7. C. T. Swift, "Admittance of a Waveguide-Fed Aperture Loaded with a Dielectric Plug," IEEE Trans. Antennas Propagat., Vol. AP-17, May 1969, pp. 356-359.

8. R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, New York, 196 I .

9. NAG Fortran Library, Mark 13, Vol. 1, DOlDAF, The Numerical Algorithms Group Limited, Sept. 1988.

10. M. Tian, D. P. Tran, and L. P. Ligthart, "Air Gap Technique for Matching the Aperture of Miniature Waveguide Antennas," Mi- crowaues Radio Freq., Feb. 1995.

Received 2-24-95

Microwave and Optical Technology Letters, 9/5, 256-260 0 1995 John Wiley & Sons, Inc. CCC 0895-2477/95

CHARACTERIZATION OF CYLINDRICAL MICROSTRIP GAP DISCONTINUITIES Hua-Ming Chen and Kin-Lu Wong Department of Electrical Engineering National Sun Yat-Sen University Kaohsiung, Taiwan 804, Republic of China

KEY TERMS Microstrip gap discontinuity, cylindrical microstrip line

ABSTRACT A rigorous full-wave analysis of microship gap discontinuities on cylin- drical surfaces is presented. Numerical results for the fiequency-depen- dent reflection and transmission coefficients and the gap capacitance and conductance are calculated using a moment-met~od calculation, in which a combination of enfire-domain sinusoidal and piecewise sinu- soidal basis fincfions is used to represent the currents on the microstrip line. The characteristics of cylindrical microstrip gap discontinuities for various cumatures and gap spacings are analyzed. 0 1995 John Wiley & Sons. Inc.

1. INTRODUCTION

Recently, the cylindrical microstrip open-end discontinuity has been studied rigorously by using a full-wave technique [ 11. The frequency-dependent characteristics of the reflection co- efficient and the equivalent terminal admittance at the cylin- drical open-end discontinuity are analyzed. These full-wave results provide useful information for the accurate design of microstrip circuits on cylindrical surfaces. Because a mi- crostrip open-end discontinuity is only a special case of the gap discontinuity, we extend in this article the full-wave modeling [l] to a cylindrical microstrip gap discontinuity. For the numerical computation, the entire-domain sinusoidal and piecewise sinusoidal (PWS) functions are used in the mo- ment-method calculation to model the currents near the gap. From the analysis, the reflection and transmission coefficients for the cylindrical gap discontinuity are calculated, and the gap discontinuity can be modeled as a r network of a gap admittance and two open-end admittances. Variations of these equivalent admittances with the curvature and the gap spacing are also presented and discussed.

2. THEORETICAL FORMULATION

Figure 1 shows the geometry of a microstrip gap discontinuity on a cylindrical body of radius a. The microstrip line has a

width W and is printed on a cylindrical substrate of thickness h(= 6 - a) and relative permittivity cr. The gap spacing of the microstrip line is S. By assuming that the width and the substrate thickness of the microstrip line are much less than the operating wavelength [2], the currents (incident, reflected, and transmitted currents) far away from the gap discontinuity can be treated as traveling waves of the fundamental propa- gation mode and can be represented by a traveling wave of the form

where p is the effective propagation constant of the mi- crostrip line [ l , 3 ,4] and f(4) is chosen to be uniform. As for the currents near the gap, we have

with

where g,"(z) and g,b(z) are PWS basis functions chosen to represent currents that are higher-order propagation modes; r and T are, respectively, the reflection and transmission coefficient from the gap discontinuity; I," and I: are the unknown expansion coefficients for the PWS function. And, to deal with only real expansion modes and eliminate current discontinuities, g ( t ) is modified to be [2]

z , "=-nd , n = 1 , 2 , 3 ,...,

(4)

(6)

z , b = n d + S , n = 1 , 2 , 3 ,...,

(7)

(8)

sinu, 0 > u > - m a ,

0 < u < m r , elsewhere.

fs"(u) = ( 0 , elsewhere,

sinu,

In Eqs. (5) and (6), d is the half length of the PWS function. The sinusoidal functions in Eqs. (7) and (8) with m / 2 circles in length represent currents that are the fundamental propa- gation mode.

Then, by applying the boundary condition that the electric field on the microstrip line must vanish and following the

260 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 9, No. 5, August 5 1995

p I substrate

Figure 1

J h i t ' microstrip line

The geometry of a cylindrical microstrip gap discontinuity

Figure 2 The equivalent circuit of a cylindrical microstrip gap discontinuity shown in Figure 1

where

In Eqs. (10)-(12), G,,(b, p, k,) is the Green's function in the spectral domain denoting the 2-directed electric field at p = b due to a unit 2-directed-current at (b, 4, I) [I]; f l p ) , g,"(k,), g,b(k,), t ( k , ) , f ( k , ) , f,"(k,), and Lb(k,) are, respec- tively, the Fourier transforms of f(4>, g,"(z>, g,"(z), f:( Pz), f:( p(z - S)), f,"( pz + ?r/2), and f:C p(z - S) - ~ / 2 ) , and are given as

fi(kz) = , - j k Z w / 2 S - ~ f s w 7

Eb(k ) = ejkzw/2k3 f s -b w. (18)

(19)

By solving Eqs. (9), the reflection coefficient r, the trans- mission coefficient T, and the unknown coefficients, 1; and I:, can be obtained, Once r and T are calculated, an equivalent circuit for the gap discontinuity as shown in Figure 2 can be obtained, where Gp + joCp and Gg + joCg are the open-end and gap admittances. These circuit elements can be expressed as

1 - r - T Gp + j w C p = zoo - r + T )

(20)

7 T

where Zo is the characteristic impedance of the microstrip line [3]. The inclusion of Gp and Gg in the equivalent circuit accounts for the radiation and surface wave losses, and Cp and Cg are mainly due to the fringing electric field at the gap discontinuity.

3. NUMERICAL RESULTS AND DISCUSSION To obtain good convergent solutions, four PWS functions (N = 4) and six circles (m = 12) of the sinusoidal functions are used in the moment-method calculation. The reflection coefficients versus frequency for different curvilinear coeffi-

MICROWAVE AND OPTICAL TECHNOLOGY LElTERS / Vol. 9, No. 5, August 5 1995 261

cients, defined to be the ratio of inner to outer radii, that is, i.e., R = a / b [3,4], are first calculated and shown in Figure 3. The results shown are with a gap spacing of 0.8 substrate thickness. It is seen that the reflection coefficient increases with increasing curvilinear coefficient (i.e., the curvature de-

n 0.025 -1

0.02 aJ 0 8 0.015 0

c"a v

.CI % 0*01

0.005

'.. \ R = 0.7 . \ !

" ' . I ' ' , , : ' . . . : . . ' . : ' . , . 1

0 5 10 15 20 25

Frequency (GHz) Figure 3 The reflection coefficient versus frequency for different curvilinear coefficients of R = 0.7, 0.8, and 0.9; E, = 9.9, h = W = 0.635 mm (or 25 mils), S = 0.8 h

-:

':

-: ::

1

0.95

0.9

0.85 -

-: h = 25 mils - S = 2.0 h

S = 0.8 h

S = 0.2 h . - - S = 0 . 4 h

0.7 0 5 10 15 20 25

Frequency (GHz) (a)

0.5 T I h = 25 mils ---SS2.0h

S = 0.8 h S = 0.4 h

- 0.3 S = 0.2 h . .

0.35

O:: 1 0.15

0 5 10 15 20 25

Frequency (GHz) (b)

Figure 4 (a) The reflection coefficient and (b) the transmission coefficient versus frequency for S = 0.2, 0.4, 0.8, and 2.0h; h = W = 0.635 mm (or 25 mils), 6, = 9.9, R = 0.8

creases) and approaches the corresponding planar result [6]. In Figure 4, we also present the reflection and transmission coefficients for different gap spacings with R = 0.8. It is observed that, with increasing gap spacing, the reflection coefficient increases; however, the transmission coefficient however decreases. This indicates that, for the case of a large gap spacing, the gap discontinuity behaves like an open-end discontinuity.

From the obtained r and T , the equivalent capacitance, Cp and Cg, for the gap discontinuity can be calculated. The results versus gap spacing for different values of R are presented in Figure 5. The planar results obtained from the quasistatic method are also shown for comparison. It is found that, when the curvature increases (i.e., the value of R decreases), the capacitance Cp increases; however, Cg de- creases. The dependence of Cp and Cg on the gap spacing also resembles the observations for the planar configuration [5]. It is also observed that the values of C, and C, for a large value of R in general agree with the quasistatic solu- tions for the planar configuration [7], except for the case of a wide gap. The discrepancy at large values of S is probably due to the radiation and surface wave losses that are not included in the quasistatic approach [5]. The results of the conductance, Gp and Gg, are also calculated and shown in Figure 6. In this case, both values of Gp and Gg increase with decreasing values of R . It seems that the curvature increases

0 ~ " " ~ : ~ ~ " : ~ ~ ~ ~ : ~ ' ' ' ~ 0.02 0.12 0.22 0.32 0.42 0.52

s (mm) Figure 5 The gap capacitance, Cp and Cg, versus gap E, = 8.875, h = W = 0.508 mm (or 20 mils), f = 1.3 GHz

0.1 0.09 n

3 0.08 E 8 0.06 - 0.07

3 0.05 8 0.04 2 0.03

0.01 3 0.02

R = 0.7 R = 0.8

h = 0.508 mm

... ..

- . R = 0 . 9

spacing;

0.02 0.12 0.22 0.32 0.42 0.52

s (mm> Figure 6 The gap conductance, Gp and Cg, versus gap spacing; E, = 8.875, h = W = 0.508 mm (or 20 mils), f = 1.3 GHz

262 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 9, No. 5, August 5 1995

the radiation and surface-wave losses. It is also found that G,,, the open-end conductance, is insensitive to the variation of the gap spacing. And Gg, the gap conductance, is relatively more gap-spacing dependent. The value of Gg is also much larger than the value of Gp, a similar result as observed in 151.

4. CONCLUSIONS

This article presents a full-wave solution of microstrip gap discontinuities on a cylindrical surface. A significant variation of the characteristics of cylindrical gap discontinuities due to the curvature variation is observed. The full-wave approach presented here is useful for the characterization of cylindrical microstrip gap discontinuities.

REFERENCES 1. K. L. Wong, H. M. Chen, and R. B. Tsai, “Full-Wave Analysis of

Cylindrical Microstrip Open-End Discontinuities,” IEEE Trans. Microwave Theory Tech., to be published.

2. R. W. Jackson and D. M. Pozar, “Full-Wave Analysis of Mi- crostrip Open-End and Gap Discontinuities,” IEEE Trans. Mi- crowave Theory Tech., Vol. MTT-33, Oct. 1985, pp. 1036-1042.

3. N. G. Alexopoulos and A. Nakatani, “Cylindrical Substrate Mi- crostrip Line Characterization,” IEEE Tmns. Microwave Theory Tech., Vol. MTT-35, Sept. 1987, pp. 843-849.

4. R. B. Tsai and K. L. Wong, “Characterization of Cylindrical Microstriplines Mounted Inside a Ground Cylindrical Surface,” IEEE Trans. Microwave Theory Tech., to be published.

5. H. Y. Yang, N. G. Alexopoulos, and D. R. Jackson, “Microstrip Open-End and Gap Discontinuities in a Substrate-Superstrate Structure,” IEEE Trans. Microwave Theory Tech., Vol. MTT-37,

6. N. G. Alexopoulos and S. C. Wu, “Frequency-Independent Equivalent Circuit Model for Microstrip Open-End and Gap Discontinuities,” IEEE Trans. Microwave Theory Tech., Vol.

7. M. Maeda, “Analysis of Gap in Microstrip Transmission Lines,” IEEE Trans. Microwave Theory Tech., Vol. M’IT-20, June 1972, pp. 390-396.

Oct. 1989, pp. 1542-1546.

M’IT-42, July 1994, pp. 1268-1272.

Received 2-16-95

Microwave and Optical Technology Letters, 9/5,260-263 0 1995 John Wiley & Sons, Inc. CCC 0895-2477/95

SENSITIVITY COMPARISON OF MSM-HBT AND p-i-n - HBT OEIC RECEIVERS Qlng Z. Liu Telecommunications Research Laboratories No. 800 Park Plaza, 1061 1-98 Ave Edmonton, Alberta Canada. T5K 2P7

KEY TERMS OEIC, optical receiuer, sensitiuity, photodetectors, and HBT

ABSTRACT The sensitiuity of heterojunction-bipolar-trans is to^ (HBT-) based OEIC receiwrs with metal-semiconductor metal photodetectors (MSM-PD) or p-i-n photodetectors (p-i-n-PD) are compared. With an accurate small- signal circuit model of an InP/InGaAs HBT, it is shown that sensitiuity of the p-i-n-HBT recewer is higher than that of the MSM-HBT receiver euen though the high-frequency equivalent input noise cumnt density is higher for the f o m r than the latter. However, the HBT receiver using

MSM-PD with light transparent conductingfingers would offer a poten- tial sensitiuity improvement of 0.5 dB over the p-i-n-HBT receiver at 10 Gb i t / s due to an increased respnsiuiv of the MSM-PD. 0 1995 John Wiley & Sons, Inc.

INTRODUCTION To meet the future requirements of very high capacity optical interconnections and high-speed long-wavelength lightwave signal transmission systems, optoelectronic integrated circuits (OEICs) have been pursued worldwide. One of the active research focuses is on the development of high-speed OEIC receivers. OEIC receivers operating above 10 Gbit/s have been reported, using p-i-n photodetector (p-i-n-PD) or metal semiconductor-metal photodetector (MSM-PD) followed by MESFET or HEW preamplifiers 11-31, Due to its ease of integration with FETs and low capacitance, the interdigitated MSM-PDs are expected to play an important role in the future design of broadband OEIC receivers using FETs.

On the other hand, impressive progress has been made on the development of heterojunction bipolar transistors (HBTs), and cutoff frequencies higher than 100 GHz have been reported [41. OEIC receivers with p-i-n-PD and InP/In- GaAs HBTs operating up to 10 Gbit/s have been demon- strated successfully [5, 61. To further optimize the perfor- mance, a small-signal circuit model of the p-i-n-HBT re- ceiver was developed to calculate the total equivalent input noise current density [7], and the accuracy of the model was verified by obtaining a good agreement between the calcu- lated and measured receiver performances reported in [5]. Recently, monolithic integration of MSM-PD and HBT has been proposed for broadband OEIC receiver design, due to the excellent high-frequency performance of both devices 18, 91.

In this article the noise contributions and sensitivities of p-i-n-HBT and MSM-HBT OEIC receivers are compared and discussed using an accurate small-signal equivalent cir- cuit model of an InGaAs HBT. We will show that a potential sensitivity improvement can be made for the MSM-HBT receiver over its counterpart with p-i-n-PD by using a MSM- PD with light transparent conducting fingers.

SENSITIVITY CALCULATION An equivalent circuit of the input stage of the p-i-n-HBT OEIC receiver reported in [5] is shown in Figure 1. In [7], analytical expressions were derived for calculating the equiva- lent input noise current density Si of the receiver front end. The equivalent input noise current power is given by

MICROWAVE AND OPTICAL

where B is the bandwidth of the receiver and

s? = s; + SZbi + qi + S i i + s& + SZi + s:,

Sji: Equivalent input noise current density of photodiode

Sbbi: Equivalent input noise current density of base resis-

Sf i : equivalent input noise current density of feedback

SBi: equivalent input noise current density of base current SCi: equivalent input noise current density of collector

series resistance

tance

resistance

current

TECHNOLOGY LElTERS / Vol. 9, No. 5, August 5 1995 263