Lowpass filters with maximally flat group delay are ideal for digital signal transmission systems, since they allow pulsed signals to rise and fall smoothly and settle rapidly to final values without distortion. Filters such as the Bessel-Thomson lowpass filter have been used in these applications, although the configuration is reflective at stopband frequencies and can cause signal distortion with impedance mismatches. A better solution is the quasi- Gaussian absorptive lowpass filter, with flat group delay in the passband, good match at all frequencies, and absorptive rather than reflective behavior for stopband signals. An effective method for creating quasi-Gaussian absorptive lowpass filters with shunt resistive-capacitive (RC) elements is through the use of dot-terminations.

Because of their sensitivity to impedance mismatches, Bessel-Thomson lowpass filters have often been used with attenuators before and after the filter to compensate for poor filter return loss while also minimizing signal distortion. Recently, quasi-Gaussian lowpass filters were developed with a Bessel-Thompson-like transmission response and low reflection in both the passband and stopband.^{1} Signals that are not transmitted through the filter are absorbed in its resistive elements of the filter, minimizing reflections and signal distortion.

The quasi-Gaussian absorptive lowpass filter has flat group delay in the passband, good match at all frequencies, and absorbs rather than reflects unwanted signals in the stopband. This filter is often called a lowpass, risetime filter and adheres to the time-frequency relationship in Eq. 1. For a filter having a desired pulse rise time, its -3-dB bandwidth can be determineda necessary parameter when designing the filter. With a lowpass rise-time filter, limiting the rise time in a digital system can eliminate high-frequency noise, overshoot, ringing, and reflections.

Other applications include telecommunications SONET/SDH compliance testing specifications that mandate the use of lowpass filters with Bessel-Thompson- like transmission characteristics. The quasi-Gaussian absorptive filter is not limited to time-domain signal applications. Often, frequency multipliers and mixers require broadband termination impedances for all harmonics and spurious combinations, respectively. At the expense of degraded dynamic range, attenuators followed by reflective filters have been used to attenuate unwanted signals while maintaining the termination impedance at the output of the multiplier or mixer. With the use of an absorptive lowpass filter, the unwanted signals can be attenuated and the desired signals passed without degrading dynamic range.

Most absorptive lowpass filters utilize a shunt RC filter topology with series inductance (L). Any energy that is not transmitted through an absorptive lowpass filter is absorbed in the resistive element. One way of creating a quasi-Gaussian absorptive lowpass filter with shunt RC elements is to use microstrip dot-terminations. Dot-terminations are typically used to terminate high-frequency, broadband microstrip circuits. The performance is acceptable at high frequencies over wide ranges but not at low frequencies or to DC since they do not have a direct ground connection.^{3} Dot-terminations consist of an input metal trace attached to a circular area of thin-film resist processed on a ceramic substrate. The resistivity of the thin-film resist matches the impedance of the metal trace to produce a low-reflection termination. A parallel-plate capacitance is formed between the circular area of the resist and the ground through the substrate material. The size of the circular area of the thin-film resist determines the lowfrequency limit of the termination. Dotterminations have been used to provide ultra-wideband terminations for optical modulators,^{5} as well as broadband microstrip couplers.^{4,6}

Earlier work^{7} on microstrip dot terminations focused on an analytic approach based on static approximations for an isolated, nongrounded thin-film dot-termination. This approach treated the dot-termination as a cascade of very short lengths of lossy transmission lines. The analytical approach showed considerable error in the reflection coefficient level when compared to three-dimensional (3D) electromagnetic (EM) simulation results. The simplified analytical model using a cascade of very short lengths of lossy transmission line could not predict the following: (1) the effects of step discontinuities between each adjacent cascaded element of different line width, (2) how the RF current spreads from the microstrip trace into and through the resistive film, and (3) the open-ended effects of each cascaded element. The author noted that when using 3D EM simulation, the current distribution in the thin-film resist is concentrated near the center of the resistive film and is not evenly distributed throughout the area of the termination.

Since analytical models do a relatively poor job in predicting the frequency response, the dot-termination structure is ideally suited to be modeled using an electromagnetic field solver. From the results of the simulated model, equivalent shunt resistance and capacitance as a function of frequency and structure geometry can be computed from the simulated one-port Zparameters.

* Figure 1* shows three structures of a microstrip dottermination. Each has a 250- mwide microstrip trace connecting to a 500- m-diameter circular area of thin-film resist on a 250- m-thick alumina substrate. The thin-film resist has resistivity of 50 Ohm/sq. The first termination in

*has its reference, X = 0, defined at the end of the microstrip trace where it meets the circumference of the circular dot. The second and third terminations show the microstrip trace pulled back from the circumference of the circular dot by 100 and 200 m, respectively.*

**Fig. 1** Analysis of these structures was performed on a full-wave field solver^{9} from 2 to 30 GHz. In all three cases, the calibrated reference plane of the field solver was set at the end of the microstrip trace where it meets the thinfilm resist. The results of the analysis for the three microstrip trace offsets are shown in * Figs. 2* and

*.*

**Fig. 3***shows that the equivalent resistance is higher in value as the microstrip trace is pulled back from the circumference of the dot. Since a small length of resist was added between the dot's circumference and the microstrip trace, dotthe resistance increases. Although not shown, the equivalent resistance can be further lowered in value by embedding the microstrip trace into the circular dot-termination. Embedding the microstrip trace into the dot's circumference allows a small portion of resist to be in parallel with each side of the trace, which reduces the equivalent resistance. Since the dot-termination diameter is held constant, the equivalent shunt capacitance shown in*

**Figure 2***is weakly influenced by the microstrip trace offset. The three sets of data in*

**Fig. 3***are closely grouped but do show increasing equivalent capacitance with frequency. With increasing frequency, the dot diameter will approach a quarter wavelength, ?g/4, and the equivalent shunt capacitance increases rapidly. Near ?g/4, the total reactance of the structure can turn inductive and the dot-termination will pass through a low-quality-factor (Q) resonance. For this reason, the dot diameters were kept less than ?g/8 at the highest simulated frequency so the simple shunt resistance-capacitance model would be valid.*

**Fig. 3**For a given shunt capacitance, dotterminations printed on high-dielectric- constant substrates are smaller in diameter relative to low-dielectric-constant substrates for the same substrate thicknesses. Furthermore, dot diameters can also be reduced by reducing the substrate thickness. Both dielectric constant and substrate thickness can be chosen so the dot-termination diameter can be kept less than ?g/8 below the highest operating frequency.

Analysis of two other dot-termination structures were performed by varying only the dot diameter while keeping the metal trace offset at -100 m (i.e., the resist protruding from the dot circumference by 100 m). These results are shown in * Figs. 4* and

*. The dot diameters that were analyzed were 400, 500, and 600 m.*

**5***shows that the equivalent shunt capacitance increases with increasing dot diameter whereas the equivalent resistance is only weakly influenced.*

**Figure 5***show the dependence on metal trace offset and dot diameters on equivalent resistance and capacitance. These graphs provide a convenient lookup table for filter designers using dot-terminations as lossy RC filter elements.*

**Figures 2-5** A lumped-element topology for a quasi-Gaussian absorptive lowpass filter is shown in * Fig. 6*. Each L and C element has a single value and does not span a range unlike most other filter prototypes. Wide ranges of element values can make a filter impossible to realize. The interior shunt RC elements were paralleled to halve the capacitance value from 2C to C and double the resistance value from R/3 to 2R/3 to make lossy RC filter elements easier to realize when synthesizing designs with dot terminations. Low resistance values require a large metal trace offset protruding into the dot-termination and makes the attachment point from the adjacent inductors to the dot structure difficult to realize (Eqs. 2-4).

The design of a quasi-Gaussian absorptive filter using dot-termination lossy RC filter elements begins by calculating lumped-element inductance and capacitance values using Eqs. 2 and 3. Parameter Zo is the characteristic impedance and ?o is the -3-dB cut-off frequency. Resistance R is simply equal to the characteristic impedance. The number of reactive filter sections is n, whereas the order of the filter is n + 1 due to the effect of the additional resistance.^{8} The number of reactive elements, n, is an odd number greater than 3 to maintain filter symmetry. Once the lumped-element values have been calculated from a given order n, the characteristic impedance, Z_{o}, and the -3-dB corner frequency, ?_{o}, the physical parameters of the dot-termination can be synthesized from equivalent RC values using graphs similar to * Figs. 2-4*. Knowing the ideal lumped-element capacitance value needed from Eq. 2, an initial estimate can be made for the circular dot diameter d using Eq. 5 and applies for d

where c = the lumped element capacitance (in pF); h = the dielectric height (in um); e_{}0 = the free-space permittivity (8.854 x 10^{-6} pF/ m); and e_{r} = the relative dielectric constant.

Continue to page 2

### Page Title

A tenth-order quasi-Gaussian lowpass filter was designed to demonstrate the effectiveness of using dot-terminations as lossy RC filter elements. The filter was designed with a -3-dB cut-off frequency of 8 GHz on 250- m-thick alumina substrate. The lumped-element values were calculated using Eqs. 2-4, C = 0.14 pF, L = 0.35 nH, and R = 50 Ohms. The inductor was synthesized using a 790 x 30 m microstrip line. The 30- m width translates to a microstrip impedance of 103 Ohms. The dot-termination geometries were synthesized using graphs similar to * Figs. 2-4*. The lumped-element values were matched to the graphs at the corner frequency of the filter. The table shows the dimensions of the 8 GHz dot-termination filter elements.

*shows the filter, simulated with a full-wave 3D simulator*

**Figure 7**^{10}with a cover height of 2000 m. The length and width of the substrate are 6610 and 2210 m, respectively.

A coaxial fixture using Anritsu's V105F coaxial spark plug connectors with a 5-mil backside pin was used to launch from the alumina circuit to a V (1.85 mm) coaxial environment. The fixture and its connectors were not deembedded from the measurement results but they are included as part of the filter measurements. An Anritsu model MS4647A 70-GHz "VectorStar" vector network analyzer (VNA) from Anritsu Company was used to measure group delay and S-parameters. Measured versus simulated results are shown in * Figs. 8* and

*. In Fig. 8, the group delay is higher for the measured value due to the lengths of each V105F spark plug connector. There is excellent agreement between the measured and simulated group delay curves. The simulated S-parameter results show a cavity resonance at 66 GHz but this resonance was not present in the measured results, although a resonance may be present above 70 GHz for the measured filter. Unlike most distributed filters, the stopband rejection is very broadband with slight return response around 60 GHz. The results for the distributed microstrip filter with dot-terminations and the lumped-element filter are in good agreement.*

**9**The dot-termination used as an RC element was chosen as a matter of convenience but is restricted to thin-film designs. Other solid, thin-film resist shapes such as ellipses can be used. Another commonly used RC element is a resistor in series with a fan stub, with the fan stub serving as the capacitive element.

Dot-terminations have been used as broadband lossy RC filter elements for absorptive filters. The equivalent lumped-element resistance and capacitance can be accurately modeled using a full-wave EM field solver and computed directly from the Z-parameters. Tabulating and graphing these results can provide a means of synthesizing dot-terminations for equivalent lumped-element resistance and capacitance values for practical designs.

**REFERENCES**

1. A. R. Djordjevic, A. G. Zajic, A. S. Stekovic, M. M. Nikolic, Z. A. Maricevic, and M. F. C. Schemmann, "On a Class of Low-Reflection Transmission- Line Quasi-Gaussian Low-Pass Filters and Their Lumped-Element Approximations," IEEE Transactions on Microwave Theory & Techniques, Vol. 51, No. 7, July 2003, pp. 1871-1877.

2. ITU-T G.957 Recommendation, Optical Interfaces for equipments and systems relating to the synchronous digital hierarchy, March, 2006, pp. 20-21, www.itu.int.

3. Bill Oldfield, "Connector and termination construction above 50 GHz," *Applied Microwave & Wireless*, April 2001, pp. 56-66.

4. Bill Oldfield, "Development of Broadband Coaxial Components for a 0.04 to 110 GHz Measurement System," IEEE MTT-S International Microwave Symposium Digest, Orlando, FL, May 1995, pp. 14711473.

5. Bill Oldfield and K. Noujeim, Anritsu Company, Morgan Hill, CA, "Microstrip Dot Termination usable with Optical Modulators," United States Patent No. 6,593,829, 2003.

6. M. Morgan and S. Weinreb, "Octave Bandwidth High-Directivity Microstrip Codirectional Couplers," IEEE MTT-S International Microwave Symposium Digest, Philadelphia, PA, June 2003, pp. 1227-1230.

7. R. R. Monje, V. Vassilev, A. Pavolotsky, and V. Belitsky, "High Quality Microstrip Termination for MMIC and Millimeter-Wave Applications," IEEE MTT-S International Microwave Symposium Digest, Long Beach, CA, June 2005, pp. 1827-1830.

8. J. Breitbarth and D. Schmelzer, "Absorptive Near-Gaussian Low Pass Filter Design with Applications in the Time and Frequency Domain,", IEEE MTT-S International Microwave Symposium Digest, Fort Worth, TX, June 2004, pp. 13031306.

9. Sonnet Professional Version 11.52, Sonnet Software, North Syracuse, NY.

10. HFSS Version 11, Ansoft Corp., Pittsburg, PA, www.ansoft.com.