Circuit Model of Multilayer Microstrip Step Discontinuity Using Single-Layer Reduction Formulation

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [INFLIBNET India Order]On: 25 August 2009Access details: Access Details: [subscription number 909277354]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

ElectromagneticsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713770615

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

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713770615

http://dx.doi.org/10.1080/02726340903098555

http://www.informaworld.com/terms-and-conditions-of-access.pdf

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:[email protected]

483

Downloaded By: [INFLIBNET India Order] At: 11:40 25 August 2009

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


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%.


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



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)



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



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



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



(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.



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)



(b)

(c)

Figure 4. (Continued).



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:

Ys D

8

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

:

�63:846

�

Z01

Z02

�6

C 323:71

�

Z01

Z02

�5

� 645:631

�

Z01

Z02

�4

C648:24

�

Z01

Z02

�3

� 344:51

�

Z01

Z02

�2

C91:494

�

Z01

Z02

�

� 8:812 for

�

Z01

Z02

�

� 1:32

�9:9804

�

Z01

Z02

�3

C 51:532

�

Z01

Z02

�2

�87:265

�

Z01

Z02

�

C 49:14 for

�

Z01

Z02

�

> 1:32

:

(16)

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



(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.



(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)



(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.

References

AWR Inc. 2004. Microwave Office—2004. El Segundo, CA.

Benedek, P., & P. Silvester. 1972. Equivalent capacitance for microstrip gaps and steps. IEEE Trans.

Microw. Theory Tech. MTT-20:729–733.

Crampagne, R., M. Ahmadpanah, & J. L. Guiraud. 1978. A simple method for determining the

Green’s function for a large class of MIC lines having multilayered dielectric structures. IEEE

Trans. Microw. Theory Tech. MTT-26:82–87.

Gopinath, A., A. F. Thomson, & I. M. Stephenson. 1976. Equivalent circuit parameters of microstrip

step change in width and cross junctions. IEEE Trans. Microw. Theory Tech. 23:142–144.

Gupta, C., & A. Gopinath. 1977. Equivalent circuit capacitance of microstrip step change in width.

IEEE Trans. Microw. Theory Tech. MTT-25:819–822.



Gupta, K. C., R. Garg, I. Bahl, & P. Bhartia. 1987. Microstrip lines and slotlines, Chap. 3, pp. 190–

198. Boston: Artech House.

Hammerstad, E., & O. Jensen. 1980. Accurate models for microstrip computer aided design. IEEE

Int. Microw. Symp. Digest MTT-S:407–409.

Hoffmann, R. K. 1987. Handbook of microwave integrated circuits, Chap. 10, pp. 278–279.

Norwood, MA: Artech House.

Norbert, K. H. L., & R. H. Jansen. 1986. The microstrip step discontinuity: A revised description.

IEEE Trans. Microw. Theory Tech. 34:213–223.

Sonnet Software Ltd. 1986–2009. Sonnet Lite Version 10.55. North Syracuse, NY: Sonnet Software

Ltd.

Thomson, A. F., & A. Gopinath. 1975. Calculation of microstrip discontinuity inductances. IEEE

Trans. Microw. Theory Tech. MTT-23:648–655.

Verma, A. K., & R. Kumar. 1998. New empirical unified dispersion model for shielded-, suspended-,

and composite-substrate microstrip line for microwave and mm-wave applications. IEEE Trans.

Microw. Theory Tech. 46:1187–1192.

Verma, A. K., & G. H. Sadr. 1992. Unified dispersion model for multilayer microstrip line. IEEE

Trans. Microw. Theory Tech. 40:1587–1591.

Verma, A. K., & G. H. Sadr. 1994. Determination of open-end discontinuity and resonance

frequency of suspended microstrip using SLR technique. Microw. Optical Tech. Lett. 7:532–535.

Verma, A. K., & H. Singh. 2005. Closed-form model of shunt capacitance of microstrip step

discontinuity. Asia Pacific Microwave Conference, Suzhou, China, 4–7 December.


Circuit Model of Multilayer Microstrip Step Discontinuity Using Single-Layer Reduction Formulation

Documents