диафрагмированные волноводные фильтры / 2557f7e0-586a-4a6d-9fda-53c945c07f90

Keywords: waveguide filters, Q factor.

Abstract

Waveguide resonators incorporating E-plane metal inserts with fins have already been demonstrated in building extracted pole sections (EPSs) for very compact and easily fabricated microwave filters. Flexibility of transmission poles (TPs) and zeros (TZs) locations for different resonant and non-resonant modes is further shown and the electromagnetic nature of these modes as well as natural frequencies of EPS sections are explained and numerically estimated. Detailed comparison with conventional waveguide resonators, overmoded cavities, state of the art waveguide resonator solutions and substrate intergerated waveguide resonators is performed at 10 GHz.

1 Introduction

Even though waveguide technology had its initial most significant development in the first half of the 20th century, it is still unparalleled when it comes to applications requiring low loss, high power and perfect EM isolation. Moreover, during time, demands for compactness, complex filter networks with very low insertion loss and high roll-off have been pushed ever closer to the obtainable physical limits. Also, new system requirements such as millimetre wave links for 5G mobile networks as well as new fabrication technologies and implementations like substrate integrated waveguide (SIW) one are always bringing freshness to the area.

In [1] Konishi and Uenakada proposed use of metal E-plane inserts for realisation of directly coupled waveguide filters [2]. E-plane waveguide technology is likewise very suitable for implementation of ridged waveguide resonators [3] and quasi-lowpass corrugated-waveguide filters [4]. In [5], the size of the conventional E-plane resonator was reduced and transmission zero introduced through the addition of metal S- shaped lines on dielectric slab. Finally, in [6] extracted pole sections using fins have been introduced. Further miniaturization was achieved by use of several fins [7], and multiplexers based on this filter structure have been designed [8]. In this paper, compact E-plain resonators with fins are explored in more details regarding their fundamental characteristics, resonant frequencies and quality factors, as well as for their power handling capability as one of the chief characteristics of resilience of these microwave components in real life applications.

2 Resonator structure

3D model of an E-plane inline resonator with fin coupled on both sides to form an EPS section is shown in Fig 1a, whereas its equivalent scheme is depicted in Fig 1b. That is, the central fin can be represented by a parallel connected serious LC circuit, the waveguide straight sections around it by equivalent transmission line sections having the same characteristic impedance as the waveguide wave impedance, and the enclosing septa as immitance invertors.

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tĨŝŶ

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Figure 1: a) 3D model of proposed E-plane inline resonator and b) its equivalent circuit

The dominant mode is based on TE101 mode, which can be observed by half sine field oscillations in x and z directions. In horizontal plane, electric field diminishes on all the side walls, while in the central part, where it is the strongest, it surround the fin and is much tighter localized than it is for the resonator without a fin. Adding the fin, which effectively meanders the EM field inside the cavity, can be represented as a continuous transformation of the top waveguide wall.

3 Resonances

First of all, the fin length will be extracted from the TZ position. Transmission zeros are inherent property of a transmission function of a network between two of its ports. They tell at which (complex) frequencies the ports are decoupled. For that reason, it does not matter how are these

two ports closed – if we inspect the transfer function relation for that pair of ports through different parameters (Z,Y,S) it will have exactly the same transfer function numerator zeros if we do not take into the account possible cancellation effect. In other words, there is a cut in the signal path. This directly implies that we can extract EPS section zeros by removing invertors, that is, just keeping the fin inside a waveguide. In practice, this can be affected by mutual coupling of the fin with other elements, in this case septa and other fins, which becomes more significant as the structure’s size reduces.

For ideally thin fin, its length can be approximately calculated by the expression:

So, the fin length is very slightly larger than the quarter wavelength. Here, Ȝ=c/f is a free space wavelength rather than the guided one. This can be explained by the fact that the fin lays in a cross section plane and not along the waveguide. And by image theory applied on waveguide walls, the fin transforms into 2D array of half-wavelength dipoles in open space. Hence in theory, at the cut-of frequency, the fin length is roughly equal to the waveguide height.

Metal insert is supposed to be thin (< 2% a), but thicker than the skin depth so that perturbation method can be applied. Then, it can be taken that insert thickness does not influence the transmission zero frequency.

Including the fin width Wfin, TZ frequency can be roughly calculated by

where fc is the waveguide cut-off frequency. Estimating Wfin, which enlarging reduces the coupling and narrows the bandwidth, Lfin can easily be found from the TZ frequency fz calculated in the ideal model to satisfy the specification.

In frequency band where higher order modes start to appear, first there is a visible shift in fin length, and at about 3Ȝ/4, secondary radiation from the fin is mostly transferred to higher modes with first index odd (odd number of half-sine oscillations along the wider cross section rectangle edge due to the location of the fin in the centre which fixes field

maximum in that position), i.e. TE11, TM11, TE30,… in the order they appear. Accordingly, TZ effect in the dominant

mode diminishes. Since for historical reasons the wider rectangle side is a bit more than double length of the shorter one, these spurious modes are further shifted to higher frequencies, i.e. more than twice the cut-off frequency of the dominant mode. E.g. for X-band WR-90 , cut-off frequencies of TE11 and TM11 modes are 16.16 GHz.

Pole resonant frequency in an unloaded resonator can be calculated starting with the ubiquitous expression for the TE101 mode in the rectangular waveguide cavity, modifying it through division with a nonlinear function larger or equal

than one, which depends on Lfin and has relatively modest steepness for small values of Lfin, but it increases afterwards. Nevertheless, of interest are only those larger values of Lfin for which this function can be linearized. Except for Wfin that can vary in a large band, small changes with all parameters fixed apart from one make roughly linear changes, meaning that partial derivatives of frequency are nearly constant in the ranges of interest.

Therefore, using least square method to approximate overdeterminate system of linear equations, for thin fin the transmission pole frequency in X band can be estimated by:

26.5 - 0.68ÂL[mm] - (2.13-0.073ÂL[mm])ÂLfin[mm]

Since Lfin is already known from satisfying TZ location, it is not difficult to calculate L from the known TP frequency.

Transmission poles from the transmission network are isolated by eigenmode solver, as they are the natural frequencies when the network ports are short circuited. There may be a confusion arising by the asymmetric properties of TPs and TZs. If we take as an example a reflexion parameter, there is indeed symmetry between the denominator and the numerator, since two different reflection parameters are just inverse one to another. But this is just an exception which does not violate the more universal property of not having symmetry between the denominator and the numerator. It is important to stress that the mathematical function through which we can observe the natural frequencies (complex in general case) is a transfer function, which in strictly a response function over a source function in the Laplace domain. This signifies that although in both denominator and numerator of a transfer function we have polynomials, there is no symmetry in the general case between them – the fundamental characteristic of a circuit lays in the zeros of the denominator, not the numerator.

3.1 Higher order modes

In [6] are as well used higher order modes, put together to form dual-mode resonator. Actually, these are not degenerate modes, but modes of very different nature, though are of similar resonant frequencies. First one is modification of TE101 mode, just as the dominant mode (for both modes it can be checked that varying a and d alters the resonant frequency, whereas changing b has only minuscular indirect effect over the fin), however, with a different mode in the volume around the fin. This can be view by comparison of the field distributions at the foot of the fin for these two modes, fig. 2. While one has electric field lines going into the corner like rays of cylindrical waves (a) to satisfy no tangent electric field component boundary condition, the other has electric field lines forming arcs centred on the fin slightly below the waveguide top wall (b). The second mode in the transmission pole pair is almost unaltered TE102 mode due to the fact that the fin is positioned where the field has its minimum. This also means that changing fin length, while having large effect