## SIW filters

Filters are a key part of any transmission channel. Indeed, the latter must satisfy a certain number of strong constraints, particularly at the electrical level: bandwidth, selectivity, insertion losses, rejections, group delay, attenuated bandwidth, return losses, etc.

Many technologies are available for making microwave filters. We present here the main technologies allowing to perform filtering functions. They can be classified into three main categories: acoustic, volume and planar technologies. The most appropriate technology should be chosen to meet the specifications of the filter. We present here a summary table of the characteristics of the different technologies.

table n ° 1 Comparison of the different technologies for producing microwave filters

This table indicates that the volume and planar technologies are quite complementary. It is therefore natural that the idea of associating them arrived, this new technology is so called “Substrate Integrated Waveguide (SIW)” [1]. It was introduced in 2001 by Ke Wu (University of Montreal). This technology consists of operating volume cavities in the thickness of the dielectric substrates while remaining compatible with planar accesses. Therefore, the SIW allows us to benefit from the advantages of these two technologies. Technically, the volume cavities are integrated into the substrate and are delimited on the upper and lower faces by metallic planes, and by rows of metallized holes (vias) on the side faces. These vias must have a diameter and a sufficiently small spacing to appear as quasi-perfect electric walls at the resonant frequency of the mode considered.

* [1] D. Deslandes et Ke Wu, « Single-substrate integration technique of planar circuits and waveguide filters », Microwave Theory and Techniques, IEEE Transactions on, vol. 51, n ^{o}. 2, p. 593-596, 2003.*

### SIW waveguide dimension and cutoff frequency

SIW technology, technically they are waveguides produced by two lower and upper metallic planes separated by a dielectric substrate, the side faces of which are replaced by rows of metallized holes which connect these two planes. Figure 1 shows a waveguide of width ** c**, integrated in a thick substrate

**,with rows of diameter metallized holes**

*h***, and spaced between them by a distance**

*d***. Several studies have been carried out to calculate the spacings and diameters of the holes. An adequate choice of parameters**

*p***and**

*d***limits the losses by radiation, that is to say the leakage of the electromagnetic field to the outside of the guide.**

*p*Unlike transmission lines which have two conductors allowing the propagation of quasi-TEM modes, rectangular waveguides have only one conductor and only allow the propagation of transverse electrical (TE) and / or transverse magnetic modes. (TM). In the direction of propagation of the electromagnetic field Oz (Oz longitudinal axis of the guide), the TE and TM modes do not have electrical components Ez and magnetic Hz, respectively.

There is a notable difference between rectangular waveguides and SIW waveguides. SIW waveguides propagate only TEm0 modes in the thickness of the substrate. Indeed, the vias are non-contiguous and do not allow the establishment on the lateral surfaces of the waveguide of surface currents oriented along the longitudinal axis Oz of the waveguide. However, these surface currents along Oz are essential for establishing the TEmn (n ≠ 0) and TM modes in the guide, their absence therefore de facto prohibits the existence of these modes. The figure shows the distribution of the current lines of the fundamental mode TE10 of a SIW waveguide for which the spacing between the vias is materialized by narrow slots on the side faces.

Figure n ° 1 SIW waveguide delimited laterally by rows of metal vias

Figure n ° 2: Current lines on the faces of a SIW waveguide: TE10 mode

As in the case of conventional rectangular waveguides, for an SIW guide the cutoff frequency of the propagation mode is fixed by its width ** c,** but with a few correction factors. The cutoff frequency of a conventional waveguide is given by the following equation (1),

**et**

*c***, width and height of the guide, respectively and**

*h***speed of light in vacuum.**

*C*_{0}Once the diameter of the vias and their spacing are determined, the following equation (2) can accurately calculate the effective width of the SIW guide. It is valid for d / p <1/3 and d/w <1/5 .

Figure 3 shows the simulation results of a SIW waveguide showing rows of vias as side faces. The structure was simulated under HFSS ™ by directly exciting the input of the guide by the mode considered. The simulation results show that the measured and calculated cutoff frequencies of 12 GHz are in good agreement.

Figure n ° 3: EM simulation of the SIW waveguide with a cutoff frequency of 12 GHz

#### SIW / microstrip transitions

It is possible to realize the SIW design and the transitions on the same substrate using conventional planar manufacturing processes such as the printed circuit. The electromagnetic fields on a microstrip line are located in the dielectric substrate and in the air. The energy on the microstrip line propagates in a quasi-TEM mode (cf. Figure n ° 4). The transition must therefore ensure a transformation from the quasi-TEM mode to the waveguide mode used in the SIW cavities. In addition, the transition must also match the characteristic impedance of the planar circuit with the input wave impedance of the waveguide which depends on the propagation mode considered.

Figure n°4 : (a) Distribution of electric field lines in a microstrip line (a) Distribution of electric field lines in a waveguide

We recall here that the fundamental mode of the waveguide is the TE10 mode, it has a maximum electric field at the center of the guide, see Figure 4. A large part of the electric fields of the two modes (TE10 of the guide and quasi-TEM of the microstrip line) are therefore collinear.

Then, there are no analytical formulas for sizing the transitions. the dimensioning is done by a succession of electromagnetic simulations in order to optimize the most suitable transition for a well determined frequency band. you can connect classic “tapping” transitions on either side of a SIW waveguide, then optimize the transmission and reflection of the device (Figure 5).

Figure 5: Optimizing a two-stage “taper” transition

#### Resonant cavities in SIW technology

A waveguide band-pass filter is produced by volume resonators coupled together via discontinuities which can be of several types:

- capacitive,
- inductive,
- post inductive,
- inductive with circular iris,
- with rectangular window,
- with variable height post.

The topology of SIW bandpass filters is similar to that of conventional waveguide filters. The resonator is the basic building block of a filter. We will therefore detail here the design in SIW technology of a resonant cavity and then assess its quality factor.

#### SIW resonator

The design of volume cavities is based on classical waveguide equations. They are constructed from sections of the waveguide by forcing new boundary conditions in the direction of propagation of the electromagnetic wave. They are usually shorted at both ends. A resonant cavity is thus obtained.

For example the fundamental mode of propagation in a rectangular waveguide is the TE._{10} mode and the propagation along Oz The new boundary conditions (short circuit) require the electric field to be zero at ** z=0** and

**. The first resonance mode occurs at the frequency for which**

*z=l***is half the guided wavelength. At this frequency, electromagnetic energy is stored in the cavity, and power can be dissipated through the metal walls of the cavity, as well as the dielectric filling the cavity. The same design is reproducible in the case of a SIW. It suffices to replace the side walls with rows of vias by introducing correction factors. It is also recalled that the height of the volume resonator is that of the thickness of the substrate. The structure is shown in Figure 6.**

*I*Figure n ° 6: Structure of a rectangular SIW cavity

Equation (3) represents the resonant frequency for a volume cavity. This expression is particularized for SIW technology by the introduction of correction factors through ωeff (cf. equation (2)) and leff (cf. equation (4)). This is due to the use of rows of vias in place of metal walls.

As an example, we have designed a cavity operating in Ku band at the frequency of 15 GHz. It is a cavity made on an alumina substrate with a permittivity of 9.9 and a thickness of 380 µm. The resonance mode is the fundamental TE mode_{101}. The quality factor of such a structure is about 274 This is an important value compared to the classical values obtained with classical planar solutions (150 maximum). Figure n ° 7 illustrates the distribution of the electromagnetic field in this resonant structure for the TE101 and TE202 modes

Figure n ° 7: Distribution of the electric field for a SIW cavity (a) in TE_{101} mode at the resonant frequency 15 GHz and (b) in TE_{202} mode at the frequency 30 GHz.

#### Design of a SIW bandpass filter

The first step in the design of the filter consists in determining the filtering function (band pass, notch), the order and the associated topology to verify the specifications, particularly from the point of view of rejections. The choice of the filtering function is initially based on a Tchebyheff approximation. Using Chebyshev filtering functions results in much better rejection than that obtained by a Butterworth filter. On the other hand, this filtering function is simpler than that of an elliptical filter.

Then the order of the filter plays on the level of losses, they are important with a high order It also affects the rejection level, which increases with the order. There is therefore a compromise to be found.

Generally speaking, filter topologies can be classified into two categories:

Direct coupling filters: this category of filters is the simplest. It consists in using only couplings between adjacent resonators. Thus, for example, resonator 2 will only be coupled with resonators 1 and 3.

Direct and cross-coupling filters: Couplings are always carried out between adjacent resonators, thus creating a direct “path” allowing energy to pass successively through all the resonators to go from the input to the output of the filter. . However, non-adjacent resonators can also be associated by creating new couplings. Thus, for example, on a 6th order bandpass filter, resonators 1 and 6 can be coupled. This allows the introduction of transmission zeros which make it possible to either control and therefore improve the rejection by the introduction of attenuation poles (or transmission zeros) on the amplitude response of the filter transmission (| S_{21}|) ; or to control the phase of the filter transmission to make it as linear as possible.

For SIW filters with resonant cavities coupled by iris, the electromagnetic fields are well confined within each cavity, resulting in a more conventional synthesis.

The first step is to size the resonant cavities. you have to define the resonance mode used. In SIW technology, the TE101 mode is used for a rectangular cavity and the TM010 mode for a cylindrical cavity. Analytical formulas make it possible to carry out a first dimensioning, it must then be validated by electromagnetic simulations.

The couplings are made by iris. To design them, we can use synthetic elements available in the literature (charts, formulas, etc.), or we can draw a shart giving the level of coupling obtained according to the opening of the iris. A set of electromagnetic simulations is therefore carried out using two cavities coupled by an iris, the width of which is varied. To design the iris filter, it is therefore sufficient to construct an SIW structure whose iris dimensions determined by the above methods correspond to the desired couplings. However, it is generally necessary to proceed by electromagnetic simulation to a series of adjustments to obtain the optimal response.

For the design of SIW post filters, the resonators are coupled by posts. The same procedure is used as for filters with iris couplings. The two approaches used for iris couplings are declined here for post couplings. The procedures are identical.

Once the filters have been set, the transitions are then introduced. These transitions may have been previously optimized on a SIW guide. However, they usually need to be touched up as well as the filter when combining the set.

Figures 8 (a) and 8 (b) show ku-band iris or post-coupled SIW filteri structures.

figure 8 : (a) SIW bandpass filter with iris and “taper” transitions and (a) SIW bandpass filter with post and “taper” transitions

#### Conclusion

The height of the substrate is important in the design of SIW filters, it determines in association with the permittivity and the central frequency the dimensions of the conductive lines, as well as the level of the losses.

Thus for SIW technology, a thick substrate therefore makes it possible to minimize losses, but undoubtedly complicates the production of vias. The determination of the optimum substrate height is often done by electromagnetic simulations. It is therefore important to call on Elliptika .** Elliptika **in order to obtain **the best solution** in terms of performance.

You will find the contact form below, as well as the other technologies that we master in our “filter” catalog.

The following figures show examples of the realization of Elliptika

SIW alumina bandpass filter surface mounted on a PCB @ 27GHz

SIW Alumina bandpass filter operating at 50 GHz

Tunable SIW bandpass filter based on MEMs

3rd order cross-coupled alumina bandpass filter

Filtre passe bande SIW réalisé en circuit imprimé sur un substrat RO 40003

Tunable SIW band pass filter based on pin diodes