Thinking Outside the Band: Absorptive Filtering Matthew A. Morgan

Thinking Outside the Band: Absorptive Filtering Matthew A. Morgan Introduction Today's high-requency radio system engineer has at his ingertips an encyclopedic body o work to draw upon or his iltering requirements rom topologies, to synthesis equations, to proven methods o implementation. Compiled rom decades o work by countless researchers, the canon o literature on ilters nevertheless ocuses almost exclusively on techniques that assume rom the outset that all circuit elements will be nominally-lossless (some ew exceptions are noted in []-[9]. This constraint serves most o the conventional ideas about what a ilter should be quite well, including the minimization o passband insertion loss and the sharpest possible transition between passband and stopband. Fundamentally, however, it demands that the out-o-band power rejected by the ilter is relected back to its source. In microwave terminology, the impedance mismatch is maximized. In a ield where impedance-matching is almost as automatic as breathing, this departure rom the norm is accepted because it only occurs out o the operating band. It is easy to all into a pattern o thinking that in-band perormance is all that matters, and that engineers should not care how their circuits behave outside o the nominal operating requency range. This is not always true. (I outo-band signal power did not matter, why ilter it in the irst place? Frequency conversion circuits, like mixers and multipliers, may respond in unpredictable ways to chance reactive terminations at their ports outside o the nominal input and output requency ranges hence the common practice o "padding" the RF and IF ports o mixers, not only to improve an oten mediocre in-band impedance match, but also to desensitize them to broadband impedance variations. Out-o-band noise and spurious CW tones may be up- or down-converted into the operating band, degrading signal-to-noise ratio and dynamic range. Further, ampliier gain rarely drops o as sharply as one would like outside o the passband, especially now with the prolieration o wide-band and multi-band consumer electronics, giving rise to excess power at uncontrolled requencies. Conventional relective ilters do nothing to dissipate these spurious signals or added noise. Instead they build up as standing waves between the ilter and the source and/or are radiated into the electronics housing, impacting the linearity, stability, and electromagnetic compatibility o the system. These eects oten go unnoticed, but they can be important in some high-perormance applications. For these reasons, it is sometimes advantageous to design a ilter that absorbs the stopband spectrum rather than relecting it. As this requires at least some lossy elements to be used, care must be taken to ensure that the low inband insertion loss and impedance-matching are not discarded in the process. The two most common ways o doing this are with balanced ilters two identical, conventional ilters coupled at the input and output with 3 db quadrature hybrids and diplexers where the output o one or more branches is terminated with a resistor. There are, however, ar simpler requency-selective circuits that possess the desired broadband impedance match, while also exhibiting greater amplitude- and phase-stability with extended requency applicability. Relectionless Filter Cell The generic relectionless ilter cell with which we will be concerned is shown in Fig.. It is a symmetric network derived by the application o even- and odd-mode analysis with the constraint that the resultant two-port will have identically zero input relection coeicient at all requencies [0]-[]. The structure shown is low-pass, but may be trivially converted to any other coniguration high-pass, band-pass, band-stop, and multi-band by well-known transormation techniques [0]. Since all o the salient eatures o this topology translate directly into those other domains, the discussion in this article will be conined to low-pass ilters or simplicity. To illustrate, let us consider the transer characteristic o the single-pole cell shown at the top o Fig.. We may begin by decomposing the excitation o a single port into two simultaneous excitations o both ports, one in which both ports are excited in phase with one another, and the other with both ports excited in anti-phase. By superposition, the net response o the network to the single-sided excitation must be a linear combination o these two. Further, when the two ports are excited in phase, a virtual open-circuit may be drawn down the symmetry plane, and the entire circuit response may be determined by a reduced circuit containing at most hal o the elements. This is called the even-mode equivalent circuit. Conversely, when the two ports are excited in anti-phase, a virtual short-circuit may be drawn down the symmetry plane to arrive at the odd-mode equivalent circuit. The input impedance o these two equivalent circuits is given by

( ( jωl R jω ( ( even j C C odd ω (a j L jωc jωl R ω. (b To simpliy the algebra, let us substitute L 0 /ω p, C Y 0 /ω p, and R 0, and normalize the requency variable to the pole requency, ω/ω p. The above expressions then become even odd j ( ( ( 0 j j j 0 (a j ( j ( ( (. (b 0 j j j 0 j ( The relectionless coeicients o the even- and odd-mode equivalent circuits are then given by 3 even 0 Γ even even 0 j ( (3a odd 0 ( Γ odd ( odd 0 3 j. (3b The ull two-port scattering parameters may then be recovered as ollows ( Γeven Γ 0 ( Γeven Γodd 3 j ( s (4a s odd s s. (4b The relectionless property is thus proven by (4a. The transer characteristic, (4b, is plotted in Fig. 3. The perection o the impedance match is o course theoretical and dependant on the quality and tolerance o the elements used. Measured circuit perormance or low-pass and band-pass prototypes consisting o multiple cascaded cells is shown in [0]. The stop-band peak occurs at 3 (relative to the pole requency and has a value o (4 j 3 -, limiting stop-band rejection per cell to 0 log(8 4.47 db. The poles and zeros may be identiied simply by substituting the complex requency -js and actoring the transer unction (4b, ( s j ( s j ( s H. (5 ( j 7 j 7 s s s 4 4 Thus, the poles are at s - and (-±j 7/4, and the zeros are at s ±j. It can be shown, with the proper requency scaling, that this pole-zero coniguration corresponds to that o a third-order Chebyshev Type II (or Inverse Chebyshev ilter with ripple actor ε (3 3-0.9. [] An individual relectionless ilter cell may not have suicient stopband rejection on its own to satisy system requirements, but the cumulative eect o several in cascade can be competitive with conventional ilter types, especially in scenarios where broad impedance-matching is required. Consider, or example, the transer characteristic o our single-pole cells when compared to matched (resistively-terminated diplexers based on 0 th - order Chebyshev (Type I and 6 th -order Butterworth prototypes, as shown in Fig. 4. A quality actor o Q 30 was assumed or the inductors. Note that the overall requency selectivity (deined in this example as the dierence in 3 db and 60 db corners, respectively is the same in all cases. The perormance o higher-order cells may be derived in a similar manner, and in general result in sharper cuto at the expense o poorer stopband rejection at the peak [0]. Thus, arbitrarily sharp cuto is available, but larger numbers o cells must be cascaded to achieve a given level o attenuation in the stopband.

Distributed Filtering Though it usually occurs without pre-meditation, the standard practice or RF designers is to build-up a system using relatively broadband components and then deine the operating band with a single ilter, in eect attenuating all o the accumulated out-o-band noise, intererers, and spurious tones all at once. In that sense, it would be preerable to place the ilter at the very end o the analog portion o the system, or just prior to a requency conversion, so that all unwanted signals and sideband noise may be iltered out beore they are digitized or irrevocably overlaid upon an image band. On the other hand, dynamic range and linearity requirements tend to encourage iltering earlier in the RF or IF chain so that sensitive ront-end components are not overdriven. To get the best o both worlds, one would like to have ilters in several places, but this is usually an ineicient and costly practice considering the relative complexity o conventional high-order ilters, as well as the potential or negative out-o-band interactions between the ilter and neighboring components discussed previously. In this light, the small, divisible cells o the relectionless ilters represent a substantial advantage. Since each o the cells are small and individually matched, they need not be adjacent in the signal path. They may, instead, be distributed throughout the system as needed without risk o poor out-o-band interactions. On the contrary, when used in this manner they usually ameliorate other subtle out-o-band eects that are not always well-understood, resulting in a less glitchy, more stable system. They also prevent out-o-band signals rom accumulating or getting ampliied, keeping the unwanted signals at a lower level throughout the chain rather than allowing them to build and then beating them back down in the inal step. Stability Apart rom the system-stabilizing eects o good broadband impedance-matching described above, the relectionless ilter itsel is also more stable in amplitude and phase than its conventional counterparts, especially near the edge o the band, which can be a boon to precision applications where system calibration and component drit is critical. This superior stability is evident rom the derivatives o the complex gain (4b when contrasted with that o traditional ilter topologies [0], but is perhaps illustrated more clearly by an example. Consider the relectionless and conventional Chebyshev ilters in Fig. 4. A yield analysis was perormed to elucidate the gain changes that result rom small variations in the component values. The result is shown in Fig. 5. This plot shows the magnitude o the net change in ilter transer characteristic (amplitude and phase between 5 C and 35 C over a normal distribution o component values, each having % tolerance and a temperature coeicient o 50 ppm/ C. Not only does the complex gain o the relectionless ilter change less with temperature (it has a lower peak, it is also much more consistent (less spread. As expected, the bulk o the variation or both ilters is concentrated near the band edge. Frequency Versatility For any given manuacturing process, there is a limited range o component values available, and these components are limited in the requency range over which they can be used eectively. Surace mount components, or example, can only be made so small in value beore the pad capacitance and/or trace inductance dominates their behavior, and can only be made so large in value beore losses, sel-resonant eects, and/or dielectric breakdown render them useless. MIM capacitors that are too small suer rom excessive tolerance error due to ringing and contributions rom the coupling traces, whereas those that are too large become overmoded. Spiral inductors, transmission lines, thin-ilm resistors components o every variety all are subject to upper and lower bounds introduced by the manuacturing process. A curious eature o the relectionless ilter cell as drawn in Fig. is that all elements o a single type resistors, capacitors, and inductors have the same value, no matter what the order. This means the relectionless ilter will generally require more moderate values, neither too big nor too small, or a given passband than that o its conventional counterparts. To highlight the point, consider that the Chebyshev diplexer in Fig. 4 required capacitors ranging rom 6.5-3 pf, and inductors rom -03 nh. The Butterworth diplexer required.6-55 pf, and -44 nh, respectively. The relectionless ilter, however, required only 9. pf capacitors, and 3 nh inductors. This reduced component spread will enhance requency versatility in at least two ways. First, the larger components required by conventional ilter topologies when compared to the relectionless ilter cell will tend to have larger parasitics and lower sel-resonant requencies. On the other end o the scale, the smallest components required as you move to higher requencies may simply be unavailable, and in any case become lost in the parasitics o the manuacturing process. One might argue that the moderate component values are a consequence o building up the composite ilter by cascading several low-order ilters rather than a single high-order ilter. There is truth in this argument, however the comparison is not entirely air, because the relectionless property is what allows them to be cascaded so eectively;

conventional ilters cannot be cascaded in this way without severe passband distortion due to mismatches in the transition region. (And again, even the higher-order relectionless ilter cells in Fig., which do have sharper cutos, still require only the single component values o each type. Conclusions The relectionless ilter cell described in this article alleviates many system problems associated with excess out-o-band gain, impedance mismatches, and component interactions. The simplest ilter cell exhibits a third-order Inverse Chebyshev response with 4.47 db peak stopband attenuation, and can be cascaded or additional attenuation as needed. They are simple to design and easy to use or much like small ixed attenuators ("pads", they can be placed anywhere in the signal path desired without ear o causing unwanted standing waves, and instead reducing them where they already exist beyond the intended requency range. Compared to conventional ilter topologies with similar cuto requencies, they exhibit an order o magnitude greater amplitude and phase stability, ensuring accurate and repeatable complex gain in calibrated systems. Finally, the component requirements are moderate in value and minimal in number, easing their implementation while improving design yield, and extending the applicability o a given component technology beyond the normal range, both above and below. Reerences [] G. Matthaei, L. Young, and E. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures. Norwood, MA: Artech House, 980. [] V. Met, Absorptive ilters or microwave harmonic power, Proc. IRE, vol. 47, no. 0, pp. 76 769, Oct. 959. [3] C. Nelson, Ferrite-tunable microwave cavities and the introduction o a new relectionless, tunable microwave ilter, Proc. IRE, vol. 44, no. 0, pp. 449 455, Oct. 956. [4] J. Breitbarth and D. Schmelzer, Absorptive near-gaussian low pass ilter design with applications in the time and requency domain, in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 004, vol. 3, pp. 303 306. [5] A. Guyette, I. Hunter, and R. Pollard, Design o absorptive microwave ilters using allpass networks in a parallel-cascade coniguration, in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 009, pp. 733 736. [6] A. Guyette, I. Hunter, R. Pollard, and D. Jachowski, Perectly-matched bandstop ilters using lossy resonators, in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 005, pp. 57 50. [7] D. Jachowski, Compact requency-agile, absorptive bandstop ilters, in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 005, pp. 53 56. [8] S. Cohn and F. Coale, Directional channel separation ilters, Proc. IRE, vol. 44, no. 8, pp. 08 04, Aug. 956. [9] M. Morgan, T. Newton, R. Hayward, and T. Boyd, Non-relective transmission-line ilters or gain slope equalization, in IEEE MTT-S Int. Microw. Symp. Dig., Honolulu, HI, Jun. 007, pp. 545 548. [0] M. Morgan and T. Boyd, "Theoretical and Experimental Study o a New Class o Relectionless Filter," IEEE Trans. Microwave Theory Tech., vol. 59, no. 5, pp. 4-, May 0. [] M. Morgan, "Relectionless Filters," U.S. Patent Application No. /476883, June 009, PCT Application No. PCT/US0/507, January 00. [] T. Lee, Planar Microwave Engineering: A Practical Guide to Theory, Measurements and Circuits, Cambridge, UK: Cambridge University Press, 004.

Figures Fig.. Low-pass relectionless ilter cell. As drawn, all resistors, capacitors, and inductors, respectively, have the same value or a given cuto requency. Fig.. Analysis o a single-pole relectionless ilter cell by even- and odd-mode equivalent circuits.

Fig. 3. Theoretical response o a single-pole low-pass relectionless ilter cell. Fig. 4. Frequency selectivity o a cascade o 4 relectionless ilter cells, compared to 0 th -order Chebyshev and a 6 th -order Butterworth diplexers. Inductor Q 30. Fig. 5. Yield simulation showing the stability o the transmission coeicient or 0 th -order Chebyshev and 4-cell Relectionless ilters.