ ⎫ ⎛ ⎞⎛ ⎞−⎪ ⎪⎛ ⎞⎢ ⎥= × + + + − +⎨ ⎬ ⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥+⎝ ⎠ ⎝ ⎠⎝ ⎠⎪ ⎪⎩ ⎭⎣ ⎦
 (16) 
 
For w/h > 3.3 
 
 
11 2
0
2
5.56 10 4 ( /16) 1 1 0.94
2 2 2 2 2
r r r
rr
w In In e e wC In In
h h
ε ε π ε ε π
π π π πεε
− ⎡ ⎤⎛ ⎞× ⎧ ⎫− + ⎛ ⎞= + + + + +⎢ ⎥⎜ ⎟ ⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭⎢ ⎥⎝ ⎠⎣ ⎦
 
 (17) 
 
where e = 2.7182818 
 
 
Model 9: Abuelma’atti [11] 
 
 
( )3 0.765 3.9
2.3553.355 1.108 0.00045
1 0.54 0.0236
wC b b
h b b
ε ⎡ ⎤⎢ ⎥= + − − + +⎢ ⎥⎣ ⎦
 (18) 
where b=h/w, the aspect ratio. 
 
Model 10: Chow and Tang [12] 
 
 
nn
r
n
r
o Wh
W
C
/1
8
2
1
9.0
⎥⎥⎦
⎤
⎢⎢⎣
⎡ ⎟⎠
⎞⎜⎝
⎛ ++⎟⎠
⎞⎜⎝
⎛= πεεε 
 
 (19) 
with n= 1.114. 
 
 
 
 
 
Model 11: A.A. Shapiro [13] 
 
1
0 1.393 0.667 1.444eff w wC In
h h
ε ε
π
−⎡ ⎤⎛ ⎞= + + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
 
 (20) 
 
where 
⎥⎥⎦
⎤
⎢⎢⎣
⎡ ⎟⎠
⎞⎜⎝
⎛ −+⎟⎠
⎞⎜⎝
⎛ +−++=
− 25.0
104.0121
2
1
2
1
h
w
w
hrr
eff
εεε 
 (21) 
 
 
Model 12: Kwok et al [14] 
1/1.081.08
1.08 1( 1)
8 8ln 1
r
o r
w wC
hh h
w
εε π ε
⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎪ ⎪⎛ ⎞ ⎢ ⎥= + + −⎨ ⎬⎜ ⎟ ⎛ ⎞⎝ ⎠ ⎢ ⎥⎪ ⎪+⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭
 (22) 
Bogatin [1] made a comparison of the first six models. 
However, the remaining six models that have been 
proposed over the last eighteen years and are included in 
this paper. In the next section we shall make a comparison 
of these models. 
 
 
3. Comparison of Models 
 
In order to compare the models 1-12 listed in the preceding 
section, we shall take recourse to the same measured values 
used by Bogatin in [1]. For the benefit of the reader, the 
details of the experimental specimens are restated from [1] 
as follows: A series of microstrips were fabricated using a 
sheet of Rogers Corporation Duroid 5880, Teflon printed 
circuit board, 0.125” thick. The manufacturer gives the 
dielectric constant εr in the range 2.178-2.222 with 2.2 as 
the nominal value. The board was clad on one side with 1 
oz copper. On the other side, strips were laid down 
approximately 18” long of widths from 0.015” to 3”. About 
9 choices of widths were made and accordingly 9 
measurements made. The strips were composed of 1-mil-
thick copper with a 2 mil thick layer of adhesive. The 
capacitance of these specimens was measured at 1KHz 
using IET Labs Inc. model IMF – 600 Impedance meter. 
The stray capacitance typically 1.0-2.0pF was measured 
with electrodes very close to the microstrip. This value was 
subtracted from each measurement. 
 
As for the uncertainty in measurements, it was stated that, 
there was less than ±1% in h, ±2% in w and ±1% in C. 
Also, any two models that agree with the data to better than 
3%, are considered equivalent. The capacitance values 
thus tabulated were used in the comparison process. We 
429
have used the norm of the relative error ||E|| as the error 
metric defined as: 
 
∑
=
⎥⎦
⎤⎢⎣
⎡ −=
N
i Meaured
CalculatedMeaured
iC
iCiC
E
1
2
)(
)()(
 
 (23) 
 
 
Figure 2 depicts the capacitance per unit length measured 
in pF per inch for all the 12 models. The measured values 
are shown by dots on each graph. Also the reader may note 
that each subplot in Figure 2 has a vertical bar whose 
height is proportional to the error to get a sense of relative 
goodness of each model. Note that for the model 2 the y 
axis limit goes all the way up to 100. Thus the error bar 
should be interpreted as highest for this model. 
10-1 101
0
4
8
14
#1
10-1 101
0
100
#2
10-1 101
0
4
8
14 #3
10-1 101
0
4
8
14
#4
10-1 101
0
4
8
14
#5
10-1 101
0
4
8
14
#6
10-1 101
0
4
8
14
#7
10-1 101
0
4
8
14
#8
10-1 101
0
4
8
14
#9
10-1 101
0
4
8
14
#10
10-1 101
0
20
40
60
#11
10-1 101
0
4
8
14
width measured in inches
C
ap
ac
ita
nc
e 
in
 p
F 
pe
r i
nc
h 
#12
 Figure 2. Capacitance per unit length measured in pF per inch in each of the 12 models. 
430
#4 #8 #9 #11
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Model number
E
rro
r n
or
m
original
composite
 
Figure 3. Showing the reduction in ||E|| by combining model 3 with other models 4, 8, 9 and 11 
 
 
Table 1 shows the error norm in all these models. As can be 
seen, Schneider’s model has the least error, and that is 
followed by Wheeler’s model. Of all, the Kaupp’s model 
has the greatest error. The reason for this has to do with the 
term in the denominator. Whenever the ratio weff/h equals 
5.98, we find a singularity because the denominator has log 
of unity. This would greatly offset the values and hence we 
get the highest error. 
 
In [1] Bogatin has made an excellent point that it is 
possible to make up a composite model taking εreff based on 
one model and C per unit length based on other model. In 
fact, the model 5 of Wheeler we have shown is one such 
with Schneider’s expression for εreff combined with it. 
 
With this idea in mind, we have tried to use Schneider’s 
expression in few other models viz., 4, 8, 9, and 11. The 
observation is indeed very interesting that the error norm 
has decreased. This is shown in the third column of the 
Table 1. The error norm is also shown by histogram plot in 
Figure 3 wherein we depict pair-wise comparison of 
models 4, 8, 9, & 11 with and without combining with 
model 3. 
 
Model 9 of Abuelma’atti coupled with Schneider’s model 3 
for εreff has the least error. Even Shapiro’s model 11 gained 
significant accuracy by this composite approach. The error 
is reduced from 4.7115 to 0.2707. 
 
 
Table 1. Various models and the error norm in 
each. Note that the model 9 of Abuelma’atti 
combined with model 3 of Schneider has the least 
error in bold face as shown by an arrow. 
Model Error norm 
||E|| 
Error norm 
||E|| (composite 
model) 
# 1 Parallel plate 1.9380 
# 2 Kaupp 13.1240 
# 3 Schneider 0.0480 
# 4 Kumar et al. 0.9632 0.8336 
# 5 Wheeler 0.0656 
# 6 Poh et al. 0.0724 
# 7 Sakurai and Tamaru 0.2982 
# 8 Edwards 0.2275 0.0760 
# 9 Abuelma’atti 0.7152 0.0267 Å 
#10 Chow & Tang 0.9865 
#11 Shapiro 4.7115 0.2707 
#12 Kwok et al. 0.0851 
 
 
 
431
4. Conclusion 
 
Capacitance per unit length of microstrip line specimens 
have been computed and compared with the measured 
values. Having a reliable closed form expression for the 
capacitance per unit length of microstrip line is very vital 
for the development of CAD tools. Several authors 
proposed models for the same. Bogatin made a comparison 
of various formulas for their accuracy against the real 
measurements. This paper is written for the furtherance of 
Bogatin’s work. In agreement with his observation, use of a 
composite model with the help of Schneider’s model has 
improved the accuracy. Other ways of combining the 
models could be examined. 
 
5. References 
 
 [1] E. Bogatin, “Design rules for microstrip capacitance,” IEEE 
Trans. Components, Hybrids, and Manufacturing Technology, 
vol. 11, no. 3, Sept. 1988, pp.253-259 
 
[2] M. N. O. Sadiku, Elements of Electromagnetics. New York: 
Oxford Univ. Press, 4th ed., 2007, pp. 236, 553. 
 
[3] H. R. Knapp, “Characteristics of microstrip transmission 
lines,” IEEE Ttans. Energy Conversion, vol. EC-16, no. 2, April 
1967, p. 185. 
 
[4] M. V. Schneider, “Microstrip lines for microwave integrated 
circuits,” Bell Systems Technical Journal, vol. 48, no. 5, May 
1969, p. 1421. 
 
[5] A. Kumar et al, “A method for the calculation of the 
characteristic impedance of microstrips,” International Journal of 
Electronics, vol. 40, no.1, 1976, p. 45. 
 
[6] H. A. Wheeler, “Transmission line properties of a strip on a 
dielectric sheet on a plane,” IEEE Trans. Microwave Theory and 
Techniques, vol. MTT-25, no. 8, Aug. 1977, p. 631. 
 
 
 
 
 
[7] S. Y. Poh, W. C. Chew, and J. A. Kong, “Approximate 
formulas for line capacitance and characteristic impedance of 
microstrip line,” IEEE Trans. Microwave Theory and Techniques, 
vol. MTT-29, Feb. 1981, p. 135-142; and the erratum in vol. 
MTT-29, no. 10, Oct. 1981, p. 1119. 
 
[8] T. Sakurai and K. Tamaru, “Simple formulas for two and three 
dimensional capacitances,” IEEE Trans. Electron Devices, vol. 
ED-30, no. 2, Feb. 1983, pp. 183-185. 
 
[9] T. C. Edwards, Foundations for Microstrip Circuit Design. 
New York: John Wiley & Sons, 1983, p.45. 
 
[10] E. Bogatin, “A closed form analytical model for the electrical 
properties of microstrip interconnects,” IEEE Trans. Components, 
Hybrids, and Manufacturing Technology, vol. 13, no. 2, June 
1990, pp. 258-266. 
 
[11] M. T. Abuelma’atti, “An improved approximation to the 
microstrip line capacitance,” International Journal of Infrared 
and Millimeter Waves, vol. 13, no. 11, Nov. 1992, pp. 1795-1800. 
 
[12] Y.L.Chow and W.C.Tang, “CAD formulas of integrated 
circuit components by Fuzzy Electromagnetics – Simplified 
formulation by rigorous derivation,” Proc. Antennas & Prop. Soc. 
Int’l. Sympo., 2000, vol.3, 16-21 July 200, pp.1566-1569. 
 
[13] A. A. Shapiro, M. L. Mecartney , and H. P. Lee,”A 
comparison of microstrip models to low temperature co-fired 
ceramic-silver microstrip measurements”, Microelectronics 
Journal, no. 33, Jan. 2002, pp. 443-447. 
 
 [14] S. K. Kwok, K. F. Tsang, and Y. L. Chow, “A novel 
capacitance formula of the microstrip line using synthetic 
asymptote,” Microwave and Optical Technology Letters, vol. 36, 
no. 5, March 2003, pp. 327-330. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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