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Circuit Model of Multilayer Microstrip Step Discontinuity Using Single-LayerReduction FormulationA. K. Verma a; Himanshu Singh a; Y. K. Awasthi a
a Microwave Research Laboratory, Department of Electronic Science, University of Delhi, South Campus,New Delhi, India
Online Publication Date: 01 August 2009
To cite this Article Verma, A. K., Singh, Himanshu and Awasthi, Y. K.(2009)'Circuit Model of Multilayer Microstrip Step DiscontinuityUsing Single-Layer Reduction Formulation',Electromagnetics,29:6,483 — 498
To link to this Article: DOI: 10.1080/02726340903098555
URL: http://dx.doi.org/10.1080/02726340903098555
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Electromagnetics, 29:483–498, 2009
Copyright © Taylor & Francis Group, LLC
ISSN: 0272-6343 print/1532-527X online
DOI: 10.1080/02726340903098555
Circuit Model of Multilayer Microstrip Step
Discontinuity Using Single-Layer
Reduction Formulation
A. K. VERMA,1 HIMANSHU SINGH,1 and Y. K. AWASTHI1
1Microwave Research Laboratory, Department of Electronic Science,
University of Delhi, South Campus, New Delhi, India
Abstract New closed-form models for the microstrip step discontinuity to compute
shunt capacitance (Cp ) and series inductance (Ls ) is reported for the substrate 2:3 �"r � 40:0 or more. The model is extended to the multilayer (composite and suspended
substrate) microstrip step discontinuity. The average deviation for normalized Cp is5%, and normalized Ls are 2.9 against the results extracted from Sonnet. For the
multilayer step discontinuity, the average deviation in the present model for Cp is5.2%. The method of moment analysis gives an average deviation of 13.53% for Cp
and 5.3% for Ls against the results of Sonnet. Comparison with Sonnet indicates thatthe static model for simple and multilayer case is valid for h=� � 0:42.
Keywords microstrip step, step discontinuity, equivalent circuit model, closed-formmodel
1. Introduction
The symmetrical step discontinuity, shown in Figure 1(a), is a popular structure in
single- and multilayer microwave integrated circuit (MIC) and monolithic MIC (MMIC)
technologies. At the lower end of the microwave, the symmetrical step discontinu-
ity is normally modeled as the equivalent T-network shown in Figure 1(b), although
the S -parameter description is also popular (Benedek & Silvester, 1972; Norbert &
Jansen, 1986; Gupta & Gopinath, 1977; Gopinath et al., 1976; Thomson & Gopinath,
1975; Gupta et al., 1987; Hoffmann, 1987). However, the circuit model is more useful
in accommodating the step discontinuity at the design stage of the circuits. Gupta et al.
(1987) have provided the design-oriented closed-form expressions to compute both the
shunt capacitance Cp and the series inductance Ls . We have extended the accuracy and
parametric range of the closed-form model for the shunt capacitance Cp (Verma & Singh,
2005). However, a closed-form model applicable to the wide range of parameters for the
series inductance Ls is not available. Also, there is no closed-form model to compute
Cp for the microstrip step discontinuity on the suspended and composite layer substrate.
This work is divided into two parts. First, we summarize more accurate expres-
sions for the shunt discontinuity capacitance Cp that are valid over the wide range of
Received 18 November 2008; accepted 7 February 2009.Address correspondence to Prof. A. K. Verma, Microwave Research Laboratory, Department
of Electronic Science, University of Delhi, South Campus, New Delhi, 110021, India. E-mail:anandv48@bol.net.in
483
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484 A. K. Verma et al.
Figure 1. Microstrip step discontinuity: (a) step discontinuity in microstrip and (b) equivalent
circuit of step discontinuity.
parameters, and we also report the new closed-form expressions to compute series step
discontinuity inductance Ls . Next, using the single-layer reduction (SLR) formulation
(Verma & Sadr, 1992), we extend the model for Cp to the step discontinuity in the
multilayer microstrip line. The SLR has already been used to compute the open-end
discontinuity in the multilayer microstrip line (Verma & Sadr, 1994). Accuracy of the
models presented is compared against results on Cp and Ls extracted from the commer-
cially available electromagnetic (EM) software (Sonnet Software Ltd., 1986–2009). Once
Cp and Ls are obtained for the step discontinuity of multilayer microstrip, its equivalent
T-circuit is constructed.
Finally, the S -parameter responses of the equivalent circuit using circuit components
of the present model are compared against the responses obtained from Sonnet. For this
purpose, the circuit simulator Microwave Office is used (AWR Inc., 2004). The good
agreement of the responses for h=� � 0:42 validates the present models. The present
models are useful for the analysis and synthesis of the microwave filter, matching network,
etc. on the simple multilayer microstrip line.
2. Step Discontinuity Models for Single-LayerMicrostrip Line
The symmetrical microstrip step discontinuity and its T-equivalent circuit are shown in
Figure 1. First, models are presented for the shunt step discontinuity capacitance Cp
and the series step discontinuity inductance Ls that are applicable to the single-layer
substrate. In the next section, it is extended to the multilayer substrate.
Shunt Step Discontinuity Capacitance Cp
It is due to the end fringe capacitance (Cf ) of the wide width (W1) microstrip #1. We
assume that the fringe capacitance per unit length (p.u.l.) at the junction is same as that of
the fringe capacitance p.u.l. Cf along one edge of microstrip #1. Thus, the approximate
value of Cp is obtained from
Cp D Cf .w1 � w2/: (1)
The above expression provides a framework for more accurate empirical modeling (Hoff-
mann, 1987). The corrected shunt step discontinuity capacitance Cp is obtained by
introducing an empirical improvement factor (IF) that depends on relative permittivity
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Multilayer Microstrip Step Discontinuity 485
and width/height ratio of the structure
Cp.corr/ D�
Cp
IF."r ; w1=h; w2=h/
�
: (2)
The empirically derived expression for the IF is summarized in this section. The fringe
capacitance (Cf ) p.u.l. of the wider width (W1) microstrip #1 with substrate thickness h
and relative permittivity "r is
Cf D1
2
�
"r;eff 1
Z01."r D 1/V0
�"0"r w1
h
�
; (3)
where V0 is the velocity of the EM wave in the free space. The effective relative
permittivity "r;eff 1 of microstrip #1 and its characteristic impedance on air substrate
Z01."r D 1/ are obtained from the closed-form expressions of Hammerstad and Jensen
(1980).
The empirical IF, IF."r ; w1=h; w2=h/ is obtained by the multidimensional logistic
curve fitting of the method of moment (MOM) reference data (Gupta & Gopinath, 1977)
and the data extracted from the commercial software (Sonnet Software Ltd., 1986–2009)
over the wide range of parameters "r , w1=h, and w2=h. The power and linear regressions
are used to obtain the following curve-fitted expressions:
IF."r ; w1=h; w2=h/ D 10y ; y D A.w1=h/�B ;
A D A1
�w2
h
�B1
; B D A2
�w2
h
�B2
;
A1 D 0:7785 C 0:0132"r; B1 D 0:4023 C 0:016"r ;
A2 D 0:632 C 0:02071"r; B2 D(
0:2491 C 0:0077"r; for 2:3 � "r < 9:6
0:2491 C 0:0077"r; for 9:6 � "r � 40:
(4)
For the multidimensional curve fitting, one variable is taken at a time, and the
customary constants generated in the process are dependent on other variables of the struc-
ture. These customary constants are again curve fitted to obtain a complete set of
expressions for the IF. The accuracy of the model is tested in the range 2:3 � "r � 40,
0:1 < w1=h < 10 against the results of the MOM (Gupta & Gopinath, 1977) and the
results extracted from the EM software (Sonnet Software Ltd., 1986–2009). Some typical
comparisons for normalized shunt capacitance are shown in Table 1. The normalization
of Cp is carried out by the line capacitance of microstrip #1. As compared against
the results of Sonnet, the results of the present model are better that the MOM-based
model of Gupta and Gopinath (1977). For the step discontinuity in the microstrip on
substrate "r D 4:0, maximum and average deviations percentages of the MOM results
are 31.3% and 18.35%, respectively; whereas for the present model, these are 16% and
8.8%. For the case of "r D 15:1, the MOM results have these deviations as 31.2%
and 8.7%; whereas for the present model, these deviation percentages are 6.2% and 3%.
The deviation of the present model decreases for the steps of larger impedance ratio step.
For "r D 40:0, the MOM results are not available. However, in this case, the present
model has a maximum deviation of 9% and an average deviation of 3.3%.
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Ta
ble
1
No
rmal
ized
dis
con
tin
uit
yca
pac
itan
ceC
pfo
rw
2=
hD
1:0
," r
D9:6
w2=
hD
1:0
" rD
4:0
" rD
15:1
" rD
40:0
W1=
hS
on
net
Go
pin
ath
Dev
iati
on
(%)
Pre
sen
t
mo
del
Dev
iati
on
(%)
So
nn
et
Go
pin
ath
Dev
iati
on
(%)
Pre
sen
t
mo
del
Dev
iati
on
(%)
So
nn
et
Pre
sen
t
mo
del
Dev
iati
on
(%)
2.0
0.0
68
0.0
47
30
.00
.05
81
4.7
0.0
64
0.0
44
31
.20
.06
06
.20
.06
30
.06
21
.5
3.0
0.1
82
0.1
25
31
.30
.15
41
5.3
0.1
28
0.1
13
11
.70
.12
15
.40
.15
60
.14
75
.7
4.0
0.2
18
0.1
66
30
.20
.18
11
6.0
0.1
70
0.1
49
12
.30
.16
05
.80
.20
40
.18
79
.0
5.0
0.2
79
0.2
26
18
.90
.26
93
.50
.22
80
.21
27
.00
.22
41
.70
.24
80
.23
16
.8
6.0
0.3
11
0.2
28
26
.60
.27
81
0.6
0.2
31
0.2
42
4.7
0.2
32
0.4
30
.27
30
.26
43
.2
7.0
0.3
35
0.3
03
9.5
0.3
06
8.6
0.2
54
0.2
69
5.0
0.2
64
3.9
30
.29
20
.29
40
.6
8.0
0.3
54
0.3
28
7.3
0.3
37
4.8
0.2
79
0.2
92
4.6
50
.27
32
.10
.30
70
.31
00
.97
9.0
0.3
86
0.3
45
10
.60
.36
55
.40
.31
10
.30
40
.22
0.3
08
0.9
00
.33
30
.32
71
.8
10
.00
.39
70
.40
00
.75
0.3
98
0.2
50
.32
30
.31
91
.27
0.3
27
1.2
0.3
42
0.3
41
0.2
9
486
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Multilayer Microstrip Step Discontinuity 487
Series Step Discontinuity Inductance Ls
It has been computed by the MOM (Thomson & Gopinath, 1975) and also by the closed-
form model of Schwinger and Saxon as reported by Hoffmann (1987). The Schwinger
and Saxon model is valid for the strip line case. Therefore, for a microstrip, we have
to take one-half of the value of the step series discontinuity inductance of the strip
line. Gupta et al. (1987) have also provided a simple expression to compute the series
step discontinuity inductance Ls that is applicable to a wide range of line parameters;
however, the results are not accurate. In this section, we empirically improve this model
by using a multiplying correction factor Y . The corrected model for Ls with correction
factor Y is
Ls.corr/ D LsY; (5a)
LS D 987h
�
1 � Z02
Z01
r
"r;eff 2
"r;eff 1
�2
: (5b)
The substrate thickness h is in mm. Low-impedance microstrip #1 has characteristic
impedance Z01 and effective relative permittivity "r;eff 1; whereas high-impedance mi-
crostrip #2 has characteristic impedance Z02 and effective relative permittivity "r;eff 2.
The step discontinuity inductance Ls could be normalized by the line inductance of
microstrip #2. Thus, the corrected normalized step discontinuity inductance Ls.corr/ is
Ln.corr/ DLs.corr/
Lw2h; (6a)
LW 2 DZ02."r D 0/
p"r;eff 2
v0
: (6b)
The characteristic impedance Z02."r D 0/ and effective relative permittivity "r;eff 2 are
computed by the closed-form expressions of Hammerstad and Jensen (1980).
The reference data on the step discontinuity inductance are obtained from the results
of the MOM (Thomson & Gopinath, 1975) and also extracted from the commercial soft-
ware (Sonnet Software Ltd., 1986–2009). The expression for the multiplying correction
factor Y is obtained by the multidimensional curve fitting of these data:
Y D Y1
�w1
h
�
�Y2
; (7)
where parameters Y1 and Y2 are given as
For 0:5 � .w1
h/ < 3:
Y1 D 5:3063�w2
h
�3
� 14:189�w2
h
�2
C 17:43�w2
h
�
� 4:551:
For 3 � .w1
h/ � 10:
Y1 D �0:6556�w2
h
�6
C 14:832�w2
h
�5
� 107:14�w2
h
�4
C 341:36�w2
h
�3
� 512:08�w2
h
�2
C 350:29�w2
h
�
� 82:261: (8)
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488 A. K. Verma et al.
Table 2
Normalized discontinuity inductance Ls for w2=h D 1:0, "r D 9:6
w1=h Sonnet Gopinath Deviation (%) Present model Deviation (%)
2.0 0.058 0.049 15.0 0.060 3.3
3.0 0.135 0.127 5.9 0.134 0.74
4.0 0.197 0.191 3.14 0.193 2.0
5.0 0.253 0.260 2.7 0.254 0.39
6.0 0.303 — — 0.289 4.62
7.0 0.347 — — 0.329 5.18
8.0 0.376 — — 0.366 2.6
9.0 0.419 — — 0.400 4.5
10.0 0.446 — — 0.432 3.1
For 3 � .w1
h/ � 10:
Y2 D �0:0485�w2
h
�6
C 0:8205�w1
h
�5
� 5:1389�w1
h
�4
C 15:196�w1
h
�3
� 22:259�w1
h
�2
C 15:582�w1
h
�
� 2:7651: (9)
Table 2 compares the normalized series Ls computed on the alumina substrate ("r D9:6) by Sonnet, the MOM of Thomson and Gopinath (1975), and the present model.
The maximum and average deviations of the MOM are 15% and 5.34%, respectively;
whereas for the present model, these are 5.18% and 2.9% respectively. The present
closed-form model with its acceptable performance is suitable for computer-aided design
(CAD) application.
3. Step Discontinuity Model for Multilayer LayerMicrostrip Line
On several occasions, components and circuits with step discontinuity are developed
in the multilayer microstrip environment, such as suspended and composite substrate
microstrips. The equivalent T-network for such cases could be extracted from the EM
simulators. However, for each structure, we have to extract the values of the elements
of the equivalent circuit. Therefore, this method is not useful for the analysis and
optimization of the circuit and components with step discontinuity. However, the above
models for Cp and Ls of the equivalent circuit, shown in Figure 1(b), can be adapted
to step discontinuity in the multilayer microstrip line shown in Figure 2. The general
multilayer step discontinuity circuit model is useful for the analysis and optimization
work. The adaptation is achieved by using the concept of the SLR formulation (Verma
& Sadr, 1992; 1994). In the SLR process, the multilayer microstrip line is reduced to the
equivalent single-layer microstrip line using Wheeler’s transformation (Verma & Sadr,
1992; 1994). The equivalent single-layer substrate maintains the total substrate thickness
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Multilayer Microstrip Step Discontinuity 489
Figure 2. Multilayer microstrip line.
of the original structure, i.e., H D h1 C h2 C � � � C hN . It also retains the original width
w of the strip conductor. However, the equivalent relative permittivity of the equivalent
single-layer substrate is strip-width/substrate-height dependent. Therefore, in the case of
step discontinuity in the microstrip on the multilayer substrate, we get two equivalent
relative permittivities—one each for the widths w1 and w2. In order to maintain one
uniform permittivity for the equivalent substrate, we have taken the geometric mean of two
permittivities. Over the equivalent single substrate with uniform permittivity and thickness
H , both microstrip lines should maintain their original characteristic impedances as that
of the step on the multilayer substrate. The new widths w0
1 and w0
2 of two strip conductors
on the equivalent single substrate can be computed from the synthesis program (AWR
Inc., 2004) in order to maintain the original characteristic impedances of the step on the
multilayer substrate.
The line capacitance, effective relative permittivity, and characteristic impedance of
the multilayer microstrip line shown in Figure 2 are computed by the variational method
in the Fourier domain by using the transverse transmission line (TTL) technique to get
the Green function of the structure (Verma & Kumar, 1998; Crampagne et al., 1978).
The results are summarized below:
1
C.w; h1; h2; : : : ; hN ; "r1; "r2; : : : ; "rN /D 1
�"0
Z
1
0
Œ Qf .ˇ/=Q�2
Yˇdˇ; (10)
"r;eff .w=H; "r;eq/ D C.w; h1; : : : ; hN ; "r1; : : : ; "rN /
C0.w; H D h1 C � � � C hN ; "r1 D "rN D 1/;
(11)
Z0.w; h1; h2; : : : ; hN ; "r1; "r2; : : : ; "rN / DZ0.w=H; "r;eq;1 D "rN D 1/
p
"r;eff .w=H; "r;eq/: (12)
The above expressions are applied to both the strip conductors having widths w1 and
w2 to compute their effective relative permittivity and characteristic impedance. The
SLR formulation reduces both microstrip lines to their respective single-layer substrate
of thickness H and equivalent relative permittivities "r;eq;k.wk=H; "r1; : : : ; "rN / (k D1, 2) while keeping their widths wk (k D 1, 2) unchanged. It is achieved by Wheeler’s
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490 A. K. Verma et al.
transformation (Verma & Sadr, 1992; Verma & Kumar, 1998):
"r;eq;k.wk=H; "r1; : : : ; "rN / D 1 C "r;eff ;k.wk=H; "r1; : : : ; "rN / � 1
q.wk=H/: (13)
The filling factor, q.wk=H/ (k D 1, 2) is obtained from
q D 1
2.1 C p/; p D
8
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
:
�
1 C 12H
wk
�
�1=2wk
H� 1;
�
1 C 12H
wk
�
�1=2
C 0:04h
1 � wk
H
i2
;wk
H< 1
: (14)
We get one uniform permittivity of the equivalent single-layer substrate by taking
the geometric mean of both equivalent relative permittivity "r;eq;k.wk=H; "r1; : : : ; "rN /
(k D 1, 2):
"r;eq.w1=H; w2=H; "r1; : : : ; "rN /
Dq
"r;eq;1.w1=H; "r1; : : : ; "rN /"r;eq;2.w2=H; "r1; : : : ; "rN /: (15)
The strip widths wk (k D 1, 2) are modified to w0
k(k D 1, 2) on the equivalent single-
layer substrate with relative permittivity "r;eq.w1=H; w2=H; "r1; : : : ; "rN / and substrate
thickness H in order maintain the original characteristic impedance of both lines on
multilayer substrates. It is carried out by using the microstrip synthesis program (AWR
Inc., 2004).
Finally, the equivalent microstrip step discontinuity is formed by two strip widths
w0
1 and w0
2 on the substrate with relative permittivity "r;eq.w1=H; w2=H; "r1; : : : ; "rN /
and substrate thickness H . The shunt discontinuity capacitance for the multilayer case,
Cp.w0
1=H; w0
2=H; "r1; : : : ; "rN / and the series discontinuity inductance Ls.w1=H; w2=H;
"r1 D "rN D 1/ are obtained using models presented in Section 2 for the single-layer sub-
strate. In the model for Cp presented in Section 2, w1 and w2 are replaced by w0
1 and w0
2,
respectively, h is replaced by H , and "r is replaced by "r;eq.w1=H; w2=H; "r1; : : : ; "rN /.
In the model for Ls presented in Section 2, h is replaced by H .
4. Discussion of Results
The normalized step discontinuity shunt capacitance Cp is computed for microstrip
structures on both the composite and suspended substrates. Figure 3(a) compares the
normalized Cp for the GaAs substrate with the polyamide passivation layer, i.e., for
cases "r1 D 12:9, "r2 D 3:5, h1 D 0:6 mm, h2 D 0:037 mm, and w2=H D 1
corresponding to the characteristic impedance Z02 D 48:63 �. Figure 3(b) compares the
normalized Cp for the GaAs substrate on the alumina substrate, i.e., for cases "r1 D 9:8,
"r2 D 12:9, h1 D 0:635 mm, h2 D 0:150 mm, and w2=H D 1 corresponding to
the characteristic impedance Z02 D 45:18 �. For both of the cases, the normalized
Cp computed by the present model is compared against the de-embedded results from
Sonnet. The illustrative numerical results are also compared in Table 3. The average
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Multilayer Microstrip Step Discontinuity 491
(a)
(b)
Figure 3. Normalized step discontinuity capacitance for: (a) "r1 D 12:9, "r2 D 3:5, h1 D 0:6 mm,
h2 D 0:037 mm and (b) "r1 D 9:8, "r2 D 12:9, h1 D 0:635 mm, h2 D 0:150 mm.
deviation of the present SLR model against the results of Sonnet is 5.44% and 3.3%
for "r1 D 12:9, "r2 D 3:5 and "r1 D 9:8, "r2 D 12:9, respectively. Figures 3(a)
and 3(b) also present these comparisons. The deviation in Cp for the multilayer case
is less than the average deviation of the MOM results for the single-layer substrate
mentioned above.
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492 A. K. Verma et al.
Table 3
Composite substrate normalized Cp
"r1 D 12:9, "r2 D 3:5 "r1 D 9:8, "r2 D 12:9
W1=h Sonnet Present model Deviation (%) Sonnet Present model Deviation (%)
0.1 0.135 0.132 2.2 0.127 0.124 2.3
0.5 0.035 0.039 5.2 0.028 0.029 3.5
0.8 0.008 0.011 9.8 0.006 0.008 3.3
2.0 0.054 0.052 3.7 0.058 0.049 2.0
5.0 0.188 0.200 6.3 0.187 0.198 5.8
Figures 4(a) through 4(c) present a comparison of the present model against Sonnet
for the discontinuity capacitance Cp on the suspended substrates, "r2 D 2:3, 9.8, and 12.9.
The illustrative numerical results are compared in Table 4. The deviations in the present
model increase with an increase in relative permittivity as the average deviations of
normalized Cp are 1.5%, 9.3%, and 13.28% for "r2 D 2:3, 9.8, and 12.9, respectively.
We note that for the suspended semiconducting substrate, the deviation is high. In order
(a)
Figure 4. Normalized step discontinuity capacitance for: (a) "r1 D 1:0, "r2 D 2:3, h1 D 0:5 mm,
h2 D 0:2 mm; (b) "r1 D 1:0, "r2 D 9:8, h1 D 0:2 mm, h2 D 0:65 mm; and (c) "r1 D 1:0,
"r2 D 12:9, h1 D 0:2 mm, h2 D 0:5 mm. (continued)
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Multilayer Microstrip Step Discontinuity 493
(b)
(c)
Figure 4. (Continued).
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494 A. K. Verma et al.
Table 4
Suspended substrate normalized Cp
"r1 D 1:0, "r2 D 2:3 "r1 D 1:0, "r2 D 9:8 "r1 D 1:0, "r2 D 12:9
W1=h Sonnet
Present
model
Deviation
(%) Sonnet
Present
model
Deviation
(%) Sonnet
Present
model
Deviation
(%)
Improved
model
Deviation
(%)
0.1 0.125 0.125 0.0 0.116 0.114 1.7 0.102 0.091 10.0 0.103 0.98
0.5 0.035 0.035 0.0 0.038 0.032 15.0 0.030 0.024 20.0 0.033 10.0
0.8 0.008 0.010 2.0 0.008 0.009 1.2 0.007 0.007 0 0.008 10.0
2 0.074 0.077 4.0 0.100 0.088 12.0 0.114 0.127 11.4 0.117 2.6
5 0.317 0.312 1.5 0.272 0.225 17.0 0.326 0.242 25.0 0.354 8.5
to improve it, we can multiply Eq. (2) by the following factor:
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The results of the improved present model for the step discontinuity of the suspended
substrate ("r2 D 12:9) is also shown in Table 4 and Figure 4(c). The improved average
deviation of normalized Cp is 6.4%.
Finally, the accuracy of the present SLR-based model for the step discontinuity
on the composite substrate microstrip and suspended substrate microstrip are tested by
constructing the equivalent T-circuit in the circuit simulator Microwave Office (AWR
Inc., 2004). The step discontinuity equivalent circuit parameters for five cases are shown
in Table 5. The S -parameter responses obtained from the equivalent circuit are compared
against the responses obtained from the EM Sonnet Simulator in Figures 5(a) and 5(b) for
the composite substrates. Figures 6(a) through 6(c) compare the S -parameter responses
Table 5
Equivalent circuit parameters of step discontinuity
Structures W1=h Cp (pF) Ls (nH)
Composite I, Figure 5(a) 0.1 0.012 0.10Composite II, Figure 5(b) 5.0 0.020 0.08Suspended I, Figure 6(a) 2.0 0.0038 0.006Suspended II, Figure 6(b) 0.5 0.005 0.007Suspended III, Figure 6(c) 5.0 0.100 0.100
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Multilayer Microstrip Step Discontinuity 495
(a)
(b)
Figure 5. Composite substrate: (a) "r1 D 12:9, "r2 D 3:5, h1 D 0:6 mm, h2 D 0:037 mm and
(b) "r1 D 9:8, "r2 D 12:9, h1 D 0:635 mm, h2 D 0:150 mm.
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496 A. K. Verma et al.
(a)
(b)
Figure 6. Suspended substrate: (a) "r1 D 1:0, "r2 D 2:3, h1 D 0:5 mm, h2 D 0:2 mm, W1=H D2:0; (b) "r1 D 1:0, "r2 D 9:8, h1 D 0:2 mm, h2 D 0:65 mm, W1=H D 0:5; and (c) "r1 D 1:0,
"r2 D 12:9, h1 D 0:2 mm, h2 D 0:5 mm, W1=H D 5:0. (continued)
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Multilayer Microstrip Step Discontinuity 497
(c)
Figure 6. (Continued).
for the suspended microstrip line on "r2 D 2:3, 9.8, and 12.9 for w1=H D 2:0, w1=H D0:5, and w1=H D 5:0, respectively. Although the SLR-based models are static ones, the
responses are still satisfactory up to 16 GHz for the wide range of w=H ratio and relative
permittivity.
5. Conclusion
We have presented accurate closed-form models for the shunt step discontinuity capac-
itance Cp and series discontinuity inductance Ls for the single-layer microstrip. The
models are adapted to the multilayer microstrip case by using the concept of the SLR
formulation. The present models are useful for development of filters, matching network,
etc., on the composite layer and suspended layer microstrip substrates.
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