EP2882033A1

EP2882033A1 - Radio-frequency resonator and filter

Info

Publication number: EP2882033A1
Application number: EP13196298.7A
Authority: EP
Inventors: Hakim Aouidad; Eric Rius; Jean-François FAVENNEC; Yann Clavet; Alexandre Manchec
Original assignee: Centre National de la Recherche Scientifique CNRS; ELLIPTIKA; Univerdite de Bretagne Occidentale; Ecole Nationale dIngenieurs de Brest ENIB
Current assignee: Centre National de la Recherche Scientifique CNRS; ELLIPTIKA; Univerdite de Bretagne Occidentale; Ecole Nationale dIngenieurs de Brest ENIB
Priority date: 2013-12-09
Filing date: 2013-12-09
Publication date: 2015-06-10

Abstract

A radio-frequency resonator comprising a plurality of coaxial line sections (CLS₁ ...CLS_n) connected in cascade, each of said coaxial line sections comprising an inner conductor (C_i) surrounded by an outer conductor (C_i+1) of tubular shape, wherein said coaxial line sections are nested within each other, the outer conductor of each said coaxial line section, except an outermost one (C_n+1), serving as the inner conductor of another one of said coaxial line sections, the resonator being characterized in that each conductor of each coaxial line section, except the outer conductor of the outermost coaxial line section, has an open-circuit end and an opposite end which is short-circuited to said outer conductor of said outermost coaxial line section, and in that said conductors of said coaxial line sections - except the outer conductor of said outermost coaxial line section - are arranged head-to-tail, their open-circuit and short-circuited ends being alternatively situated on opposite sides of the resonator. Such a resonator achieves a better tradeoff between the conflicting requirements of reduced size and good electrical performances.

A radio-frequency filter comprising a plurality of such radio-frequency resonators mounted adjacent to each other and coupled by openings in the outer conductors of their respective outermost coaxial lines, forming coupling irises.

Description

The invention relates to a radio-frequency resonator, and more specifically to a stepped-impedance coaxial radio-frequency resonator. The invention also relates to a radio-frequency filter comprising a plurality of coupled resonators.
The expression radio-frequency designates frequency comprised between 3 kHz and 300 GHz. The invention applies more specifically to frequencies belonging to the UHF (Ultra-High-Frequency) band, i.e. comprised between 300 MHz and 3 GHz, although it can also apply to frequencies belonging to the VHF (Very-High-Frequency) band, i.e. comprised between 30 MHz and 300 MHz, and/or to the SHF (Super-High-Frequency) band, i.e. comprised between 3 GHz and 30 GHz, and/or millimetric band, i.e. comprised between 30 GHz and 300 GHz.
In the fields of electronics and telecommunications there is an ever-increasing need for radio-frequency resonators and filters having a small size, good electric performances (low insertion losses, flat amplitude and linear phase in the pass-band, high rejection outside the pass-band, tunability, etc.) and low cost. Size and mass are particularly critical parameters in comparatively low-frequency applications, i.e. in the VHF and UHF bands. Moreover, size and mass reduction usually comes at the expense of increased insertion losses and decreased quality factor. Figure 1 shows qualitatively the size - insertion losses tradeoff for the main available technologies in the UHF Band.
Lumped element resonator and filters, which can be used up to the C-band (4 - 8 GHz) are the most compact ones, but are plagued by high insertion losses and have very reduced quality factors (Q= 10 - 50). At the other end of the spectrum, waveguide (or "volume") resonator and filters have very high quality factors (up to 10.000 or more) and low insertion losses, but they are very bulky. Microstrip, coaxial and dielectric resonators offer intermediate performances. Within the coaxial technology, in particular, one can distinguish between uniform air coaxial resonators, with better electrical properties, and uniform dielectric coaxial resonators, more compact but with lower Q and higher insertion losses. Coaxial stepped impedance resonators, constituted by a plurality of coaxial line sections of different characteristic impedance connected in cascade, offer an interesting tradeoff between conflicting electrical and mechanical requirements. The coaxial line sections can be partially or wholly loaded with high-dielectric-constant ceramics for further miniaturization, but again at the expenses of increased losses and reduced Q. Coaxial stepped impedance resonators are described, in particular, by the following papers:

M. Makimoto and S. Yamashita "Compact Bandpass Filters Using Stepped Impedance Resonator" Proceedings of the IEEE, Vol. 67, No. 1, January 1979;
S. Yamashita and M. Makimoto "The Q-Factor of Coaxial Resonators Partially Loaded with High Dielectric Constant Microwave Ceramics" , IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-31, No. 6, June 1983; and
S. Yamashita and M. Makimoto "Miniaturized Coaxial Resonator Partially Loaded with High-Dielectric-Constant Microwave Ceramics", IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-31, No. 9, September 1983;

US Patent 4,059,815

Document US 4,292,610 describes a stepped-impedance coaxial resonator having a particularly compact structure. As illustrated on figure 2, such a resonator comprises an outer, hollow conductor 1, an inner conductor 2 coaxially mounted within the outer conductor and having a first end short-circuited to a wall of the outer conductor and a second end spaced from the wall of the outer conductor, and an intermediate hollow conductor 3 coaxially mounted between said inner conductor and said outer conductor. Intermediate conductor 3 has a closed end short-circuited to the second end of the inner conductor, and an open end. Conductors 1, 2 and 3 form three coaxial line sections I, II and III having different geometries and therefore, in general, different characteristic impedances Z₁, Z₂ and Z₃. Interestingly, sections II and III are nested with each other. A drawback of this resonator is its lack of design flexibility, as the electrical lengths and the characteristic impedances of the different coaxial line sections are not independent from each other. Moreover, the number of cascaded different coaxial line sections cannot be changed, and the resonator is not easily made tunable.
The invention aims at curing, at least in part, some or all the above-mentioned drawbacks of the prior art, and more particularly at providing a new resonator structure achieving a better tradeoff between the conflicting requirements of reduced size and good electrical performances.
An object of the present invention, allowing achieving this aim, is a radio-frequency resonator comprising a plurality of coaxial line sections connected in cascade, each of said coaxial line sections comprising an inner conductor surrounded by an outer conductor of tubular shape, wherein said coaxial line sections are nested within each other, the outer conductor of each said coaxial line section, except an outermost one, serving as the inner conductor of another one of said coaxial line sections, the resonator being characterized in that each conductor of each coaxial line section, except the outer conductor of the outermost coaxial line section, has an open-circuit end and an opposite end which is short-circuited to said outer conductor of said outermost coaxial line section, and in that said conductors of said coaxial line sections - except the outer conductor of said outermost coaxial line section - are arranged head-to-tail, their open-circuit and short-circuited ends being alternatively situated on opposite sides of the resonator.
According to different embodiments of the invention:

The radio-frequency resonator may comprise a conductive shell of generally cylindrical shape having a lateral surface, a first base surface and a second base surface, said lateral surface constituting the outer conductor of the outermost coaxial line; a conductor of tubular or rod-like shape, called central conductor, constituting the inner conductor of an innermost coaxial line, said central conductor being contained within said conductive shell and having a first end short-circuited to said first base surface a second end facing said second base surface; and at least one tubular conductor, contained within said conductive shell and containing said central conductor, having an end contacting one said base surface and an opposite end facing the other base surface.
The radio-frequency resonator may comprise exactly two said coaxial line sections.
The radio-frequency resonator may comprise exactly three said coaxial line sections.
An inner conductor of an innermost coaxial line section may be mounted slidably along its longitudinal axis, in such a way that it can be partially extracted from the resonator.
An inner conductor of a coaxial line section may be electrically connected to a variable capacitor.
The outer conductor of the outermost coaxial line section may have a rectangular (including square as a particular case) cross-section.
The radio-frequency resonator may further comprise an excitation rod extending transversally with respect to said coaxial line sections, said excitation rod having a distal end extending to the outside of the resonator and a proximal end contacting or facing a conductor other than said outer conductor of the outermost coaxial line section

Another object of the invention is a radio-frequency filter comprising a plurality of such radio-frequency resonators mounted adjacent to each other and coupled by openings in the outer conductors of their respective outermost coaxial lines, forming coupling irises. In particular, said radio-frequency resonators may be identical to each other.
It is important to note that the expression "coaxial line" should be interpreted broadly, as indicating any transmission line section comprising an elongated conductor ("core") surrounded - without direct contact - by a tubular conductor ("shielding"), both conductor having constant cross-sections (i.e. being "cylindrical" in the general sense of the term, not limited to circular-base cylinders) and parallel longitudinal axis. This definition includes e.g. "eccentric coaxial lines", which are not exactly "coaxial" in geometrical terms.
Additional features and advantages of the present invention will become apparent from the subsequent description, taken in conjunction with the accompanying drawings, wherein:

Figure 3 is a simplified representation of the structure of a resonator according to an embodiment of the invention, comprising an arbitrary number N of cascaded coaxial sections;
Figure 4, is an electrical scheme of the resonator of figures 3 ;
Figure 5 is a simplified representation of the structure of a resonator according to an embodiment of the invention, comprising exactly two cascaded coaxial sections;
Figures 6A - 6D are plots showing how different parameters characterizing the resonator of figure 5 depend on the characteristic impedance ration of its two cascaded coaxial sections;
Figure 7 is a simplified representation of the structure of a resonator according to an embodiment of the invention, comprising exactly three cascaded coaxial sections;
Figure 8 allows comparing the size and the unloaded quality factor Q, of the resonators of figures 5 and 7, of an air uniform coaxial resonator and of a dielectric uniform coaxial resonator (ε_r = 90);
Figure 9 is the electrical scheme of a second-order Chebyshev band-pass filter according to an embodiment of the invention;
Figure 10 is an elevation view of a practical implementation of the filter of figure 9;
Figures 11A and 11B are plots of the measured and simulated values of the scattering parameters S₁₁ and S₁₂ of the filter of figure 10;
Figure 12 is the electrical scheme of a sixth-order Chebyshev band-pass filter according to an embodiment of the invention;
Figures 13A and 13B are plots of the simulated values of the scattering parameters S₁₁ and S₁₂ of the filter of figure 12;
Figures 14A and 14B illustrates two tunable resonators according to alternative embodiments of the invention; and
Figure 15 shows plots of the measured and simulated values of the scattering parameters S₁₁ and S₁₂ of a second-order tunable filter according to an embodiment of the invention.

The physical structure of a quarter-wave radio-frequency resonator according to the invention is represented on figure 3. In such a resonator, n>1 coaxial line sections are nested within each other (""matryoshka" structure), disposed head-to-tail - i.e. with open-circuit and short-circuit ends on alternate sides of the resonator. From another point of view, the resonator comprises (n+1) cylindrical conductors C₁ - C_n+1 whose cross-sections have diameters indicated by d_i, with i=1 to (n+1). The innermost or central conductor C₁ can be either tubular (i.e. hollow) or rod-like (i.e. solid); the outermost conductor C_n+1 has a lateral surface and two opposite base surfaces, forming a conductive shell. All the conductors, except the outermost one, have an open-circuit end and an opposite end which is short-circuited to the outermost conductor C_n+1. Conductors are arranged head-to-toe in the sense that their open-circuit and short-circuited ends are alternatively situated on opposite sides of the resonator (on figure 3, odd-numbered conductors have their left end short-circuited to the left base surface of the outermost conductor, while even-numbered conductors have their right end short-circuited to the right base surface of the outermost conductor). It can be understood that conductors C₁ and C₂ form a first coaxial line section having C₁ as the inner conductor (or "core") and C₂ as the outer conductor (or "shielding"); similarly, C₂ and C₃ form a second coaxial line section having C₂ as its core and C₃ as its shielding, and so on. Every conductor is then simultaneously the core of a coaxial line section and the shielding of another coaxial line section - except the central conductor, which is only a core, and the outermost conductor, which is only a shielding.
The n coaxial line sections are connected in cascade, and the last coaxial line section (implemented by the C_n and C_n+1 conductors) is shorted. Therefore, the structure of figure 3 can be represented by the electric scheme of figure 4, wherein each coaxial line section CLS₁ - CLS_n is represented by a quadripole (two-port network) characterized by its characteristic impedance Z_i and its electrical length θ_i with i=1 - n, and Z_SC is the impedance of the short-circuit termination. This representation neglects the coupling sections, of physical length I_ε<<L (L being the physical length of the resonator) and characteristic impedance Z'_i. It should be noted that the discontinuities at the end of each tubular conductor are very favorable to the coupling of TEM modes propagating along the coaxial line sections.
In the present example it is assumed that all the coaxial line sections are concentric and have circular cross-section but, as discussed above, this is not essential.
TEM modes can be excited inside the resonator by different means known in the art. For example a rod (not represented here, but see CR₁, CR₂ on figure 10) can extend transversally to the axis of the coaxial line sections, having a "distal" end connected to the outside of the resonator through an opening in the lateral surface of the outermost conductor, and a "proximal" end contacting - or being spaced from - another conductor. The distal end can be connected to the core of a RF feeding coaxial cable, the shielding of which is connected to the outermost conductor. Alternatively, TEM modes can be also excited e.g. by an iris, or by a "current loop" obtained by forming a loop with the core of the RF feeding coaxial cable, whose distal end contacts the outermost conductor.
Let L be the physical length of all the coaxial line sections, ε_ri its relative dielectric constant of the i-th section, d_i and D_i the diameters of the conductors forming its core and shielding, respectively (it will be noted that the i-th conductor, with i>1, has an inner diameter equal to D_i-1 and an outer diameter equal to d_i; if the thickness of the conductor is negligible, D_i-1≅d_i; this approximation was used when discussing figure 3). Then, at frequency f, the electrical length θ_l is given by: $θ_{i} = β_{i} L$

where $β_{i} = \frac{2 π \sqrt{ε_{ri}}}{λ}$

is the propagation constant (λ₀=c/f being the wavelength in vacuum, c being the light speed in vacuum), and - assuming losses negligible - the characteristic impedance Z_i is given by: $Z_{i} = \frac{1}{2 π} \cdot \sqrt{\frac{μ_{0}}{ε_{ri} . ε_{0}}} \ln \frac{D_{i}}{d_{i}}$

ε₀ and µ₀ being the electrical permittivity and the magnetic permeability of vacuum.
As it is well known in the art of electric networks, each two-port network can be represented by an ABCD matrix, or chain matrix; the overall chain matrix of the resonator, [ABCD] is obtained by orderly multiplying the individual chain matrices of the different coaxial line sections, and $(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix})$
matrices representing the crossed connections between them: $\begin{matrix} [ABCD] = [\begin{matrix} \cos θ_{1} & j Z_{1} \sin θ_{1} \\ j \frac{\sin θ_{1}}{Z_{1}} & \cos θ_{1} \end{matrix}] \cdot [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] \cdot [\begin{matrix} \cos θ_{2} & j Z_{2} \sin θ_{2} \\ j \frac{\sin θ_{2}}{Z_{2}} & \cos θ_{2} \end{matrix}] \\ \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot [\begin{matrix} \cos θ_{i} & j Z_{i} \sin θ_{i} \\ j \frac{\sin θ_{i}}{Z_{i}} & \cos θ_{i} \end{matrix}] \cdot [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \\ \cdot [\begin{matrix} \cos θ_{n - 1} & j Z_{n - 1} \sin θ_{n - 1} \\ j \frac{\sin θ_{n - 1}}{Z_{n - 1}} & \cos θ_{n - 1} \end{matrix}] \cdot [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] \cdot [\begin{matrix} \cos θ_{n} & j Z_{n} \sin θ_{n} \\ j \frac{\sin θ_{n}}{Z_{n}} & \cos θ_{n} \end{matrix}] \end{matrix}$
The A, B, C and D elements of [ABCD] allow determining the impedance as seen in the ○○' plane of the open circuit end of the central conductor C₁: ${Zr}_{1} = \frac{V_{1}}{I_{1}} = \frac{Z_{sc} A + B}{Z_{sc} C + D} = \frac{B}{D}$

This, in turn, allows determining the frequency at which the resonance condition Zr_i=∞ is satisfied.
Generalization of equations (1) to (5) to the case of coaxial sections having different length I_i is straightforward.
A detailed analysis of the electrical properties of the general structure of figure 3 is cumbersome; therefore, such an analysis will only be carried out for the case of a two-section (i.e. three-conductor) resonator, illustrated on figure 5. This turns out to be the preferred embodiment of the invention. It should be noted that, in the structure of figure 5, the conductive shell formed by the outermost conductor C₃ need not be completely closed; in particular one or both of its base surfaces might comprise openings.
It follows from equations 1 to 5 that, for n=2: ${Zr}_{1} = Z_{1} \frac{j Z_{2} \tan θ_{2} + j Z_{1} \tan θ_{1}}{Z_{1} - Z_{2} \tan θ_{2} \tan θ_{1}}$
The resonance condition Zr₁=∞ is satisfied when: $Z_{1} - Z_{2} \tan θ_{2} \tan θ_{1} = 0$
By defining M as the characteristic impedance ratio M=Z₂/Z₁, equation (7) becomes: $\frac{1}{M} = \tan θ_{1} \tan θ_{2}$
If θ₁=θ₂=θ₀ (same-length sections, filled with a same dielectric e.g. air), then: $\tan θ_{0} = \sqrt{\frac{1}{M}}$

and the resonance frequency (more precisely: the fundamental resonance frequency) F₀ is given by: $F_{0} = \frac{c}{2 π L \sqrt{ε_{r}}} \tan^{- 1} \sqrt{\frac{1}{M}}$
Figure 6A illustrates, the relationship between θ₀ and M expressed by equation (9).
The first spurious resonance is given by: $F_{S 1} = (\frac{π}{\tan^{- 1} \sqrt{\frac{1}{M}}} - 1) F_{0}$
Figures 6B illustrates the relation between the ratio F_S1/F₀ and characteristic impedance contrast M.
Figures 6C and 6D have been obtained by considering the case of a particular implementation wherein:

central conductor C₁ is a cylindrical rod with d₁=7mm, made of brass;
conductor C₂ is a 1-mm thick aluminum cylinder whose internal diameter D₂ is varied between 7.2 to about 34 mm;
conductor C₃ is also made of aluminum; it has a square cross section, with a 34 mm side (rectangular geometry - of which a square cross-section is a particular case - is useful for implementing filters, as it will be discussed later);
the resonator is air-filled;
the length of the coaxial sections formed by C₂ and C₃ is 30.6 mm and that of the coupling sections is 2mm; and
the length of the coaxial sections formed by C₁ and C₂ is 28.6 mm.

On figure 6C, two curves illustrate, the relationship between the ratio F_S1/F₀ and M. One of said curves correspond to theoretical results, the other one to numerical electromagnetic simulations. The difference between theoretical and simulation results is small, and can be traced back to the coupling sections neglected by the analytical theory. A third curve illustrates, the relationship between the ratio L/L₀ and M, where L₀ is the length of the inventive resonator and L that of an equivalent uniform quarter-wave air coaxial resonator. When M increases, the resonant frequency decreases, and L increases while L₀ remains constant; as a consequence (L/L₀) increases, as shown by the figure.
Figures 6D illustrates the unloaded quality factor Q as a function of M.
Inspection of figures 6A - 6D shows that:

To minimize the electrical (and therefore physical) length, and to maximize the F_S1/F₀ ratio, it is necessary to choose a high characteristic impedance contrast M (Z₂>>Z₁), which can be achieved by reducing D₁ (and therefore d₂) while maintaining d₁ and D₃ constant; the intermediate conductor C₂ is then only slightly larger than the central conductor C₁ and much smaller than C₃.
To minimize the F_S1/F₀ ratio, it is necessary to choose a low characteristic impedance contrast M (Z₁>>Z₂), which can be achieved by increasing D₁ (and therefore d₂) while maintaining d₁ and D₂ constant; the intermediate conductor C₂ is then much larger than the central conductor C₁ and slightly smaller than C₃
Q is maximal for M=1 (Z₂=Z₁).

In the case M≅49 (corresponding to Z₁=1.7Ω, Z₂=82.9Ω, D₁-d₁=100 µm), F₀=220 MHz and F_S1=4.7 GHz, i.e. F_S1/F0≅21. A uniform quarter-wave, air-filled coaxial resonator having the same resonant frequency F₀ would have a length of 341 mm. The invention allows then a length reduction of a factor 11.14.
In the case M≅0.1 (corresponding to Z₁=87.3Ω, Z₂=8.2Ω), F₀=2 GHz and F_S1=2.9 GHz, i.e. F_S1/F₀=1.45. The length reduction with respect to an uniform quarter-wave coaxial resonator is small; however, the availability of two close resonant frequencies can be useful to implement dual-band filters.
Figure 7 is similar to figure 5, except in that it illustrates a three-section (and then four-conductor) resonator, whose properties depend on two characteristic impedance contrast factors, M1=Z₃/Z₂ and M2=Z₂/Z₁.
Figure 8 allows comparing at 435 MHz the lengths and unloaded quality factors of: a two-section air-filled resonator according to an embodiment of the invention (cf. figure 5), a three-section air-filled resonator according to another embodiment of the invention (cf. figure 7), an uniform quarter-wave air-filled coaxial resonator with characteristic impedance Z_c=77Ω corresponding to the optimal unloaded quality factor Q=3 100, and an uniform quarter-wave dielectric coaxial resonator with ceramic relative permittivity Er=90 and loss tangent tg(δ)=10^-3. All the resonators have a square cross section with a side length of 34mm. Both the two- and the three-section resonators have a substantially higher quality factor than the uniform dielectric resonator, and only a slightly greater length. From a different point of view, they are much shorter than the uniform air-filled resonator, at the expense of a moderate reduction of the quality factor.
The application of the inventive filters to the realization of band-pass filters will now be discussed. Extrapolation to the case of pass-band band-stop filters is straightforward.
Figure 9 shows the electrical scheme of a second-order band-pass filter comprising a parallel connection of two identical 2-section resonators R1 and R2 between admittance inverters. K₀₁ and K₂₃ are the external coupling and K₁₂ the coupling between the resonators. The values of these parameters are determined by a conventional Chebyshev synthesis.
Figure 10 illustrates a possible and non limitative, physical implementation of the filter of figure 9. The outermost conductors C₃ ¹, C₃ ² of both resonators are constituted by a common aluminum casing AC, partitioned by an internal wall SW which does not extend on the whole length of the resonators so as to form, with the back wall of the casing, a coupling iris Cl. Two tuning screws TS₁, TS₂ are provided to adjust the coupling strength, and therefore the coupling K₁₂. The intermediate conductors C₂ ¹, C₂ ² of both resonators are constituted by respective aluminum tubes fixed to the face wall of the aluminum casing (not shown, to make the inner structure of the filter visible) and not contacting the back wall; their central conductors C₁ ¹, C₁ ² are constituted by brass rods fixed to the back wall of the casing and not contacting the face wall. Two excitations rods ER₁ and ER₂ extend transversally; their proximal ends contact the intermediate conductors C₂ ¹, C₂ ² while their distal ends exit the resonator through respective openings in the aluminum casing. The distal ends of the excitation rods are connected - trough conventional connectors - to the cores of two coaxial cables, whose shielding are connected to the aluminum casing and to the ground thorough respective conducting chocks (visible on the sides of the casing). The dimensions of the different elements are:

diameters of the central conductors: d₁ ¹=d₁ ²= 7mm;
spacing between the central and the intermediate conductors: 500 µm;
lengths of the intermediate conductors C₂ ¹, C₂ ²: 30.6 mm and that of the coupling sections is 8 mm;
thickness of the intermediate conductors C₂ ¹, C₂ ²: 1 mm;
lengths of the central conductors C₁ ¹ C₁ ²: 32.8mm;
inner dimensions of the aluminum casing: 34 mm ;
outer dimensions of the aluminum casing with back and frontal walls: 86 mm x 50 mm x 43 mm.

A smaller size (e.g. a reduction of the longitudinal dimension from 43 to 25 mm) could be achieved using more sophisticated manufacturing techniques, such as electroforming.
The requirements of the 2^nd order filters are the following: central frequency F₀=435 MHz, relative pass bandwidth of 4.6% and 0.01 dB ripple within the pass band.
Figures 11A and 11 B show the frequency dependence of the scattering parameters S₁₁ and S₁₂, obtained from simulation and from measurement. Figure 11A is a narrow-band representation, while figure 11B is a large-band one, showing the first spurious frequency F_S1, with F_S1=3.8GHz i.e. (F_S1/F₀)≅9.
The requirements of the 6^th order filter are the following: central frequency F₀=435 MHz, relative pass bandwidth of 6.8%, matching level better than 20 dB within the pass band.
Figure 12 shows the electrical scheme of a 6^th order filter comprising six identical two-section resonators, with coupling constants K₁₂, K₂₃, K₃₄, K₄₅, K₅₆ and K₆₇. The K₁₆ coupling is important as it allows introducing two transmission zeros near the pass band.
Physically, the resonators can be arranged in a 3x2 matrix. Figures 13A and 13B are, respectively, narrow-band and large-band representations of the frequency dependence of scattering parameters S₁₁ and S₁₂. Figure 13A also shows (thick line and dotted line) the required filter specification.
Resonators according to the invention, and therefore filters built from them, can be made tunable in several ways, two of which are illustrated on figures 14A and 14B.
In the embodiment of figure 14A, an electrical motor EM is used to move the central conductor C₁ in an axial direction, thus changing the length of the portion of it which is contained within the resonator. This changes the physical - and electrical - length of the first coaxial line section, and therefore the resonance frequency. This allows broad, but slow, tunability. Figure 14B shows an alternative embodiment, wherein a variable capacitor (implemented e.g. by a varactor diode) is connected between the central conductor C₁ (or an intermediate conductor, such as C₂) and the ground. This allow faster tuning, but in a narrower range, and reduces the quality factor. It is clear that different tuning mechanisms (e.g. those of figures 14A and 14B) can be combined in a same device.
Figure 15 shows the frequency dependence of the scattering parameters S₁₁ and S₁₂ of a tunable filter, for 5 different values P1 - P5 of the central frequency, spanning the 435 MHz - 1.63 GHz range.

Claims

A radio-frequency resonator comprising a plurality of coaxial line sections (CLS₁ ...CLS_n) connected in cascade, each of said coaxial line sections comprising an inner conductor (C_i) surrounded by an outer conductor (C_i+1) of tubular shape, wherein said coaxial line sections are nested within each other, the outer conductor of each said coaxial line section, except an outermost one (C_n+1), serving as the inner conductor of another one of said coaxial line sections, the resonator being characterized in that each conductor of each coaxial line section, except the outer conductor of the outermost coaxial line section, has an open-circuit end and an opposite end which is short-circuited to said outer conductor of said outermost coaxial line section, and in that said conductors of said coaxial line sections - except the outer conductor of said outermost coaxial line section - are arranged head-to-tail, their open-circuit and short-circuited ends being alternatively situated on opposite sides of the resonator.
A radio-frequency resonator according to claim 1, comprising:
- a conductive shell (C_n+1) of generally cylindrical shape having a lateral surface, a first base surface and a second base surface, said lateral surface constituting the outer conductor of the outermost coaxial line;

- a conductor of tubular or rod-like shape, called central conductor (C₁), constituting the inner conductor of an innermost coaxial line, said central conductor being contained within said conductive shell and having a first end short-circuited to said first base surface a second end facing said second base surface; and

- at least one tubular conductor (C₂), contained within said conductive shell and containing said central conductor, having an end contacting one said base surface and an opposite end facing the other base surface.
A radio-frequency resonator according to any of the preceding claims, comprising exactly two said coaxial line sections.
A radio-frequency resonator according to any of claims 1 or 2, comprising exactly three said coaxial line sections.
A radio-frequency resonator according to any of the preceding claims, wherein an inner conductor (C₁) of an innermost coaxial line section is mounted slidably along its longitudinal axis, in such a way that it can be partially extracted from the resonator.
A radio-frequency resonator according to any of the preceding claims, wherein an inner conductor (C_i) of a coaxial line section is electrically connected to a variable capacitor (VC).
A radio-frequency resonator according to any of the preceding claims, wherein the outer conductor of the outermost coaxial line section has a rectangular cross-section.
A radio-frequency resonator according to any of the preceding claims, further comprising an excitation rod (ER₁, ER₂) extending transversally with respect to said coaxial line sections, said excitation rod having a distal end extending to the outside of the resonator and a proximal end contacting or facing a conductor other than said outer conductor of the outermost coaxial line section
A radio-frequency filter comprising a plurality of radio-frequency resonators (R1, R2) according to any of the preceding claims mounted adjacent to each other and coupled by openings (CI) in the outer conductors of their respective outermost coaxial lines, forming coupling irises.
A radio-frequency filter according to claim 9, wherein said radio-frequency resonators (R1, R2) are identical to each other.