Resemblance between gap waveguides and hollow waveguides

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Published in IET Microwaves, Antennas & PropagationReceived on 4th April 2013Revised on 14th June 2013Accepted on 25th June 2013doi: 10.1049/iet-map.2013.0178

T Microw. Antennas Propag., 2013, Vol. 7, Iss. 15, pp. 1221–1227oi: 10.1049/iet-map.2013.0178

ISSN 1751-8725

Resemblance between gap waveguides and hollowwaveguidesHasan Raza, Jian Yang, Per-Simon Kildal, Esperanza Alfonso

Department of Signals and Systems, Chalmers University of Technology, S-41296 Gothenburg, Sweden

E-mail: [email protected]

Abstract: There is a lack of suitable numerical ports for the gap waveguides. This study studies the resemblance between theridge/groove gap waveguide and the conventional hollow ridge/rectangular waveguide, respectively, by using numericalanalysis. The dispersion diagrams and characteristic impedances are compared and found to be similar within the stopband ofthe parallel-plate modes. Therefore conventional hollow waveguides can be used as ports in numerical analysis of gapwaveguides. In addition, this means that the geometry of circuit components in hollow ridge/rectangular waveguides can beused as a good starting point when designing ridge/groove gap waveguide components.

1 Introduction

Gap waveguide technology is a new way to make low-losscircuits at millimetre and submillimetre waves, as initiallyproposed in [1] and experimentally verified in [2]. The gapwaveguide exists in three different major configurations: theridge, groove and microstrip gap waveguides [3], and thepresent paper deals with the two former ones. Some gapwaveguide components have actually already been designedand measured, such as the gap waveguide power divider in[4], the rat race ridge gap waveguide hybrid in [5, 6], andthe high Q groove gap waveguide filters in [7, 8]. However,the analysis and design were difficult to perform becausethere are no gap waveguide ports in available commercialcomputer codes. Still, the experience is that normalrectangular waveguide ports work to some degree whendesigning groove gap waveguide components, provided theport dimensions are correctly chosen, but no documentationof this is yet available. Therefore the purpose of the presentpaper is to investigate how accurate the hollow waveguideport available in commercial codes is when applied as aridge/groove gap waveguide port for analysing ridge/groovegap waveguide components. We expect that there is such aresemblance between the classical hollow rectangularwaveguide and the groove gap waveguide, and between thehollow ridge waveguide [9] and the ridge gap waveguide.The study is also relevant to finding good initial geometriesfor gap waveguide components before the numerical designand optimisation, because resemblances between hollowwaveguides and gap waveguides would mean that theirwaveguide components should have similar geometries.This engineering aspect should not be under-emphasised.The ridge and groove gap waveguides can be realised

without having any dielectric material inside it, and withouthaving electrical contact between the top and the bottommetal plates, where the latter contains a quasi-periodic

texture normally in the form of metal pins surrounding theridges or grooves forming the actual waveguides. Thistexture creates a high surface impedance which ideallyshould be a perfect magnetic conductor, so that a stopbandis generated for the parallel-plate modes that otherwisewould propagate in the gap between the two plates, as longas the spacing between the upper and lower plates issmaller than a quarter wavelength. A stopband is defined asthe frequency band within which no parallel-plate modescan propagate over the periodic pins. Therefore propagatingwaves are prohibited in this parallel-plate waveguide, exceptalong the metal ridges or grooves between the metal pins.The stopband of parallel-plate modes can also be used forpackaging of both passive and active microwave circuits,realised with microstrip or co-planar waveguides, to avoidresonant modes such as demonstrated for packaging ofmicrostrip lines in [10], microstrip baluns in [11] and activemicrostrip circuits with MMIC amplifier chains in [11].The propagating wave along the metallic ridge or

groove within the periodic texture is a quasi-transverseelectromagnetic (TEM) mode or quasi-TE mode [12],respectively, as illustrated in Fig. 1. We will limit our studyto ridge and groove gap waveguides where the stopband isrealised by square metal pins [1], but the method is alsovalid for other realisations of the stopband such as thosedescribed in [13], and it can be extended to microstrip gapwaveguides such as those described in [14, 15].The key parameters for controlling the stopband are the

period of the pins, the height of the air gap between the topof the pins and the upper smooth plate (referred to as thepin gap height), and the height and width of the pins [13].The period of the pins must be smaller than half thewavelength at the centre frequency. The distance betweenthe ground of the lower plate and the upper plate (i.e. thepin height plus gap height) determines the upper-endfrequency of the stopband and must be smaller than the half

1221& The Institution of Engineering and Technology 2013

Fig. 1 Top (upper) and front (lower) view of basic structures ofridge gap waveguide (left) and groove gap waveguide (right)

Grey region is the air gap and the terms STOP and GO are used to depict thestop region and propagation region, respectively

Fig. 2 Cross-sections with dimensions of the simulated ridge andgroove gap waveguides in Fig. 1 (left), and of the equivalent hollowridge and rectangular waveguides (right)

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wavelength at centre frequency. The height of the pinsdetermines the lower-end frequency of the stopband [13].The modes in a ridge gap waveguide propagate in the air

gap between the ridge and the top plate, so the ridge gapwaveguide has lower ohmic loss than the microstrip line,where the modes propagate in the substrate. The advantageof the gap waveguide over the conventional hollowrectangular waveguide is that the former does not requirehigh-precision manufacture at high frequency, whereas thelatter does. The normal is that hollow waveguides andwaveguide components are manufactured from two metalpieces that must be joined together with good conductivemetal contact, and this imposes a difficulty, whereas for gapwaveguides there is no such requirement since the upperand the lower plates do not need to touch each other. Theperiodic texture creates a very effective stopband of leakingparallel-plate waves and modes.The purpose of this paper is to obtain the propagation

characteristics of the ridge/groove gap waveguides bycomparing it with the classical ridge/rectangular waveguides[1]. The comparison is carried out by numerical analysis interms of dispersion diagrams and characteristic impedances.

2 Dispersion diagram

First, the dispersion diagrams of the ridge/rectangularwaveguides and the ridge/groove gap waveguides areobtained by using the Eigenmode solver in CST microwavestudio, where the periodic structure is assumed infinitelylong in the direction of propagation. In this section, thematerial used for the simulation is the perfect electricconductor with zero surface roughness. The cross-sectionalgeometries of the gap waveguide simulation models aresummarised in Fig. 2, where the dimensions are chosen inagreement with those in [13] in order to obtain a targetedstopband from 10 to 20 GHz. Note that the width and theperiodicity of the pins play significant roles to determinethe stopband [13].In Fig. 2, we also define the dimensions of the

corresponding hollow ridge/rectangular waveguide relativeto the dimensions of the ridge/groove gap waveguideby putting the metal side walls of the former at the firstpins’ wall of the latter. Later on, we will see that thepropagation characteristics of such corresponding ridge/hollow waveguide are very similar to the ridge/groove gapwaveguides within the stopband of the periodic pins.Therefore we refer to the corresponding ridge/hollowwaveguide as the equivalent hollow waveguide of the gapwaveguide.


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The simulated dispersion diagrams of the ridge gapwaveguide and its equivalent ridge waveguide are shown inFig. 3 for four different gap heights between the ridge andthe top metal plate (referred to as the ridge gap height), thatis, h = 0.5, 1, 2, 3 mm. The pin heights and the pin gapheight remain the same for all cases. We see that thestopbands of these gap waveguides are almost the same,from 11 to 22 GHz. The fundamental mode (solid curvewithin stopband) of the ridge gap waveguide with a smallridge gap height is seen to have a dispersion curve (referredto as the ridge gap dispersion curve) very close to that of aTEM mode (the light line). However, the ridge gapdispersion curve moves away from the light line when theridge gap height increases. Nevertheless, the ridge gapdispersion curve is very close to the dispersion curve of thefundamental mode in the equivalent ridge waveguideswithin the stopband of the gap waveguide. The curvesoutside the stopband are lower/higher cut-off modes similarto that of normal rectangular waveguide modes.Similarly, we compare in Fig. 4 the dispersion diagrams

between a groove gap waveguide and its equivalent hollowrectangular waveguide. The stopband for this groove gapwaveguide is from 11 to 19 GHz. The dispersion diagramsof the fundamental propagating modes in both thewaveguides are similar within the stopband.In summary, the dispersion diagram of a ridge/groove gap

waveguide can be approximated by that of its equivalentridge/hollow waveguide, which can be easily obtained byeither analytical expressions or empirical formulas.

3 Direct transition between hollowwaveguide and gap waveguide

The simulations in the present section have been done byusing both CST and HFSS, two of the most usedcommercial electromagnetic solvers, on ridge/groove gapwaveguides, with the dimensions shown in Fig. 2 and alength of 91.5 mm (see Fig. 5a).The direct transition from ridge/groove gap waveguide to

hollow ridge/rectangular waveguide is a simple andlow-cost way to convert gap waveguides to conventionalwaveguides and therefore very useful in measurementsituations. We see such transitions in Fig. 5, showing a

T Microw. Antennas Propag., 2013, Vol. 7, Iss. 15, pp. 1221–1227doi: 10.1049/iet-map.2013.0178

Fig. 5 Port configuration for calculating the S-parameters of thegap waveguides

a Ports directly attached to ridge/groove gap waveguideb Ports attached to the equivalent hollow waveguide transitions

Fig. 3 Dispersion diagram of ridge gap waveguide with differentridge gap heights

Size of the pins and the pin gap heights are the same for all cases. The twoblack vertical lines mark the beginning and end of the stopband, and thedashed black line is the dispersion diagram of the equivalent hollow ridgewaveguidea ridge gap height, h = 0.5mmb ridge gap height, h = 1mmc ridge gap height, h = 2mmd ridge gap height, h = 3mm

Fig. 4 Dispersion diagrams of groove gap waveguide andequivalent rectangular waveguide (dashed black line)

Two black vertical lines mark the beginning and end of the stopband

Fig. 6 Illustration of the parametric study of the port location andwidth

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direct connection of a numerical or mechanical port, and aseparate mechanical transition to the rectangular waveguide

IET Microw. Antennas Propag., 2013, Vol. 7, Iss. 15, pp. 1221–1227doi: 10.1049/iet-map.2013.0178

built-in with the gap waveguide hardware. The latterreduces some of the flexibility of the gap waveguide, so theformer approach should be used. We will now make anumerical study to see how optimum the dimensions of theequivalent hollow waveguides are.First, we consider the ridge gap waveguide, and make a

parametric study on k1 and k2 which defines the locationof the interface (see Fig. 6). The best result appears whenk1 = p/2, where p is the spacing between the walls of the


Fig. 7 S-parameters at different port locations and width, using

a CSTb HFSS

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pins and k2 is equal to the distance between wall of the pinsand the ridge (in this example, it is also p), as shown inFig. 7. This is then also the best waveguide port for gapwaveguide simulations in CST and HFSS. Note that theresults provided by CST and HFSS are very similar.Similarly, the best waveguide port for groove gap

waveguide is found to coincide with the equivalent hollowwaveguide, shown in Fig. 5b, after a parametric study.Figs. 8 and 9 show the S-parameters for the ridge and

groove gap waveguide cases shown in Fig. 5a with the bestequivalent waveguide port. The ridge gap waveguides inFig. 8 have different ridge gap heights h as defined inFig. 2. We see that the reflection coefficient S11 by usingthe equivalent waveguide port is below –35 dB over mostof the stopband if the ridge gap height is smaller than orequal to 1 mm, but it increases when the ridge gap heightincreases. For the groove gap waveguide, S11 is below−30 dB over most of the stopband, except in the beginningof the stopband. Thus, the matching between the gapwaveguide and the equivalent waveguide port is good,except at the low end of the stopband. The periodic natureof the reflection coefficient comes from the two interferingreflections, from the ports at both ends of the waveguides.This means that the reflection coefficient from each portinterface is 6 dB lower than the peaks of the combinedreflection coefficient S11 [16].


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The simulation models also include the ohmic loss byusing the lossy copper (σ = 5.8 × 107 S/M ) in the models.The transmission coefficient S21 for the 91.5 mm long gapwaveguide is about –0.1 dB within the part of the stopband,where the reflection coefficient is low.

4 Characteristic impedance

The characteristic impedance of the ridge gap waveguide isdetermined mainly by the ridge width and the ridge gapheight [17]. There were several methods available tocalculate the approximate value of the characteristicimpedance of ridge gap waveguide [18, 19]. In this section,we introduce a new numerical method to have an accuratecalculation of the characteristic impedance for a ridge gapwaveguide.We know from the previous section that the reflection

coefficient of the direct transition is below − 30 dB. Thisindicates that the equivalent ridge waveguide is wellmatched to the ridge gap waveguide, and we can use, as thefirst approximation, the simulated characteristic impedanceof the equivalent ridge waveguide as that of the ridge gapwaveguide. Then, we apply a correction to the first-orderapproximated characteristic impedance by using thesimulated reflection coefficient of the two-port network

T Microw. Antennas Propag., 2013, Vol. 7, Iss. 15, pp. 1221–1227doi: 10.1049/iet-map.2013.0178

Fig. 8 S-parameters of the ridge gap waveguide for different ridge gap heights, using the best equivalent hollow waveguide ports in

a CSTb HFSS

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shown in Fig. 5a, the gap waveguide under test with itsequivalent waveguide ports at the two ends. The reflectioncoefficient r of one transition can be written as

r = Zo − ZportZo + Zport

(1)

where Zport is the characteristic impedance of the equivalentridge waveguide, and Zo is the characteristic impedance ofthe ridge gap waveguide. The S-parameter S11 of thetwo-port network in Fig. 5 can be estimated by using thetheory of small reflections in [16] as long as |r| < 0.2

S11 ≃ r 1− e−j2bl( )(2)

where l is the length of the ridge gap waveguide. Using (1)and (2), we have

Zo ≃1− e−j2bl( )+ S111− e−j2bl( )− S11

Zport (3)

We simplify the above calculation by (i) calculating the


values of Zo at the frequencies fi, where S11 has its peaks, by

Zo(fi)≃2+ S11peak(fi)

2− S11peak(fi)Zport (4)

(ii) the impedance curves over the whole band is obtained byinterpolation and extrapolation. Fig. 10 shows the impedancecalculation for ridge gap waveguides with different ridge gapheights while the ridge width is kept the same in all cases.From Fig. 10, it is observed that with the lower air gaps, the

gap waveguide impedance and the port impedance aresimilar. Re-examining Fig. 8, it can be observed that thereflection coefficient for that particular air gaps is below−35 dB as seen in both CST and in HFSS. From this, wecan say that the equivalent ridge waveguide is well matchedto the ridge gap waveguide. Similarly Figs. 3a and b alsoshow similar dispersion diagrams of the fundamental modesof both the ridge waveguide and ridge gap waveguidewithin the stopband. This explains why both have similarpropagation characteristic, but only within the stopband ofthe parallel-plate modes. From Fig. 10, it is also observedthat the discrepancy between the gap waveguide impedanceZo and the port impedance Zport (the approximation of Zo)increases with the increase of the ridge gap height,


Fig. 10 Characteristic impedance of ridge gap waveguide fromHFSS port model and our corrected result (Zo)

Fig. 9 S-parameters of the groove gap waveguide using the best equivalent hollow waveguide ports in both CST and HFSS

Fig. 11 Wave impedance in groove gap waveguide from HFSSport model and our corrected result (Zo)

Fig. 12 Comparison of the characteristic impedance of the ridgegap waveguide by using the present method, the stripline modelmethod in [18] and the circuit model V/I and P/I2 method in [19]

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especially at the low end of the stopband. Re-examiningFig. 3, it can be observed that the difference between thedispersion diagrams of the fundamental modes of both theridge waveguide and ridge gap waveguide increases withthe increase of the ridge gap height, especially at the lowend of the stopband. This explains this impedancediscrepancy.The same procedure is also applied to groove gap

waveguide, where the wave impedance for the waveguide iscalculated through the equivalent waveguide portimpedance Zport and the S-parameter S11. The results areshown in Fig. 11.The characteristic impedance of the ridge gap waveguide

was previously calculated by several methods, such as thehalf-stripline-model method in [18, 20], the circuit model V/I and P/I2 method in [19]. Fig. 12 compares the presentmethod to those previous methods on the characteristic

Table 1 Characteristic impedance in Ω of the ridge gap waveguide in Fig. 12, obtained by different methods at three frequencies

Characteristic impedance (Ω) 12 GHz 15 GHz 18 GHz

P/I2 method [19] 57.5 50.2 47.9V/I method [19] 60.0 53.5 51.9half-stripline model method [18] 62.0 62.0 62.0port-impedance method in the paper 48.6 46.9 46.0port-impedance-with-correction method in the paper 50.5 47.7 46.8



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impedance for the ridge gap waveguide with the dimensionsshown in the figure. Table 1 lists, for clarification purpose,some digit values of them obtained by these differentmethods at three frequencies. It seems that the presentmethod has a better accuracy.
5 Conclusions

We have defined an equivalent hollow rectangular waveguidefrom the geometry of the groove gap waveguide, and anequivalent hollow ridge waveguide from the geometry ofthe ridge gap waveguide. We have found that within thestopband of the parallel-plate modes, the dispersion diagramand the characteristic impedance of the gap waveguides canbe determined from the equivalent hollow waveguides,except near the lower end of the stopband. The effect of thediscontinuity at the numerical port is over most of thestopband a return loss smaller than − 30 dB in aback-to-back configuration, meaning that each port is betterthan − 36 dB. This discrepancy is so small that it will bevery difficult to verify by measurements, and this is notneeded either because the main intention of this paper is tostudy the hollow waveguides as ports in numerical work.The discrepancy increases for the ridge gap waveguidewhen the height of the air gap above the ridge increases(for fixed pin gap height), and when the pin gap heightincreases, so that the extent of the stopband decreases.We have introduced a method to determine the

characteristic impedance of the ridge gap waveguide fromthe simulations using hollow waveguide ports, and we havecompared with the previous published methods. The resultsshow characteristic impedances that are much closer to theimpedance of the equivalent hollow waveguide (as achievedfrom the commercial programme) than they are to theprevious introduced analytical stripline approximation forridge gap waveguides. The results are also closer to thehollow waveguide approximation than the previousnumerical studies based on integration of the fields andpower densities over the cross-section of the waveguide,and the present approach is more rigid because the formeris not based on a proper definition of a reference port.Thus, the overall conclusion is that we can use a hollow

waveguide port when analysing and designing the gapwaveguide components, provided the port dimensions areproperly chosen and the stopband is large enough, andprovided we are not too close to the start of the stopband.This resemblance between gap waveguides and hollowwaveguides can also be used to find initial topologies andgeometries for gap waveguide components to be used inlarge integrated gap waveguide circuits, because the finaloptimised result should be similar in shape to the hollowwaveguide equivalent.

6 Acknowledgments

This work has been supported by the Swedish ResearchCouncil VR and Pakistan’s NESCOM scholarship program.


7 References

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Resemblance between gap waveguides and hollow waveguides

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