## Unparalleled Availability, Design and Manufacturing Capability to 6 GHz

Jacques and Pierre Curie discovered the piezoelectric effect in 1880-1881, but it would take another century for the predecessor to the modern-day ceramic coaxial resonator-based filter to be designed and described.^{1} Over the course of that century, and in the decades that followed, advances in materials science and manufacturing technology enabled the primitive ceramics discovered in the 1940s^{2} to evolve into the compact, temperature-stable, rugged, and complex bandpass filter solutions of the 21st century. Mini-Circuits boasts one of the strongest portfolios of ceramic coaxial resonator filters anywhere in the world, with unparalleled availability to 6 GHz.

Ceramic coaxial resonators boast two parameters: quality factor (Q) and dielectric constant (ϵ_{r}) that are so impressively high as to explain why these resonators have become standard building blocks for bandpass and bandstop filters for over four decades. While Q and ϵ_{r} both vary as a function of the ceramic material utilized, Q ranges in the hundreds to as high as 1000 and ϵ_{r} spans from slightly less than 10 to nearly 100. High-Q leads to sharp resonant behavior (steep roll-off for filters) and high-ϵ_{r} reduces the electrical length of the resonator by the square root of ϵ_{r}. This means that the length of a resonator constructed of ceramic with an ϵ_{r} of 100 will be one tenth that in free space. Using the equation for free space wavelength λ = c/f, the wavelength at 1 GHz is equal to 0.3 meters, and the quarter-wave length 7.5 cm. Contrast that length with the 1 GHz quarter-wave length of a resonator constructed with a ceramic of dielectric constant ϵ_{r} of 90 using the equation^{5}:

l = 300/(4*f*√ ϵ_{r}) where l is in mm and f is in GHz,

l (mm) = 300/(4*1*√90) = 7.9 mm = 0.79 cm.

The dramatic difference in size achieved by utilizing a ceramic coaxial resonator is apparent in the results of these calculations. The size and selectivity advantages that ceramic coaxial resonators yield in filter applications are significant, but the list of advantages doesn’t end there.

Modern silver-plated, ceramic materials are also very predictable with respect to their temperature dependency. The temperature coefficient of resonant frequency Ꚍ_{f} of ceramic coaxial resonators is typically between 0 and 10 ppm per ⁰C. For example, even at a Ꚍ_{f} of 10 ppm per ⁰C, a filter centered at 1000 MHz would drift just a total of 0.6 MHz between 25⁰C and 85⁰C. Consequently, the center frequency stability of ceramic coaxial resonator filters with respect to operating temperature is generally exceptional, and easily eclipses that of lumped element filters. Additionally, the high-Q (low loss tangent) of the ceramic materials leads to very low insertion loss and little ripple in the passband for bandpass filters. Due to their low insertion loss, the power handling of ceramic coaxial resonator filters is generally specified in Watts. It’s not surprising to see such a filter capable of withstanding a 10W input signal continuously.

Ceramic coaxial resonator filters are very rugged. Fired at temperatures exceeding 1000⁰C and plated with silver, these painted rocks^{3} can withstand very harsh environments and are only affected by the presence of condensing humidity. In fact, once the water droplets evaporate, the filter returns to normal performance levels. Finally, ceramic coaxial resonator filters are far more cost-effective than their counterparts utilizing other technologies with comparable performance. Mini-Circuits’ line of ceramic coaxial resonator filters is a comprehensive selection with many models already in stock. This broad portfolio of existing designs enables Mini-Circuits to furnish affordable, custom designs with fast turnaround time for special requirements. Mini-Circuits has ceramic coaxial resonator filter design capability, design velocity, and product availability found nowhere else in the industry.

**Figure 1: Mini-Circuits’ line of ceramic coaxial resonator bandpass filters in stock.**

Test data for greater than 100 models of ceramic coaxial resonator filters is available on the Mini-Circuits website. These models are shown in Figure 1 as existing designs. Also superimposed in Figure 1 is the design capability Mini-Circuits enjoys in the ceramic coaxial resonator filter space.

## Show Them What You’re Made of

In ceramic coaxial resonator filter construction, the resonators are soldered to a ground plane, positioned side-by-side. Each resonator is a quarter-wave shorted transmission line that takes advantage of the piezoelectric effect of the ceramic material to both reduce size (often by a factor of 5-10) and boost Q (typically into the hundreds) as compared to a traditional microstrip transmission line. Energy is coupled from one resonator to the next through coupling capacitors. each resonator combined with its associated coupling capacitor is referred to as a “section.” Figure 2 shows an example of a 5-section Mini-Circuits ceramic coaxial resonator bandpass filter with the cover removed. Figure 2 illustrates the five resonator sections including the capacitors that couple energy between the resonators themselves. On the lower left-hand side of the figure is an LTCC cleanup filter, and castellations are shown on the right and left for input and output, respectively.

**Figure 2: A Mini-Circuits 5-section ceramic coaxial resonator filter with cover removed.**

A cavity filter with equivalent performance would be many times the size, and a lumped element filter would not be nearly as temperature-stable, even if it were temperature-compensated and hand-tuned.

A comparison of a ceramic coaxial resonator filter and its lumped element counterpart is intriguing, but even more interesting, and perhaps more informative is a discussion of the theoretical lumped element *equivalent* of a ceramic coaxial resonator, and the general design of a bandpass filter utilizing these types of resonators. Fundamentally, the resonator length (l), width (W), inner diameter (d) and the dielectric constant of the ceramic material (ϵ_{r}) determine the characteristic impedance (Z_{0}) of the coaxial resonator. Figure 3 shows the physical dimensions, circuit symbol and lumped element equivalent of the ceramic coaxial resonator. As shown in Figure 3, the length of a shorted resonator utilized in bandpass filter construction is approximately a quarter wavelength, so that it becomes resonant within the frequency band that the filter is designed to pass.

**Figure 3: Ceramic coaxial resonator physical properties, circuit symbol and lumped element equivalent ^{4,5}**

For shorted, quarter wave resonators, the formula that relates the length of the resonators to their corresponding resonant frequencies is^{5}:

f = 300/(4*l*√ϵ_{r}) or, by rearranging: l = 300/(4*f*√ ϵ_{r})

where f is frequency in GHz and l is length in mm.

The characteristic impedance (Z_{0}) of the quarter wave resonator is defined by^{5}:

Z_{0} = (60/√ϵ_{r})(ln(1.079(W/d))) and Q = 60(W√f/25.4)

where Z_{0} is the ceramic coaxial resonator impedance in Ohms, f is frequency in MHz, W is width in mm, and d is resonator inner diameter in mm. Q is, of course, dimensionless.

The formulas that govern the values of the circuit elements that comprise the lumped element equivalent are then given by^{5}:

L = 2Z_{0}/∏^{2}f, C = 1/8Z_{0}f, and R = 4Z_{0}Q/∏

where Z_{0} is the ceramic coaxial resonator impedance in Ohms, f is frequency in Hz, L is inductance value in H, C is capacitance in F, and R is resistance in Ohms.

Designing a filter that takes advantage of the high dielectric constant (ϵ_{r}) of the ceramic material and the high Q of the coaxial resonator structure requires the use of filter theory and a knowledge of poles and zeros. For a filter similar to that shown in Figure 2 (capacitively-coupled with shunt quarter-wave ceramic coaxial resonators), it is possible to use the J-inverter method to complete the design.^{4,6} This and other design methodologies are well published and understood, and beyond the scope of this article. Mini-Circuits utilizes many advanced design and simulation techniques for quick turnaround of custom ceramic coaxial filter designs to address a myriad of complex filter applications.

## Outline Drawing and Interface Methodology

The CBP-2400A+ ceramic coaxial filter is a similar, 5-section filter to that shown in Figure 2 and is housed in the KU1513 package, which has a footprint of just 1.040 in. (26.24 mm) long and 0.55 in. (13.97 mm) wide, with a height of 0.185 in. (4.70 mm), including the metal cover. The KU1513 package is shown in the case outline drawing for the CBP-2400A+ filter, and a bottom view is shown in Figure 4.

**Figure 4: KU1513 package bottom view and footprint dimensions for Mini-Circuits’ CBP-2400A+ ceramic coaxial resonator filter.**

In addition to the input and output traces (pins 1 and 10, respectively), there are a total of 14 castellations designed to be reflowed to the ground plane (along with the ground plane of the KU1513 package itself). The bottom view in Figure 4 clearly shows the key areas to be reflowed at the next higher level of assembly. The metal lid limits radiation out the ends of the resonators and also limits stray coupling into or out of the filter itself.

## What’s under the Hood?

Ceramic coaxial resonator filters exhibit outstanding filter performance for their respective sizes, which is the reason for their increasing popularity since the first of the modern-style, plated, ceramic resonators were described at the outset of the 1980s.^{1} Take, for instance the performance of the CBP-2400A+ shown in Figure 5. The filter roll-off is very steep for a filter that measures just 1.040 in. (26.24 mm) long, 0.55 in. (13.97 mm) wide, and with a height of only 0.185 in. (4.70 mm),. Additionally, the mid-band insertion loss, while not apparent from the plot, is just 1.01 dB at 25⁰C.

**Figure 5: Mini-Circuits’ CBP-2400A+ ceramic coaxial resonator filter frequency response.**

The CBP-2400A+ has a specified bandwidth of 400 MHz (roughly 16.7%), spanning 2200-2600 MHz. While the actual BW exceeds the specification, the lower skirt still furnishes 60 dB of rejection at approximately 1500 MHz, and the upper skirt provides 40 dB of rejection at 3700 MHz. The ultimate rejection below 1000 MHz for the CBP-2400A+ is greater than 80 dB, and the ultimate rejection above 3800 MHz exceeds 50 dB. The performance of the CBP-2400A+ shows the advantages of ceramic coaxial resonator bandpass filters, steep upper and lower skirt slopes and exceptional stopband rejection as well, considering that the filter insertion loss across the passband is approximately 1 dB at 25⁰C.

## The Final Section

The prevalence of ceramic coaxial resonator filters is not surprising due to their many advantages over cavity and lumped element solutions. Smaller size, lower cost, superb ruggedness, and excellent temperature stability are some of the most important metrics that make the choice of a ceramic coaxial resonator filter a good one. In addition to those advantages, the electrical performance of ceramic coaxial resonator filters is second to none in a given package size in terms of low insertion loss, steep roll-off and high stopband rejection. Look beyond Mini-Circuits’ leading portfolio of over 150 existing ceramic coaxial resonator filters and you will find quick-turn, custom filter design capabilities that will furnish solutions with industry-leading velocity and affordability.

Browse Mini-Circuits’ full selection of ceramic coaxial resonator filters.

## References

- United States Patent 4223287, Nishikawa, Toshio; Ishikawa, Youhei; Tamaura, Sadahiro; Matsumoto, Haruo; 09/16/1980, Murata Manufacturing Co., Ltd (Kyoto, JP), Electrical filter employing transverse electromagnetic mode coaxial resonators – Murata Manufacturing Co., Ltd. (freepatentsonline.com)
- The History of Ceramic Filters, Satoru Fujishima, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Volume 47, No. 1, January 2000, uf509.dvi (psu.edu)
- Comnav Engineering, Frequently Asked Questions (FAQ) – Ceramic Filters, “How do ceramic filters act under extreme environmental conditions?”, ComNav Engineering | FAQs (comnav-eng.com)
- Ceramic coaxial resonator filter in a CubeSat system, Odette Sandrine Bakam Nguenouho, December 2017, 208008616-Bakam Nguenouho-Odette Sandrine-MEng-Electrical-Engineering-Eng-2018.pdf (cput.ac.za)
- Exxelia TEMEX 07/2015 pp. 143-144 epsilon-21-1-4-v1.pdf (exxelia.com)
- Matthaei, Young and Jones, 1980. Microwave filters, impedance matching networks and coupling structures. Dedham, MA: Artech House

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