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bandstop filers with extended upper bassbands

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
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Bandstop Filters With Extended Upper Passbands
Ralph Levy, Life Fellow, IEEE, Richard V. Snyder, Life Fellow, IEEE, and Sanghoon Shin, Member, IEEE
Abstract—Previous microwave distributed bandstop ?lters have had their second harmonic response centered at no more than three times the fundamental bandstop mid-band frequency, due to the use of quarter-wave resonators. This limitation has now been removed by the use of compound resonators having shorter electrical length. Some con?gurations incorporate lumped capacitors, resulting in additional design degrees of freedom and wider passbands. The new theory has been developed to apply to both wide and narrow stopbands. Example ?lters having upper passband widths of up to six times the fundamental bandstop center frequency are presented. Index Terms—Bandstop ?lters, ?lters, inhomogeneous ?lters, microwave ?lters, mixed lumped distributed, noncommensurate lines, redundant lines.
Fig. 1. New commensurate bandstop ?lter. (Stubs are connected in shunt to the main lines.)
I. INTRODUCTION ANDSTOP ?lters are frequently employed to reject narrow-to-broad frequency bands located within a wide passband. Up to now, bandstop ?lters comprised of distributed elements have encountered a severe restriction on the extent of the upper passband imposed by the periodicity of the distributed elements [1]. This causes the stopbands to repeat at odd multiples of the fundamental stopband center frequency. In particular, the bandstop ?lters using quarter-wavelength resonators as previously described [1] give a ?rst upper stopband center frequency located at three times the fundamental stopband. This paper describes how this restriction may be overcome and how the ?rst upper stopband center frequency may be raised to a much higher multiple, e.g., as high as six times the fundamental stopband center frequency.1 This paper addresses the design of distributed bandstop ?lters normally realized in coaxial, stripline, or microstrip form. In [2], a capacitive-loading technique applicable to extending the width of the upper passband was proposed, with a good illustration given in [2, Figs. 1–3]. Other important references include [3], which presents a modern synthesis design technique, and [4], the results from which will be used later in this paper. Modern single-variable synthesis is used to obtain the basic rational functional form of the transfer function of a new class of commensurate-line bandstop ?lters having the desired broader upper passbands. Although exact synthesis could almost certainly be performed, a relatively new modern synthesis technique is employed, which may be termed “synthesis by
B
optimization.” This method obtains a new transfer function, using a combination of exact synthesis and approximation. This transfer function is used as a good starting point for optimization, ensuring that the optimization proceeds ef?ciently and rapidly. The optimization gives designed transfer functions having the correct number of passband ripples, for example, as predicted from the theoretical transfer function. The circuits are then modi?ed to incorporate lumped capacitors and short capacitive lines (“short” implies noncommensurate). Optimization is required also because some of the circuits employ both distributed (commensurate and noncommensurate lines) and/or lumped elements, and, presently, direct two-variable polynomial-based element extraction is not available. The combination of network synthesis and both circuit and E-M based optimization is shown to yield practical circuits having stopband widths as high as 25% of the rejection center frequency, and passbands as wide as six times the fundamental rejection-band center frequency. Example designs are given, including those capable of operating at power levels high enough to function well in dif?cult co-site scenarios (i.e., transmitters for one system located in close proximity to wideband receivers). II. THEORY A. Basic Filter Circuit The new bandstop ?lter prototype circuit is shown in Fig. 1. It consists of “compound” stubs, the th one consisting of a unit element having a relatively high impedance and a lower . These comimpedance open-circuited stub of impedance pound stub (resonant) elements are spaced by a pair of transmission lines of varying impedances and with each line being of the commensurate length. If this electrical length at resonance is 45 and the compound stubs are each of uniform impedance, , then the conventional bandstop ?lter prototype i.e., is recovered [1], [3]. The second harmonic or spurious resonance occurs when has increased to , i.e., the ratio of the center frequencies of the ?rst spurious to the fundamental ?rst harmonic is ( . When is 45 , then this ratio is 3, but when is less, the ratio increases, e.g., if is 36 , the ratio increases to 4.
Manuscript received October 24, 2005; revised February 28, 2006. R. Levy is with R. Levy Associates, La Jolla, CA 92037 USA. R. V. Snyder and S. Shin are with RS Microwave Inc., Butler, NJ 07405 USA (e-mail: r.snyder@http://www.wendangku.net/doc/282d1ff0b9d528ea81c779de.html). Digital Object Identi?er 10.1109/TMTT.2006.875804
1The work of Snyder and Shin [2], presented at the 2005 IMS Symposium, employed a capacitive loading technique to extend the passband, using approximate techniques rather than exact synthesis. Independent work had been carried out by the ?rst author, and it was decided to combine these efforts.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
The input impedance of the compound stub looking out from the main through line is
(1)
case. It has been found that it is necessary to include these extra unit elements in the general case in order to obtain equiripple well-matched passbands. These additional unit elements will be further discussed in Section V. The transfer matrix of the th cell is
Dividing by and expressing in terms of the Richards variusing able , obtained by substituting for (2) we then obtain where
(5)
(6) (3) The fundamental resonance condition at and are constrained by the equation maintained if is where is given by (3). Multiplication of the three matrices in (5) leads to the cell transfer matrix (7), shown at the bottom of this page. given by (3) and (6), the polynomial Substituting for form of the cell matrix becomes
(4) This condition applies to Chebyshev ?lters, but is relaxed in the case of bandstop ?lters that have an elliptic type of response where the loss poles are distributed across the stopband at various frequencies. In order for the ?lter prototype circuit to be commensurate, then, as stated previously, the connecting lines should also consist of a cascade of two unit elements of electrical length . In general, this cascaded pair no longer has a total length of 90 at resonance and, thus, does not directly approximate an ideal impedance inverter. The question then arises, can the ?lter be matched in the entire passband regions, especially that between the fundamental resonance and the ?rst harmonic? The answer is af?rmative, since the commensurate nature of the circuit means that the transfer function of the ?lter is the ratio of two rational polynomials, and, in theory, this may be synthesized to give equal ripple passband response. B. Polynomial Formation The rational polynomial form of the transfer function is derived in the following section. The ?lter is treated as a cascade of n unit cells. A typical cell (index r) consists of a compound stub connected to unit elements of impedance and on each side, as indicated in Fig. 1. Here, the subscript denotes a “main line” element. Note that the main line includes end unit and , which are not present in the convenelements , i.e., the uniform stub tional bandstop ?lter having
(8) where , and are simple polynomials in , each of degree indicated by their respective suf?ces, polynomials being even and the and with the and polynomials odd functions of The overall transfer matrix of the -cell ?lter is given by multiplication of matrices of the form given by (8), leading to the transfer matrix
(9) The and polynomials remain even, and the and polynomials odd functions of . It is seen that the real frequency loss poles are given by equating the product in the denominator to zero, i.e., the th such pole is given by
or (10) as in (4).
(7)

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The re?ection coef?cient is given by the well-known formula
(11)
(a)
zeros of distributed along the In general there are ) of the complex plane, real frequency axis (where . Note that this range corresponding to the range includes two stopbands corresponding to the loss poles occur. Hence, there are three passband regions ring at and occurring in the ranges 0 to to , in to 180. Symmetry considerations imply that there and are zeros in the range 0 to , zeros in to , and zeros in to 180. An interesting exception occurs when the ?lter is electrically symmetric, i.e., when
Symmetry condition
(12)
given by In this case, the degree of the numerator of to . The “missing” loss zero is (11) reduces from the central one in the to region, which then has zeros. Additionally, since in the symmetrical case the degree of the denominator in (11) is one greater than that , i.e., at of the numerator, one of the zeros will occur at , which is the mid-point of the the central passband to . Examples will be presented to demonstrate these characteristics. C. Synthesis by Optimization It is now fairly obvious that formal synthesis techniques should exist for these commensurate bandstop ?lter networks. However, such a development is quite a time-consuming task, especially since it will be shown that the simplest case having a Chebyshev all-pole response, i.e., with all of the loss poles coincident at one frequency in the fundamental stopband, does not give an entirely satisfactory result, and more complicated elliptic function responses are more desirable. Although formal synthesis programs have not yet been written, the certainty of their existence encouraged the development of an optimization technique using commercial optimizers.2 This has enabled various ?lter responses to be investigated, and ?lters having excellent equiripple passband responses have been derived having either Chebyshev or pseudoelliptic function stopband response. The process is facilitated by knowledge of the zero responses, as described in Section II-B. Hence the optimization method gives rapid designs having a variety of responses and is initially much simpler to develop than formal synthesis programs. An initial design may be obtained commencing from an exact prototype bandstop ?lter having uniform 90 shunt open-circuited stubs, as in [1]. Each uniform shunt stub is then replaced
2For example, “Touchstone,” which is no longer commercially available but remains in widespread use.
(b) Fig. 2. (a) Conventional prototype bandstop ?lter. (Impedance values shown,  = 45 at resonance). (b) Performance of conventional bandstop ?lter.
by a compound stub by equating the reactance slope parameters of the two circuits, which for the 90 stub is [1, Fig. 5.08-1] (13) where the stub impedance is denoted by . The reactance slope parameter for the compound stub is given by differentiation of (1), and, at the resonant angle , this leads to (14) where (4) is used to give (14) in remarkably simple form. It is . seen that (14) degenerates to (13) when The new ?lter design now proceeds by equating (13) and (14), giving (15) is, of course, given by (4). and the value of The above procedure is applied to each stub of the exact prototype ?lter. It will be illustrated here by presenting an example stubs. The basic prototype has a fractional bandwith width of 33.95% and a ripple VSWR of 1.2:1 or a return loss of 20.83 dB, and the circuit is depicted in Fig. 2(a). The performance of this bandstop ?lter with the electrical length of the at the fundamental resonance, half unit element being is shown in Fig. 2(b). It is seen that the second harmonic occurs

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
(a)
(a)
(b) Fig. 3. (a) Chebyshev prototype bandstop ?lter prior to optimization.  25:7 . (b) Performance of Chebyshev ?lter prior to optimization.
(b)
=
at the expected three times the fundamental bandstop mid-band frequency, i.e., the ratio of the frequencies of the second to the ?rst harmonic is the normal 3:1. It is now desired to double this ratio to 6:1, which means that the commensurate angle must be reduced from 45 to 25.7 . Application of (15) and (4) leads to the preliminary circuit shown in Fig. 3(a) that has the response shown in Fig. 3(b). The ratio of the second to ?rst harmonic stopbands is , as desired, but the equiripple characteristics are distorted, but not to a severe degree, as the return loss remains better than 10 dB. In some cases, the performance obtained may be acceptable without further optimization, as demonstrated further in Section IV. The ideal match may be recovered almost exactly by optimization, allowing the impedances of the main lines, i.e., of Fig. 1, and the impedance of the stubs, i.e., the of the Fig. 1, to vary. In the Chebyshev case, the stub impedances are constrained by (4) so that the resonances occur at . The resulting circuit is shown in Fig. 4(a), and the response is shown zeros in Fig. 4(b). Note that there are the predicted in the main central passband. Fig. 4(c) is an expanded view of ) 0–1.6, Fig. 4(b) for the range of normalized frequency ( zeros in the and it is seen that there are the expected lower passband. The ?lter characteristic is symmetrical about ). 3.5 times normalized frequency ( A disadvantage of the new ?lter is the rather poor upper stopband skirt as depicted clearly in Fig. 4(b) and (c). This may be contrasted with the perfectly symmetrical response of the prototype ?lter shown in Fig. 2(b). Considerably improved characteristics may be obtained by allowing the loss poles to spread
(c) Fig. 4. (a) Optimized Chebyshev ?lter.  = 25:7 . (b) Performance of optimized Chebyshev ?lter. (c) Expanded view of (b), 0 < f =f o < 1:6.
out at different frequencies across the stopband, resulting in a pseudoelliptic ?lter having equiripple stopband rejection. Such a design is shown in Fig. 5(a), where the circuit was constrained to be electrically symmetric, and the rejection level was set to be 40 dB. The performance shown in Fig. 5(b) indicates that the ripple level is not absolutely ideal, which is rarely the case for passan optimization, but it is very close to this. The band re?ection zeros (or minima in this case) are retained in the range between the two stopbands. An expanded view of the lower pass and stop bands is shown in Fig. 5(c). The stopband ) of 0.94, 1, loss poles occur at normalized frequencies (

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(a)
(a)
(b)
(b)
(c) Fig. 5. (a) Pseudoelliptic symmetric bandstop ?lter.  = 25:7 . (b) Performance of pseudoelliptic symmetric bandstop ?lter.  = 25:7 . (c) Expanded view of Fig. 5(b), 0 < f =f o < 1:6.
(c) Fig. 6. (a) Full elliptic asymmetric bandstop ?lter.  = 25:7 . (b) Performance of full elliptic asymmetric bandstop ?lter.  = 25:7 . (c) Expanded view of Fig. 6(b), 0 < f =f o < 1:6.
and 1.1, in agreement with the values for the stub impedances given in Fig. 5(a), with application of (4). The lower passband shows four zeros clearly, but it is fairly obvious that the “lost” zero is either coincident with one of the four shown or would appear with further optimization. This type of slightly nonoptimal behavior is frequently demonstrated in optimization techniques, where the optimization terminates when a speci?cation is obtained closely rather than giving perfect agreement with a design theory. Fig. 5(c) in particular demonstrates the considerably improved upper stopband rejection compared with that shown in Fig. 4(c).
In another transformation of the original prototype of Fig. 2(a), a full elliptic function ?lter was designed by allowing the circuit to become electrically asymmetric so that the ?ve loss poles were allowed to separate distinctly across the stop band. The circuit is shown in Fig. 6(a), and the performance is given in Fig. 6(b) and (c). Here we see that the tenth minimum has appeared, which is in agreement with the theory of presented in Section II-B. The expanded view of Fig. 6(c) demonstrates the ?ve loss poles occurring at normalized fre) of 0.90, 0.91, 0.96, 1.04, and 1.09. There quencies (

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Fig. 7. Shunt compound stub equivalence relating: (a) a unit element cascaded with the series-connected compound stub, (b) the compound stub replaced by a series connection of open and short-circuited stubs connected in shunt with the main line, and (c) the parallel-coupled-line realization.
Fig. 8. Series compound stub equivalence relating: (a) a unit element cascaded with the shunt-connected compound stub, (b) the compound stub replaced by a parallel connection of open and short-circuited stubs connected in shunt, with the main line, and (c) the parallel-coupled-line realization.
are four distinct minima in the lower passband, and it is apparent that the ?fth one is “hidden” at about 0.4 normalized frequency. In a similar fashion, an optimization procedure is employed in the design of the wide-stopband, wide-passband ?lter (i.e., 24% stopband width, with more than 5.5:1 upper passband width), discussed as Example 2. With the initial circuit topology based on commensurate circuits (such as those illustrated in Figs. 6–10), implemented as capacitively loaded shunt resonators parallel to a stepped center conductor, the stopband upper and lower slopes are unacceptably asymmetric. Because the shunt transmission line resonators are capacitively loaded, it was hypothesized that providing similar capacitive loading on the series transmission-line portions parallel to the shunt resonators (i.e., on the through lines, clearly implementing a physically more symmetrical con?guration) would also achieve better electrical symmetry, while still achieving asymmetric loss pole placement. The resulting circuit resembles the commensurate line con?gurations derived with exact theory earlier in this paper, but includes the addition of short (i.e., lengths less than ) “redundant” capacitive sections. Several sections of the initial topology, development of the modi?ed topology, and response characteristics are illustrated in Section V. While the
argument for such a circuit modi?cation might be considered heuristic, the insight resulting in this “modi?ed topology” has proven to be effective. The modi?cation was performed with addition of short low-impedance (i.e., capacitive) transmission lines (with lengths less than 7 at fo) preceding each coupled section. These additional degrees of freedom, during optimization, allow the asymmetric placement of loss poles and the consequent achievement of essentially symmetrical attenuation slopes, both above and below the center of the rejection band. Use of the “redundant” short low-impedance lines also results in a reduction of the initially synthesized very large difference and and, thus, facilitates implementation between without very thin line sections or very small gaps. These details will be further illustrated in Section V covering example designs, but it is to be noted that the starting point for circuit modi?cation and optimization is the commensurate network resulting from exact synthesis as presented herein, with the addition of the aforementioned capacitive sections. D. Realization of Bandstop Filters of Narrow Bandwidth Using Parallel-Coupled Lines Filters having rather broad stop bandwidths may be realized directly if the impedance levels are not too high. However, in

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Fig. 9. (a) Schematic of a ?ve-section bandstop ?lter having short-circuited shunt stubs. (b) Schematic of a ?ve-section bandstop ?lter having open-circuited shunt stubs.
many cases (e.g., narrower stop bandwidths and/or small values of ), it is necessary to use resonators having loose couplings to the main through line in order to give realizable impedances. This is certainly the case with the examples presented in Section II-C. It has been found that it is convenient to use one of two types of equivalent circuits of the compound stub combined with an adjacent unit element as shown in Figs. 7 and 8. Fig. 7(a) shows the shunt stub case, which is easily converted into the circuit of Fig. 7(b). The exactly equivalent parallel-coupled-line realization of this is shown in Fig. 7(c). The equivalence between the circuits of Fig. 7(b) and (c) has been given in [4, Fig. 4]. In the present representation, the ideal :1 transformer of [4] has been absorbed into the circuit elements of Fig. 7(b). Another simpli?cation is to consider only symmetrical coupled lines, so that and . Application of the equations given in [4] then leads to the design equations given in Fig. 7. The dual circuits to those given in Fig. 7 are of equal interest and are shown in Fig. 8. This dual case starts from the series-connected version of the prototype bandstop ?lter, which is shown in Fig. 8(a). For a given bandstop ?lter speci?cation, the impedances of Fig. 7(a) are equal to the admittances in Fig. 8(a). This circuit is converted into that of Fig. 8(b) with the element values shown. The ?nal conversion into the parallel coupled line form with is shown in Fig. 8(c), with the equations being derived from the expressions given in [4, Fig. 3]. It is seen that the topological differences between the circuits of Figs. 7(c) and 8(c) are the reversals of the open and short-circuited ends of the lower coupled lines on the left and of the shunt stubs on the right. However, there are also large differences in the impedance levels. The shorted stub of Fig. 7(c) tends to have a high impedance, while the open-circuited stub of Fig. 8(c) has a low impedance. Later, it will be shown that the open-circuited stubs may be replaced by lumped capacitors. In addition to being an important modi?cation from design and construction points of view, this substitution also gives a signi?cant increase in the width of the upper passband. It enables better element values to be achieved since the band-edge angle may be increased.
(a)
(b) Fig. 10. (a) Parallel-coupled-line n = 5 Chebyshev ?lter with open-circuited shunt stubs,  = 36 , bandwidth 11.11%. (b) Filter of (a) with the stubs replaced by lumped capacitors.
Schematic diagrams of ?ve-section ?lters of the two classes are shown in Fig. 9(a) and (b). The shunt lines between the par-

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
TABLE I SHUNT NETWORK PARAMETERS [SEE FIG. 9(a)]
allel-coupled-line resonators as introduced by the prototype circuit of Fig. 1 are very useful to space these from each other to reduce cross couplings, enabling thick short-circuiting walls to ground to be incorporated. These are useful also in an optimization procedure, where their impedances and lengths are useful and effective variable parameters. III. CIRCUIT PARAMETERS AS A FUNCTION OF BANDWIDTH AND Reference has been made to the effect of the bandwidth on the impedance levels within the and band-edge angle bandstop ?lter structures. This will now be illustrated by examples having two values of the bandwidth, namely, 11.11% and 33.95%, and two values of , which are 25.7 and 36 . These give ratios of the second to ?rst harmonic bandstop resonances of 6:1 and 4:1, respectively. There are also the two types of parallel-coupled-line realizations to consider, and both are demonstrated. The results are summarized in Tables I and II. The various impedance values are de?ned mainly by reference to Fig. 9, and the even- and odd-mode impedances are derived and of Figs. 7 and 8 using the formulas from
(16)
The values of the various impedances in the stub versions of the ?lters are normalized to unity, but the values of Zoe, Zoo, and Zs are normalized to 50 in order to give a more direct impression of the realizability of the coupled-line versions. The impedances of the shunt stub protototypes are given in Table I, and, in Table II, the same numerical values appear as the for the series networks, but since they are dual networks they are now admittances. Signi?cant items that arise include the following. 1) In Table I, it is seen that the values of the stub impedances are much higher for the smaller value of and become higher for the narrower bandwidth. High values of stub impedances mean a lack of support for the coupled line in air-line coaxial realizations. 2) Table II shows that the shunt open-circuited stubs have for smaller values of , making them lower impedance more dif?cult to realize directly. Section IV explains how this problem may be alleviated by replacing them with lumped capacitances to ground. 3) Table I indicates that a direct realization of the ?lters using shunt stubs is feasible for wide bandwidth and the larger . In both tables, it is seen that the Zoe and Zoo values are less realizable for broad than narrow bandwidths. 4) The prototype circuits have physical symmetry about the center of the circuit, but the parallel-coupled-line realizations are slightly asymmetric since the individual coupled line sections are themselves physically asymmetric.

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TABLE II SERIES NETWORK PARAMETERS [SEE FIG. 9(b)]
IV. REPLACEMENT OF OPEN-CIRCUITED STUBS BY LUMPED CAPACITORS As stated above in 2), it is often advantageous to replace the open-circuited stubs in the realization of Figs. 8 or 9(b) by lumped capacitors. A capacitance value is selected to give the same susceptance to ground as the open-circuited stub at the mid-band frequency of the bandstop region, i.e., at the electrical length . It is important to note that such a selection provides a good starting point for optimization, reducing the time required to optimize this mixed (both lumped and distributed variable) circuit. with As an example, we consider the case and the 11.11% bandwidth, as this is the second design given in Table II. The performance of the initial fully distributed ?lter with stubs is shown in Fig. 10(a). This is an example where optimization may not be required, as the worst return loss ripple in the main central passband is better than 15 dB and mainly better than the 20-dB return loss level of the original prototype having uniform 90 lines. The performance after replacing the stubs with lumped capacitors is shown in Fig. 10(b). The return loss has degraded to about 13 dB, but the most interesting feature is the much widened central passband, where the second harmonic has been ) of 4 to over 5. increased from the normalized frequency ( Thus, rather than having the wider central passband, if the objective were to realize a 4:1 ratio using the lumped capacitors,
then it would be possible to increase , giving more realizable element values. The result of optimization on this mixed lumped and distributed circuit is shown in Fig. 11(a), where the return loss has improved to 18 dB over the central region up to ?ve times ), and the stopband has been alnormalized frequency ( lowed to become elliptic, as shown more clearly by the expanded plot of Fig. 11(b). Inclusion of the additional main-line transmission-line elements preceding and following the coupled-line sections allows for signi?cant improvement in both upper stopband slope and passband return loss. This will be shown in Section V. V. EXAMPLE DESIGNS A. Example 1 This is an example of a ?ve-section pseudoelliptic bandstop ?lter of the symmetrical pseudoelliptic type as shown in Fig. 5(a). In terms of the electrical length , the ?rst harmonic , and the passband edges for 20-dB return was at loss were at 30.5 and 44.5 , i.e., a fractional bandwidth of 37.33%. The wide bandwidth and rather large value of meant that the impedances as shown in Fig. 12(a) were realizable directly without requiring a parallel-coupled-line realization. The ripple level in the stopband was 68 dB from to , which is a 68-dB fractional bandwidth of 16%. The edge of the second harmonic passband , giving an upper passband extending from was at

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(a)
(a)
(b) Fig. 12. (a) Pseudoelliptic ?lter of Example 1.  = 37:35 . (b) Theoretical and measured performance of the ?lter shown in (a). Solid line: theoretical data; dotted line: measured data.
(b) Fig. 11. (a) Filter of Fig. 10(b) optimized to give improved return loss and elliptic-function response. (b) Expanded view of the ?lter of Fig. 11(a), 0:6 < f =f o < 1:3.
to , i.e., a ratio of 3.045:1. This is a , where considerable improvement over the case with the passband edges for the same fractional bandwidth would be and , giving a ratio of only 2.37:1. at The ?lter design giving the impedance values is shown in Fig. 12. The ?lter was built in suspended substrate stripline using 0.01-in-thick Rogers Duroid 5880. The stopband was centered at about 5.65 GHz, and the upper passband extended to beyond 18 GHz. The speci?ed return loss up to 18 GHz was 10 dB, and the initial measured results compared with the theory are shown in Fig. 12(b). Thus, it achieved the speci?ed performance with no iterations being required. The practical return loss may be improved using a ?eld-based optimization routine. B. Example 2 Fig. 13 illustrates development of a 13-section pseudoelliptic bandstop ?lter example using 21 long coupled lines, loaded with parallel-plate lumped capacitors. Fig. 13(a) shows the
Fig. 13. Design approach for example 2. (a) Several sections of coupled line con?guration with open circuit stubs replaced by loading capacitors and no redundant transmission lines between the coupled lines, i.e., “initial topology” as per Figs. 6–10 (see text). (b) Several sections of coupled-line con?guration with open circuit stubs [as in Fig. 9(b)] replaced by loading capacitors and with the addition of redundant transmission lines (Zm Zm Zm ), i.e., “modi?ed topology” (see text). (c) Asymmetric rejection response for no redundant transmission line [from Fig. 13(a) and (d)]. Symmetric rejection response due to the redundant transmission lines [from Fig. 13(b)].

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(a) Fig. 14. 13-section parallel-coupled-line pseudoelliptic band-rejection ?lter with parallel-plate lumped capacitors for ?lter from example 2.
initial topology, which is a commensurate circuit with asymmetric loss poles, derived as in Figs. 6–10. Fig. 13(b) shows the modi?ed topology, in which the “redundant” low-impedance lines have been included in the main line of the ?lter. Fig. 13(c) shows the asymmetric response resulting from the use of Fig. 13(a), while Fig. 13(d) shows the result of incorporating the “redundant” lines and optimizing. Fig. 14 presents photographs of the implemented ?lter. The desired ?rst notch is centered at 1.08 GHz with 24% bandwidth, at the 45-dB rejection level. The second-harmonic passband is at 6 GHz, with a resulting extended passband ratio of 5.56:1. However, due to the impedance mismatch at the end of passband near 6 GHz, the effective passband ratio is 5:1. It is important to note that the coupled-line lengths were reduced slightly (during optimization) to allow for the effects over the passband of the lead inductance required to implement the physical connection for the lumped capacitors. Each coupled-line section is connected through 6 long noncommensurate transmission lines for improved matching. Fig. 15 displays the measured performance for this noncommensurate and mixed-variable example. It is interesting to note that the measured return loss performance is actually better than the “theoretical” performance. The so-called theoretical performance is based upon circuit simulation, using the tabulated impedance and length data in Table III. This shows that the theoretical model using simple TEM approximation (as in Table III) is inadequate, because it does not fully incorporate the effects of tuning screws and other tuning elements. However, full analysis of the entire structure using electromagnetic simulation is very time consuming, and the results are satisfactory. The design was accomplished with an initial synthesized structure analyzed using Genesys,3 followed by co-simulation in Ansoft HFSS,4 (electromagnetic analysis and parameter extraction), followed
3Genesys, 4HFSS,
(b) Fig. 15. (a) Measured performance of 13-section parallel-coupled-line pseudoelliptic bandstop ?lter with plate lumped capacitors,  = 21 , bandwidth 22%, passband ratio = 5 : 1. (b) Expanded view of Fig. 14(a). Solid line: measured data; dotted line: theoretical data.
by optimization at the circuit level in Ansoft Designer.5 The design was implemented as a machined, air-slab structure and is intended for rejection of the military MIDS/JTIDS passband, at high power levels (300-W peak with a 120-W average), to alleviate certain communication co-site interference issues.
VI. CONCLUSION Bandstop ?lters of narrow or moderate rejection bandwidth having upper passbands which are much wider than those previously reported are presented, enabling systems designers to specify such designs. The new design technique is based on an exact analytical response function for the ?lters which could
5Designer,
by Eagleware. by Ansoft.
by Ansoft.

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
TABLE III (FOR FIGS. 13–14) LUMPED LOADING CAPACITORS, 24% BANDWIDTH, 50-
TERMINATIONS
be used as the basis for a conventional single-variable exact synthesis to obtain element values. However, an almost equivalent design procedure which may be termed “synthesis by optimization” has been used, giving essentially identical and certainly very acceptable results. Both directly coupled stub and two types of parallel-coupled-line designs are described. One of the latter may be modi?ed by replacing its shunt open-circuited stubs by lumped capacitors, further extending the upper passband range. The procedure presented herein called “synthesis by optimization” obtains satisfactory ?lter response without the need for two-variable exact synthesis. The power-handling ability of the bandstop ?lters may be limited by the impedance levels of the open-circuited stubs or the physical design of the lumped capacitors that may be used in their place in some designs. The physical design of such capacitors includes choice and thickness of dielectric, as well as location within the physical structure. An example has been presented showing the application of the new design technique to ?lters capable of handling at least 300-W peak, 120-W average power in the 960–1220-MHz frequency range. Passband return loss values of a maximum of 10 dB have been attained in practice over a 6:1 passband ratio of the second to ?rst stopband center frequencies. There is no intrinsic limitation on the passband return loss based on the initial synthesis or modi?ed topology. In fact, levels of less than 15 to 20 dB are attained over much of the passband region, and further work on both matching stub inclusion and connector transition design is ongoing, with the goal of improving this return loss.
[3] O. P. Gupta and R. J. Wenzel, “Design tables for a class of optimum microwave bandstop ?lters,” IEEE Trans. Microw. Theory Tech., vol. MTT-18, no. 7, pp. 402–404, Jul. 1970. [4] H. C. Bell, “L-resonator bandstop ?lters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2669–2672, Dec. 1996. [5] J. Reed and G. J. Wheeler, “A Method of analysis of symmetrical four port networks,” IRE Trans. Microw. Theory Tech., vol. MTT-4, pp. 246–252, Oct. 1956. Ralph Levy (SM’64–F’73–LF’99) received the B.A. and M.A. degrees in physics from Cambridge University, Cambridge, U.K., in 1953 and 1957, respectively, and the Ph.D. degree in applied sciences from London University, London, U.K., in 1966. From 1953 to 1959, he was with GEC, Stanmore, U.K., where he was involved with microwave components and systems. In 1959, he joined Mullard Research Laboratories, Redhill, U.K., where he developed a widely used technique for accurate instantaneous frequency measurement using several microwave discriminators in parallel known as digital IFM. This electronic countermeasures work included the development of decade bandwidth directional couplers and broad-band matching theory. From 1964 to 1967, he was a member of the faculty of The University of Leeds, Leeds, U.K., where he carried out research in microwave network synthesis, including distributed elliptic function ?lters and exact synthesis for branch-guide and multiaperture directional couplers. In 1967, he joined Microwave Development Laboratories, Natick, MA, as Vice President of Research. He developed practical techniques for the design of broad-band mixed lumped and distributed circuits, such as tapered corrugated waveguide harmonic rejection ?lters, and the synthesis of a variety of microwave passive components. This included the development of multioctave multiplexers in SSS, requiring accurate modeling of inhomogeneous stripline circuits and discontinuities. From 1984 to 1988, he was with KW Microwave, San Diego, CA, where he was mainly involved with design implementations and improvements in ?lter-based products. From August 1988 to July 1989, he was with Remec Inc., San Diego, CA, where he continued with advances in SSS components, synthesis of ?lters with arbitrary ?nite frequency poles, and microstrip ?lters. In July 1989, he became an independent consultant and has worked with many companies on a wide variety of projects, mainly in the ?eld of passive components, especially ?lters and multiplexers. He has authored approximately 70 papers and two books, and holds 12 patents. Dr. Levy has been involved in many IEEE Microwave Theory and Techniques Society (IEEE MTT-S) activities, including past editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (1986–1988). He was chairman of the Central New England and San Diego IEEE MTT-S chapters, and was vice-chairman of the Steering Committee for the 1994 IEEE MTT-S International Microwave Symposium (IMS). He was the recipient of the 1997 IEEE MTT-S Career Award.
REFERENCES
[1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures. New York: McGraw-Hill, 1964, see Ch. 12. [2] R. V. Snyder and S. Shin, “Parallel coupled line notch ?lter with wide spurious-free passbands,” in IEEE MTT-S Int. Microw. Symp. Dig., 2005, Paper TU4A-3, CD ROM.

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Richard V. Snyder (S’58–M’63–SM’80–F’97– LF’05) received the B.S. degree from Loyola-Marymount University, Lost Angeles, CA, in 1961, the M.S. degree from the University of Southern California, Los Angeles, in 1962, and the Ph.D. degree from the Polytechnic Institute of New York, Brooklyn, NY, in 1982. He is President of RS Microwave, Butler, NJ. He teaches and advises at the New Jersey Institute of Technology (NJIT), Newark. He is a Visiting Professor with The University of Leeds, Leeds, U.K. He was previously Chief Engineer for Premier Microwave. He has authored 69 papers and two book chapters. He holds 17 patents. His interests include E-M simulation, dielectric and suspended resonators, high power notch ?lters, and active ?lters. Dr. Snyder served the North Jersey Section as chairman and 14-year chair of the IEEE Microwave Theory and Techniques (MTT)-Antennas and Propagation (AP) chapter. He is currently chair of the North Jersey Electron Device Society (EDS) and Circuits and Systems (CAS) chapters. He served as general chairman for the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) 2003 International Microwave Symposium (IMS2003), Philadelphia, PA. In January 2005, he began a three-year term as an elected member of the Administrative Committee (AdCom). Within the AdCom, he serves as vice-chair of the TCC, vice-chair of the IMSCC, and chair of the Standards Committee. He is a member
of the American Physical Society, the American Association for the Advancement of Science (AAAS), and the New York Academy of Science. He is a reviewer for IEEE MTT-S publications and Microwave Journal. He is active in the IEEE MTT-S Speaker’s Bureau and the three above-mentioned AdCom committees. He served seven years as chair of MTT-8 and continues in MTT-8/TPC work. He was a two-time recipient of the Region 1 Award. He was the recipient of the 2000 IEEE Millennium Medal.
Sanghoon Shin (S’98–M’02) was born in Seoul, Korea, in 1967. He received the B.S. degree from Hanyang University, Seoul, Korea, in 1993, the M.S. degree in electrical engineering from the Polytechnic University of New York, Brooklyn, in 1996, and the Ph.D. degree in electrical engineering from the New Jersey Institute of Technology (NJIT), Newark, in 2002. In 2002, he joined RS Microwave Inc., Butler, NJ, where he is currently a Research Engineer. His research interest has focused on analysis and design of RF and microwave ?lters.