Synthesis of Filters and Filtering Antennas for Micro and Millimeter Waves Applications

Abstract

Filters and antennas are the closest building blocks to the air interface in modern wireless communications systems. Filters allow the transmission of signals in a target frequency range and attenuate those that operate in the unwanted range. Antennas help radiate signals within their operating range. This article is a summary of the author’s doctoral thesis and focuses on the development of new synthesis-based methods for the design of these two building blocks and the integration between them, in other words, filtennas. The main advantage of the synthesis-based design is that the expected frequency response of the final prototype is approximated in advance. Therefore, the selection of the best solution that satisfies the given requirements can be done through fast but accurate circuit simulations. When the best solution is found, then the actual prototype is designed according to the synthesized circuit.

This work was supported by the European Union’s Horizon 2020 Research Program 5G STEP FWD under Grant Agreement No. 722429.

1 Synthesis-Based Filter Design

This section presents an overview of advanced filter synthesis techniques described in [4,5,6]. All the techniques described here follows the same principle: they start with the well-known coupling-matrix synthesis, suitable circuit or matrix transformations are then applied to get the required topology.

Fig. 1

Synthesis of extracted-pole filters by matrix rotations. Black circles: unit capacitance in parallel to frequency-independent (FI) susceptances (resonating nodes). Circles with ($\times$) are FI susceptances (non-resonant nodes), and with ($+$) are unit conductance (source, load). Lines are admittance inverters (couplings)

Fig. 2

Fully canonical stop-band filter synthesis using old and new method

1.1 Accurate Synthesis of Extracted-Pole Filters

This method relies in the well-known accuracy of the coupling matrix synthesis method. To understand the method, let us work with synthesis of a filter with 20 dB of return loss and transmission zeros placed at $[\infty$, $-1.889$, 1.1512, 1.702] rad/s. Then, the first step is to synthesize a coupling matrix using any of the methods already available in the literature [7]. Then, the circuit is transformed into the arrow form [7] as shown in Fig. 1a (without the blue coupling). After that, each transmission is extracted by means of matrix rotations. The first rotation create a coupling (dashed blue line in Fig. 1a) in such a way that the last three resonators form a triplet that contains the first zero to be extracted. Then, successive rotations push this triplet towards the position where it is required to locate the extracted pole (Fig. 1b and c). After that, a delta-to-star circuit transformation is performed, obtaining the circuit in Fig. 1d. This must be done with each of the transmit zeros, the output of this algorithm is shown in Fig. 1e. Note that the scattering parameters shown in Fig. 1f are preserved throughout the transformation. Figure 2 shows the synthesis of a fully canonical stopband filter, it is evident that the synthesis with the old method [1] produces a response destroyed by round-off errors. Instead, the new method produces a response that matches the ideal equiripple response.

1.2 Synthesis of Cascade-Block Filters

This section presents the overview of the unified analytical method for the synthesis of cascaded n-tuplets prototype filters including non-resonating nodes (NRNs) and extracted pole blocks. This method helps to overcome the issues of accuracy, computation time, and uncertainty of optimization methods used to synthesize some topologies, particularly those that include singlets, doublets, or mixed topologies. To understand the method, let us work with synthesis of a filter with 20 dB of return loss, and withe transmission zeros located at $-3$, 2, $\infty$, $-0.1+0.79i$, $-0.1-0.79i$, $\infty$, 3 and $-2$ rad/s. Complex transmission zeros helps to group delay equalization. The method begins with previously described extracted-pole synthesis arbitrarily defining the transmission zeros, as shown in Fig. 3a. Then, a filter topology transformation is applied by grouped node blocks to obtain the desired topology, as shown in Fig. 3b. Finally, some matrix rotations are applied to remove redundant couplings obtaining the circuit in 3c. The scattering parameters and group delay shown in Figs. 3d and 3e are preserved throughout the topology transformation. Note that this is a circuit of mixed topology: singlet—extracted-pole—quadruplet—doublet, which was not possible to synthesize analytically with the synthesis techniques available before the publication of the papers [4, 6].

Fig. 3

Synthesized circuit: singlet (x3). Physical Filter: $TE_{201}$ cavity (x3). Unitary inverters are 90$^\circ$ phase shift. Black circles are resonators denormalized with C $=$ $1/(2 \pi \, BW)$, L $=$ $1/(C*(2 \, \pi \, F_c)^2)$. BW is bandwidth. Dimensions in mm: $W_a$ $=$ 22.86, h $=$ 10.16, $C_{y}$ $=$ 40.806, $C_{x1}$ $=$ 20.484, $C_{x2}$ $=$ 21.047, $C_{x3}$ $=$ 20.372, $S_x$ $=$ 3, $S_{y1}$ $=$ 14.699, $S_{y2}$ $=$ 14.654, $S_{y3}$ $=$ 14.294, $O_{1}$ $=$ 18.6255, $S_{2}$ $=$ 14.723, $O_{3}$ $=$ 19.298, $W_{L1}$ $=$ 20.0738, $W_{L2}$ $=$ 20.4047

1.3 Synthesis-Based Filter Design

This subsection intends to show the flexibility of the previously presented methods to synthesize different topologies that actually implements the same filtering function. These topologies are implemented into waveguide technology.

The filtering function for this example requires a passband from 9.966 to 10.045 GHz with return loss of 16 dB. The required attenuation is 75 dB between 9.8 and 9.82 GHz and more than 30 dB between 10.16 and 10.18 GHz. To fulfill these specifications a fully-canonical function with transmission zeros are placed at 9.823, 10.17 and 9.8 GHz. As state before, there are several synthesizable circuits that could implement this filtering function. One of them is the singlet-singlet-singlet topology. To proceed with the design, first the circuit shown in the top part of Fig. 3 is synthesized. Then each singlet is implemented with a $TE_{201}$ cavity by optimizing it according to the corresponding circuit block. This figure also shows the scattering parameters of the designed $TE_{201}$ cavities together with those of the corresponding equivalent singlets. Once all the blocks are designed, they are joined by 90$^\circ$ waveguide lines as shown in Fig. 3.

A second synthesizable topology for this filtering function is the extracted-pole—singlet extracted-poles shown in the top part of Fig. 4. The singlet is implemented as before with a $TE_{201}$ cavity. The extracted-poles are implemented as stubs since there is no non-resonant nodes in the block. As in the previous topology, all the full-wave blocks are optimized to have the scattering parameters equal to those of the corresponding circuit blocks. Then, they are assembled using waveguides whose electrical length is defined by the circuit.

Finally, Fig. 5 shows instead the scattering parameters of both designs: singlet-singlet-singlet and extracted-pole -singlet- extracted-pole. It can be seen that the full-wave simulations are in good agreement with the circuit simulations. Also both topologies implement the same filtering function.

Fig. 4

Synthesized circuit: extracted-pole—singlet—extracted-pole. Physical Filter: stub—$TE_{210}$ cavity—stub. Dimensions in mm: $W_a$ $=$ 22.86, h $=$ 10.16, $C_{x1}$ $=$ 19.54, $C_{y1}$ $=$ 17, $C_{x2}$ $=$ 21.622, $C_{y2}$ $=$ 40.806, $C_{x3}$ $=$ 23.04, $C_{y3}$ $=$ 13.59, $S_{x}$ $=$ 2.98, $S_{y}$ $=$ 10.166, $O_{1}$ $=$ 17.304, $W_{L1}$ $=$ 5.754, $W_{L2}$ $=$ 3.603

Fig. 5

Scattering parameters of synthesized circuits (solid line) and of the designed physical filters (dotted, dotted-dashed lines)

Fig. 6

Equivalences between synthesized circuit and designed antenna array. Extracted-poles: $\left( Q,\,J,\,f_r,\,R\right) \,=\,$ $\left( 25,\,0.3536\,\Omega ^{-1},\,27\,\textrm{GHz}, 0.0086\,\Omega \right)$. Transmission lines: $TL_{01}/\,\,TL\,/TL_{Op}=$ $\left( T_{\sigma },\,\,T_{\delta },\,\, Z_{ref},\,\, f_{ref},\,\, \theta _{ref} \right) =\left( 100\, \sqrt{\textrm{Hz}},\,\,0.002,\,\, 78 \,\Omega ,\,\, 27\,\textrm{GHz},\, \, \frac{\pi }{2}/ \pi / 0 \, \textrm{rad} \right)$. Source S and $TL_S$ impedance: $Z_{S}=50 \,\Omega$. PCB substrate: $\epsilon _r=3.66$, $\tan {\delta }=0.004$, Physical antenna dimensions in mm: $h_{cond}=0.04$, $h_{diel}=0.254$, $w_{s}=0.52$, $w_{tl}=0.2$,$d_{01}=1.86$, $L_a=2.74$, $d_c=3.46$, $W_a=2$, $O_a=0.32$, $S_a=0.73$, $d_{op}=0.28$

2 Synthesis-Based Antenna Design

This section present a general overview of the synthesis and design methodology of a series-fed proximity-coupled antenna array. Further details can be found in [3]. This type of antenna was first presented in [11] and then further exploited in different works [10]. These works provide some guidelines to get an initial prototype, but then the design procedure is mainly based on full-wave optimization. This optimization could be time consuming, particularly for relative high number n of elements (e.g. n $=$ 8).

That is why, I have proposed a synthesis-based design method. The method starts with the synthesis of an equivalent circuit of the antenna array. This circuit allows to estimate the antenna return loss and antenna radiation pattern. At this point, a screening can be done to verify which circuit best fit the antenna requirements. Once, the best synthesized circuit is chosen, the actual antenna can be designed by using the divide-to-conquer approach. That is, each base element of the antenna array is designed according to the corresponding block of the synthesized circuit. Once all the blocks are designed, they are putting together by means of transmission lines according to the synthesized circuit.

For a better understanding let us analyze the example shown in Fig. 6. This figure shows the circuit synthesized by means of the method described in [3]. Let us call the block highlighted by the dotted rectangles the base antenna/circuit block. For this example, the first three blocks are the same, the last one differs only due to the open-ended line. Therefore, for the actual design of the antenna, the physical parameters $L_a$, $W_a$ and $S_a$ of the base antenna block are optimized for the best fit in magnitude to the S parameters of the basic circuit block. Then, $O_a$ is optimized for the best fit in phase. Since the last block has an open line, $d_{op}$ is optimized to match the S-parameters of the corresponding circuit block in magnitude and phase. Figure 7a shows the two-port scattering parameters for the first three circuit blocks and those for the optimized antenna blocks. Figure 7b shows instead the one-port scattering parameter for the last circuit/antenna block.

Once all the blocks have been designed, the complete antenna array is built by joining all the antenna blocks by means of suitable microstrip transmission lines. To verify the accuracy of the design, Fig. 7c shows the $S_{11}$ parameter and the broadside antenna gain G of the whole synthesized circuit and those of the designed antenna array. Figure 7d shows instead the antenna array radiation pattern computed using both the circuit and the designed array. All these figures show a very good fit the circuit and full-wave simulations. It is noticeable that the synthesized circuit not only estimate accurately the scattering parameters but also the far-field behavior of the actual antenna array. This result was possible without any full-wave optimization of the entire circuit, which makes the design procedure easier and less time-consuming.

Fig. 7

Scattering parameters of the Synthesized Circuit blocks in solid lines, corresponding full-wave response of the Antenna blocks in dotted lines

3 Synthesis-Based Filtenna Design

This section present a general overview of the synthesis and design methodology of a circularly polarized coaxial horn filtering antenna (filtenna). Further details can be found in [2, 8, 9]. This solution tries to help to the interference mitigation between units on a space system. The filtenna must allow the transmission with 19 dB of gain and 15 dB of return loss in the portion of the Ka-band reserved for the communication with the earth (25.5–27 GHz). The filtenna must also attenuate 20 dB the band reserved for improved sensing of snow and ice thickness (35.5–36 GHz).

Fig. 8

Coaxial Horn Filtering Antenna with stubs in the horn body (left), and in the coaxial core (middle) . Gray: horn body. Orange: coaxial core. Green: Teflon supports. On the right is the synthesized stop-band circuit

The proposed filtering antenna, shown in Fig. 8, consist of integrating a filtering function into the flare of the horn antenna. So that, there is no need for more space for the filter part. The design methodology consist of (a) first design a standard horn antenna that fulfills the specified gain and return loss; (b) synthesize a stop-band circuit that fulfill the specified attenuation; (c) integrate this filtering function into the horn flare of the designed horn antenna.

In order to integrate the filtering function into the flare horn, we first tried using standard circular stubs. However, it was found that the filtering behavior of these stubs had a bad performance in the stop band because of high-order modes are exited in the horn flare ($TM_{11}$ specifically). To overcome this problem, a coaxial core is inserted such that the $TM_{11}$ cutoff frequency is located as far as possible from the stopband at the position where the coaxial stubs will be placed. In other words, a coaxial core is inserted into the horn in order to have a suitable area to implement the filtering function with stubs. This coaxial core should also be designed to have good return loss ( ${>}$15 dB) and antenna gain comparable to the antenna gain without it. Finally, the stubs are placed into the suitable area and tuned according to the synthesized circuit.

Two options were envisioned for the stubs: (a) the stubs carved into the horn antenna body (external stubs), (b) the stubs excavated into the coaxial core (internal stubs) as shown in Fig. 8. The last one has the advantage that the horn body is standard, and all the complexity of the filtering function relies only on the coaxial core. This allows also to tune the stopband by just changing the coaxial core. Figure 9a shows the full-wave simulations of both designs as well as the transmission parameter of the synthesized circuit. It can be seen a very good match between the normalized broadside gain of the designed filtering antennas and the $|S_{21}|$ parameters of the synthesized circuit. Also, the return loss is higher than 15 dB as required. Figure 9b shows instead the radiation pattern in the centers of the passband and stopband. This verifies the 30 dB of difference between the passband and stopband co-polarization and cross-polarization gain in all the directions (except in the radiation nulls) and not only at the broadside. This figure also shows more than 19 dB of gain in the passband as requested.

Fig. 9

Frequency Response of the designed Coaxial Horn Filtering Antenna.

4 Conclusion

This paper presents first a general overview of novel filter synthesis techniques that are more precise and completely analytical for extracted-pole and cascaded-block filters including non-resonating nodes. All these synthesis methods follows the same principle: starting with the well-known coupling-matrix synthesis, suitable circuit or matrix transformations are then applied to obtain the required topology. Then, the paper provides a summary of novel synthesis-based designs of filters, antennas, and filtering antennas. These designs starts with the synthesis of an equivalent circuit that estimate accurately the full-wave behavior, then the actual full-wave prototype is designed by block using the divide-to-conquer technique. Finally, the prototype are assembled by means of transmission lines. The results have shown a very good agreement between the synthesized circuits and the full-wave simulations of the actual prototypes.