I. INTRODUCTION
There is a continually growing need for small-sized microwave filters with good electrical performances (insertion losses, amplitude flatness and sufficient phase linearity in the passband, wide band rejection, power handling, and a wide tunability dynamic if the structure is a tunable one), low sensitivity and, if possible, easily adjustable in frequency and low cost. The bulk characteristics and associated mass are serious constraints when working at low frequency, particularly in the ultra high frequency (UHF) band, where wavelengths are large. However, attention should also be paid to the quality factor, which decreases with size. For UHF band, the air-filled coaxial resonator offers an interesting compromise [Reference Herman and Espenschied1]. It has good quality factors, around 3000, but is less bulky than traditional waveguides. It is still possible to reduce the size of coaxial resonators using high permittivity ceramic versions. Thus, dielectric constant values of about 90 make it possible to divide their length by a factor of almost 10. However, the size reduction is accompanied by a significant drop in the quality factor, partly due to the loss tangent values of ceramics, which are usually around 10−3.
In this paper, we present a miniature two-section stepped impedance resonator (SIR) coaxial resonator, which we use here in air versions. This solution is based on the well-known SIR principle, which makes it possible to reduce lengths by adjusting the levels of two different characteristic impedance line sections in cascade. The stepped impedance enables spurious harmonics to be kept at a relatively good distance. Miniaturization and distancing of spurious harmonics become more important when the contrast of the characteristic impedances becomes stronger. Finally, by working in air, the limitations imposed by the materials in terms of dielectric loss tangents are removed, enabling good quality factors to be maintained.
Considering the two coaxial sections of identical physical length, the shielding of the inner coaxial section becomes the outer section of the central core. The two-sections are cascaded by alternating open- and short-circuit conductor terminations. In this simple manner, an effective coupling zone is created between the two-sections. The stepped impedance characteristics are then adjusted by altering the internal and external diameters of the two coaxial sections. In this configuration the second cylindrical conductor is simultaneously the central core of one coaxial section (with its outer wall) and the shielding of the other coaxial section (with its inner wall).
Many solutions exist to implement SIR in coaxial resonators and have been widely described in the literature over several decades [Reference Makimoto and Yamashita2–Reference Sagawa, Makimoto and Yamashita6]. These are therefore essentially geometric changes in transversal planes. The re-entrant coaxial or combline resonator has also been widely described in the literature [Reference Wang and Zaki7, Reference Wu, Mansour and Wang8], for making post-tunings or temperature compensation. The re-entrant character has also been exploited over the whole length of the conductors, leading to geometries not very distant from our structure, but without taking benefit of SIR effect [Reference Kartik, Hiroji, Kenneth, Lawrence, Andrew and Steven9–Reference Wenzel12]. Finally, as SIR is used, we consider the coaxial structures, proposed in [Reference Song, Fan and Zhang13, Reference Makimoto and Yamathita14], to be the most original, but they notably have miniaturization factors far inferior to those we propose.
Furthermore, this paper shows how the proposed solution could be useful for industrial applications, where flexible solutions are often sought. By flexibility, we mean that once it is built, the filter can be adjusted to address different standards within a large frequency range, without the need of a new design and associated fabrication. This property can also lead to the conception of tunable filters [Reference Aouidad, Favennec, Rius, Clavet and Manchec15].
II. TWO-SECTION SIR COAXIAL RESONATOR
The physical structure of the two-section SIR coaxial resonator is shown in Fig. 1(a). The two coaxial sections, with identical physical lengths l, are fitted inside one another in a head-to-tail arrangement: alternately presenting an open circuit at one end and a short circuit at the other, thus forming a quarter-wave resonator overall. Starting from the first section, until the second one with potentially different electrical characteristics, the ground conductor of the first section becomes the central core of the second. Practically, this is achieved by fitting cylinders on increasing diameter inside one another. Three cylinders are needed here.
Fig. 1. Two-section SIR coaxial resonator. (a) Physical representation. (b) Electrical schema.
Finally we consider two coaxial sections with relative permittivities ε r1 and ε r2, the characteristic impedances and associated electrical lengths of the sections will be Z 1, θ 1 for one section and Z 2, θ 2 for the other.
Remembering that the electrical length θ i is defined by
$${\theta _i} = {\beta _i}l,$$
and that the propagation constant is
$${\beta _i} = \displaystyle{{2\pi \sqrt {{\varepsilon _{ri}}}} \over {{\lambda _0}}},$$
λ 0 being the wavelength in a vacuum.
In the case where there are no losses, the characteristic impedances Z 1 and Z 2 of the two coaxial sections are obtained by the following formulas [Reference Vizmuller16]
$${Z_1} = \displaystyle{1 \over {2\pi}} \cdot \sqrt {\displaystyle{{{\mu _0}} \over {{\varepsilon _0}{\varepsilon _{r1}}}}} {\ln}\left( {\displaystyle{{{D_1}} \over {{d_1}}}} \right),$$
$${Z_2} = \displaystyle{1 \over {2\pi}} \cdot \sqrt {\displaystyle{{{\mu _0}} \over {{\varepsilon _0}{\varepsilon _{r2}}}}} {\ln}\left( {\displaystyle{{{D_2}} \over {{d_2}}}} \right).$$
D 1 and d 1 are the interior diameter of the metallic cylinder C 2 and exterior diameter of metallic cylinder C 1, respectively. In the same manner, D 2 and d 2 are the interior diameter of the metallic cylinder C 3 and exterior diameter of metallic cylinder C 2, respectively.
The coupling of one coaxial section to another, in transverse electromagnetic mode (TEM), is done at their extremities, where the short circuit of one will lie beside the open circuit of the next. Such a discontinuity is naturally favorable for coupling a TEM mode of a section to another TEM mode in the next section.
By ignoring the coupling areas, consisting of the sections of physical length l ε , which is small in comparison with the coaxial sections, we can adopt a relatively simple electrical representation for the resonator overall (Fig. 1(b)); this consists in cascading two quadrupoles, representing each of the coaxial sections. A crossed connection is made by linking the hot spot and ground of one section to the ground and hot spot of the other section, respectively. This crossed connection is represented electrically by a transformer of ratio m = −1. This model implies that we have also ignored the effects of the capacitive and inductive discontinuities which would necessarily be produced at the open and short circuit extremities of the different sections, respectively. Such an electrical representation has been first proposed in [Reference Wenzel12].
In the general case the rings can be filled with dielectric materials that may or may not be different. In the case when the resonator is filled with air, meaning that the cylinders are empty, to simplify the calculations and taking into consideration that the lengths lε are negligible, we can suppose that the electrical lengths θ 1 and θ 2 are identical, while in the other case they can be different.
By choosing the cylinder diameters carefully, we are able to obtain highly contrasted impedance characteristics, giving particularly attractive electrical responses, and different according to the combinations of characteristic impedances. Indeed, as illustrated in Fig. 2, from the same contrast of impedances we can obtain very low or very high Z 2/Z 1 ratio. In one case, characteristic impedance Z 2 could be the highest, in another case it could be Z 1. The kind of frequency behavior that we find here corresponds very well to that found in stepped impedance structures, notably in microstrip, on which numerous papers have been written [Reference Yen, Wollack, Doiron, Papapolymerou and Laskar17, Reference Zhang and Kevin18]. For a given size, if the characteristic impedance ratio Z 2/Z 1 increases, the frequencies of the fundamental mode and the first harmonic tend to move away from each other; the harmonics are considered spurious. However, if the characteristic impedance ratio Z 2/Z 1 decreases, the frequencies, fundamentals, and harmonics move closer. Furthermore, in the first case, the major interest is the reduction of dimensions of resonators or filters [Reference Zhang and Kevin19]. In the second case it may allow dual-band electrical responses to be realized [Reference Apriyana and Ping20].
Fig. 2. Two contrasting cases for the characteristic impedances Z 1 and Z 2. (a) Z 2/Z1 ≫ 1. (b) Z 2/Z 1 ≪ 1.
A) Fundamental frequency
The two-section SIR coaxial resonator can be represented by expression (5), which does not take into account the short-circuit. This is a succession of two matrix chains,
$$\eqalign{& \quad \quad \quad \quad \quad \quad \quad \quad \quad \left[ {ABCD} \right] = \cr & \left[ {\matrix{ {\cos {\theta _1}} & {j{Z_1}\sin {\theta _1}} \cr {j\displaystyle{{\sin {\theta _1}} \over {{Z_1}}}} & {\cos {\theta _1}} \cr}} \right].\left[ {\matrix{ { - 1} & 0 \cr 0 & { - 1} \cr}} \right].\left[ {\matrix{ {\cos {\theta _2}} & {j{Z_2}\sin {\theta _2}} \cr {j\displaystyle{{\sin {\theta _2}} \over {{Z_2}}}} & {\cos {\theta _2}} \cr}} \right],} $$
from which we can easily trace back to the input equivalent impedance in the OO’ plane of the two-section SIR coaxial resonator, open circuit plane of section 1, [Reference Makimoto and Yamashita21, Reference Sagawa, Makimoto and Yamashita22]
$$Z{r_1} = {Z_1}\displaystyle{{j{Z_2}\tan{\theta _2} + j{Z_1}\tan{\theta _1}} \over {{Z_1} - {Z_2}\tan{\theta _2}\tan{\theta _1}}}.$$
The resonance condition in the OO’ open circuit plane is then obtained when Z r1 = ∞, or Y r1 = 0.
$$Y{r_1} = \displaystyle{1 \over {Z{r_1}}} = - j{Y_1}\displaystyle{{{Z_1} - {Z_2}\tan{\theta _1}\tan{\theta _2}} \over {{Z_1}\tan{\theta _1} + {Z_2}\tan{\theta _2}}} = 0.$$
Taking into account the previous equation, Yr 1 will be zero if
$${Z_1} - {Z_2}\tan{\theta _1}\tan{\theta _2} = 0.$$
Defining the ratio M as the ratio Z 2/Z 1 of characteristic impedances, we obtain:
$$M = \displaystyle{{{Z_2}} \over {{Z_1}}},$$
$$\displaystyle{1 \over M} = \tan{\theta _1}\tan{\theta _2}.$$
In the particular case where θ 1 = θ 2 = θ, and by using (7) and (10), Yr 1 becomes
$$Y{r_1} = - j{Y_1}\displaystyle{{1 - M\tan^{2}\theta} \over {\tan\theta {\kern 1pt} (M + 1)}}.$$
The resonance condition Yr 1 = 0, corresponding to θ = θ 0, is thus written
$$\tan^{2}{\theta _0} = \displaystyle{1 \over M},$$
or
$${\theta _0} = {\tan ^{ - 1}}\sqrt {\displaystyle{1 \over M}}. $$
The resonance frequency is thus obtained by writing:
$${F_0} = \displaystyle{c \over {2\pi l}}\tan^{ - 1}\sqrt {\displaystyle{1 \over M}}. $$
Figure 3 shows the electrical length θ 0 at the resonance, as a function of the ratio of characteristic impedance M, obtained from equation (13). It illustrates well some antagonistic behavior for extreme values of M.
Fig. 3. Electric length θ 0 as a function of the characteristic impedance ratio M.
B) Transmission zeros
The condition expressing transmission zeros is obtained when Zr 1 = 0, or Yr 1 = ∞.
From (11) we will have Yr 1 = ∞, for θ = θ z , when
$$\tan {\theta _z} = \infty. $$
Thus, we obtain
$${\theta _z} = \left( {2n + 1} \right)\displaystyle{\pi \over 2}.$$
The Fz n frequencies (n = 0, 1, 2…) of the transmission zeros corresponding to θz n (n = 0, 1, 2…), are obtained from (16)
$$F{z_n} = \left( {2n + 1} \right)\displaystyle{c \over {4l}}.$$
The frequencies of transmission zeros depend directly on the length l of the resonator and thus remain fixed regardless of the characteristic impedance ratio M = Z 2/Z 1.
C) Harmonic frequencies
Equation (11) has several solutions, the first θ 0 corresponds to the fundamental mode, the other solutions θs p (p = 1, 2,…) correspond to harmonics with θs p > θ 0.
We know that
$${\tan ^2}\left( {\pi - {\theta _0}} \right) = {\tan ^2}{\theta _0}.$$
Otherwise, for the first harmonic θs 1, Yr 1 = 0, and:
$${\tan ^2}\theta {s_1} = {\tan ^2}{\theta _0} = \displaystyle{1 \over M}.$$
So, for the first harmonic [Reference Sagawa, Makimoto and Yamashita22], we obtain
$$\theta {s_1} = \left( {\pi - {\theta _0}} \right).$$
The frequencies of the other harmonics corresponding to θ = θs p (p = 2, 3…), are obtained from the expression (21).
$$\theta {s_p} = \left( {p\pi - \theta {s_{p - 1}}} \right).$$
The harmonic frequencies Fs p (p = 1, 2,…) are specified by equations (22) and (23), obtained
$$F{s_{(2k - 1)}} = \left( {\displaystyle{{k\pi} \over {{{\tan} ^{ - 1}}\sqrt {1/M}}} - {{\left( 1 \right)}^{(2k - 1)}}} \right){F_0},$$
$$F{s_{(2k)}} = \left( {\displaystyle{{k\pi} \over {{{\tan} ^{ - 1}}\sqrt {1/M}}} - {{\left( 1 \right)}^{(2k)}}} \right){F_0},$$
with k = 1, 2,…
From (22) and (23) we calculate the first two harmonic frequencies by taking k = 1:
$$F{s_1} = \left( {\displaystyle{\pi \over {{{\tan} ^{ - 1}}\sqrt {1/M}}} - 1} \right){F_0},$$
$$F{s_2} = \left( {\displaystyle{\pi \over {{{\tan} ^{ - 1}}\sqrt {1/M}}} + 1} \right){F_0}.$$
Figure 4 shows the ratio of the frequency of the first harmonic and the frequency of the fundamental mode obtained from equations (14) and (24), depending on the ratio of characteristic impedance M, again illustrating the different behaviors for the extreme values of M.
Fig. 4. Ratio of the frequency of the first harmonic on the frequency of the fundamental mode according to the characteristic impedance ratio M.
D) Electromagnetic study of a two-section SIR coaxial resonator
Highly contrasting characteristic impedances can be obtained in two ways, depending on whether we consider Z 1 < Z 2 or the inverse. In the first case, cylinders C 1 and C 2 were chosen with small diameters and relatively close, while the diameter of cylinder C 3 was much larger. In the second case, cylinders C 2 and C 3 were chosen with large diameters and relatively close, while the diameter of cylinder C 1 remained small. To enhance the contrast in the characteristic impedance according to different cases, rings A 1 or A 2, can be loaded with dielectric materials.
Recalling that cylinder C 1 can be filled, it can thus be either a tube or a metal rod.
In Fig. 5, we present the characteristics of a two-section air-filled SIR coaxial resonator, considering a practical case in direct relation with what will be later used when we present our experimental results on second order filter. The central section of characteristic impedance Z 1 is made with a metallic rod C 1 of diameter d 1 = 7 mm, integrated into a hollow cylinder C 2 with a variable interior diameter of D 1. C 2 has a low thickness, equal to 1 mm. If this thickness remains moderate, it has almost no impact on the electromagnetic behavior. It may, however, add a degree of freedom when working to obtain the required characteristic impedances. To reduce the complexity of the mechanical realization stage, we can realistically imagine that the minimum width of ring A 1 should not be lower than 100 µm. Diameter D 1 would therefore vary from 7.2 to about 34 mm. This last value was chosen to maintain acceptable form factors with regard to the frequencies we are working with. This first coaxial section is integrated into a rectangular metal housing with a side a = 34 mm, which constitutes the shielding of the second coaxial section. We chose this form for A 2 because it fits better into a vision filter, or all the resonators are arranged next to one another. Section A 2, however, remains the seat of a TEM mode.
Fig. 5. 3D representations of two-section coaxial SIR resonators with two sections. (a) Case where M ≫ 1. (b) Case where M ≪ 1.
To calculate the characteristic impedance of such a configuration for A 2 we used the following expression [Reference Omar and Miller23].
$${Z_2} = \displaystyle{1 \over {2\pi}} \cdot \sqrt {\displaystyle{{{\mu _0}} \over {{\varepsilon _0}{\varepsilon _{r2}}}}}\ln \left( {1.079\displaystyle{a \over {{d_2}}}} \right).$$
We chose to compare the performances of our structure with that of a uniform coaxial resonator. By ignoring the coupling areas, which we set here at 2 mm, we chose the size of our two-section air-filled SIR coaxial resonator by taking L 1 = 28.6 and L 2 = 30.6 mm. This length of 30.6 mm, without considering the open extremity, corresponds to a uniform air-filled quarter-wave coaxial resonator, at a resonance frequency of 2.45 GHz. Figure 5 shows a two-section SIR coaxial resonator in two different configurations, corresponding to strong impedance contrast characteristics. In case (a), the ratio M is greater than 1 and in case (b) the inverse is the case.
Considering the above dimensions, we plotted the ratio of the first harmonic frequency to the fundamental mode, Fs 1/F 0, on the characteristic impedance ratio M = Z 2/Z 1, and the length of a uniform air-filled coaxial resonator equivalent to the resonance frequency, normalized on that of our coaxial SIR resonator, or L/L 0.
The results are presented in Fig. 6. The points on the curves, are the results of an electromagnetic simulation, the variation of the ratio M = Z 2/Z 1 having been obtained by varying the diameter D 1 of cylinder C 2. For the Fs 1/F 0 ratio, we also present the associated theoretical curve.
Fig. 6. Ratio of the first harmonic frequency to that of the fundamental mode (Fs 1/F 0) (theory and electromagnetic simulation) and length of a uniform air-filled coaxial resonator, equivalent to the resonance frequency, normalized on that of our SIR coaxial resonator (L/L 0) (electromagnetic simulation) as a function of M = Z 2/Z 1.
Two cases can be distinguished by the order of magnitude of the ratio M.
When M = Z 2/Z 1 increases, the resonant frequency of the fundamental frequency decreases and the harmonic frequency increases. For example, for an M ratio of M = 49 (Fig. 6), corresponding to Z 1 = 1.7 and Z 2 = 82.9 Ω and obtained with an A 1 ring of 100 µm, the frequency of the fundamental mode F 0 is equal to 220 MHz and the first harmonic frequency F S1 is situated at 4.7 GHz, or 21 times F 0. To obtain such an F 0 resonance frequency with a uniform air-filled coaxial resonator, the value of the quarter wavelength would need to be 341 mm. By considering this length, the miniaturization factor would thus be 341/30.6 = 11.14.
When the ratio M = Z 2/Z 1 is low, i.e. when the diameter d 2 approaches the width of conductor C 3 (Fig. 5(b)), the resonance frequencies of the fundamental and harmonic move closer. For example, for M = 0.093, corresponding to Z 1 = 87.3 and Z 2 = 8.2 Ω, obtained with a d 2 diameter similar to the cross-section of the conductor C 3, the frequency of the fundamental mode F 0 is equal to 2 GHz, and the first harmonic F S1 is found at 2.9 GHz, or 1.45 times F 0. In this case, very little miniaturization of the two-section SIR coaxial resonator is seen with two-sections compared with the uniform coaxial resonator. This reduction factor here is 1.22. However, the electrical responses obtained may allow us to consider the design of dual band resonators and filters [Reference Apriyana and Ping20].
It can be seen that the electromagnetic results are in good agreement with the theoretical expressions. Figure 6 plots the theoretical curve according to Fs 1/F 0 as a function of M, derived from equations (14) and (24). The differences can be explained by the coupling sections, which we have neglected theoretically.
For each point, the electromagnetic simulations also made it possible to extract unloaded quality factors. Figure 7 shows the evolution of these coefficients as a function of the characteristic impedance ratio M = Z 2/Z 1. The quality factors are between 1000 and 3000 and remain sufficiently stable over a large range of characteristic impedance ratios. We can see that the optimum configuration is obtained for M = 1 [Reference Yamashita and Makimoto5]. In passing, we can note that in this particular configuration, where the continuity of the characteristic impedance is maintained, the first harmonic is 3 times F 0, for a length reduction of 2. Finally, this is a two-section SIR coaxial resonator, uniform in terms of characteristic impedances.
Fig. 7. Unloaded quality factor plotted on M = Z 2/Z 1.
Figure 8 compares the results we obtained, at a resonance frequency of 435 MHz, with the performances of two other uniform coaxial resonators, one in air and the other loaded with high permittivity dielectric.
Fig. 8. Relative unloaded quality factors and lengths for different coaxial resonators.
The uniform air-filled coaxial resonator, with a square section with a side equal to 34 mm, has a characteristic impedance of 77 Ω so as to correspond to the optimal quality factor (3100) [Reference Yamashita and Makimoto5]. We chose the same section for our dielectric resonator.
Figure 8 presents the unloaded quality factors for each resonator, as a function of their length. We note that the length improvement is significant for the two-section SIR coaxial resonator. We also note that the quality factors are affected, but nevertheless remain well above 1000. To resume this example, the decrease in the quality factor is of the order of 2 for a length reduction of the order of 5.5. Regarding the uniform dielectric coaxial resonator, the dielectric permittivity was chosen large enough (here 90) so that the size reduction was comparable. The loss tangent was chosen realistically; at 10−3 this inevitably has an impact on the quality factor, here estimated at 1000.
For the same structures, Fig. 9 compares the ratio of the frequency of the first harmonic to the fundamental mode, as a function of the physical length of the resonators. For a uniform resonator this is 3. The distancing of the first harmonic is far greater for the two-section SIR coaxial resonators, with a ratio of the order of 10 times F 0.
Fig. 9. Ratio of the first harmonic frequency to the fundamental mode as a function of the lengths of the different coaxial resonators.
III. DESIGN AND IMPLEMENTATION OF A SECOND ORDER FILTER WITH TWO-SECTION SIR COAXIAL RESONATORS
A) Synthesis and design
In this part of the paper we present the conception and realization of a second order filter based on two two-section SIR coaxial resonators. A classic Tchebychev synthesis was used to calculate the coupling coefficient K between the two resonators and the external quality factor Qe [Reference Matthaei, Young and Jones24].
$$k{\kern 1pt} {{\kern 1pt} _{i,i + 1}} = \displaystyle{{FBW} \over {\sqrt {{g_i}{g_{i + 1}}}}}, \quad Q{e_1} = \displaystyle{{{g_0}{g_1}} \over {FBW}},\quad Q{e_n} = \displaystyle{{{g_N}{g_{N + 1}}} \over {FBW}},$$
i = 1, 2,…, n−1
The g i are components of the prototype lowpass Tchebyshev filter, FBW is the relative passband, and n is the filter order.
The slope parameter b i is expressed from [Reference Matthaei, Young and Jones24] by
$${b_i} = \displaystyle{{{\omega _0}} \over 2}{\left. {\displaystyle{{d{B_i}} \over {d\omega}}} \right \vert _{\omega = {\omega _0}}},$$
where B i is the susceptance of the ith resonator. After derivation of formula (11) we obtain
$${b_i} = {\theta _0}{Y_{1i}} = \displaystyle{{{{\tan} ^{ - 1}}\sqrt {1/{M_i}}} \over {{Z_{1i}}}}.$$
The specifications used here means that the filter must be centered on 435 MHz with a relative passband of 4.6% and a 0.01 dB ripple in the passband. Taking into account the specifications, we calculated from (27); Qe 1 = Qe 2 = 9.7614, K 12 = 0.1056.
The two-section SIR coaxial resonators that we will deal with here have largely the same form factor as that used in Part II. To obtain a resonance frequency F 0 = 435 MHz, and to ease fabrication constraints and, as a consequence, minimize sensitivity, we chose an A 1 ring thickness of 500 µm, leading to characteristic impedances of Z 1 = 8 Ω and Z 2 = 77.9. In these conditions, by taking l ε = 6 mm we obtain, for L 1 and L 2: L 1 = 24.8 and L 2 = 30.6 mm. From Fig. 6, by taking M at about 12, we can estimate that the first harmonic specific to the resonator will appear at around 4.3 GHz, or at 10 times F 0. However, in the built version, the construction of the coupling iris modifies the electromagnetic behavior of the resonators and causes a shift in the value of this frequency.
Figure 10 shows the ideal electrical schema and Fig. 11 gives its electrical responses, around the passband and over a wide band. It may be noted in passing that the frequency of the second harmonic, located at 5.2 GHz, corresponds to what gives the relation (25).
Fig. 10. Ideal electrical schema of a symmetrical second order filter.
Fig. 11. Ideal electrical response of the second order filter. (a) Around the passband. (b) Over a wide band.
B) Filter fabrication
The filter is made entirely of aluminum, except for the C 1 cylinders, which are made of brass. These are standard commercial metallic rods. The aluminum used here corresponds to the 7000 series, its electrical conductivity is of the order of 1.4 × 107 s/m. Figure 12(a) presents a three-dimensional (3D) view of the filter without its upper metal cover. The in/out excitations are realized with a metallic rod, shown in yellow on Fig. 12(a) in direct contact with cylinder C 2. The contact position is adjustable along cylinder C 2 and makes it possible to obtain the required external coupling. The coupling between the two resonators is realized through an iris, and two metallic coupling screws are added here to adjust the coupling if necessary. They are also presented in yellow on the diagram.
Fig. 12. Second order filter. (a) 3D representation without cover. (b) Photograph of the filter; (c) photograph of the filter and its cover, on which the C 2 cylinders are fixed.
Figures 12(b) and 12(c) show photographs of the filter. Its dimensions are 86 × 50 × 43 mm3, which is low compared with the state of the art. By using more precise fabrication methods, which are also more costly, e.g. electrical discharge machining, we can also obtain A 1, ring dimensions of 100 µm. In this case, we obtain a height of 25 mm, or an additional reduction in total volume of 43/25 = 1.7.
C) Electromagnetic simulation and filter measurement
Figure 13 presents the experimental results and the electromagnetic simulations of the structure. We see a very good agreement, both around the passband and in wide band. The filter is correctly matched around 20 dB and the level of insertion losses measured is 0.45 dB, compared with 0.4 dB by simulation. The discrepancy can be attributed to inaccuracies in the measurement and calibration. The wide band measurement illustrates the distancing of the harmonic frequencies particularly well.
Fig. 13. Simulations and measurements of the second order filter. (a) Around the passband. (b) In a wide band.
D) Flexibility and associated tunability
In addition to miniaturization aspects and removal of spurious harmonics, which are the most important benefits of two-section SIR coaxial resonators, these resonators also offer good potential in terms of flexibility and, therefore, associated tunability. In fact, if one imagines making C 1 rods mobile, by making it possible to adjust their position in the C 2 cylinder, we modify the resonance frequency of the resonators and, therefore, of the filter. The tuning range is very wide as it can sweep frequencies from 435 MHz to 1.63 GHz. It might even be made to go up to 1.7 GHz by completely removing the rods, but this would require the excitation to be altered.
Figure 14 presents the experimental results obtained by manually manipulating the two C1 metal rods. We gradually removed the rods and measured the electric response of the filter for some of the positions. The coupling screws were not used, which explains why the filter passband is not controlled here. It should also be noted that we did not change the excitation position. We chose a position that made it possible not to exceed 15 dB for the matching level, at the ends of the tuning band. As shown, the level of the losses is not significantly impacted by the position of the C1 rods. In fact, the bulk of the losses are mainly caused by the access devices, and not by the two resonators. Finally, the measured data and simulations are in agreement with one another. Indeed, we electromagnetically simulated the extreme states, 1 and 5, which are also shown in Fig. 14.
Fig. 14. Simulation and measurement of a second order filter for several positions of the C 1 cylinders.
We obtained a wide range of frequency deviation of about 4 times F 0, which makes it possible to use the same resonator to make filters able to present a large diversity of frequency responses, as long as we know how to control the couplings. The coupling screws, in particular, were foreseen for this use.
This wide tuning range can be exploited to obtain tunable filters, since we know how to solve the problem of tuning elements. The manual manipulation shows that a solution consisting of using movable rods, which could, for example, be driven by stepper motors, were all conceivable. However, it is clear that the mechanical tunability solutions suffer greatly in terms of response time.
To overcome this problem, the solutions are quite possible based on discrete components such as positive intrinsic negative (PIN) diodes, varactors [Reference Aouidad, Favennec, Rius, Clavet and Manchec15] or microelectromechanical systems (MEMS), this time connecting the open end of cylinder C 1 to additional electrical lengths. However, these tuning ranges will not be as large and the limited quality factors of these tuning elements will diminish the electrical performance of the filters.
IV. CONCLUSION
Our paper describes an interesting topology of coaxial resonators with stepped impedance. This has the advantage of greatly reducing the bulk of microwave filters. The example presented here is five times smaller than their conventional counterparts based on uniform air-filled coaxial resonators. It is possible to reduce size still further if one is able to use very narrow rings by way of adapted fabrication procedures. We also show that the size reduction has an impact in terms of quality factors and thus of losses, but it is not really crippling. Additionally, spurious harmonics are significantly distanced. Finally, we have shown that the solution presented is easily adjustable in frequency and promising with regard to tunability. The prototype validates the general concept, as there is good agreement between experimental measures and electromagnetic simulations. We have worked here with two-section SIR coaxial resonators, but it would be certainly interesting to study the behavior of n-section SIR coaxial resonators.
ACKNOWLEDGEMENT
This work was partially supported by the ANRT under project number 2011/0759.
Hakim Marc Aouidad was born in Tabouda centre, Algeria in 1985. He received his Master degree in Electronics and Telecommunication from the “Université de Bretagne Occidentale”, Brest, France in 2011 and Ph.D. degree in Electronics in 2014 at the Lab-STICC laboratory (Laboratoire en Sciences et Techniques de l'Information, de la Communication et de la Connaissance) of the “Université de Bretagne Occidentale”. He is currently R&D engineer with Protecno, where he is involved in PCB design.
Eric RIUS was born in Auray in 1966. He received his Ph.D. in Electronics in 1994 at the University of Brest. In 1995, he was appointed as Assistant Professor and 10 years later Professor, always at the University of Brest. His main scientific interest is the study and design of front-end radio and especially everything about filters frequencies, in the centimetric and millimetric frequency ranges. He has published more than 160 papers, has been cited over 1400 times and has an h-index of 17. He was the TPC Chair of the European Microwave Conference (EUMC) in Paris in 2010. It was also, in 2010–2011, guest editor of the International Journal of Microwave and Wireless Technologies (Cambridge University Press, Special Issue on EuMW 2010 (IJWMT)). He obtained a CNRS delegation in 2010–2011 and during this period he performs a Visiting Professor at Nanyang Technological University (NTU) at Singapore for 4 months, from January to April 2011. He is the scientist director and coordinator for two ANR projects, RF-IDROFIL (2009–2013) and COCORICO (2012–2015).
Jean François Favennec received his Ph.D. degree in Electronics from the University of Brest, France in 1990. In 1991, he became an Assistant Professor within the Ecole Nationale d'Ingénieurs de Brest. He essentially teaches electromagnetic theory and microwaves. He currently conducts research within the Lab-STICC laboratory (Laboratoire en Sciences et Techniques de l'Information, de la Communication et de la Connaissance). His research activities concern the modeling and design of passive devices for microwave applications and are mainly focused on filters.
Alexandre Manchec was born in Quimper in 1978. He received his Ph.D. in Electronics in 2006 at the Lab-STICC laboratory (Laboratoire en Sciences et Techniques de l'Information, de la Communication et de la Connaissance) of the “Université de Bretagne Occidentale”, Brest, France. He is Co-founder and joint manager of Elliptika since February 2008. His R&D activities principally concern the definition of specific solutions in planar technology for microwave and millimeter-wave applications. He has published more than 40 technical papers, and has been cited over 200 times.
Yann Clavet was born in Mont-de-Marsan, France, in 1979. He received his Master of Science in Engineering from the Ecole Nationale d'Ingénieurs de Brest, France, in 2002 and the Ph.D. degree in Electronics from the University of Brest, Brest, France, in 2006. He is co-founder and manager of ELLIPTIKA since February 2008. His main area of interest concerns the design and optimization of microwave components.
