同轴腔滤波器中的交叉耦合

1368 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 51, NO. 4, APRIL 2003 Cross-Coupling in Coaxial Cavity Filters—A Tutorial Overview J. Brian Thomas, Member, IEEE Abstract—This paper presents a tutorial overview of the use of coupling between nonadjacent resonators to produce transmission zeros at real frequencies in microwave filters. Multipath coupling diagrams are constructed and the relative phase shifts of multiple paths are observed to produce the known responses of the cascaded triplet and quadruplet sections. The same technique is also used to explore less common nested cross-coupling structures and to pre- dict their behavior. A discussion of the effects of nonzero electrical length coupling elements is presented. Finally, a brief categoriza- tion of the various synthesis and implementation techniques avail- able for these types of filters is given. Index Terms—Coaxial cavity filter, cross-coupling, nested cross- coupling, tutorial. I. INTRODUCTION L IFE IS really simple, but we insist on making it compli-cated”—Confucius. This paper will attempt to provide a simplified, qualitative, and intuitive understanding of an impor- tant and complex topic in the field of microwave filters: cross- coupling, especially cross-coupling in coaxial cavity filters. The ever increasing need for capacity in cellular and personal communications systems (PCSs) has led to more stringent re- quirements for basestation filters and duplexers. The transmit and receive bandpass filters composing basestation duplexers may have required rejection levels greater than 100 dB on one side of the passband and, at the same time, have very mild re- jection requirements on the opposite side [1]. The technique of cross-coupling to produce asymmetric frequency responses has become popular for these applications because it concentrates the filter’s ability to provide rejection only over the band where it is needed. This “concentration of rejection” means the filter’s response is trimmed of unnecessary rejection anywhere it is not absolutely required, increasing the overall efficiency of the de- sign. The tradeoff in slope between the bandpass filter’s upper and lower skirt is optimized for the particular requirement. Fig. 1 shows a filter response with and without two transmission zeros on the upper skirt. Using the technique of cross-coupling to pro- duce transmission zeros, the rejection above the passband is in- creased, while rejection below the passband it is relaxed. This can reduce the number of resonating elements required to meet a specification and this, in turn, reduces the insertion loss, size, and manufacturing cost of the design, though at the expense of topological complexity and, perhaps, development and tuning time. Manuscript received January 31, 2002. The author is with the Engineering Department, Baylor University, Waco, TX 76703 USA. Digital Object Identifier 10.1109/TMTT.2003.809180 Fig. 1. Two possible filter responses, with (dashed line) and without (solid line) finite transmission zeros produced by cross-coupling. Note the increase in rejection above the passband and the relaxation in rejection below. This asymmetrical response concentrates a filter’s ability to provide rejection only where it is required. Not only has the industry seen electrical requirements be- come more stringent, but mechanical packaging requirements have become less flexible due to basestation miniaturization and multiple-sourcing considerations. The choices of overall size and shape and constraints in connector locations play a vital role in determining a filter’s layout, topology, and internal structure. The differences can be substantial between a filter with con- nectors on the same surface versus opposite surfaces, all other parameters being equal. Cross-coupling provides an additional degree of flexibility in these design scenarios. When all things are considered, the use of cross-coupling produces a superior design for many requirements. It is not surprising then that these circuits have been the subject of the field’s best and brightest for some time [2]. However, despite the prodigious numbers of expert-level publications available, the nonspecialist RF engineer may be left with the impression that cross-coupling is unapproachably complex: likely a mix- ture of Maxwell’s equations and Voodoo magic. The intent of this paper is to provide a general understanding of some fundamental cross-coupling techniques by using mul- tipath coupling diagrams to illustrate the relative phase shifts of multiple signal paths. This technique can also be used to under- stand, and aid in the design of, less common topologies using nested cross-couplings. Section II reviews the simplified phase relationships of fundamental components in the equivalent circuit of coaxial cavity filters. Although the technique of cross-coupling can be 0018-9480/03$17.00 © 2003 IEEE THOMAS: CROSS-COUPLING IN COAXIAL CAVITY FILTERS 1369 Fig. 2. Prototype equivalent circuit for combline or coaxial cavity filter. Shunt inductor/capacitor pairs represent individual resonating elements and the series inductors represent the dominantly magnetic coupling between resonators. used with other types of filter realizations, (such as dielectric resonators, microstrip, or waveguide) special attention will be given to coaxial cavity filters because of their dominant role in wireless basestation filter applications. Section III illustrates the multipath coupling diagram ap- proach to describe the operation of well-known cascaded triplet (CT) and cascaded quadruplet (CQ) sections. The techniques of Section III require slight modification for realizations other than coaxial cavities; however, these are beyond the scope of this tutorial. In Section IV, less common nested structures are explored, similar to the design of [3], where a five-section dielectric resonator filter with three transmission zeros is described. Section V gives a very broad description of the various imple- mentation techniques in use today. Special focus will be given to methods accessible to most RF engineers. The conclusions are summarized in Section VI. II. PHASE RELATIONSHIPS Combline and coaxial cavity filters may be represented by the prototype equivalent circuit of Fig. 2 [4]. Although simple lumped components are being used to represent three-dimen- sional structures with complex field patterns, nonetheless, they are useful and illustrative for purposes of this tutorial. The shunt inductor/capacitor pairs represent individual res- onating elements and the series inductors represent the domi- nantly magnetic coupling between resonators. The total coupling between adjacent resonators has both magnetic and electric components. However, these are out of phase with each other; the total coupling is the magnetic coupling less the electric coupling [5]. This is the reason a tuning screw placed be- tween the open ends of two resonators increases the coupling between them (see Fig. 3). The screw decreases the electric coupling and, hence, increases the total coupling (less is sub- tracted from the total). By the same reasoning, a decoupling wall between the shorted ends of two resonators decreases the overall coupling by decreasing the magnetic coupling. For a more rigorous treatment of the coupling phenomenon, see [6]–[9]. The off-resonance (away from the passband) behavior of the components of Fig. 2 is utilized to produce the destructive in- terference resulting in transmission zeros and, therefore, needs to be understood. Let the phase component of the -parameters and be denoted and , respectively. Consider the series inductor of Fig. 4 as a two-port device. A signal en- tering port 1 will undergo a phase shift upon exiting port 2. This is , and it tends toward 90 . The fact that the magnitude of is quite small at this point is not problematic in that the Fig. 3. Coupling fields between adjacent resonators. Total coupling may be affected by decoupling walls and/or tuning screws. Fig. 4. Primary coupling between coaxial cavity resonators may be modeled as a series inductor. When considered as a two-port device, the phase of S (� ) approaches �90 . off-resonance behavior is what is of concern. It should be em- phasized that, although this phase shift only approaches 90 and, in general, may be much less, for purposes of general un- derstanding, the approximation of 90 is quite useful. Thus, (for series inductors). (1a) The shunt inductor/capacitor pairs of Fig. 2 (resonators) can also be thought of as two-port devices. However, the phase shift at off-resonance frequencies is dependent on whether the signal is above or below resonance (see Fig. 5). For signals below the resonant frequency (below the passband), the phase shift tends toward 90 . However, for signals above resonance, the phase shift tends toward 90 . This behavior is due to the simple fact that below resonance, the resonator is dominantly inductive and an inductor in shunt is the dual of a capacitor in series. Similarly for frequencies above resonance; the resonator is dominantly 1370 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 51, NO. 4, APRIL 2003 Fig. 5. Coaxial cavity resonators may be modeled as shunt inductor/capacitor pairs. When considered a two-port device, phase � approaches �90 away from resonance, and passes through 0 at the resonant frequency. capacitive and a shunt capacitor is the dual of a series inductor. Thus, (for resonators below resonance) (1b) (for resonators above resonance). (1c) Although there are no series capacitors in Fig. 2, the series capacitor is an important cross-coupling device. The phase shift is (for series capacitors). (1d) These phase shifts should not be confused with coupling co- efficients commonly found in the literature, which may be of opposite sign as the phase shift of (1a)–(1d). III. MULTIPATH COUPLING DIAGRAMS A. CT With Inductive Cross-Coupling Consider the three resonator structure of Fig. 6(a) and (b), which represents a CT section using an inductive cross-coupling between resonators 1 and 3. The resonators of the equivalent circuit (Fig. 2) have been replaced by circles, but the inter- resonator inductors remain shown. Using the relationships of Section II, the phase shifts can be found for the two possible signal paths. Path 1–2–3 is the primary path, and path 1–3 is the secondary path that follows the cross-coupling. When summing the phase-shift contributions of the individual components, the contributions from resonators 1 and 3 are not required. Both paths share a common beginning and ending location; only the contribution of circuit elements internal to resonators 1 and 3 need to be considered. (Indeed, 1 and 3 need not even be resonators; the signals can be combined at the input or output of the filter itself (see [3]).) Furthermore, resonator 2 must be considered both above and below resonance. Table I shows these results. (a) (b) Fig. 6. (a) Multipath coupling diagram for CT section with inductive cross-coupling and possible frequency response including transmission zero (solid line). (b) Physical representation of CT section of (a). Below resonance, the two paths are in phase, but above res- onance, the two paths are 180 apart. This is exactly the case at one frequency only (here, approximately 2030 MHz), but is approximately the case for frequencies in the region (approxi- mately 2020–2040 MHz). This destructive interference causes a transmission zero or null on the upper skirt. Stronger coupling between 1 and 3 causes the zero to move up the skirt toward the passband. Decreasing the coupling moves it farther down the skirt. This type of cross-coupling can be realized by a window between the cavities in the same way the primary coupling be- tween resonator 1 and 2 or between 2 and 3 is realized. This is THOMAS: CROSS-COUPLING IN COAXIAL CAVITY FILTERS 1371 TABLE I TOTAL PHASE SHIFTS FOR TWO PATHS IN A CT SECTION WITH INDUCTIVE CROSS-COUPLING [SEE FIG. 6(a)] Fig. 7. Multipath coupling diagram for CT section with capacitive cross-coupling and possible frequency response including transmission zero (solid line). Also shown is the standard Chebyshev response without cross-coupling (dashed line). advantageous in that no additional components are required [see Fig. 6(b)]. B. CT With Capacitive Cross-Coupling In Fig. 7, the inductive cross-coupling between resonator 1 and 3 has been replaced with a capacitive probe. The phase shifts for the two possible signal paths are given in Table II. Again, path 1–2–3 is the primary path and is no different than in Table I. Path 1–3 is the secondary path and now has a 90 (positive) phase shift. Thus, for a capacitive cross-coupling, the destruc- tive interference occurs below the passband. C. CQ With Inductive Cross-Coupling In Fig. 8, the four-resonator scenario known as the CQ is shown with an inductive cross-coupling. The primary path in this case is 1–2–3–4; the secondary path 1–4, therefore, by- passes two resonators. As Table III shows, transmission zeros are not produced at any real frequencies above or below the passband. However, zeros at imaginary frequencies can be pro- duced, which have the effect of flattening the group delay over the passband. These types of responses are useful in extremely linear systems using feed-forward amplifiers. The flattening of TABLE II TOTAL PHASE SHIFTS FOR TWO PATHS IN A CT SECTION WITH CAPACITIVE CROSS-COUPLING (SEE FIG. 7) Fig. 8. Multipath coupling diagram for CQ section with inductive cross-coupling and possible frequency response. TABLE III TOTAL PHASE SHIFTS FOR TWO PATHS IN A CQ SECTION WITH INDUCTIVE CROSS-COUPLING (SEE FIG. 8) the group delay also has the effect of flattening the insertion loss. Midband losses increase slightly while band-edge rolloff effects are decreased.1 These effects are not apparent from this analysis. See [10] and [11] for a more detailed analysis of filters with transmission zeros at imaginary frequencies. D. CQ With Capacitive Cross-Coupling Replacing the inductive element between resonators 1 and 4 with a capacitive probe, the other type of CQ is formed. This topology is particularly interesting, as Table IV shows, because transmission zeros are produced both above and below the pass- band (see Fig. 9). 1R. Wenzel, “Designing microwave filters, couplers and matching networks,” a video tutorial published by Besser Associates, Los Altos, CA, 1986. 1372 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 51, NO. 4, APRIL 2003 TABLE IV TOTAL PHASE SHIFTS FOR TWO PATHS IN A CQ SECTION WITH CAPACITIVE CROSS-COUPLING (SEE FIG. 9) IV. NESTED STRUCTURES It has been shown that two signal paths can be combined to produce a transmission zero. In this section, nested structures having three or more signal paths will be explored. Consider first the circuit of Fig. 10. The outer path 1–2–3 combines with 1–3 to form one transmission zero. Simultaneously, the interior path 1–3–4 combines with the innermost path 1–4 to produce a second transmission zero. Both are on the upper skirt. See Table V. In the same way, the circuit of Fig. 11 produces two transmis- sion zeros on the lower skirt. Key to its function is the capaci- tive cross-coupling between resonators 1 and 3 (see Table VI). These two circuits are especially useful in duplexer designs due to their similar topology and symmetry in response. The multipath coupling diagram approach can also be used to understand the operation of some very interesting structures described in [3], although in it, the filters are realized with di- electric resonators instead of coaxial resonators. Two possible coaxial realizations of a five-section three-transmission-zero filter are shown in Fig. 12(a) and (b). These are the only two configurations for achieving all three zeros on the same side of the passband. Other combina- tions exist that produce two zeros above and one below, for example. Tables VII and VIII show the phase relationships of Fig. 12 (a) and (b), respectively. Of particular interest is the circuit of 12(b) because it may be realized using all window couplings. For some filter topologies having more than two possible signal paths [as in Fig. 12(a)], the question may be raised as to why the outermost and innermost paths do not also combine to produce an additional transmission zero. To produce a can- cellation, the two paths must not only be opposite in phase, but equal in magnitude. A signal at an off-resonance frequency will be partially attenuated by every resonator in its path, therefore, the outermost and innermost paths will be of vastly different magnitude, regardless of phase. As a working verification of this nested cross-coupling method, see Fig. 13. This eight-resonator filter was developed using the topologies of Figs. 10 and 11 combined. Resonators 1–4 produce both transmission zeros below the band, and resonators 5–8 produce both zeros above the band (special thanks to J. Roberds, Commercial Microwave Technology Inc., Rancho Cordova, CA, for this measured data). V. FILTER IMPLEMENTATION There is a multitude of important topics in the field of filter design and implementation such as the determination of the number of cavities and transmission zeros required to meet a (a) (b) Fig. 9. (a) Multipath coupling diagram for CQ section with capacitive cross-coupling and possible frequency response. Transmission zeros are produced both above and below the passband (solid line). Compare to standard Chebyshev response (dashed line). (b) Physical representation of CQ section of (a). Here, the capacitive probe is formed from a length of semirigid coaxial cable with the outer conductor and insulator removed from the ends. specification, stopband requirements, and resonator design for proper frequency and operation, high power-handling con- siderations, and others. The reader is referred to [5], [12], and [13] as sources for designing microwave filters. These sources present much general material over a range of technical levels. Limiting the scope of this tutorial to understanding the rela- tionship between a cross-coupled filter circuit’s topology and THOMAS: CROSS-COUPLING IN COAXIAL CAVITY FILTERS 1373 Fig. 10. Nested cross-coupling to produce two high-skirt transmission zeros (solid line). Compare to standard Chebyshev response (dashed line). TABLE V TOTAL PHASE SHIFTS FOR THREE PATHS OF THE CIRCUIT IN FIG. 10 its frequency response, a natural quantitative question arises; to what degree and in what sense are the various resonators coupled to one another for a given response? The answer to this question contains all the information required to describe the bandwidth and presence of any transmission zeros (at ei- ther real or imaginary frequencies). These couplings can be de- scribed by coupling coefficients or coupling bandwidths, where the coupling bandwidth equals the coupling coefficient multi- plied by the design bandwidth. For an resonator filter, these bandwidths can be arranged in an coupling-bandwidth matrix. Each coupling bandwidth is readily measurable with a vector network analyzer and is, therefore, immediately useful in the development and production tuning of microwave filters (see [14] for a unified approach to the technique of using cou- pling coefficients with coupled resonator filters). The methods for obtaining the coupling-bandwidth matrix re- quired for a given frequency response may be broadly catego- rized into the following three categories: 1) direct synthesis; 2) low-pass prototype transformations; 3) optimization methods. Fig. 11. Nested cross-coupling to produce two low-skirt transmission zeros (solid line). Compare to standard Chebyshev response (dashed line). TABLE VI TOTAL PHASE SHIFTS FOR THREE PATHS OF THE CIRCUIT IN FIG. 11 A. Direct Synthesis The direct analytical synthesis of coupling co