A current-matrix model for metallic and dielectric postwall waveguides

Authors


Abstract

[1] Waveguide structure integration in planar substrates for use in microwave components has received considerable attention in recent years. Waveguides with side walls consisting of cylindrical posts (postwall waveguides or PWWGs) are of interest, since they are compatible with standard PCB fabrication technology and exhibit low loss. In this paper we present an electromagnetic model for PWWG building blocks, whose characteristics are described entirely in terms of equivalent electric and magnetic surface currents at predefined port interfaces consistent with Lorentz's reciprocity theorem. Introducing input and output surface currents, we determine the response of a block for a given port excitation. The expansion of the currents in terms of suitable bases results in a matrix that relates input and output currents. The scattering parameters of a building block are determined by expressing waveguide modes in terms of these bases. This facilitates the future integration of PWWG components in a microwave circuit simulator. We validate our model by comparing the results for simulated and measured uniform PWWGs implemented with metallic and dielectric posts.

1. Introduction

[2] In the nineties, postwall waveguides (PWWGs) were proposed as an alternative to classical transmission lines at microwave frequencies [Furuyama, 1994; Hirokawa and Ando, 1998; Uchimura et al., 1998; Deslandes and Wu, 2001]. Such waveguides are generally embedded in substrates, e.g., printed circuit boards (PCBs), and belong to the broader class of substrate-integrated waveguides (SIWs). The side walls of a PWWG are composed of cylindrical posts, which together with optional metallic top and bottom plates enclose a rectangular region similar to the rectangular (metallic) waveguide. Consistent with classical planar transmission-line structures, PWWGs are relatively easy and inexpensive to manufacture, offer integration with other circuits of a microwave front-end, and are less bulky than classical (metallic) waveguides. Furthermore, thanks to their waveguide-like characteristics PWWGs have in general lower losses than other PCB transmission-line structures, in particular above 30 GHz [Coenen, 2010, section 1.2]. For these reasons PWWGs have attracted substantial interest during the last decade.

[3] In our paper we focus particularly on the electromagnetic analysis of PWWGs. Not surprisingly the initial analysis methods found their origin in the optical domain with dielectric-post PWWGs evolving from photonic bandgap materials [Yablonovitch, 1987] and where one utilizes the Bloch-Floquet theorem and the plane wave expansion method to study periodic guiding structures [Benisty, 1996]. Such an approach is adopted in early PWWG work [Hirokawa and Ando, 1998] and by Pissoort and Olyslager [2004]. In the microwave community, however, lumped-element circuit models are most often used in the design of microwave components. Therefore, efforts were undertaken to identify suitable layouts for a circuit model of PWWGs [Bray and Roy, 2003; Bozzi et al., 2003, 2006]. Parallel to this development, early PWWG studies stressed the determination of transmission characteristics, in particular the propagation constant, the loss factor, and the equivalent waveguide width [Coenen, 2010, chapter 3]. Beside the periodic-structure approach, several full-wave approaches were employed [Cassivi et al., 2002; Xu et al., 2003; Deslandes and Wu, 2006; Bozzi et al., 2008] as well as discretization and averaging techniques applied to the permittivity and conductivity [Ranjkesh et al., 2005; Patrovsky and Wu, 2005; Cassivi and Wu, 2005].

[4] More recently, the development of PWWG analysis techniques has more and more conformed to the practice of microwave component design, in which large structures are decomposed into smaller building blocks whose separately computed scattering parameters at predefined ports are cascaded to obtain the overall performance. The BI-RME method [Cassivi et al., 2002; Bozzi et al., 2008] offers such a procedure based on Generalized Admittance Matrices that relate modal currents and voltages at the PWWG ports. Outside these ports, the structure is laterally enclosed by fictitious metal walls, thus assuming negligible radiation loss. The approach described by Arnieri and Amendola [2008] and Arnieri et al. [2009] also considers the responses at designated ports, which include coaxial ports, waveguide ports, and slots. Instead of enclosing the PWWG by fictitious walls, the admittance matrix corresponding to equivalent magnetic current sources is calculated by expanding the Green's function of the parallel-plate waveguide in cylindrical waves and by adding separately the contribution of the posts to the corresponding integral equation. Finally using the approach described by Van de Water et al. [2005] the equivalent sources describing a specific building block are transferred to its entire boundary by linear embedding of the related Green's operator. The blocks are cascaded in a 2-D or 3-D lattice to obtain the field distribution of the overall structure.

[5] Following Coenen [2010] and Coenen et al. [2010], we present in this paper a model for PWWG building blocks that is based on the same principles as those of Van de Water et al. [2005], in particular Lorentz's reciprocity theorem. However we characterize the PWWG building block entirely in terms of electric and magnetic input and output surface currents at predefined port interfaces, as in the BI-RME and Arnieri and Amendola [2008]. Hereby we wish to facilitate the future integration of PWWG components in an existing circuit simulator. In section 2 we describe the details of our model, in particular how we relate the input and output currents using a current matrix, and in section 3 we derive expressions for the scattering parameters of the dominant TE10 mode at the ports. To validate our model, we compare in section 4 the propagation characteristics obtained by our model with those obtained by a commercial tool (Ansoft HFSS, version 11.1) and those obtained from measurements of uniform PWWGs of both metallic and dielectric posts. Finally, in section 5, we present the conclusions of our work.

2. An Electromagnetic Model for PWWG Building Blocks

[6] Figure 1 shows a section of a uniform PWWG with a single row of posts per sidewall, two parallel port interfaces, and the global coordinate system {ix, iy, iz} used in our analysis. Taking this example as a reference for our model, we assume that the dielectric or metallic posts and the surrounding dielectric are linear homogeneous media. Moreover, we assume the time harmonicity ejωt of the electromagnetic field and denote the spatial electric and magnetic fields by E and H. For application as transmission lines, PWWGs and classical rectangular waveguides are generally designed to have unimodal behavior and thus their heights are much smaller than their widths and the wavelength. Consequently, the electromagnetic field is uniform in the height (or y) direction. In a classical rectangular waveguide, only the first few TEm0 and TMm0 modes can propagate; the other TE and TM modes are significantly attenuated. Decomposing the field in a PWWG in TE and TM constituents, we obtain the following TE conditions: (1) The field does not depend on the y coordinate, (2) Ex vanishes at the top and bottom plates, provided that they are perfectly conducting, and (3) Ez = 0. The consequence of conditions 1 and 2 is that Ex = 0 and, hence, E = Ey(xt)iy, where xt = xix + ziz. The magnetic field is thus in the transverse plane and wave propagation can only occur in the (x, z) plane, not along the posts. For the TM constituent, condition 3 is Hz = 0 and condition 2 applies to both Ex and Ez. Consequently, Ex = Ez = 0 and Ey depends only on z. Therefore no independent TM solutions are obtained and, additionally, the boundary condition Ey = 0 cannot be satisfied at the posts. The three listed conditions are therefore imposed on the total field in a PWWG, limiting our focus to modal behavior similar to that of the TEm0 modes in a rectangular waveguide.

Figure 1.

Top view of a section of a uniform PWWG with a single row of posts per sidewall and the notation used to specify the port interfaces.

[7] The port interfaces of a PWWG building block are segments indicated by port(i), i = 1, 2, with widths dport(i). We assume that the radiation leakage is such that we can enclose the building block by a contour including these interfaces and that the tangential fields at this contour vanish except at these interfaces. Exciting the building block by a field incident on its port interfaces, we describe this field in terms of electric and magnetic input surface currents {Jin(i), Min(i)} at those interfaces as a consequence of Lorentz's Reciprocity Theorem. Given that the input currents induce a scattered field in the building block, we calculate the induced output surface currents {Jout(i), Mout(i)} as the sum of this scattered field and the direct radiation from port interface to port interface. To this end we employ integral expressions derived from the same theorem.

2.1. Integral Expressions Derived From Lorentz's Reciprocity Theorem

[8] We consider an excitation field {Eexc, Hexc} that is uniform in the y coordinate as above. This field is incident on an object that is also uniform in y and is described by the area Ω in the xz plane with boundary ∂Ω, as depicted in Figure 2. This field induces a field {Eint, Hint} in Ω and a scattered field {Esct, Hsct} outside Ω, i.e., its complement equation image. Let {equation image, equation image} be the field induced by the point source equation image = δ(equation imagetxt)i in equation imaget, where i is a unit vector. We first apply Lorentz's Theorem to the excitation field in Ω and the field of the point source, and subsequently to the scattered field in equation image and the field of the point source. Subtracting the two results we obtain in the absence of volume sources,

equation image

with surface currents Jsurf and Msurf,

equation image
equation image

Here the restriction ∣∂Ω to ∂Ω is applied from the outside of the boundary. We note that in the application of Lorentz's Theorem to the excitation field inside Ω, the restriction is applied from the inside to ∂Ω. Since {Eexc, Hexc} exists in an environment without Ω present, the field is continuous across ∂Ω and can be restricted from either side. Applying Lorentz's Theorem to the interior field {Eint, Hint} in Ω and the field of the point source, we obtain the same expression as (1), but with the terms in the right-hand side replaced by −i · Eint(equation imaget) and 0, respectively. Here the total tangential field is assumed continuous across ∂Ω, whereby Jsurf and Msurf equal nt × Hint∂Ω and nt × Eint∂Ω, respectively. This expression can be viewed as the interior equivalent state in Love's equivalence principle, where the field in Ω is entirely described by its tangential components on ∂Ω while the field outside Ω is zero.

Figure 2.

Graphical representation of the area Ω in two dimensions.

2.2. Application to PWWG Building Blocks

[9] Applying the above to a PWWG building block, we define ∂Ω as a contour that encloses the block, that includes the port interfaces, and that satisfies the preceding leakage assumption. The input and output surface currents {Jin(i), Min(i)} and {Jout(i), Mout(i)} are the cross products of the interface normals nt(i) and the fields in equation image and Ω, respectively, restricted to the port interfaces. From our field assumptions we have M = Mxix and J = Jyiy.

[10] The fields {Ein(i), Hin(i)} in Ω generated by the input currents {Jin(i), Min(i)} at port interface i can be calculated from the case Ω \ ∂Ω in (1). The field equation image = equation imageyiy is the fundamental solution to the Helmholtz equation in two dimensions for a source located in equation imagetequation image Ω, i.e., Δequation imagey + k12equation imagey = jωμ0δ(equation imagetxt). Consequently,

equation image

and by Maxwell's equation curlxequation image = −jk1ζ1equation image,

equation image

where ζ1 = equation image. Substituting these expressions in (1) and employing the parameter representation xport(i)(s) = cport(i) + ixsdport(i)/2, we obtain

equation image

for xt ∈ Ω, where Θ = ∣xtcport(i)ixsdport(i)/2∣. The corresponding magnetic field follows as in (4),

equation image

To calculate the output surface currents we need to restrict the input field in Ω to the port interfaces. To this end, we calculate the z derivative in (5) by

equation image

and we calculate the derivatives of the first curl in (6) in the same way after interchanging the derivatives and the integral. After a similar interchange in the second curl, we rewrite the second derivative with respect to z by substituting ΔxH0(2)(k1Θ) = −k12H0(2)(k1Θ) for xt ∈ Ω. Then,

equation image

In the second integral we replace ∂/∂x by ∂/∂s′ with factor −2/dport(i) and we apply integration by parts with Mxin(i)(±1) = 0. Substituting the results for the first and the second curl in (6) we obtain

equation image

Having obtained expressions for the electric and magnetic input fields, we focus on the next step of our analysis, which concerns the scattering of the input fields at the posts. To calculate the scattered field, we consider the description of the scattering at a set of posts for a given excitation field, where we closely follow Elsherbeni and Kishk [1992].

2.3. Electromagnetic Scattering by a Set of Posts

[11] For lossless posts, Ey is governed by the homogeneous 2-D Helmholtz equation and the boundary condition Ey = 0 at the post surfaces. We describe these surfaces in the xz plane by ℓp: xt(equation image) = cq + aqir(ϕ), where ir(ϕ) = cosϕiz + sinϕix and q = 1,…,Q with Q the number of posts. Applying separation of variables we obtain two Sturm-Liouville type differential equations from which the solutions for Ey follow: Jn(kr)ejnϕ and Yn(kr)ejnϕ, nequation image.

[12] Each post in Figure 1 can be viewed as the domain Ω in Figure 2, where we identify media 1 and 2 in Figure 1 with equation image and Ω in Figure 2. Let {Eqsct, Hqsct} be the field generated by post q and assume the same field composition as in section 2.1. Then, expansions of Eq,yexc, Eq,ysct, and Eq,yint are straightforwardly derived as expansions of the solutions of the Helmholtz equation. Given that the excitation and interior fields must be bounded at r = 0 and that the scattered-field expression must represent outgoing waves for r → ∞, we arrive at

equation image
equation image

The expansion of Eq,yint is given by (10a) with Bq,nexc and k1 replaced by Bq,nint and k2. The total field satisfies at each ℓp the boundary conditions

equation image
equation image

If the posts are perfectly conducting, Eyint = 0 and Hϕint should be replaced by Jy. To calculate the mutual coupling between posts we express the tangential electric field at post p generated by post q in the polar coordinates of post p. Analogously to Elsherbeni and Kishk [1992], we obtain for pq

equation image

applying Graf's theorem, where ϕpq is the angle between cqcp and iz. To evaluate Hqsct at ℓp we replace ap by r in (12), obtaining an expression for Eq,ysct expressed in (r, ϕ) coordinates related to ℓp. Substituting this expression in jωμHϕ = ∂Ey/∂r and evaluating the result at r = ap we obtain for pq

equation image

[13] For metallic posts, Bq,nint = 0 since Eyint = 0. The coefficients Aq,n are then determined from the boundary condition for the electric field. Substituting (10a), (10b), and (12) in (11a) and invoking the orthogonality of {ejmequation image}equation image on [−π, π] we obtain

equation image

where Bpexec = (…, Bp,−1exc, Bp,0exc, Bp,1exc,…)T, Ap = (…, Ap,−1, Ap,0, Ap,1,…)T, and

equation image

The matrices Cpq have infinite size and are in practice truncated by taking m, n = −N,…,N. For dielectric posts we apply both boundary conditions (11) to arrive at the same matrix equation, but with the factor of δmn in the case p = q replaced by

equation image

2.4. Calculating the Output Surface Currents

[14] To calculate the scattered field induced by excitation at port interface i1, we apply the preceding analysis to equation image. The electric field of post q induced by this field follows from (10b),

equation image

where (r, ϕ) corresponds to xt by xt = cq + rir(ϕ). The coefficients equation image are obtained from (14), where

equation image

and

equation image

These coefficient expressions follow from the expansion (10a) of equation image, where equation image is given by (5), with z derivative (7). The ϕ and r components of the scattered magnetic field are found by applying Maxwell's equations to (17),

equation image
equation image

The corresponding output surface currents at the port interfaces i = 1, 2 are given by

equation image
equation image

where the restrictions are applied from the inside of Ω to the port interfaces, χ(i) is defined by χ(1) = −1 and χ(2) = 1 such that nt(i) = χ(i)iz, and equation image and equation image are the sums of the scattered fields of the posts for which

equation image
equation image

For a specific value of s, with xt = cport(i) + ixsdport(i)/2 on port interface i, we evaluate the output currents as follows. For the scattered fields in (22a) and (22b), we determine rq(i)(s) and ϕq(i)(s) of xt in the parameter representation of post q. The restriction of equation image to port interface i at s follows then from (17) and the restriction of equation image follows from equation image, (20a), and (20b). For the electric input field in (21b) we obtain

equation image

where we employ (5), with z derivative (7), and

equation image

For the magnetic input field in (21a) we obtain

equation image

from (9), where we replace ∂/∂x by d/ds with factor −2/dport(i). Having calculated the cross products (22), (23), and (25) we deal with the output currents (21) induced by port interface i1. Choosing i1 = 1, 2, we obtain the output currents induced by the input surface currents of both port interfaces.

2.5. Current Matrices

[15] In our calculational procedure we expand the currents in terms of triangle functions,

equation image

where ν = in, out, i = 1, 2, and where

equation image

Λ(s) = (1 − ∣s∣)1[−1,1](s), Δexp(i) = 2/(Nexp(i) + 1), sn(i) = −1 + nΔexp(i), n = 1, 2,…, Nexp(i)). The coefficient vector (Delout(1), Dmagout(1), Delout(2), Dmagout(2))T of the output surface currents is related to the coefficient vector Delin(1), Dmagin(1), Delin(2), Dmagin(2))T of the input surface currents by the current matrix, which is a 2 × 2 block matrix with blocks

equation image

Each block Tequation image with u,v = mag, el represents the coupling between port interfaces i1 and i, where interface i1 is excited. It follows straightforwardly that Tv(i),u(i1) = (G(i))−1equation imageequation image, where G(i) is the Gram matrix of the triangle functions at port interface i and equation imageequation image is the matrix of the inner products of these functions and the output currents of type v at port i. Each of these currents is generated by an input current of type u at port i1 represented by a triangle function at that port. An output current is determined by substituting the triangle function at port i1 in (23) or (25) and by evaluating the expression (21a) or (21b) by the procedure described in section 2.4. Subsequently, the corresponding components of equation imageequation image are determined by taking inner products of this output current and the triangle functions at port i. For each triangle function, the resulting expression incorporates single integrals that represent the interactions between this triangle function and the scattered fields of the posts in (22a) or (22b), and double integrals that represent the interactions between this triangle function and the triangle function of the input current at port i through (23) or (25). We compute the single (regular) integrals by a composite Simpson rule. Moreover, we rewrite the double integrals as single integrals that involve convolutions of the triangle functions and their derivatives. These convolutions are continuous and piecewise differentiable functions that can be evaluated analytically. If port i and port i1 concern the same port, the single integrals have a logarithmic singularity that results from the Hankel kernels. Extracting the singularity and calculating the corresponding (singular) integral analytically, we compute the remaining regular integral numerically. For details of the described calculation procedure, see Coenen [2010, Appendix D].

3. Scattering Parameters

[16] The current matrices introduced in the previous section are adequate for characterizing the electromagnetic coupling between building blocks through their ports. In order to connect PWWG structures to an external system, we require the wave scattering parameters. To determine these parameters, we couple the building-block ports to rectangular waveguides of width wg(i) described only by the TE10 mode. Thus we assume that the frequency is such that higher order modes are evanescent and that we are sufficiently distant from discontinuities in the PWWG to ensure negligible contribution from these modes. For each of the two ports, we specify the electric and magnetic input (surface) currents Jin(i) and Min(i) by taking the tangential components of the magnetic and electric fields, respectively, of the TE10 mode at the port interfaces. Moreover, following Oseen's extinction theorem, we define Jin(i) = 0 and Min(i) as twice the corresponding tangential component, such that the fields for z < 0 are extinguished and the fields for z > 0 represent the TE10 mode traveling in the positive z direction. Consequently, we only require the tangential electric component of the TE10 mode at port(i). This component is given by

equation image

where f(i)(s) is the modal field distribution

equation image

evaluated at z = 0 in a rectangular waveguide with width wg(i) that is centered at port (i). Taking the −χ(i)z direction as the direction of propagation, Min(i) is given by

equation image

The corresponding expansion coefficients Dmag,nin,(i) are calculated by projecting Min(i) on the basis of triangle functions. Then,

equation image

where the inner products Wm(i) = 〈Λm(i), f(i)[−1,1] and the Gram matrix G(i) can be calculated analytically [see Coenen, 2010, section 3.5].

[17] Let us consider the case that only port(i1) is excited. Then, the generated y component of the electric field at port(i) is equation image and the scattering parameter equation image is the ratio between the projection of equation image on the TE10 mode and the projection of the TE10 mode onto itself,

equation image

Substituting (29) and (31) in this expression and requiring equal phase and power of the modes at both ports, we obtain

equation image

where equation imageequation image are the normalized expansions coefficients.

4. Uniform Postwall Waveguides

[18] To validate our model, we consider in this section three cases of PWWG transmission lines with uniform postspacing: (1) metallic posts, low-permittivity substrate, (2) metallic posts, high-permittivity substrate, and (3) dielectric posts, high-permittivity substrate. The PWWGs have been designed and manufactured for operation around 10 GHz with an operational bandwidth similar to that of standard WR 90 waveguide, and such that losses due to radiation are small, see Table 1. We first discuss the general aspects that apply to all three cases, followed by a discussion of the specific aspects and results of each separate case. For each of the three cases we determine the propagation constant from both measured and simulated data using multiline calibration [Marks, 1991]. To this end, we design for each case a set of seven uniform PWWG transmission lines of different lengths. Next, we evaluate the corresponding propagation constant by an eigenvalue analysis and a least squares fit applied to the (measured or simulated) scattering parameters of pairs of transmission lines in the set. Our choice of line lengths, as presented below per case, is driven by the calibration error, which can be reduced by enlarging the line-length differences of the transmission line pairs. Moreover, the shortest line must be long enough to attenuate the evanescent modes sufficiently ensuring that the TE10 mode is dominant. The scattering parameters of the simulated lines are determined as described in the previous sections by defining port interfaces at the ends of the lines. We assumed perfectly conducting metals for all simulations. For the measurements we have connected the PWWG transmission lines to transition structures that connect via coaxial cable to a vector network analyzer. The network analyzer has been calibrated using a thru-reflect-line (TRL) calibration where the reference planes are at the connectors between the coaxial cable and the PWWG circuit board. For further details on the general manufacturing and measurement aspects, see Coenen [2010, chapter 4, Appendix E].

Table 1. Specifications for the Three Considered PWWG Configurations
ParameterSymbolValueUnit
Case 1Case 2Case 3
Relative permittivity of PCBɛr,13.559.809.80
Relative permittivity of postsɛr,21.00
Post radiusa0.550.651.50mm
Post spacingdz2.02.573.30mm
Interwall post spacingdx4.09mm
Number of rows per sidewallL113
Waveguide widthwg12.638.918.91mm
Waveguide heighthg1.5243.813.81mm
Effective waveguide widthwg, eff12.148.05mm
Cutoff frequencyfco6.555.95GHz
First stop band frequencyfstop39.7818.6314.51GHz
Loss tangent of PCBtan δ10.00270.00200.0020

4.1. Case 1

[19] The PWWGs of case 1 have been manufactured in a single layer of Rogers RO4003C circuit board. The PWWG excitation structure covers the full bandwidth of the TE10 mode and consists of a grounded coplanar waveguide that excites a slot in the top ground plane of the PWWG. The differences in length between the shortest PWWG transmission line and the other lines are 1, 3, 5, 9, 23 and 45 unit cells. For the simulations with our method we use 3 expansion coefficients per post, i.e., setting N = 1 in (17). Figure 3 shows the dispersion and attenuation for the measured and simulated PWWGs. These quantities relate linearly to the real and imaginary parts of the complex propagation constant kp of the PWWG. The dispersion is Re{kp}/k0 and the attenuation (in dB/m) is 20 Im{kp}/ln 10. The dispersion curves in Figure 3a resulting from simulations with HFSS and our method almost coincide and the measured dispersion curve is within a few percent of the simulated curves. The simulated attenuation curves in Figure 3b also match well but the measured attenuation is slightly higher. We calculate the relative differences in the real and imaginary parts of the propagation constants determined from our simulation results and those determined from the HFSS results and the measurements. Except close to the cutoff frequency, these differences are fairly constant in the band of operation. Therefore, we present them by single numbers, which are listed per case in Table 2 normalized to the propagation constant k0 in free space. For case 1, we observe that for the simulation results the relative differences of both the real and the imaginary parts are small. The relative difference related to the real part of the measurements is higher (10−2) than for the simulations (10−4), which is explained by the slight frequency shift of the measured dispersion curve in Figure 3a. We attribute this shift to material (for the dielectric constant Δɛrr ≈ 10−2) and manufacturing tolerances (for the waveguide width Δ(2a)/wg ≈ 10−2) that are of the same order as the relative difference with the measurements. For the imaginary part, the order 10−1 corresponds to 1 dB of difference in the attenuation around 10 GHz. This is of the same order as the difference observed in Figure 3b between simulations and measurements. We attribute the difference to the copper losses that are present in the measured samples but were not taken into account in the simulations.

Figure 3.

Simulated and measured (a) dispersion and (b) attenuation as a function of frequency for the PWWG of case 1.

Table 2. The Orders of the Relative Differences in the Real and Imaginary Parts of the Normalized Propagation Constanta
CaseHFSSMeasurement
RealImaginaryRealImaginary
  • a

    For the real part, the difference is ∣Re{kpkp,ref}∣/Re{kp,ref} with kp the propagation constant determined by HFSS or by measurements and with kp,ref the propagation constant determined by our method. For the imaginary part, Re is replaced by Im.

110−410−210−210−1
210−310−210−2>1
310−310−110−2>1

4.2. Case 2

[20] The PWWGs of case 2 have been manufactured in two layers of Rogers TMM10i circuit board, where the top layer holds a microstrip line that couples to the PWWG through a resonant slot in a common ground plane. The PWWG excitation structure covers the bandwidth of the TE10 mode only partially and therefore three transitions are required to obtain results for the full bandwidth of the TE10 mode. The differences in length between the shortest PWWG transmission line and the other lines in the manufactured set are 1, 2, 4, 6, 8 and 29 unit cells. For the simulations we use the same differences in number of unit cells and we use 3 expansion coefficients per post in our method, i.e., setting N = 1 in (17). The spread in these differences is less than in case 1, which can reduce the accuracy of the multiline calibration. Figure 4 shows the simulated and measured dispersion and attenuation, where the measured dispersion plot is composed of the results for the three microstrip to PWWG transitions that together cover the TE10 bandwidth. We observe a good match between the measured and simulated dispersion and between the attenuation simulated by our method and by HFSS. We have not plotted the measured attenuation, since its measurement noise is of a higher order than the attenuation itself. This is illustrated by the relative differences in Table 2. For case 2, we observe that the relative differences related to the HFSS simulations are comparable to those in case 1, but the relative differences related to the measurements are one order higher than in case 1. The relative difference in the imaginary part of the normalized measured propagation constant is larger than 1 which indicates that the attenuation is below the measurement floor. The large error in the measurement results is mainly caused by the coaxial-to-microstrip connectors utilized in the measurement setup. These connectors matched badly and had a poor repeatability.

Figure 4.

(a) Simulated and measured dispersion and (b) the simulated attenuation as a function of frequency for the PWWG of case 2.

4.3. Case 3

[21] The PWWGs of case 3 have been realized in the same circuit board as the PWWGs of case 2. Instead of one row of metallic posts per side wall as in cases 1 and 2, the PWWGs of case 3 have three rows of dielectric posts per side wall to achieve sufficient isolation. The same excitation structure has been applied as in case 2, but an extra tapered PWWG transition with metallic posts has been inserted that matches the dominant mode of the PWWG of case 2 (metallic posts) to the dominant mode in the PWWG of case 3 (dielectric posts). The length differences in unit cells between the shortest PWWG transmission line and the other lines in the manufactured set are the same as in case 2. For the simulations with our method we use 5 expansion coefficients per post, i.e., setting N = 2 in (17). Figure 5 shows the simulated and measured dispersion and attenuation. For the same reasons discussed in case 2, we have not plotted the measured attenuation. We observe that the dispersion curves obtained by HFSS and our method coincide and that the measurement results deviate a few percent. The differences between the simulated attenuation curves are larger than in case 2, especially for the lowest frequencies. An explanation for this effect might be lower reflectivity of the side walls for low frequencies, which results in more power being scattered back from the boundaries of the substrate in the HFSS simulations. In our method the background medium is homogeneous and such scattering does not occur.

Figure 5.

(a) Simulated and measured dispersion and (b) the simulated attenuation as a function of frequency for the PWWG of case 3.

5. Conclusions

[22] We developed an electromagnetic model starting from Lorentz's reciprocity theorem that enables us to calculate the fields in PWWG building blocks with metallic and dielectric posts. This model facilitates the description of such a block in terms of a current matrix that relates the input and output currents at predefined port interfaces by means of their expansion coefficients with respect to suitable bases. We limited ourselves to parallel port interfaces and employed bases of triangle functions; other interface orientations and bases can be implemented following the same approach. Defining standard waveguide TE10 modes at the port interfaces, we showed that the scattering parameters of a block can be obtained by straightforward vector calculations. Hereby, our current-matrix formulation facilitates the determination of key PWWG characteristics as well as future interfacing with a microwave circuit simulator. To validate our model, we designed and manufactured three different types of uniform PWWG transmission lines, whose propagation constants we determined using multiline calibration applied to their measured and simulated scattering parameters. Comparison of the results obtained by our model and those obtained by Ansoft HFSS and measurements demonstrated an acceptable agreement for practical design.

Acknowledgments

[23] This work was supported by the Dutch Ministry of Economic Affairs within the scope of the WiComm project as part of the Freeband Communication Program.

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