microstrip antenna using cavity method, Electron Lett 25 (1989),391–392.

6. P. Gray-Balmaz, J.A. Encinar, and J.R. Mosig, Analysis of multilayerprinted array by a modular approach based on the generalized scatter-ing matrix, IEEE Trans Antennas Propagat AP-48 (2000), 26–33.

7. D.M. Pozar, Analysis of an infinite phased array of aperture-coupledmicrostrip patch, IEEE Trans Antennas Propagat AP-37 (1989), 418–425.

8. N.C. Karmakar and M.E. Bialkowski, Circularly polarized aperture-coupled circular microstrip patch antennas for L-band applications,IEEE Trans Antennas Propagat AP-47 (1999), 933–939.

9. S.D. Targonski, R.B. Waterhouse, and D.M. Pozar, Design of wide-band aperture-stacked patch microstrip antennas, IEEE Trans Anten-nas Propagat AP-46 (1998), 1245–1251.

10. D.M. Pozar, A reciprocity method of analysis for printed slot and slotcoupled microstrip antenna, IEEE Trans Antennas Propagat AP-34(1986), 1439–1446.

11. P.L. Sullivan and D.H. Schaubert, Analysis of an aperture coupledmicrostrip antenna, IEEE Trans Antennas Propagat AP-34 (1986),977–984.

12. J.-F. Zurcher and F.E. Gardiol, Broadband Patch Antennas, MA:Artech House, 1995, pp 45–47.

13. Z.-F. Liu, P.-S. Kooi, L.-W. Li, M.-S. Leong, and T.-S. Yeo, A methodfor designing broad-band microstrip antennas in multilayered planarstructures, IEEE Trans Antenna Propagat AP-47 (1999), 1416–1420.

14. A. Akdagli and K. Guney, Effective patch radius expression obtainedusing a genetic algorithm for the resonant frequency of electrically thinand thick circular microstrip antennas, IEE Proc Microw AntennasPropag AP-147 (2000), 156–159.

15. S. Zhong, Microstrip Antenna Theory and Application, P.R. China:Xi’an DianZi Technology University, 1991, pp 4–9.

16. M. Haneishi and S. Yoshida, A design method of circularly polarizedrectangular microstrip antenna by one-point feed, K.C. Gupta and A.Benalla, Microstrip Antenna Design, MA: Artech House, 1988, pp313–321.

© 2002 Wiley Periodicals, Inc.

FULL-WAVE MODELLING OF OFFSETOVERLAP AND GAP DISCONTINUITYFOR ASYMMETRIC STRIPCONDUCTORS IN MULTI-LAYERED MICENVIRONMENT

Rajiv Kumar Shukla and Animesh BiswasDepartment of Electrical EngineeringIndian Institute of TechnologyKanpur, India 208 016

Received 2 May 2002

ABSTRACT: A generalised model for characterising the frequency de-pendent properties of general offset overlap and gap discontinuity ofasymmetric strip conductors in multi-layered MIC environment is pre-sented in this paper. The transverse resonance technique is applied intransformed domain to extract the equivalent PI-circuit parameters ofthe discontinuity. The incorporation of options for offset between thestrips and asymmetry in strip widths make the present model verygeneral and useful for different structures in multi-layered substrateenvironment. © 2002 Wiley Periodicals, Inc. Microwave Opt TechnolLett 35: 247–251, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10570

Key words: multi-layered MIC; strip gap discontinuity; MIC circuits

INTRODUCTION

The computer aided design (CAD) and optimisation of number ofcomponents and filters in MIC environment requires the accuratemodelling of the coupling structures. The strips printed on bothside of the substrate can provide the coupling structures with awide range of coupling. Specially, very tight coupling can berealised by employing overlapping strips on both sides of thedielectric substrate, which makes it possible to realise bandpass aswell highpass filters with reduced insertion loss and wide bands[1].

In this paper, a full-wave analysis for the modelling of generaloffset overlap and gap discontinuity of asymmetric strips, printedon opposite sides of dielectric substrate in multilayered MICenvironment, is presented. The suspended substrate line is a pop-ular choice for millimetre-wave circuits. The present analysis isvery general in nature, as it can simulate frequency-dependentequivalent circuit parameters of the strip discontinuity in sus-pended substrate, substrate-superstrate, and multi-substrate envi-ronment with the options of offset and asymmetry of the stripconductors. Full-wave modelling of this type of discontinuity isreported by Schwab et al. [2], and that in finline by Schiavon et al.[3], but the effects of offset and asymmetry are not considered. Wepresent the results in order to show the effects of offset andasymmetry between coupled strips on the equivalent PI-circuitparameters.

THEORY

A full-wave analysis of the discontinuity is made based on themodal analysis in conjunction with generalised transverse reso-nance technique [3, 4]. The electric and magnetic fields are ex-panded in terms of hybrid modes in each region; then, the appli-cation of boundary conditions at all the interfaces leads to Green’sdyadic.

This is followed by the application of the Galerkin procedure inthe transform domain, leading to a set of homogeneous equations.The condition for nontrivial solutions of the resulting homoge-neous system constitutes the characteristic equation of the struc-ture. This is a function of frequency and line lengths l1, l2.

The longitudinal and cross sectional views of the general dis-continuity and its equivalent PI-circuit is shown in Figure 1. Theelectric and magnetic fields in the longitudinal direction of the

Figure 1 (a) Longitudinal and cross-sectional views of the multi-layeredsubstrate cavity housing the discontinuity. (b) Equivalent PI-network cir-cuit of the discontinuity

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 35, No. 3, November 5 2002 247

three regions can be expressed in terms of hybrid modes asfollows:

Ex1 �

n�1

� m�0

�

Amn1cos��mn1�x � h1 � d/2�� � sin��ny� � cos��mz�

Hx1 �

n�0

� m�1

�

Bmn1sin��mn1�x � h1 � d/2�� � cos��ny� � sin��mz�

Ex2 �

n�1

� m�0

�

�Amn2sin��mn2x�

� A mn2cos��mn2x�� � sin��ny� � cos��mz�

Hx2 �

n�0

� m�1

�

�Bmn2cos��mn2x�

� B mn2sin��mn2x�� � cos��ny� � sin��mz�

Ex3 �

n�1

� m�0

�

Amn3cos��mn3�x � h1 � d/2�� � sin��ny� � cos��mz�

Hx3 �

n�0

� m�1

�

Bmn3sin��mn3�x � h1 � d/2�� � cos��ny� � sin��mz�

(1)

Where,

�mn1 � ��k02r1 � �n

2 � �m2 � �mn2 � ��k0

2r2 � �n2 � �m

2 �

�mn3 � ��k02r3 � �n

2 � �m2 �

�n �n

b�m �

m

�l1 � l2 � s�k0 � ��0�0 (2)

The transverse electric and magnetic field components can bederived from Eq. (3).

�E� ti

H� ti� �

1

�k02ri � �mni

2 � ��

� xj��0

�1x�

j�0ri�1x�

�

�x���tEx

�tHx�

(3)

The homogeneous set of equations obtained by application ofGalerkin procedure in the transformed domain is given in Eq. (4).

r�1

2 p�1

� q�1

�

Cpr n�0

� m�0

�

PmPnG11L1mn� p,q,r�L1mn

�i, j,k� � r�1

2 p�1

� q�1

�

Dpr n�0

� m�0

�

QmQnG12L2mn� p,q,r�L1mn

�i, j,k� � r�1

2 p�1

� q�1

�

Epr n�0

� m�0

�

PmPnG13L3mn� p,q,r�L1mn

�i, j,k�

� r�1

2 p�1

� q�1

�

Fpr n�0

� m�0

�

QmQnG14L4mn� p,q,r�L1mn

�i, j,k� � 0

r�1

2 p�1

� q�1

�

Cpr n�1

� m�1

�

PmPnG21L1mn� p,q,r�L2mn

�i, j,k� � r�1

2 p�1

� q�1

�

Dpr n�1

� m�1

�

QmQnG22L2mn� p,q,r�L2mn

�i, j,k� � r�1

2 p�1

� q�1

�

Epr n�1

� m�1

�

PmPnG23L3mn� p,q,r�L2mn

�i, j,k�

� r�1

2 p�1

� q�1

�

Fpr n�1

� m�1

�

QmQnG24L4mn� p,q,r�L2mn

�i, j,k� � 0

r�1

2 p�1

� q�1

�

Cpr n�0

� m�0

�

PmPnG31L1mn� p,q,r�L3mn

�i, j,k� � r�1

2 p�1

� q�1

�

Dpr n�0

� m�0

�

QmQnG32L2mn� p,q,r�L3mn

�i, j,k� � r�1

2 p�1

� q�1

�

Epr n�0

� m�0

�

PmPnG33L3mn� p,q,r�L3mn

�i, j,k�

� r�1

2 p�1

� q�1

�

Fpr n�0

� m�0

�

QmQnG34L4mn� p,q,r�L3mn

�i, j,k� � 0

r�1

2 p�1

� q�1

�

Cpr n�1

� m�1

�

PmPnG41L1mn� p,q,r�L4mn

�i, j,k� � r�1

2 p�1

� q�1

�

Dpr n�1

� m�1

�

QmQnG42L2mn� p,q,r�L4mn

�ijk� � r�1

2 p�1

� q�1

�

Epr n�1

� m�1

�

PmPnG43L3mn� p,q,r�L4mn

�i, j,k�

� r�1

2 p�1

� q�1

�

Fpr n�1

� m�1

�

QmQnG44L4mn� p,q,r�L4mn

�i, j,k� � 0 (4)

248 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 35, No. 3, November 5 2002

The characteristic equation is solved for three pairs of resonantlengths to compute three unknown parameters of the equivalentcircuit for a given frequency.

Current Distribution on the Strip ConductorsThe basis functions for the longitudinal and transverse currents onthe strip conductors are chosen so that the singular behaviour of

the currents at the metallic edges is automatically incorporated.The current density distributions on the strip conductors in thetransverse direction are chosen as follows:

Jzu� y� � ��w1

2 �2

� � y � P1�2�1/ 2

Jzl� y� � ��w2

2 �2

� � y � P2�2�1/ 2

Jyu� y� � � y � P1���w1

2 �2

� � y � P1�2� 1/ 2

Jyl � y� � � y � P2���w2

2 �2

� � y � P2�2� 1/ 2 (6a)

and in the longitudinal direction are:

Jyu� z� �

p�1

P

apcos�2p

�g1� z Jz

u�z�� p�1

P

bpsin�2p

�g1� z

Jyl �z��

p�1

P

a pcos�2p

�g2� �z � Lr� Jz

l�z��p�1

P

b psin�2p

�g2� �z � Lr�

(6b)

The current transforms can be written as follows:

L1mn � 0

b 0

Lr

Jyu� y, z�cos��ny�cos��mz�dy � dz

L2mn � 0

b 0

Lr

Jzu� y, z�sin��ny�sin��mz�dy � dz

L3mn � 0

b 0

Lr

Jyl � y, z�cos��ny�cos��mz�dy � dz

L4mn � 0

b 0

Lr

Jzl� y, z�sin��ny�sin��mz�dy � dz (7)

In the above equations superscript l denotes the lower conductorstrip and u denotes the upper strip conductor.

NUMERICAL RESULTS

The software is developed based on the above theory and analysis.The results are generated to study the effects of variation indifferent structural parameters, on the discontinuity model. Theimportant parameters to be studied are the offset between thestrips, and the asymmetry in the widths of the strips.

Comparing the simulated results with the published data vali-dates the theory. Figure 2 shows the comparison of present theory(lines) with the results reported by Schwab [2] (marker points).The present results follow the similar behaviour as reported in [2].

There is slight deviation in the transition region from gap tooverlap, which may be due to a dominant open-end effect.

Figure 3 shows the effect of offset between strips on the seriesand shunt capacitance’s of the equivalent PI-circuit of the discon-tinuity in suspended substrate environment.

Figure 4 shows the effect of offset between strips on theparameters of equivalent PI-circuit of the discontinuity in sub-strate-superstrate environment.

It is observed from the above results for symmetric strips thatthe series capacitance Cse decreases with increasing offset be-tween the strips. This implies that the coupling between the stripsdecreases with increase in offset between the strips, whereas theshunt capacitance Csh1 and Csh2 increase with an increase inoffset, due to the increase in fringing effect. The exception is in the

Figure 2 Variation of S12 of offset layered strip overlap/gap disconti-nuity in suspended substrate environment with overlapping: a � b � 5.0mm, d � 0.254 mm, h1 � h2 � 2.373 mm, w1 � w2 � 1.75 mm, r1 �r3 � 1.0, r2 � 2.2, P1 � P2 � 2.5 mm, frequency � 10 GHz

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 35, No. 3, November 5 2002 249

case of Csh1 for suspended substrate environment, which startsdecreasing for large offsets.

Figure 5 shows the effect of asymmetry of strips widths on theseries and shunt reactance’s of the equivalent PI-circuit of thediscontinuity in suspended substrate environment.

Figure 6 shows the effect of asymmetry strips widths on theparameters of equivalent PI-circuit of the discontinuity in sub-strate-superstrate environment.

The results for asymmetric strips show that the nature of shuntreactance of strip with narrow width is changing from capacitive to

Figure 3 Variation of reactance parameters of offset layered strip overlap/gap discontinuity in suspended substrate environment with strip offsetparameters. (a) Overlap � 0.25 mm, (b) Gap � 0.25 mm: a � 5.0 mm, b � 10.0 mm, d � 0.254 mm, h1 � h2 � 2.373 mm, w1 � w2 � 1.75 mm,r1 � r3 � 1.0, r2 � 2.2, P1 � 5.0 mm, frequency � 10 GHz

Figure 4 Variation of reactance parameters of offset layered strip overlap/gap discontinuity in substrate-superstrate environment with strip offsetparameters. (a) Overlap � 0.25 mm. (b) Gap � 0.25 mm: a � 5.0, b � 20.0 mm, d � 0.254 mm, h2 � 254 mm, w1 � w2 � 2.54 mm, r1 � 1.0,r2 � 9.6, r3 � 2.2, P1 � 10.0 mm, gap � 0.25 mm, frequency � 10 GHz

Figure 5 Variation of reactance parameters of offset layered asymmetric strip overlap/gap discontinuity in suspended substrate environment with stripwidths ratio. (a) Overlap � 0.25 mm. (b) Gap � 0.25 mm: a � 5.0 mm, b � 10.0 mm, d � 0.254 mm, h1 � h2 � 2.373 mm, w � 1.75 mm, r1 �r3 � 1.0, r2 � 2.2, P1 � P2 � 5.0 mm, frequency � 10 GHz

250 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 35, No. 3, November 5 2002

inductive as the width ratio is increasing. The reason for thisbehaviour may be due to the change in nature of disturbed elec-tromagnetic fields by the discontinuity from electric to magneticwith increasing width ratio.

CONCLUSION

The full-wave CAD model for generalised offset and gap discon-tinuity for asymmetric strips in multi-layered medium is presentedin this paper. The results for specific cases are compared withavailable data for the purpose of validation of the present theory.This model is suitable for computer-aided design and optimisationof microwave circuits, such as inter-digital filters with advancedfiltering characteristics, which require complex cross-couplingsbetween printed elements.

REFERENCES

1. G. Matthaei, L. Young, and E.M.T. Jones, Microwave Filters, Imped-ance Matching Networks, and Coupling Structures, McGraw Hill Co.,New York, 1960.

2. W. Schwab and W. Menzel, On the design of planar microwave com-ponents using multilayer structures, IEEE Trans Microwave TheoryTech MTT-40 (1992), 67–72.

3. G. Schiavon, P. Tognolatti, and R. Sorrentino, Full-wave analysis ofcoupled finline discontinuities, IEEE Trans Microwave Theory TechMTT-36 (1998), 1889–1894.

4. A. Biswas and V.K. Tripathi, Modeling of asymmetric and offset gapsin shielded microstrips and slotlines, IEEE Trans Microwave TheoryTech MTT-38 (1990), 818–822.

© 2002 Wiley Periodicals, Inc.

COMMENTS ON “LOW-NOISEONE-PORT MICROWAVETRANSISTOR AMPLIFIER”

P. Gardner and D. K. Paul

Received 2 May 2002

© 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 35:251–252, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10571

Venguer et al. [1] reported a low noise transistor based reflectionamplifier, and described it as “a new type of a microwave reflec-tion amplifier with very low noise properties . . . ” In making thisclaim, they failed to acknowledge significant earlier work in thesame field.

The concept of using a GaAs MESFET in a reflection modeamplifier to achieve high gain at high frequency was reported aslong ago as 1979 [2], and a power amplifier using the same ideawas described in the same year [3]. A design method for low-noise, negative-resistance, reflection-mode, transistor amplifiers,along with a practical example operating in X-band, was describedin 1991 [4]. The potential benefits of this type of amplifier in apulsed radar system were investigated and discussed in 1992 [5].An application achieving low noise in the mm-wave region wasreported in 1993 [6]. A more general theory of the low-noiseoperation of such circuits was given in [7], showing that differentloss-less embeddings of the same transistor, forming differentnegative resistance amplifier configurations, would result in thesame optimum noise measure.

In [1], the authors reported an opportunity to achieve lowernoise figure in negative resistance mode than in conventionaltransmission mode. In [7], it was shown that the optimum noisemeasure is the same for a reflection amplifier as for a transmissionamplifier using the same transistor. This indeed implies that alower noise figure (or equivalently a lower noise temperature) canbe achieved in a reflection amplifier, but only at the expense ofreducing the gain.

Thus the key points in [1], apart from the approximate noiseanalysis and the specific design example at a low microwavefrequency, had been reported previously. The authors of [1] failedto acknowledge the earlier work in [2–7], and the referees failed tonotice this omission.

Figure 6 Variation of reactance parameters of offset layered asymmetric strip overlap/gap discontinuity in substrate-superstrate environment with stripwidths ratio. (a) Overlap � 0.25 mm. (b) Gap � 0.25 mm: a � 5.0, b � 20.0 mm, d � 0.254 mm, h2 � 254 mm, w1 � 2.54 mm, r1 � 1.0, r2 �9.6, r3 � 2.2, P1 � P2 � 10.0 mm, gap � 0.25 mm, frequency � 10 GHz

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 35, No. 3, November 5 2002 251