Figure 1 : Cross section of the shielded
symmetrical bandline.
y
Discontinuity
angleθ
x
Thickness w
Shield (r
b
)
Dielectric (ε
r1
)
Symmetrical
b
and
(r
o
)
Dielectric (ε
r2
)
Rigorous Analytical Expressions for the Effective Dielectric
Constants of the Shielded Symmetrical Bandline
Nasreddine Benahmed
University of Tlemcen, Algeria
ABSTRACT
This article is a continuity of the reference [1] and it presents a set of accurate closedforms
formulas for the effective dielectric constants of the shielded symmetrical bandline. This formulas
are based on rigorous analysis by finite element method (FEM) [2], method of moment (MoM) [3]
and curves fitting techniques.
The good coherence of the two numerical methods (FEM and MoM) [1] allows to generate
rigorous analytical solutions for a widerange of discontinuity angles and are suitable for all
shielded symmetrical bandlines which have an outerinner conductors radius ratio between 2 and 6.
These expressions can be easily implemented in CAD simulation tools, to design many
components as RF resonators, RF couplers [1], filters, transmission lines,… for wireless
communication and probes for material characterization [4].
INTRODUCTION
The electrical properties of a lossless
shielded symmetrical bandline with a quasi
TEMmode can be described in terms of even
(Z
oe
, ε
effe
) and odd (Z
oo
, ε
effo
) mode impedances
and effective dielectric constants, and its
primary parameters [L] and [C].
A variety of numerical techniques are
available to accurately determine the
characteristic impedance, the effective dielectric
constant and the primary parameters of the
shielded symmetrical bandline. But they are
timeconsuming and too tedious for use in
circuit design, where closedform analytical
models are to be preferred. By applying FEM
and MoM analyses along with curvefitting
strategies, it is possible to develop these closed
form expressions for determining the
characteristic impedance, the effective dielectric
constant and primary parameters of the shielded
symmetrical bandline.
In [1], a set of closedform equations was
developed to determine the characteristic
impedances and the primary inductance and
capacitance matrices (the [L] and [C] matrices,
respectively). In order to complete the study, we
present rigorous analytical expressions for the
effective dielectric constants of the shielded
symmetrical bandline having an outerinner
conductors radius ratio between 2 and 6.
SHIELDED SYMMETRICAL BAND LINE
The line is assumed to be lossless with inner
conductors of radius r
o
, negligible thickness w,
a discontinuity angle θ and an outer shield of
radius r
b
. Dielectric materials with permittivities
ε
r1
and ε
r2
are placed respectively inside the
bands and between the bands and the shield.
NUMERICAL RESULTS
The numerical results for the effective
dielectric constant of the shielded symmetrical
bandline using the FEM and MoM methods are
shown in figures 2 to 4. These results
demonstrate the excellent coherence between
the FEM and MoM methods.
Figure 3 : Effect of the discontinuity angle on the
even mode effective dielectric constant using MoM.
0 20 40 60 80 100 120 140 160 180
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
(
ε
r1
=2.3 and
ε
r2
=1)
r
b
/r
o
=2
r
b
/r
o
=3
r
b
/r
o
=4
r
b
/r
o
=5
r
b
/r
o
=6
(
ε
r2
=2.3 and
ε
r1
=1)
r
b
/r
o
=2
r
b
/r
o
=3
r
b
/r
o
=4
r
b
/r
o
=5
r
b
/r
o
=6
Even mode effective dielectric constant
MoM results
Discontinuity angle (°)
0 20 40 60 80 100 120 140 160 180
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
(
ε
r1
=2.3 and
ε
r2
=1)
r
b
/r
o
=2
r
b
/r
o
=3
r
b
/r
o
=4
r
b
/r
o
=5
r
b
/r
o
=6
(
ε
r2
=2.3 and
ε
r1
=1)
r
b
/r
o
=2
r
b
/r
o
=3
r
b
/r
o
=4
r
b
/r
o
=5
r
b
/r
o
=6
Even mode effective dielectric constant
FEM Results
Discontinuity angle (°)
Figure 2 : Effect of the discontinuity angle on the
even mode effective dielectric constant using
FEM
DERIVATION OF ANALYTICAL
MODELS
1.EVEN MODE EFFECTIVE DIELECTRIC
CONSTANT
The even mode effective dielectric constant
(ε
effe
) of the shielded symmetrical bandline can
be expressed by the relations (1) and (2) for
°
<
<
≤
≤
180062
θ
andr
.
• For
1/
12
≥
rr
ε
ε
(
)
(
)
3.2
011
−
+
=
ba
oeffreffe
ε
ε
ε
(1)
• For
1/
12
<
rr
ε
ε
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−−+= 3.2
1
1
0
121
b
a
oeffrreffe
εεεε
(2)
Where:
26
10978.500126.001474.1 θθ
−
−−=
o
a
12
/
rro
b
ε
ε
=
2
21*1
θθεε bb
effeff
++=
53801.1/)2(
*
01061.03.2
−−
+=
r
eff
eε
2784
2
.1/)2(4
1
00103.0100074.3
−
−
−
−=
r
eb
38
2666
2
1029407.8
1022252.11062897.51011425.4
r
rrb
−
−−−
−
+−−=
ob
rrr/
=
=
2.ODD MODE EFFECTIVE DIELECTRIC
CONSTANT
For
°<<
≤
≤
180062
θ
and
r
the odd
mode effective dielectric constant (ε
effo
) is
expressed by the relations (1) and (2), where:
41037
253
1007707.95340710.3
103888.51053.351371.0
θθ
θθ
−−
−−
−+
−+=
o
a
12
/
rro
b
ε
ε
=
3
3
2
21*1
θθθεε bbb
effeff
+++=
94195.0/)2(
*
07192.065706.1
−−
+=
r
eff
eε
12538.1/)2(4
1
00171.01009284.2
−−
−
+=
r
eb
00823.1/)2(6
2
2475.1105713.1
−
−
−
−−=
r
eb
90595.0/)2(89
3
1037349.1109738.1
−−−
−
+=
r
eb
ob
rrr/
=
=
周攠牥污瑩癥牲潲整睥敮⁴桥畭敲楣慬e
慮搠瑨攠慮慬祴楣慬敳畬瑳牥敳猠瑨慮′┠楮a
睩摥慮来Ⱐ楮摩捡瑩湧⁴桥潯搠慣捵牡捹映瑨攠
灲潰潳敤硰牥獳楯湳潲⁴桥桩敬摥搠
獹sme瑲楣慬慮摬楮攮≥
=
=
=
=
䙩杵牥″›⁅晦散琠潦⁴桥楳捯湴楮畩瑹湧汥渠瑨攠
潤搠o潤攠敦晥捴楶攠摩敬散瑲楣潮獴慮琮o
ε
r1
=2.3
ε
r2
=1
ε
r1
=1
ε
r2
=2.3
20 40 60 80 100 120 140 160 180
1.50
1.55
1.60
1.65
1.70
1.75
1.80
FEM MoM FEM MoM FEM MoM
r
b
/r
o
=2
r
b
/r
o
=2
r
b
/r
o
=3
r
b
/r
o
=3
r
b
/r
o
=4
r
b
/r
o
=4
r
b
/r
o
=5
r
b
/r
o
=5
r
b
/r
o
=6
r
b
/r
o
=6
r
b
/r
o
=2
r
b
/r
o
=2
r
b
/r
o
=3
r
b
/r
o
=3
r
b
/r
o
=4
r
b
/r
o
=4
r
b
/r
o
=5
r
b
/r
o
=5
r
b
/r
o
=6
r
b
/r
o
=6
Odd mode effective dielectric constant
(FEM and MoM results)
Discontinuity angle (°)
CONCLUSION
This article presents a set of accurate closed
form formulas for the dielectric constants (ε
effe
,
ε
effo
) of the even and odd modes of the shielded
symmetrical bandlines.
The expressions obtained from the finite
element method and the moments method, are
valid in a wide range of the discontinuity angle
and the outerinner conductors radius ratio.
REFERENCES
1. N. Ben Ahmed and M. Feham, Analyzing EM
parameters for shielded bandline, Microwaves
& RF, March 2006, pp.8692.
2. N. Ben Ahmed and M. Feham, Finite Element
Analysis of RF couplers with Sliced Coaxial
Cable, Microwave Journal, Vol.2 N°2, 2000,
pp.106120.
3. A.R. Djordjevic, D.Darco, M.C. Goran, T.K.
Sarkan, Circuit Analysis Models for
Multiconductors Transmission Lines, Artech
Housse, 1997.
4. N. Ben Ahmed and M. Feham, Design NMR
probes at high frequencies, Microwaves & RF,
Vol. 41, No. 2, 2002,
pp.77103
.
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