Analytic Modelling of a Planar Goubau Line with Circular Conductor
AAnalytic Modelling of a Simplified PlanarGoubau Line
Tobias Schaich, Daniel Molnar, Anas Al Rawi, Mike PayneAugust 20, 2020
Abstract
This paper analyses the surface wave mode propagating along a simpli-fied planar Goubau line consisting of a perfectly conducting circular wireon top of a dielectric substrate of finite thickness but infinite width. Anapproximate equation for the propagation constant is derived and solvedthrough numerical integration. The dependence of the propagation con-stant on various system parameters is calculated and the results agreewell with full numerical simulations. In addition, the spatial distributionof the longitudinal electric field is reported and excellent agreement withthe numerical simulation and previous studies is found. Moreover, valida-tion against experimental phase velocity measurements is also reported.Finally, insights gained from the model are considered for a Goubau linewith a rectangular conductor. These results present the first step towardsan analytic model of the planar Goubau line.
Surface waves (SW) on circular conducting wires have been of theoretical inter-est since their discovery by Sommerfeld in 1899 [1]. However, due to the largelateral extent of the fields at low frequencies, practical applicability seemedlimited at first. Goubau discovered that by coating wires in dielectric or cor-rugating the wire’s surface, the fields’ lateral confinement could be drasticallyenhanced [2]. These coated wires, named Goubau lines in later years, showedlow loss and weak dispersion. Hence, they were discussed as an alternative totraditional, two-conductor transmission lines. Recently, interest in surface wavetechnology has re-emerged at the GHz-THz frequency range where it presentsa promising alternative to current waveguide technology [3, 4]. Furthermore,by introducing sub-wavelength corrugations, spoof surface plasmon polaritonsemerge which have tunable properties and can exhibit sub-wavelength lateralconfinement [5–7]. These new technologies have been discussed as solutions toproblems such as signal integrity in integrated circuits and backhaul solutionsfor the network standard 5G [8, 9].In many cases it is favourable to print a conductor design on a substrate us-ing established printed circuit board fabrication processes such as etching. Thisled to the invention of a planar Goubau line (PGL) consisting of a thin rectan-gular conducting strip on a substrate [10–13]. Multiple electronic componentshave been proposed for PGLs including broadband loads, power dividers and1 a r X i v : . [ phy s i c s . c l a ss - ph ] A ug 𝑎𝑧 𝑥 𝜖 𝑦ℎ𝜖 𝜖 𝑏 Figure 1: Cross sectional illustration of the simplified planar Goubau line. Awire of radius a is at a height h over a dielectric slab with permittivity (cid:15) and thickness b . The surrounding medium has permittivity (cid:15) . A cartesiancoordinate system ( x, y, z ) is defined with its origin placed on the substrate inline with the centre of the conductor.frequency selective filters [14–16]. Additionally, application of PGLs in terahertzspectroscopy has been established [17].Despite these advances, only numerical and experimental studies have beenpublished on the PGL to date and no analytic theory or model exists [18, 19].A difficulty in the exact treatment is presented by the presence of sharp cor-ners which introduce lightning rod effects as reported in Ref. [12]. Therefore,a simplified model of the Goubau line in which the rectangular conductor isexchanged for a conductor with circular cross section will be considered in thispaper. For ease of notation the simplified system will be referred to as a PGL aswell. A related system - the single conductor above a semi-infinite conductingearth - has been extensively studied (see [20] and references therein). We willdraw parallels to this system where appropriate.The paper is structured as follows: First, we discuss the wave created byan infinitesimally small current filament above a substrate. Then, the finitethickness of wire is incorporated to derive a characteristic equation for the sys-tem and applicability criteria for this approach are discussed. Using the derivedequation, the dependence of the propagation constant on input parameters is es-tablished and some field patterns are reported. We validate our results againstnumerical data obtained through the finite element method and experiment.Finally, we draw parallels to the Goubau line with rectangular conductor. The system we are investigating consists of a perfectly conducting wire of radius a with its centre located at a height h above a substrate of thickness b . Figure 1shows the cross section of the system and defines the coordinate system ( x, y, z ) located on the surface of the substrate with its origin in line with the centreof the wire. We assume the wire and substrate are uniform in the z -directionwhich will be the direction of wave propagation. Additionally, the substrateextends infinitely far in the x -plane. It has a dielectric constant of (cid:15) and isimmersed in a medium with dielectric constant (cid:15) with (cid:15) < (cid:15) . For most2ractical applications the surrounding medium is air whose dielectric constantcan be approximated as the dielectric permittivity of vacuum (cid:15) . All materialsare assumed to be non-magnetic and have permeability equal to the magneticconstant µ .We start our derivation by postulating a time harmonic current density (cid:126)J that replaces the wire. Its form, which is motivated by the current found inregular surface waveguides such as the Goubau line, is given by: (cid:126)J = Iδ ( x ) δ ( y − h ) e iωt − iβz ˆ z (1)with angular frequency ω , time t , propagation constant β , current amplitude I , unit vector ˆ z and Dirac delta function δ . This approach is only strictlyvalid in the case where the wire is infinitesimally small but it can also givereasonable results for thin wires. The physical conditions under which wiresmay be considered thin will be given later when discussing the characteristicequation. The presented current density can be interpreted as the exact currentdistribution averaged across the cross section of the wire. For now, we note thatas we are not resolving the exact current distribution inside the wire, the nearfield close to the wire will deviate from an exact solution. However, at distancesmuch greater than the wire radius, the distribution of current inside the wireshould have an insignificant effect on the electromagnetic fields.Next, we assume that the total field can be separated into a transversemagnetic (TM) and transverse electric (TE) component which are characterisedby having no longitudinal magnetic or electric field, respectively. Thus, we mayexpress the total electric and magnetic field, (cid:126)E and (cid:126)H , as (cid:126)E = (cid:126)E T M + (cid:126)E T E (cid:126)H = (cid:126)H T M + (cid:126)H T E (2)We expect the fields to be in phase with the current density and should alsocontain a factor e iωt − iβz which will be implicitly assumed but omitted for clarity.Faraday’s and Ampere’s law then take the form (cid:126) ∇ × (cid:126)H T M = − iω(cid:15) j (cid:126)E T M + Iδ ( x ) δ ( y − h )ˆ z (cid:126) ∇ × (cid:126)E T M = iωµ (cid:126)H T M (3) (cid:126) ∇ × (cid:126)H
T E = − iω(cid:15) j (cid:126)E T E (cid:126) ∇ × (cid:126)E
T E = iωµ (cid:126)H T E (4)where (cid:15) j takes either the value (cid:15) or (cid:15) . From the defining property of TEand TM modes, it may be shown that all field components may be calculatedfrom the longitudinal component of the magnetic and electric field, H z and E z , respectively [21]. Taking the curl of the second equation in (3) and usingFaraday’s and Gauss’ law in combination with the continuity equation as wellas standard vector calculus identities, we arrive at (cid:16) ∂ ∂x + ∂ ∂y + γ j (cid:17) E T Mz = − iωµ I γ j k j δ ( x ) δ ( y − h ) (5)where we have introduced k j = ω (cid:15) j µ and the new variable γ j = k j − β with j ∈ { , } . In a similar fashion, we may manipulate equation (4) to arrive at (cid:16) ∂ ∂x + ∂ ∂y + γ j (cid:17) H T Ez = 0 (6)3s the boundaries along the dielectric substrate extend infinitely along the x-direction, it is convenient to introduce a Fourier transform and its inverse as E T Mz = 12 π (cid:90) ∞−∞ ˜ E T Mz e − iξx dξ ˜ E T Mz = (cid:90) ∞−∞ E T Mz e iξx dx (7)Similarly, ˜ H T Ez is the Fourier transform of H T Ez . Generally, we signify functionsin Fourier space by a tilde. The transformed equations (5) and (6) become: (cid:16) ∂ ∂y + u j (cid:17) ˜ E T Mz = − iωµ I γ j k j δ ( y − h ) (8) (cid:16) ∂ ∂y + u j (cid:17) ˜ H T Ez = 0 (9)with u j = γ j − ξ . Solutions to these equations are readily available [22]. Weimpose the condition that fields should decay towards infinity and find ˜ E T Mz = − ωµ I γ k e iu | y − h | u + C e iu y y ≥ C e iu y + C e − iu y − b ≤ y ≤ C e − iu y y ≤ − b (10) ˜ H T Ez = C η e iu y y ≥ C η e iu y + C η e − iu y − b ≤ y ≤ C η e − iu y y ≤ − b . (11)where u is defined such that it has positive imaginary part and C to C are constants yet to be determined. We introduced the free space impedance η = (cid:112) µ /(cid:15) so all constants have the same dimensions.At the interface between the dielectric substrate and air, the tangential com-ponents of (cid:126)E and (cid:126)H must be continuous. Introducing A = − ωµ I γ k e iu h u , theseboundary conditions may be expressed in the following matrix equation Q C C C C C C C C = A − k √ ε u γ A βξγ A (12)with relative dielectric permittivities ε j = (cid:15) j /(cid:15) and matrix Q which can befound in the appendix. Q may be inverted to find expressions for the constants C to C . We note here that the TE and TM modes are coupled through thecontinuity of H x and E x at the boundaries between substrate and air. Pure TMsolutions with C , C , C and C equal to zero cannot fulfil the matrix equation.Hence, the resultant electromagnetic field will be hybrid in nature contrary tothe Goubau mode for the dielectric coated cylinder.4 Characteristic Equation
So far we have discussed the exact solution for an arbitrary, infinitesimal cur-rent filament carrying a known current wave at a height h above a substrate.However, in most practical cases, we want to find the propagation constant ofthe wave carried by an extended wire of finite size. This is still a formidabletask even with the possibility of formally expressing the electromagnetic fieldsgiven any current distribution by convoluting the calculated fundamental solu-tion with the source term [22].As a means of characterising the propagating mode, we introduce the ef-fective refractive index n eff which is related to the propagation constant via β = n eff k . In order to formulate an approximate characteristic equation,we assume the wire is a perfect electrical conductor (PEC). This is valid formany metals in the GHz to THz frequency range if the radius is much largerthan the skin depth. As a PEC, the tangential electric field should be zero atits surface. In particular, the z -component of the electric field must be zero.Imposing this condition at any point on the wire’s surface, gives an equation forthe approximate propagation constant if the wire is sufficiently thin [23]. Hence,the characteristic equation may be expressed as E z ( x = a, y = h ) = 0 (13)In a study on the validity of this approach for a wire in air above a semi-infinite earth, Pogorzelski and Chang showed that reasonable results are ob-tained if the contribution of azimuthal currents in the wire can be neglected [24].Furthermore, it was shown in the same work that given k h (cid:28) and | γ | h (cid:28) the contribution due to the first order azimuthal terms in the effective refractiveindex of the wave scales as ∆ n eff n (0) eff = g ( n eff ) n (0) eff ∂E z /∂n eff | x +( y − h ) = a (cid:12)(cid:12)(cid:12) n eff = n (0) eff (14)where ∆ n eff is the correction due to higher order terms, n (0) eff is the zero ordereffective index and g ( n eff ) is a function of the effective refractive index and thegeometry given by g ( n eff ) = n eff (cid:16) a h (cid:17) (cid:16) (cid:15) − (cid:15) (cid:15) + (cid:15) (cid:17)(cid:104)(cid:16) a h (cid:17) + 11 − i (cid:15) − n eff (cid:15) (1 − n eff )( (cid:15) + (cid:15) ) (cid:105) − (15)Hence, the correction to the effective refractive index ∆ n eff due to azimuthalcurrents can be neglected if the absolute value of the right hand side in Equa-tion (14) is small. In our case the higher order contributions should be evensmaller because the dielectric is only of finite thickness. Therefore, Equation(14) provides an estimate on the obtainable accuracy when using the thin wireapproximation to calculate the propagation constant.In order to express and solve the characteristic equation, we focus on theamplitude C which is required for calculating E z in real space above the sub-strate via the inverse Fourier transform (7). Solving the matrix equation (12),we find that it may be written in the form C = A ( − F ( ξ )) (16)5here F is a complicated function of ξ whose complete form is given in theappendix. It has some noteworthy properties. First it only depends on ξ reflecting the mirror symmetry of the system with respect to the plane x = 0 .Furthermore, for | ξ | (cid:29) | γ | and | ξb | (cid:29) it asymptotically behaves as F ( ξ ) ∼ k γ ξ + u u k u + k u u ∼ k γ (cid:16) − β k + k (cid:17) (17)which can be shown as all exponential terms e iu b in F ( ξ ) will be very small.This expression is identical to the one obtained by Wait for the case of a wireabove a semi-infinite earth [23]. In fact, in the corresponding limit of b → ∞ the conditions on ξ can be relaxed. This is due to all contributions involving e iu b becoming infinitely fast oscillating or zero so that they can be neglectedin the inverse Fourier transform in eq. (7).Let us reiterate the Fourier transform of E z , which is of the following form ˜ E z = − ωµ I γ k (cid:16) e iu | y − h | u − e iu ( y + h ) u + e iu ( y + h ) u F ( ξ ) (cid:17) (18)To calculate the inverse Fourier transform, we use the identity (cid:90) ∞−∞ e iu | ( y ± h ) | u e − iξx dξ = − iK (cid:0) − iγ (cid:112) x + ( y ± h ) (cid:1) (19)where K is the zeroth order modified Bessel function of the second kind [25].Furthermore, for a real number V for which V b (cid:29) and V (cid:29) | γ | we mayreplace u with iξ and F ( ξ ) with its asymptotic form. Hence, the inverse trans-form gives (cid:90) ∞ V F ( ξ ) e iu ( y + h ) − iξx u dξ ∼ − ik γ (cid:16) − β k + k (cid:17) Γ (cid:0) , ( ix + y + h ) V (cid:1) (20)where Γ is the incomplete Gamma function [26]. A similar expression is obtainedfor the integration from −∞ to − V as F ( ξ ) only depends on ξ . In the regionfrom -V to V, no analytic expression for the integral was found but it may becalculated numerically. Note that, in general, the Cauchy principal value of theintegral needs to be taken because the integrand may contain poles.The field may then be expressed as E z = − iωµ I γ πk (cid:104) K (cid:0) − iγ (cid:112) x + ( y + h ) (cid:1) − K (cid:0) − iγ (cid:112) x + ( y − h ) (cid:1) − k γ (cid:0) − β k + k (cid:1) Re { Γ(0 , ( ix + y + h ) V ) } − i (cid:90) V cos( ξx ) F ( ξ ) e iu ( y + h ) u dξ (cid:105) (21)where Re indicates that the real part of the expression in brackets should betaken. In general, the characteristic equation (13) must be solved numerically due tothe integral containing F ( ξ ) . However, if the wire is located very far above the Taking into account the different definitions of u and u used by Wait .2 0.4 0.6 0.8 1.0Height above Substrate h [mm]1.001.051.101.151.201.251.30 n e ff ModelCOMSOL 0 5 10 15 20 25 30Frequency f [GHz]1.001.051.101.151.201.251.301.351.40 n e ff ModelCOMSOL1.0 1.5 2.0 2.5 3.0 3.5 4.0Relative Permittivity of Substrate r ,2 n e ff ModelCOMSOL 1 2 3 4 5 6 7 8Substrate Thickness b [mm]1.01.11.21.31.41.5 n e ff ModelCOMSOLTE Substrate Mode
Figure 2: Sweep of different parameters showing their effect on the effective re-fractive index n eff of the propagating mode obtained with the presented modeland the finite element solver Comsol Multiphysics. While sweeping one pa-rameter, all other parameters were kept at their nominal values a = 0 . mm, b = 1 . mm, ε = 3 , h = 0 . mm and f = 10 GHz.substrate such that | γ h | (cid:29) , the integral may be neglected due to the strongexponential damping. In fact, the characteristic equation is then dominated bya single term K ( − iγ a ) = 0 (22)This is the characteristic equation for a surface wave on a perfectly conductingcylinder surrounded by air which has been shown not to support any boundsolutions [27].For all other cases we have solved equation (13) using Wolfram Mathemat-ica. Due to the system being lossless, any bound mode will have a real n eff with √ ε < n eff < √ ε . Note that a large value of n eff generally indicatesa stronger confinement of the wave to the wire and substrate. We examinethe effects of the substrate thickness and dielectric constant, signal frequencyand the wire’s height above the substrate on the propagation constant by vary-ing their values but keeping all other parameters constant. Nominal parametervalues are a = 0 . mm, b = 1 . mm, ε = 1 , ε = 3 , h = 0 . mm and fre-quency f = 10 GHz (cf. Fig 1). Our results are validated against finite elementnumerical solutions obtained with COMSOL Multiphysics R (cid:13) [28]. Details onthe simulation are given in the methods section. Figure 2 shows the results ofthe parameter sweeps. It can be seen from the figure that the effective refrac-tive index and in turn the propagation constant crucially depends on all input7arameters.For instance, the height of the wire above the substrate influences how muchelectromagnetic energy can travel inside the dielectric substrate. In general,if the conductor is further away from the substrate, less energy travels in thedielectric. This means that the propagating mode has an effective refractiveindex closer to that of the surrounding medium. Consequently, as the conductorapproaches the substrate, the effective refractive index increases as shown in Fig.2. Good agreement between our model and results obtained with Comsol canbe seen. However, at small heights Comsol produces an effective index which isslightly higher than predicted by our method.In fact, the results obtained with Comsol in Figure 2 seem to systematicallylie above the results of our model. On the one hand, this deviation may beexplained by our model neglecting the exact current distribution in the wireleading to errors such as those predicted in Equation (14) which were on theorder of 1-3% throughout the sweep. On the other hand, as the height becomesvery small, Comsol is forced to use elongated mesh elements between the wireand the substrate which is generally not recommended.The effect of frequency on the PGL mode is shown in the second plot of Fig.2. At high frequencies the wave localises close to the conductor similar to theclassical Sommerfeld or Goubau line. This leads to an increase in n eff whichensures a fast transverse decay in the surrounding air. Additionally, one canthink of increasing the frequency as localising more of the wave energy inside thesubstrate which slows down the wave. Thus, the PGL is generally dispersive.At very low frequencies, the wire cannot be approximated as a PEC any moreas the skin depth becomes similar to the wire radius. Hence, only values of therefractive index above 1 GHz are reported. Results obtained with our modeland Comsol are within a few percent and both show the same general trendalthough Comsol predicts a slightly higher mode index.Increasing the dielectric constant of the substrate slows the wave down. Thiseffect looks to be nearly linear in the magnitude of the effective dielectric con-stant. Agreement between the model and Comsol is good at small permittivitiesbut the results deviate increasingly with the dielectric constant of the substrate.Finally, varying the substrate thickness influences the amount of energy thattravels inside the substrate. Increasing the thickness slows the wave down lead-ing to a higher effective refractive index. Figure 2 also includes the effectiverefractive index of the TE substrate mode as Gacemi et al. reported that withincreasing substrate thickness the PGL mode mixes and ultimately merges intothis mode [19]. Indeed the Comsol result aligns nicely with the TE mode andpast a thickness of 7.5 mm only the substrate mode was detected in Comsol. Onthe other hand, our model produces results beyond this thickness. However, anoticeable change in the behaviour n eff with the substrate thickness is observedafter intersecting with the n eff of the substrate mode at around 6 mm. In fact,we were not able to obtain physical field patterns for values of the refractiveindex after this point. Hence, we believe that while results for larger substratethicknesses can be calculated they do not hold any physical relevance.Due to the dependence on geometrical parameters, the planar Goubau linecan be designed to have a high or low effective refractive index signifying astrongly or weakly confined mode respectively. Clearly, a mode which is morelocalised near the substrate will experience increased dielectric loss. Hence, atrade-off between loss and field extent will need to be made. This can to some8 - - -
20 0 20 40 - - [ mm ] y [ mm ] Log | E z | - - - - - - - - - - - Figure 3:
Top Left:
Plot of E z obtained through Comsol. Top Right:
Plot of E z calculated with the presented analytic model. Excellent agreement betweenthe two field profiles is observed. Bottom:
Contour plot of log( | E z | ) . The fieldis elongated along the substrate and exponentially decays away from the wire.degree be mitigated by using low-loss substrates for instance quartz or plastics. Once the characteristic equation has been solved, equation (7) may be computedat every point in space to calculate the distribution of the z-component of theelectric field. Plots of the resulting field are shown in Fig 3 using the parametersgiven in the previous section. The top two plots show the longitudinal electricfield calculated with Comsol and with our model. Both plots are in excellentagreement with each other. It is interesting to observe that the field changes signbetween opposite sides of the dielectric. This behaviour was also found in thesimulations presented by Horestani et al. for the PGL but was not discussed [16].We emphasise that the sign change is unique to the PGL and not found for theclassic Goubau line where the mode is cylindrically symmetric. This shows thatdespite some similarities between the PGL and the classic Goubau line such asan exponential decay at large distances, the presence of a single sided substratesubstantially alters the Goubau mode. It breaks the cylindrical symmetry ofthe system which in turn means no pure TM mode can propagate. As a resultonly a hybrid mode exists on the PGL.The logarithmic contour plot at the bottom of Figure 3 shows that E z decaysexponentially away from the wire at distances much greater than the wire radius.9 Frequency [GHz] n e ff MeasurementTheory
Figure 4: Measured and theoretical effective refractive index of the PGL modeover a FR4 substrate. Error bars in the theory correspond to the uncertaintydue to first order azimuthal currents as given in eq. (14).This can also be shown directly from our expression for E z in equation (21).There, we can neglect the integral for | γ y | (cid:29) due to the exponential damping.As the incomplete Gamma function is small for large argument, the field isdominated by the modified Bessel functions which have an exponential decayfor large, real argument. Thus, the field drops off exponentially with a decayconstant | γ | . This profile is consistent with the ones reported for planar Goubaulines in Refs. [13, 16, 29]. To validate our results experimentally, we measured the effective refractive indexof the PGL mode as a function of frequency. This is achieved with a simplesetup. Using scaled versions of the planar launchers discussed by Akalin etal. , we excite a Sommerfeld surface wave on a single copper wire [1, 10]. S-Parameters are obtained with a Vector Network Analyser (VNA). Then, weintroduce a dielectric slab of finite length l into the path and suspend the wireonto it using tape such that the dielectric represents the substrate discussedin our model. In transmission, the substrate will cause a phase delay ∆ ϕ dueto the increased refractive index of the now propagating PGL mode relative tothe Sommerfeld mode. The delay can be measured with the VNA. Under theassumption that the Sommerfeld wave travels approximately at the speed oflight, the phase delay is given by ∆ ϕ = k l (cid:0) n eff − (cid:1) (23)This is now easily solved to give the effective refractive index of the PGL mode.Note that as we only measure the difference in phase that is introduced by thesubstrate, this measurement is independent of SW launching as long as a SWis propagating.The measurement results are shown in Figure 4 together with theoreticalpredicted values. Error bars were added to the theoretical values according to10quation (14) estimating the higher order azimuthal current effects. Excellentagreement between theory and measurement is observed. Above 20 GHz theeffect of azimuthal currents increases drastically reducing the accuracy of thepresented theory. Error estimates for the measurement were omitted in thefigure for clarity as they were significantly smaller than the theoretical ones. The presented simplified system has many similarities with the standard planarGoubau line such as the reported scaling behaviours with geometrical param-eters and frequency. One difference is that in a real system, conductor anddielectric will be lossy leading to a complex effective refractive index. However,the main differences are that the conductor of a standard planar Goubau linehas a rectangular cross section and the substrate is of finite width. In manycases the width of the substrate can be neglected as the electromagnetic fieldsof the bound Goubau mode decay exponentially. Thus, the edges do not have astrong influence on the field pattern. However, the shape of the conductor hasbeen shown to strongly influence the effective refractive index [12].It is beyond the scope of this work to derive a complete theory to incorporatethese effects. However, we will try to give some qualitative arguments to describethe observed trends within our framework. If we consider a conductor of finitethickness but variable width, then most of the electric fields and currents willbe localised near the edges due to the lightning rod effect. Hence, a naturalmodel is two parallel wires on the substrate which are located at the edges ofthe rectangular conductor carrying coupled surface waves. This type of couplinghas recently been studied for Sommerfeld wires and it was shown to reduce theeffective refractive index [30]. This behaviour is consistent with the observationsin Ref. [12].
This paper presents a theoretical investigation of a simplified Goubau line con-sisting of a cylindrical wire above an infinitely wide substrate. To this end,the electromagnetic field of an infinitesimal current filament above a substratehas been derived in Fourier space. By incorporating the finite width of a re-alistic wire, an approximate characteristic equation was derived. Estimates tothe applicability of this equation were given before exploring the influence ofgeometrical parameters and frequency on the wave characteristics. It was foundthat, depending on the operating frequency and setup, both weakly and stronglyconfined surface waves can propagate allowing one to tune the geometry depend-ing on the desired application. Furthermore, the field profile for a particular setof parameters was calculated. All results agreed well with numerical simulationsand previously reported experimental studies. Finally, the derived model wasexperimentally tested and excellent agreement between theory and measurementwas found.This work is one of the first attempts at analytically modelling the planarGoubau line. Although only a simplified version was discussed, the behaviourwith frequency and other parameters was found to agree with the previous11eports for a standard planar Goubau line with rectangular conductor. Hence,it may be considered as an important step towards a better understanding of thePGL. The limitations of the presented model with respect to a realistic planarGoubau line with rectangular conductor were discussed in the last section andsome qualitative arguments were made to incorporate the effect of the conductorgeometry. In conclusion, we expect these insights to help better understandand utilise planar Goubau lines in printed circuit board designs for challengingapplications such as terahertz spectroscopy or high frequency circuitry.
Methods
In our experimental setup a 15 cm long, 1.6 mm thick, dielectricslab made from FR4 epoxy in conjunction with a 0.5 mm annealed copper wirewere used. The relative permittivity of the substrate is given as 4.55 by themanufacturer. S-parameters were measured by a 8722D VNA from AgilentTechnologies.
Author Contributions
TS conceived the theoretical model and any calcu-lation based on it. Furthermore, he carried out the experiment. DM was re-sponsible for numerical modelling and simulation with Comsol. Additionally,DM and TS jointly interpreted the resulting data and drafted the manuscript.AM and MP crucially revised the manuscript and supervised the research. Allauthors approve the current version of the manuscript.
Competing Interests
The authors declare no competing interests.
Acknowledgements
This work was supported by the Royal Society GrantsIF170002 and INF-PHD-180021. Additional funds were provided by BT plc andHuawei Technologies Co., Ltd. The authors thank the Royal Society, BT andHuawei for these funds.
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