dP

dz� �2�P0

� � Prad/ 2P0. (14)

3. NUMERICAL RESULTS

For antenna applications, interest lies mostly in the case of single-beam radiation. Therefore, the case where only the n � �1 spaceharmonic of the TE01 mode radiates into free space is investigated.

In order to verify the effectiveness and suitability of the presentapproach, we investigate the leakage properties of three gratingantennas. Antenna 1, proposed in 1986 [1], is shown in Figure1(b). Antenna 2 is the proposed omni-directional antenna which isobtained by gloving a uniform shell layer to Antenna I, shown inFigure 1(c). Antenna 3 is the new omni-directional leaky waveantenna with double dielectric gratings, shown in Figure 1(a).

Figure 2(a) shows the effect of different antenna structures onthe leakage constant. For comparison, the same overall size ofthese antennas are chosen. The solid lines represent the exactresults of the rigorous mode matching method [2, 6, 7], while thedashed lines are obtained by present perturbation analysis, whichagree quite well with the exact results. The increase of the thick-ness of grating layer (tg1) can be viewed as adding more pertur-bation to the structure. Because of the strong perturbation effect onthe guided wave caused by the two grating layers, the leakageconstant � of the double grating antenna continues to be largerthan that of either single antenna, with or without a shell, for theregion of tg1 � 0.25�, and especially in the region of tg1 �0.06�, where the leakage constant of the double grating antenna ismore than ten times those of the single grating antennas. It is seenthat the leakage constant of the double grating antenna has beenreached when the maximum leakage constant of the single grating,tg1, is very small. This clearly indicates that under conditions ofthe same radiation strength, the length of the double gratingantenna can be considerably reduced.

Figure 2(b) shows variations of the leakage constant with thedielectric constant of the outer grating layer. When the dielectricconstant of the outer grating layer is small, the leakage constantincreases as the dielectric constant becomes larger. Nevertheless,leakage decreases as the thickness increases further, after �2 � 2.8.This is because the outer grating layer concentrates the fields of thesurface wave in the inner grating layer, in this case, the perturba-tion effect of the inner grating layer on the guided wave isstrengthened. However, as the thickness of outer grating layerfurther increases, the fields of the surface wave enter the shell andouter grating layers and become weaker in the inner grating layer,leakage becomes weaker as well.

4. CONCLUSIONS

In this paper, the leakage characteristics of a new omni-directionalleaky-wave antenna with multi-layer structure are analyzed usingthe improved perturbation method. The analysis of the electromag-netic fields are described in terms of a radial transmission-linenetwork, which brings considerable physical insight into the over-all behavior of the leaky-wave antenna. The expression for theleakage constant is given in a closed form so that the designprocedure of the antenna becomes very convenient. It is found thatthe introduction of the double gratings increases the leakage con-stant considerably, as well as the efficiency of the antenna, whichthus arises effectively while retaining all the same advantages ofthe original omni-directional antenna presented in [1]. This prop-erty is of essential significance for the case where size and weightlimitation of a system is strictly required.

REFERENCES

1. S.T. Peng, X. Shanjia, and F.K. Schwering, Omni-directional periodicrod antenna, Dig IEEE Int Antennas Propagat Symp, 1986, pp. 697–700.

2. X. Shanjia, M. Jianhua, S.P. Peng, and F.K. Schwering, A millimeterwave omni-directional circular dielectric rod grating antenna, IEEETrans AP AP-39 (1991), 883–891.

3. X. Shanjia and M. Zhewang, Improved perturbation analysis of milli-meter-wave omni-directional periodic dielectric rod antenna, Int J In-frared Millimeter Waves 11 (1990), 727–743.

4. X. Shanjia and M. Zhewang, Network analysis of eigenvalue problemfor radically inhomogeneous cylindrical dielectric waveguides, J Infra-red Millimeter Waves 10 (1992), 185–193.

5. X. Shanjia and G. Hong, Double dielectric grating leaky-wave antenna—improved perturbation analysis, Int J Infrared Millimeter Waves 10 (1989),1–9.

6. J. Hengzhen and X. Shanjia, Mode matching analysis for an omni-directional antenna consisting of circular rod corrugations gloved witha dielectric shell for millimeter wave applications, (Submitted to IEEETransactions on Antennas and Propagation).

© 2003 Wiley Periodicals, Inc.

AN EFFICIENT IMPLEMENTATION OFTHE PML FOR TRUNCATING FDTDDOMAINS

Omar Ramadan1 and Abdullah Y. Oztoprak2

1 Computer Engineering DepartmentEastern Mediterranean UniversityGazi Magusa, Mersin 10, Turkey2 Electrical and Electronic Engineering DepartmentEastern Mediterranean UniversityGazi Magusa, Mersin 10, Turkey

Received 17 June 2002

ABSTRACT: An efficient implementation of the perfectly matchedlayer (PML) is presented for truncating lossy and lossless finite-dif-ference time-domain (FDTD) computational domains. The implemen-tation is based on the stretched coordinate PML formulations and onthe derivation of the wave equation in the PML region. Significantsavings in computational time and memory storage requirements areachieved. In addition, the FDTD implementation of the new formula-tions is simpler than the conventional PML formulations. Two dimen-sional numerical tests have been carried out to validate the proposedformulations. © 2003 Wiley Periodicals, Inc. Microwave OptTechnol Lett 36: 55– 60, 2003; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/mop.10669

Figure 2 (a) Variation of normalized leakage constant �/k0 with tg forthree type of antennas; (b) variation of normalized leakage constant �/k0

with �2 for double dielectric grating antenna shown in Fig. 1(a)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003 55

Key words: finite-difference time domain; perfectly matched layer;wave equation

1. INTRODUCTION

Since the work of Yee [1], the finite-difference time-domain(FDTD) method has been widely used for solving Maxwell’s curlequations [2]. The FDTD method has also been extended forsolving the scalar wave equation [3]. It has been shown in [3] thatthis method, which is called the wave-equation FDTD (WE-FDTD), is both mathematically and numerically equivalent toYee’s algorithm in source free domains. In addition, unlike Yee’sFDTD algorithm, the WE-FDTD method allows computing anysingle field component without the necessity of computing otherfield components. Therefore, significant savings in the computa-tional time and memory storage are achieved [3].

An important issue for solving open region problems using theFDTD method is the truncation of the computational domains. Theperfectly matched layer (PML), introduced by Berenger [4, 5], hasbeen shown to be an effective tool for truncating lossless FDTDdomains. Recently, 1D modal PML formulations, based on theWE-FDTD method, have been published for truncating losslesswaveguide structures [6]. For lossy media, however, these formu-lations can not be applied directly.

In this paper, new PML formulations are introduced for trun-cating lossless as well as lossy FDTD domains. These formulationsare based on the stretched-coordinate PML approach [7] and on thederivation of the wave equation in the PML region. This method,as in the case of WE-FDTD, requires the solution for a single fieldcomponent in the PML region. Therefore, significant savings inmemory storage and computational time requirements areachieved. In addition, the FDTD implementation of the presentedformulations is simpler than the conventional PML formulations.Numerical tests have been carried out in two dimension to validatethe proposed formulations.

2. FORMULATIONUsing the stretched coordinate PML formulations [7], the frequen-cy-domain-modified Maxwell’s equations in the PML region canbe written as

�s � E � �j��oH (1)

�s � H � j��o�1 � /j��o�E, (2)

where is the conductivity of the FDTD domain and

�s � ax

1

Sx

x� ay

1

Sy

y� as

1

Sz

z, (3)

where Sx, Sy, and Sz are the stretched coordinate variables in thex, y, and z directions and chosen as [7]:

S� � 1 � � /j��o, �� � x, y, z�, (4)

where � is the conductivity profile in the PML region along the� direction. Due to the frequency dependence of the stretchedcoordinate variables, the FDTD implementation of Eqs. (1) and (2)necessitate splitting all of the field components [7]. Hence, ingeneral, it is required to solve for 12 field subcomponents and sixadditional integral terms that couples the conductivity of the com-putational domain and the conductivity of the PML � [7].Below, alternative formulations based on the WE-FDTD algorithmare proposed for the FDTD implementation of Eqs. (1) and (2).

Taking the curl of Eq. (1),

�s � �s � E � �j��o�s � H (5)

and substituting Eq. (2) in Eq. (5), we obtain

�s � �s � E � �c�2� j��2�1 � /j��o�E, (6)

where c is the speed of light defined as c � 1/��o�o. Using thefollowing curl identity

�s � �s � E � �s��s � E� � �s2E, (7)

and assuming a source-free PML region �s � E � 0, Eq. (6) canbe written as

�s2E � c�2� j��2�1 � /j��o�E. (8)

Equation (8) represents the modified Helmholtz wave equation inthe PML region. Rewriting the operator �s

2 as

�s2 � �

��x,y,z

1

S�

�

1

S�

�, (9)

Eq. (8) can be written in terms of the Ez field component, forexample, as

c�2� j��2�1 � /j��o� Ez � ���x,y,z

1

S�

�

1

S�

�Ez (10)

or

c�2� j��2�1 � /j��o�

� Ez � ���x,y,z

j�

j� � �/�o

�

j�

j� � � /�o

�Ez. (11)

To achieve good electromagnetic wave absorption, the selec-tion of S� (� � x, y, z) depends on three different PML regions:face, edge, and corner regions [5, 7] as shown in Figure 1. First,consider as an example the x-face PML region. In this region, thestretched coordinate variables are chosen as Sx � 1 � x/j��o,and Sy � Sz � 1 [5, 7]; hence, Eq. (11) can be written as

Figure 1 The PML regions

56 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003

c�2� j��2�1 � /j��o� Ez �j�

j� � x /�o

x

j�

j� � x /�o

xEz

� � 2

y2 �2

z2�Ez, (12)

which can be rearranged as

c�2j��1 � /j��o� Ez �1

j� � x/�o

x

j�

j� � x/�o

xEz

�1

j� � 2

y2 �2

z2�Ez. (13)

Introducing the following auxiliary variables

Fzx �1

j� � x /�o

xPzx, (14)

Pzx �j�

j� � x /�o

xEz, (15)

and

Iyz �1

j� � 2

y2 �2

z2�Ez, (16)

Eq. (13) can be written as

c�2� j� � /�o� Ez � Fzx � Iyz. (17)

Transforming Eq. (17) from the frequency domain into the timedomain using the relation j� 3 /t, we obtain

c�2�

t� /�o�Ez � Fzx � Iyz, (18)

where the auxiliary variables Fzx and Iyz are computed upontransforming Eqs. (14)–(16) into the time domain, using the rela-tions j� 3 /t and 1/j� 3 0

t , as

�

t� x /�o�Fzx �

xPzx, (19)

�

t� x /�o�Pzx �

2

xtEz, (20)

and

Iyz � �0

t � 2

y2 �2

z2�Ez dt. (21)

Following the standard Yee’s algorithm for descretizing the timederivative in Eq. (18), and averaging the term (/�o) Ez in time att � n � 1 and t � n values, Ez can be written in FDTD form as

Ezi, j,k�1/ 2

n�1 � GaEzi, j,k�1/ 2

n � Gb�Fzxi, j,k�1/ 2

n�1 � Iyzi, j,k�1/ 2

n�1 �, (22)

where Ga � (1 � �t/ 2�o)/(1 � �t/ 2�o), Gb � 1/(1 ��t/ 2�o), �t is the time step, and the auxiliary variables Fzx

n�1

and Iyzi, j,k�1/ 2

n�1 are obtained by discretizing Eqs. (19)–(21) as

Fzxi, j,k�1/ 2

n�1 � gzx1�i� Fzxi, j,k�1/ 2

n � gzx2�i��Pzxi�1/ 2, j,k�1/ 2

n�1 � Pzxi�1/ 2, j,k�1/ 2

n�1 �,

(23)

Pzxi�1/ 2, j,k�1/ 2

n�1 � gzx1�i � 1/ 2� Pzxi�1/ 2, j,k�1/ 2

n � gzx2�i � 1/ 2�

� �Ezi�1, j,k�1/ 2

n�1 � Ezi�1, j,k�1/ 2

n � Ezi, j,k�1/ 2

n�1 � Ezi, j,k�1/ 2

n �, (24)

Iyzi, j,k�1/ 2

n�1 � Iyzi, j,k�1/ 2

n � �c�t/��2

� �Ezi, j�1,k�1/ 2

n � Ezi, j�1,k�1/ 2

n � Ezi, j,k�3/ 2

n � Ezi, j,k�1/ 2

n � 4Ezi, j,k�1/ 2

n �, (25)

where �x(i) � (�tx(i))/ 2�o, gzx1(i) � (1 � �x(i))/(1 ��x(i)), gzx2(i) � (c�t/�)/(1 � �x(i)), and � � �x � �y � �zis the cell size in the x, y, and z directions, respectively. Similarexpressions can be obtained in the other face-PML regions. Itshould be pointed out that direct implementation of Eq. (22)

TABLE 1 Memory Storage and Computational Time Requirements Per Cell in the Face, Edge, and Corner PML Regions for theWE-PML and for the Stretched-Coordinate PML Formulations of [7]

PML RegionMethod

Face Edge Corner

[7]

WE-PML

[7]

WE-PML

[7]

WE-PML

Case I Case II Case I Case II Case I Case II

Addition 44 16 32 56 19 38 60 21 42Multiplication 19 7 14 26 11 22 30 14 28Storage 12 5 10 16 7 14 18 8 16

Figure 2 Local error for the PML/computational domain interface aty � �25� as observed at time 100�t obtained using the stretched-coordinate PML and the WE-PML formulations for PML[8, 2, 0.001] andlossless FDTD domain

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003 57

requires five storage elements per cell in the face PML region:Ez

n�1, Ezn, Fzx

n�1, Pzxn�1, and Iyz

n�1.Next, consider the xy-edge PML region, as shown in Figure 1.

In this region, the stretched coordinate variables are chosen asS� � 1 � �/j��o, (� � x, y), and Sz � 1. Then, the Helmholtzwave equation shown in Eq. (11) becomes

c�2� j��2�1 � /j��o�

� Ez � ���x,y

j�

j� � � /�o

�

j�

j� � � /�o

�Ez �

2

z2 Ez. (26)

Eq. (26) can be rearranged as

c�2j��1 � /j��o� Ez � ���x,y

1

j� � � /�o

�

�

j�

j� � � /�o

�Ez �

1

j�

2

z2 Ez, (27)

which can be written as

c�2� j� � /�o� Ez � ���x,y

Fz� � Iz, (28)

where the auxiliary variables Fz�, (� � x, y), and Iz are given by

Fz� �1

j� � � /�o

�Pz�, (29)

Pz� �j�

j� � � /�o

�Ez, (30)

and

Iz �1

j�

2

z2 Ez. (31)

Eq. (28) can be written in the time domain as

c�2�

t� /�o�Ez � �

��x,y

Fzx � Iz, (32)

where the variables Fz�, (� � x, y), and Iz are written in timedomain as

tFz� � � /�oFz� �

�Pz�, (33)

tPz� � � /�oPz� �

2

�tEz, (34)

and

Iz � �0

t 2

z2 Ezdt. (35)

Eq. (32) can be written in the FDTD form as

Ezi, j,k�1/ 2

n�1 � GaEzi, j,k�1/ 2

n � Gb� ���x,y

Fz�i, j,k�1/ 2

n�1 � Izi, j,k�1/ 2

n�1 � , (36)

Figure 3 Global error in the computational domain obtained using thestretched-coordinate PML and the WE-PML formulations for PML[8, 2,0.001] and lossless FDTD domain

Figure 4 Local error for the PML/computational domain interface aty � �25� as observed at time 100�t obtained using the stretched-coordinate PML and the WE-PML formulations for PML[8, 2, 0.001] andlossy FDTD domain with the parameters �r � �r � 1 and � 0.01

Figure 5 Global error in the computational domain obtained using thestretched coordinate PML and the WE-PML formulations for PML[8, 2,0.001] and lossy FDTD domain with the parameters �r � �r � 1 and �0.01

58 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003

where Fz�i, j,k�1/ 2

n�1 (� � x, y) and Izi, j,k�1/ 2

n�1 can be obtained bydiscretizing Eqs. (33)–(35) in a similar manner as in Eqs. (23)–(25). Similar expressions can be obtained in the other edge PMLregions. The implementation of Eq. (36) requires seven storageelements per cell in the edge PML region: Ez

n�1, Ezn, Fzx

n�1, Pzxn�1,

Fzyn�1, Pzy

n�1, and Izn�1.

Finally, in the corner-PML regions, all of the stretched coor-dinate variables overlap and chosen as S� � 1 � �/j��o, (� �x, y, z), and hence the wave equation in Eq. (11) can be rearrangedas

c�2j��1 � /j��o�Ez � ���x,y,z

1

j� � � /�o

�

j�

j� � � /�o

�Ez,

(37)

which can be written as

c�2� j� � /�o� Ez � ���x,y,z

Fz�, (38)

where the auxiliary variables Fz�, (� � x, y, z), are given by

Fz� �1

j� � � /�o

�Pz�, (39)

Pz� �j�

j� � � /�o

�Ez. (40)

Eq. (38) can be written in the time domain as

c�2�

t� /�o�Ez � �

��x,y,z

Fzx, (41)

where the variables Fz�, (� � x, y or z) are given by

tFz� � � /�oFz� �

�Pz�, (42)

tPz� � � /�oPz� �

2

�tEz. (43)

Eq. (41) can be descretized as

Ezi, j,k�1/ 2

n�1 � GaEzi, j,k�1/ 2

n � Gb ���x,y,z

Fz�i, j,k�1/ 2

n�1 , (44)

where Fz�i, j,k�1/ 2

n�1 , (� � x, y, z), are obtained by discretizing Eqs.(42) and (43) along x, y, and z directions as in Eqs. (23)–(24). Theimplementation of Eq. (44) requires eight storage elements per cellin the corner PML region: Ez

n�1, Ezn, Fzx

n�1, Pzxn�1, Fzy

n�1, Pzyn�1,

Fzzn�1, and Pzz

n�1.It is interesting to notice that for truncating lossless FDTD

domains, that is, for � 0, the coefficients Ga and Gb are reducedto unity. It should also be mentioned that in the FDTD computa-tional domain we can use either the WE-FDTD algorithm (forsource free regions) or the Yee’s algorithm [3]. In the latter case,two field components are required for each cell inside the PMLregions, because at the boundary between the FDTD domain andthe PML region there are two tangential field components.

Computational time and memory storage requirements, per cellin the face, edge, and corner PML regions, are presented in Table

1 for the wave-equation PML (WE-PML) formulations presentedin this paper and the stretched coordinates PML formulations of[7]. Two cases were considered for the new formulations: Case Iis for truncating the WE-FDTD method, and Case II is for trun-cating the Yee’s FDTD algorithm. As seen from Table 1, for bothcases, the WE-PML requires less memory storage and computa-tional time compared with the stretched coordinate PML. In thecase of truncation the WE-FDTD (Case I), significant reduction inthe memory storage and the computational time is achieved.

3. NUMERICAL STUDY

Numerical tests have been carried out in a two dimensional domainfor the TM case. A point source excites a 100� � 50� isotropichomogeneous computational domain at its center with �t � 25psand � � �x � �y � 2c�t. The excitation pulse used is thederivative of the pulse used in [8]. This pulse was preferred, as ithas low numerical grid dispersion [9]. The reference FDTD solu-tion, having no reflection errors from the domain boundaries, iscalculated using a much larger computational domain (400� �400�).

The computational domain was terminated by eight PML layerswith a quadratic conductivity profile and a maximum theoreticalreflection coefficient of 10�5 at normal incidence, expressed asPML[8, 2, 0.001] in Berenger’s notation [4]. Figures 2 and 3 showthe local and the global errors, as defined in [8], for lossless FDTDdomain. The local error is for the PML/computational domaininterface at y � �25� as observed at time 100�t. The resultsshown are for the WE-PML formulations and for the stretchedcoordinate PML formulations. Clearly, it can be seen from Figures2 and 3 that the WE-PML gives very similar error performance asthe stretched coordinate formulations. Very small difference in thelocal error results has been observed. This is expected as the newformulations involve discretization of higher order space deriva-tives. The same test has been carried out for lossy FDTD domainwith the parameters �r � �r � 1, and � 0.01. Figures 4 and 5show the local and the global errors for this test. Similar to thelossless case, the WE-PML shows similar absorbing performanceas the stretched coordinate PML formulations.

4. CONCLUSION

New, simple, and efficient PML formulations have been intro-duced for truncating lossless and lossy FDTD computational do-mains. The method is based on the stretched-coordinate PMLformulations and on deriving the wave equation in the PMLregion. This method requires solutions only for one or two fieldcomponents in the PML regions. Hence, significant reductions inthe computational time and memory storage requirements areachieved. Good absorbing performance has been observed for bothlossless as well as lossy domains.

REFERENCES

1. K.S. Yee, Numerical solution of initial boundary value problems in-volving Maxwell’s equations in isotropic media, IEEE Trans AntennasPropagat 14 (1966), 302–307.

2. A. Taflove, Computational electrodynamics: The finite-difference time-domain method, Artech House, Boston, London, 1995.

3. P.H. Aoyagi, J.-F. Lee, and R. Mittra, A hybrid Yee algorithm/scalar-wave equation approach, IEEE Trans Microwave Theory Tech 41(1993), 1593–1600.

4. J.-P. Berenger, A perfectly matched layer for the absorption of electro-magnetic waves, J Comput Phys 114 (1994), 185–200.

5. J.-P. Berenger, Three-dimensional perfectly matched layer for the ab-sorption of electromagnetic waves, J Comput Phys 127 (1996), 363–379.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003 59

6. K.Y. Jung and H. Kim, An efficient formulation of a 1-D modal PMLfor waveguide structures, Micro Opt Technol Lett 21 (1999), 48–51.

7. Q.H. Liu, An FDTD algorithm with perfectly matched layer for con-ductive media, Micro Opt Technol Lett 14 (1997), 134–137.

8. P.A. Tirkas, C.A. Balanis, and R.A. Renaut, Higher order absorbingboundary conditions for the finite-difference time-domain method,IEEE Trans Antennas Propagat 40 (1992), 1215–1222.

9. O.M. Ramahi, Complementary operators: A method to annihilate arti-ficial reflections arising from the truncation of the computational do-main in the solution of partial differential equations, IEEE Trans An-tennas Propagat 43 (1995), 697–704.

© 2003 Wiley Periodicals, Inc.

DC-2.1 GHz CMOS MULTIPLEFEEDBACK TRANSIMPEDANCEAMPLIFIERS WITH HIGH DYNAMICRANGE AND LINEARITY

Ming-Chou Chiang,1 Ping-Yu Chen,1 Shey-Shi Lu,1 andChin-Chun Meng2

1 Department of Electrical EngineeringNational Taiwan UniversityTaipei, Taiwan2 Department of Electrical EngineeringNational Chung-Hsing UniversityTaipei, Taiwan

Received 4 June 2002

ABSTRACT: A CMOS wideband transimpedance amplifier with multi-ple-resistive feedback loops is reported for first time. This amplifier wasfabricated by the 0.35-�m CMOS process. Multiple-feedback techniquewas used to achieve terminal impedance matching and wide bandwidth.The experimental results showed that a transimpedance gain of 51.7dB and a 3-dB bandwidth of 2.1 GHz were obtained. High input P1dB

of �5 dBm and input IP3 of 1 dBm were also measured. The high dy-namic range and high linearity achieved were attributed to the multiplefeedback technique used. © 2003 Wiley Periodicals, Inc. Microwave OptTechnol Lett 36: 60–61, 2003; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/mop.10670

Key words: wideband; multiple feedback; CMOS; transimpedance am-plifier

1. INTRODUCTION

Transimpedance amplifiers are widely used in optical-fiber com-munications as the preamplifier. The resistive feedback techniqueis the most popular method to achieve the required wide band-width. Recently, the multiple feedback technique has been pro-posed and used in the design of wideband amplifiers [1] to achievewide bandwidth and impedance matching simultaneously. Onekind of transimpedance amplifier that also utilizes the multiplefeedback loops is the so-called Kukielka configuration [2], whichhas been implemented by the silicon bipolar [3], GaAs HBT [4],and InP HEMT processes [5] with excellent performance. Becauseof the advances in CMOS technology, it is now possible for CMOScircuits to be operated in the GHz range. However, to our knowl-edge, there is still no report of CMOS transimpedance amplifiersusing Kukielka configuration in the literature. Therefore, in thispaper we present the first CMOS transimpedance amplifiers usinga multiple-resistive feedback technique and fabricated by 0.35-�mprocess. The experimental results showed that a transimpedancegain of 51.7 dB and a 3-dB bandwidth of 2.1 GHz were

achieved. High input 1-dB compression point of �5 dBm andinput IP3 of 1 dBm were also obtained.

2. PRINCIPLE OF CIRCUIT DESIGN

The circuit topology of the transimpedance amplifier with multiplefeedback loops is shown in Figure 1. The input stage consisting ofa single transistor M1 with local series feedback (Rs1) drives theoutput stage composed of a transistor with local shunt (Rf 2) andlocal series feedback (Rs2). There is an overall shunt-series feed-back loop composed of resistors Rs2 and Rf 1. Clearly, this ampli-fier can be approximated by a two-pole system with open-looppoles of �p1, and �p2. Local series feedback resistor Rs1 is usedto adjust �p1, while local shunt Rf 2 and series resistors Rs2 areused to adjust �p2 so that the two poles are brought to be coinci-dent. Then the global feedback resistor Rf 2 is selected for arequired loop gain to attain the maximally flat condition. However,since this amplifier tends to give over-damped characteristics [2],source peaking capacitors Cs1 and Cs2 [2, 6] are connected inparallel with Rs1 and Rs2, respectively, to overcome this problem.

3. CIRCUIT IMPLEMENTATION AND MEASURED RESULTS

The CMOS matched-impedance wide-band transimpedance am-plifier was fabricated by 0.35-�m process provided by the TaiwanSemiconductor Manufacturing Corporation (TSMC). The die pho-tograph of the finished circuit is shown Figure 2. Note that thecircuit (excluding the patterns for testing) only occupies a verysmaller area of 500 �m � 450 �m because no inductor was used.

Figure 1 The ac schematics of the Kukielka amplifier

Figure 2 The die photograph of the CMOS matched-impedance ampli-fier. The chip size is only 450 �m � 500 �m, excluding the testing patterns

60 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 1, January 5 2003