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Technique—Application to Filter Design. Marc Le Roy, André Pérennec, Serge Toutain and Léon Claude Calvez. Abstract—In this paper, an original approach ...
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 9, SEPTEMBER 1999

The Continuously Varying Transmission-Line Technique—Application to Filter Design Marc Le Roy, Andr´e P´erennec, Serge Toutain and L´eon Claude Calvez

Abstract—In this paper, an original approach to analyze and optimize continuously varying transmission lines (CVTL’s) is used to design planar microwave filters. The method is based on exact solutions of the telegrapher’s relations of nonuniform transmission lines. The CVTL scattering parameters are investigated in detail in the frequency domain. In order to validate the method, several different CVTL’s are built using microstrip and coplanar-waveguide technology. Measured data are presented and compared with theory over a wide frequency range. Index Terms— Nonuniform transmission lines, microwave filters.

I. INTRODUCTION

N

ONUNIFORM transmission lines have been widely studied in recent years, and their applications, such as impedance-matching devices, pulse transformers, resonators, directional couplers, and filters have been described by many authors. For commonly used nonuniform transmission lines, i.e., linear [1], exponential [2], [3], parabolic [4], [5], cosinesquared [5], and hyperbolic transmission lines supporting a TEM mode, the theory is well established in frequency and time domain. The common method of analysis for all these structures is usually based on the equivalent TEM transmission-line model. In the past, the analysis of widthmodulated microstrip periodic structure was carried out by Nair and Mallick [6]. More recently, Roberts and Town [7] proposed an original method using the theory of inverse scattering to design a microwave filter with smooth variation of its profile. The filter designed by this method seems to show distinct advantages over other widespread techniques (given maximally flat or Chebyshev response function). However, this method is complicated and requires numerical techniques that cannot avoid truncation errors. Moreover, a lack of generality prevents it from being used in applications other than filter designs. This paper presents a new continuously varying transimission-line (CVTL) design method, applied here to filtering functions (low pass). However, this technique can

be expanded to other systems such as microwave amplifiers, active filters, coupled lines, etc. Moreover, the analysis is easier to implement than a numerical electromagnetic approach and takes less computation time. The global variation of the line depends on a set of discrete points. The continuous shape of the variation is then ensured by cubic spline interpolation and, between two consecutive discrete points, the variation of the characteristic impedance follows a cubic polynomial. The telegrapher’s equations of the part of the line (between two consecutive points) are solved with two expansions in power series. The frequency-domain scattering parameters of a lossless transmission line with such a variation are then obtained by the boundary conditions. As a result, the -parameters matrix of the whole CVTL is easily computed. Applied to a low-pass filter, the performances are optimized since the line shape is described by a finite number of discrete points. This method is applicable to each guiding structure whose fundamental mode is TEM or quasi-TEM. The method is illustrated with examples of nonuniform microstrip [8] and coplanar-waveguide (CPW) filters designed by this approach. II. FORMULATION

LINE SHAPE

The first step of the design is to describe the characteristic impedance variation of the line. A cubic spline interpolation was chosen for various reasons. First, it allows one to create a curve passing through a set of discrete points that minimize the undulations between two points. Second, the cubic spline expression is generally sufficient to describe or approximate a large class of variations or undulations. Cubic spline interpolation can also be applied to regular or irregular spaces between discrete points. It also ensures the continuity of the first and second derivatives. Thus, the impedance variation between two discrete points is described by the following cubic polynomial: (1) with

Manuscript received June 16, 1998. M. Le Roy, A. P´erennec, and L. C. Calvez are with the Laboratory for Electronics and Communication Systems (LEST), UMR Centre National de la Recherche Scientifique (CNRS 6616), University of Brest, 29285 Brest, France. S. Toutain was with the Laboratory for Electronics and Communication Systems (LEST), Centre National de la Recherche Scientifique (CNRS), University of Brest, 29285 Brest, France. He is now with the Electronics and Computing Engineer School, Electronics and Computing Systems, Centre National de la Recherche Scientifique (CNRS 63), 44306 Nantes, France. Publisher Item Identifier S 0018-9480(99)06593-X.

OF THE

and

0018–9480/99$10.00  1999 IEEE

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The solution of the voltage equation is given as follows: (8) , where and are two constants and the following two power series expansions:

represent

(9) and

Fig. 1. Scattering-parameter configuration of a cubic transmission line.

(10) where varies from zero to . is the line length between and , which represent the ordinate of the th th discrete points used to define the cubic spline and and are the second derivative of the interpolation. at these respective points. impedance variation

The current relation is expressed by means of the two solutions of the voltage relation (8) (11)

III. SCATTERING PARAMETERS IN FREQUENCY DOMAIN is the propagation velocity through the line, and and are the first derivatives of and , respectively. The convergence of these four expansions is ensured by the Fuchs theorem [9]. Two recurrence relations of order five can be obtained by including (9) and (10) into (8) and then into (7), and by equating terms of the same degrees. and values The series expansions are computed until and . are negligible with respect to the summations The accurate knowledge of the voltage and current along the line leads to the determination of its scattering parameters. and in relation (8) and (11) are determined by the boundary conditions [2], [3]. Thus, at the output end where

A. Cubic Polynomial Transmission Line A nonuniform transmission line with a cubic polynomial variation of its characteristic impedance can be described as a set of scattering parameters. These scattering parameters link and to the reflected waves the incident waves and , as illustrated in Fig. 1 (2) (3) Here, a lossless nonuniform line is considered. It extends to ; this line being bounded by two from and are, respectively, different reference lines. the characteristic impedances of these lines at the left- and right-hand sides. The cubic line has a characteristic impedance that varies in accordance with (1). This relation can also be expressed in the form of

(12) where relation,

. From this is expressed as follows:

(4)

(13)

with (5)

, and similarly for with at the input end, we have

,

, and

; thus,

and (14)

(6) is equal to at the input end of the nonuniform and are the inductance and transmission line, where . Using the TEM transmission-line model capacitance at and current and (4)–(6), the frequency-domain voltage along this line are defined as

(7)

where . as

is the input impedance, which is expressed by using is defined as the reflection coefficient at the input end

(15) The cubic line is terminated with a matching transmission line , then the reflected wave of characteristic impedance vanishes. In this case, relation (3) indicates that is to , which are, respectively, equal to the ratio of

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and (incident wave) without forgetting the normalized impedance factor (16)

(17) Using this relation and the , and expressed from , ,

At the input end, coefficient

is equal to the

ratio, as follows:

is directly

(18) scattering reflection Fig. 2. Flowchart of the design procedure.

(19) equals . Following the reciprocity principle, calculation. By The same method is applied to the reflection coefficient at the using boundary conditions, the output impedance are determined from output end and the ratio. The -parameter is expressed as a function of the , , , and for in the same way as it was done . for The reference impedance must be changed before using -parameters in a global system. Changing-impedance the reference formulas are used to define each section of the line . with a common reference, e.g., B. Characterization and Optimization of the Whole Line To determine the global scattering parameters of the CVTL defined by cubic spline interpolation, each scattering matrix of a cubic polynomial line is transformed by using chain matrices, and then cascaded to be converted again into scattering parameters. The cubic spline interpolation is also modified at the CVTL input and output ends to smooth the access lines’ connections. At this step, the CVTL analysis method based on scattering parameters is complete. According to the approach used in the design process, the previous theory has to be coupled with an optimization algorithm to connect the global shape of the line to the objective. This theory is well suited for optimization methods because the shape of the line is described by a set of discrete points. A computer program based on the described technique associated with an optimization algorithm has been written to design an optimum low-pass filter with regard to the objective that is set. The optimization allows one to express -parameter of the line in function of the position the and value of the discrete points, respectively. The input data file consists of the following. is the number of discrete points. The impedance 1) and the position (relation (1) for varying from one

to ) being the components of the optimization vector. They can be generated by the following different ways. a) These data are given by the user by approximately taking the size of undulations corresponding to the size of classical synthesis elements. and values are randomly calculated. b) These values centered on 50 alternate above The and below this value to help the starting of optimization convergence and fix the filter order. is the characteristic to achieve, is the 2) number of frequency points where the error between and will be evaluated. The flow chart of the design of a low-pass filter is shown in Fig. 2. Our optimization method is based on the Levenberg–Marquardt and More algorithm [10]. It uses the least-square criteria. Two solutions are available for the optimization of the discrete point abscissa. a) A coefficient, which is a multiplier of the global line length, is also optimized; acts as a dilatation factor. b) Each distance between two discrete points is simultaneously optimized, while, of course, slightly increasing optimization duration. This second solution also doubles the number of freedom degrees and brings a further interest for the spurious response of the -parameter out of the filter bandpass. The number of freedom degrees associated with numerous random generations of the input vector gives different transmission-line shapes for frequency responses that can be equivalent. The choice of the shape is then determined by the more feasible profile. IV. RESULTS

AND

DISCUSSION

A. CVTL Filter in Microstrip Technology To validate the design theory by experiment, a stopband filter was first optimized and carried out. The center frequency of the filter is 3 GHz, with a 2-GHz bandwidth. The

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(a)

(b)

(c) Fig. 3. (a) Impedance variation of a stopband filter described by 11 discrete points. (b) Shape of the microstrip filter (not to scale). (c) of the stopband filter.

optimization goal was to reach 0.001 dB in the band for (decibels), and 30 dB out of the band. Fig. 3(a) shows the smooth impedance profile optimized by the proposed method. The filter was made in microstrip technology using Wheeler and Hammerstadt design relations [13]. The effect of strip thickness and frequency (dispersion) on the effective permittivity was taken into account. The substrate used was Isoclad GR6, which has a relative permittivity of 6 0.15 and a thickness of 0.635 mm. The shape of this microstrip line is given in Fig. 3(b). The comparison between the measured and simulated reis displayed in Fig. 3(c). The measurements sponses of were carried out with an appropriate thru-reflection line (TRL) calibration kit, which allowed us to accurately measure the phase parameter for a uniform line. An average of the substrate permittivity in the considered bandwidth was then established at 6.4. Furthermore, precise measurements of the substrate thickness gave a value of 0.665 mm. The results presented in Fig. 3(c) take these values into account. As shown in Fig. 3(c), the agreement between the theoretical and measured response is excellent. This first realization validates the method used to analyze and optimize CVTL and filters. B. CVTL Filters in CPW Technology This new theory makes the design of other kinds of filters possible. For example, low-pass filters can be developed under the condition of a TEM or quasi-TEM model. For all the filters to be designed in CPW technology, the coplanar strip and slot profiles are given for an alumina substrate permittivity of 9.6 and a thickness of 0.635 mm. The determination of the strip and slots widths was made with the Wen [11]

S11 -parameter

and Fouad-Hanna [12] design relations including dispersion due to the frequency influence on the effective dielectric constant and on the characteristic impedance. The effect of metallization thickness ( –5 m) on the characteristic impedance and the effective dielectric constant was taken into account empirically by defining effective values of strip and slots widths, as in [13]. These values were compared with those available on existing software, i.e., HP-MDS. As all measurements to be presented in CPW technology were carried out “on wafer,” 50- matched taper transitions were used. To minimize the transition influence on the filter measurements, a linear taper was computed for the slot and a quadratic taper for the strip, as described in [14]. The matched transition can then be included in the TRL calibration kit, thanks to its precise continuity in impedance. One of our first aims was to compare the performances of filters characterized by smooth profile from our model with other filters described by common techniques, i.e., equivalent distributed synthesis from the Chebyshev lumped filter. Filter responses were compared in a 1–15-GHz frequency range. Two different filters using the CVTL technique were optimized. These two filters were GHz (Figs. 4 and 5). The second filter designed at (Fig. 5) and Chebyshev filter [Fig. 6(a)–(c)] were designed to get similar performances, i.e., zeros and poles numbers (with ripples in the band at a same the same number of sections), rate, similar cut off frequencies, and, particularly, identical impedance limits. They were also designed on the same substrate. After optimization, the simulation results show that the selectivity is higher for the Chebyshev filter and its length is slightly smaller than the CVTL one. Globally, in the 1–15-GHz range, the three filters measured responses [Figs. 4(c), 5(c), and ] are in very close agreement with and 6(c) for

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(a)

(b)

(c) Fig. 4. (a) Impedance variation of the first low-pass filter described by 16 discrete points. (b) Shape of the CPW strip and slots (not to scale). (c) 10 GHz). Scattering parameters of the first low-pass filter (fc

=

(a)

(b)

(c) Fig. 5. (a) Impedance variation of the first low-pass filter described by 14 discrete points. (b) Shape of the CPW strip and slots (not to scale). (c) S -parameters of the second low-pass filter (fc = 10 GHz).

the simulated ones. An accurate conformity between the zeros and poles positions of the simulated and measured responses of the CVTL filters was also ensured. The losses in the band and the deviation between theory and measurement at the cutoff frequency are higher for the Chebyshev filter (Fig. 6(c), dB dB at 8.8 GHz, 460-MHz frequency deviation) dB dB at 8.8 than for the CVTL one (Fig. 5(c), GHz, 68-MHz frequency deviation). This frequency shift is

created by the parasitic elements brought by abrupt discontinuities. The higher losses are confirmed by the power results from the -parameter measurements. A comparison of filters with higher cutoff frequencies, i.e., 40–50 GHz, would show more significant differences between the two types of concept because the CVTL reduces the influence of discontinuity on the design and avoids making effective length corrections to take sharp impedance discontinuities into account.

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(a)

(b)

(c) Fig. 6. (a) Impedance variation of the Chebyshev low-pass filter described by 30 discrete points. (b) Shape of the CPW strip and slots. (c) of the Chebyshev filter (fc 10 GHz).

=

(a)

S -parameters

(b)

(c) Fig. 7. (a) Impedance variation of the low-pass filter with a decreasing modulation size described by 27 points. (b) Shape of the CPW strip and slots. (c) S -parameters (fc = 10 GHz).

The good behavior of CVTL low-pass filters in CPW technology associated with the number of degrees of freedom of the optimization enables filters to be designed with a larger attenuation band. The optimization goal was to eliminate the -parameter as far as possible. spurious response of the The first filter designed presents a 10-GHz cutoff frequency and an impedance profile with a varying modulation [see

Fig. 7(a) and (b)]. This profile was obtained [see Fig. 7(a)] with a large filter order and by optimizing alternatively the discrete points values and the distances between discrete points. This solution, of course, increases the filter length and the dielectric, ohmic, and radiation losses at these frequencies. The simulated and measured responses shown in Fig. 7(c) are in good agreement, even up to 40 GHz. The frequency

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(a)

(b)

(c) Fig. 8. (a) Impedance variation of the low-pass filter (fc

= 20 GHz) described by 30 points. (b) Shape of the CPW strip and slots. (c) S -parameters.

positions of the zeros and poles are also in accordance with theory. The second filter designed with the same goal as the previous filter shows a 20-GHz cutoff frequency. The spu-parameter was optimized to appear rious return of the at 70 GHz. The impedance [see Fig. 8(a)] and strip conductor [see Fig. 8(b)] profiles combine different sizes and magnitudes of undulations. This was made possible by optimizing both the position and magnitude of the discrete points that describe the impedance variation. As illustrated by the losses in the bandwidth, this solution enables us to obtain a shorter filter. The simulated and measured transmission and reflection parameters are compared in Fig. 8(c). The agreement is good between 1–50 GHz. A frequency shift between simulated and measured responses can be evidenced at the cutoff frequency. This difference may be partly due to the poor quality of the gold for this filter. It might be possible to integrate the dielectric and conductor and -parameters [14] in the losses by including the telegrapher’s relations. The time required to compute the simulated response of a filter on an HP715 workstation never exceeded 1 s, even for an impedance variation defined by 27 discrete points and for 200 frequency points, as in Fig. 8(a). V. CONCLUSION A new method for CVTL design applied to microwave filters has been proposed. The efficiency and accuracy of the method exposed is validated, even over a 1–50-GHz broad band. The approach proposed makes it possible to quickly analyze and optimize complicated structures with simple wellknown functions, which make the method particularly useful for computer-aided design (CAD) applications. This technique can be easily implemented in a CAD environment and could

replace the conventional step-by-step segmentation process in simulation softwares. Despite that the CVTL design is less selective, the method described presents different advantages over the usual techniques. First, the structure shape has a greater potential [increased power handling capability, i.e., Figs. 7(a) and 8(a)] than common lines. Second, nonuniform lines reduce accuracy problems associated to large impedance discontinuities, and should find applications at higher frequencies. However, physical limitations must be applied to avoid a breaking down of the TEM-model brought by too large profile contrasts. In order to design band-pass filters, this method will be expanded to characterize nonuniformly coupled lines. This global approach may also be applied to design impedancematching lines for multistage amplifiers [15]. REFERENCES [1] K. Lu, “An efficient method for analysis of arbitrary nonuniform transmission lines,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 9–14, Jan. 1997. [2] C. W. Hsue and C. D. Hechtman, “Transient analysis of nonuniform high-pass transmission lines,” IEEE Trans. Microwave Theory Tech., vol. 38, pp. 1023–1030, Aug. 1990. [3] C. W. Hsue, “Time-domain scattering parameters of an exponential transmission line,” IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1891–1895, Nov. 1991. [4] P. Bouchard and R. J. Gagn´e, “Transient analysis of lossy parabolic transmission lines with nonlinear loads,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 1330–1333, June 1995. [5] M. J. Ahmed, “Impedance transformation for exponential, cosinesquared, and parabolic tapered transmission lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 67–68, Jan. 1981. [6] N. V. Nair and A. K. Mallick, “An analysis of a width-modulated microstrip periodic structure,” IEEE Trans. Microwave Theory Tech., vol. MTT-32, pp. 200–204, Feb. 1984. [7] P. P. Roberts and G. E. Town, “Design of microwave filters by inverse scattering,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 739–743, Apr. 1995. [8] M. Le Roy, A. P´erennec, L. C. Calvez, and S. Toutain, “A new design of microwave filters by using continuously varying transmission lines,” in IEEE MTT-S Symp. Dig., 1997, pp. 639–642.

LE ROY et al.: CVTL TECHNIQUES—APPLICATION TO FILTER DESIGN

[9] A. Angot, Compl´ements de Math´ematiques a` l’Usage des Ing´enieurs de ´ l’Electrotechnique et des T´el´ecommunications, 6th ed. Paris, France: Masson, 1982. [10] J. J. More, “The Levenberg–Marquardt algorithm: Implementation and theory,” in Lecture Notes in Mathematics 630. Berlin, Germany: Springer-Verlag, 1978. [11] C. P. Wen, “Coplanar waveguide: A surface strip transmission line suitable for non reciprocal gyromagnetique devices applications,” IEEE Trans. Microwave Theory Tech., vol. MTT-17, pp. 1087–1090, Dec. 1969. [12] C. Veyres and V. Fouad-Hanna, “Extension of the application of conformal mapping techniques to coplanar lines with finite dimensions,” Int. J. Electron., vol. 48, no. 1, pp. 47–56, 1980. [13] K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines, 2nd ed. Norwood, MA: Artech House, 1996. [14] A. H. Hammade, A. B. Kouki, and F. M. Ghannouchi, “A CADsuitable approach for the analysis of nonuniform MMIC and MHMIC transmission lines,” IEEE Trans. Microwave Theory Tech., vol. 44, pp. 1614–1617, Sept. 1996. [15] A. P´erennec, R. Soares, P. Jarry, P. Legaud, and M. Goloubkoff, “Computer-aided design of hybrid and monolithic broad-band amplifiers for optoelectronic receivers,” IEEE Trans. Microwave Theory Tech., vol. 37, pp. 1475–1478, Sept. 1989.

Marc Le Roy was born in Brest, France, on November 21, 1970. He received the Ph.D. degree in electronics from the University of Brest, Brest, France, in 1999. He is currently with the Laboratory for Electronics and Communication Systems (LEST), Centre National de la Recherche Scientifique (CNRS 6616), University of Brest. His research activities deal with the time- and frequency-analysis method applied to design microwave matching networks and filters.

Andr´e P´erennec was born in Quimper, France, on November 8, 1960. He received the Ph.D. degree in electronics from the Universit´e de Bretagne Occidentale, Bretagne, France, in 1988. He is currently the Maˆıtre de Conf´erences at the Laboratory for Electronics and Communication Systems, Centre National de la Recherche Scientifique (CNRS 6616), University of Brest, Brest, France. His research interests are in the active and passive device microwave area and include amplifier design (low-noise, very broad-band, and power), and filters. He has also been involved with optoelectronic devices, such as laser command circuit and transimpedance amplifiers, and has published 36 papers on these subjects.

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Serge Toutain received the engineering final diploma from the Electronics and Applications National High School (ENSEA), Paris, France, in 1970, and the Ph.D. degree from the University of Lille, Lille, France, in 1976. In 1970, he joined IEMN, University of Lille. He was a Professor at the Telecommunication National High School (ENST), Bretagne, France, and head of the Electronic and Telecommunication Laboratory (LEST) from 1982 to 1998. He joined the Electronics and Computing Engineer School, Electronics and Computing Systems, Centre National de la Recherche Scientifique (CNRS), University of Nantes, Nantes, France, in October 1998, and is currently a Professor in electronics, microwaves, and telecommunications. His main research interests concern the modelization and design of microwave passive and active devices for radio communications.

L´eon Claude Calvez was born in Pludual, France, in 1943. He received the Licenci´e e` s Sciences Physiques degree from the University of Rennes, Rennes, France, in 1964, and the Docteur e` s sciences Physiques degree from the University of Brest, Brest, France, in 1973. Upon receiving his degree, he became an Assistant Professor and then a Professor of electrical sciences and signal theory at the University of Brest. His research interests are mainly in the area of integral and discrete transforms, transients, circuit and system theory, modeling, simulation, and optimization.