Closed form expressions for the modal dispersion ... - IEEE Xplore

www.ietdl.org Published in IET Microwaves, Antennas & Propagation Received on 13th December 2009 Revised on 12th March 2010 doi: 10.1049/iet-map.2009.0613

In Special Issue on Microwave Metamaterials: Application to Devices, Circuits and Antennas ISSN 1751-8725

Closed form expressions for the modal dispersion equations and for the characteristic impedance of a metamaterial-based gap waveguide A. Polemi1,* S. Maci2 1

Department of Information Engineering, University of Modena, Italy, Via Vignolese 905, 41100 Modena, Italy Department of Information Engineering, University of Siena, Via Roma 56, 53100, Siena, Italy *Department of Chemistry, Drexel University, 3241 Chestnut Street, 19104 Philadelphia, PA, USA E-mail: [email protected], [email protected] 2

Abstract: In a recent sequence of papers, a parallel-plate ridge gap waveguide has been introduced, that consists of a metal ridge in a metamaterial magnetic conductor surface, covered by a metallic plate at a small height above it. The gap waveguide is relatively simple to manufacture, especially at millimetre and sub-millimetre wave frequencies when compared with other solutions. The metamaterial surface is designed to provide a frequency band where parallel-plate modes are in cut-off, thereby allowing for a confined gap wave to propagate along the ridge. In a previous work, the authors have presented an approximate analytical solution for the confined quasi-TEM dominant mode of the ridge gap waveguide, when the surrounding metamaterial surface is in the form of a bed of nails. In this study, the authors investigation continues by providing an analytical expression of the modal dispersion equation of the first higher order ridge mode and of the characteristic impedance of the dominant mode. As in the previous paper, the field problem is divided in three regions, the central region above the ridge and the two surrounding side regions above the nails. Transverse mode-matching applied to a few modes representation in each region, results in a closed form expression of the dispersion equation of the first higher order mode. After summarising the formulation for the dominant quasi-TEM mode, the dispersion equation of the first higher order mode is derived, in order to give a criterion to maximise the unimodal bandwidth. Furthermore, three different closed form expressions of the dominant mode characteristic impedance are derived and compared with approximate expressions already used in literature.

1

Introduction

Recently, the effectiveness of a new gap waveguide technology for millimetre and submillimetre waves has been demonstrated [1, 2]. In this range of frequencies, this solution presents advantages compared to existing technologies like hollow rectangular waveguide (HRW) and microstrip lines. Indeed, HRW manufactured in two parts and joined together, suffers from problems of poor electrical contacts. Furthermore, microstrip lines, as open structures, are susceptible to cross-talk an unintentional couplings. An inverted microstrip line overcomes some of these IET Microw. Antennas Propag., 2010, Vol. 4, Iss. 8, pp. 1073 – 1080 doi: 10.1049/iet-map.2009.0613

disadvantages, but still presents losses in the dielectric at higher frequencies. On the other hand, microstrip lines are affected by losses for increasing frequency, limited power handling capability, and spurious resonances when encapsulated. Other solutions like substrate integrated waveguide (SIW) [3] exhibit undesired losses in the substrate at increasing frequencies. Therefore there is still need to find new technological solutions for waveguides above 30 GHz that have low losses and are cheap to manufacture. The waveguide introduced in [1, 2] and sketched in Fig. 1 is realised as a narrow gap between parallel metal plates. 1073

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Figure 1 Geometry of the ridge gap waveguide embedded in a bed of nails One of the plates presents a periodic texture of metal pins interrupted by a metal ridge; this ridge forms the transmission line along which a quasi-TEM wave can propagate. An experimental demonstration of this local quasi-TEM wave is reported in [4] but for the case of several parallel ridges with grooves in between. The reason why the wave field is confined along the ridge is because the textured surface prevents global parallel plate modes from propagating [5]. The gap waveguides can be realised without dielectrics, thus having advantages w.r.t microstrip lines, and there is no need for any metal connection between the two plates, which is complicated to realise practically at submillimetre frequencies. In addition, the gap waveguide can be designed to be completely packaged, without being affected by spurious cavity mode resonances, thus resulting in an attractive packaging technology. This packaging capability has been described in [6]. The structure is enclosed in a metallic box whose walls are distant form the ridge and leave at least 4 – 5 row of nails periodicity, namely where the evanescent field is practically negligible. This ensures a behaviour which is really close to the infinite ideal parallel plate configuration at least in the bandgap frequency range of the bed of nails. Therefore the encapsulated structure is not affected by spurious resonances in the same frequency range. Quasi-TEM mode propagation, by using metamaterial surfaces, is not a novelty. It is indeed true that many authors have devised the property of the quasi-TEM mode propagation in rectangular waveguides with PEC top and bottom walls, and metamaterial surfaces lateral walls. These metamaterial walls simulate perfectly magnetic surfaces [7] or hard surfaces [8], and they are often used to miniaturise the transverse dimensions [9]. However, the bandwidth of these waveguides result to be quite small. In contrast, in the solution presented here, the quasi-TEM bandwidth can reach an octave bandwidth [4]. A theoretical analysis of the gap waveguide has been investigated in [10, 11]. In particular, in our previous paper [11] the dispersion characteristics of the fundamental quasi-TEM mode has been modelled in analytical form, in order to take the dispersion effects under control and to design the structure in the appropriate frequency band. 1074 & The Institution of Engineering and Technology 2010

To this end, a suitable modal expansion is assumed in the ridge region and in the two lateral bed-of-nails regions; next, a point matching continuity at the interfaces between the three regions is established to obtain an analytical form of the dispersion equation. Results have been compared with full wave simulations, showing an excellent agreement for the first fundamental mode in that region in which the pin surface can be approximated as homogeneous through the method presented in [5]. By inspection of the full wave complete dispersion diagram, it was seen that the upper limit of the operating frequency range was due in some cases to the upper-edge band-gap of the bed of nails, and in some other cases to the cut-off of the first higher order mode supported by the ridge width. In the present paper, the dispersion equation of this mode is introduced in the modal field expansion. In particular, Section 2 shows explicit calculations for this odd mode, and presents a numerical validation through a full wave analysis. This new analysis demonstrates that, in order to maximise the unimodal bandwidth, one should use a ridge width w , w0(d, h), where d is the height of the nails and h is the size of the gap. The function w0(d, h) is defined through a trascendental closed form equation. In addition, a closed form approximation of the waveguide characteristic impedance is derived in Section 3. Since the structure is dispersive, different definitions of the characteristic impedance can be adopted [12]. All of them resort to a small (but non-negligible) correction of the solution for an ideal non dispersive TEM mode. The new analytical features introduced here complete the analytical investigation of the modal solutions for the gap waveguide, allowing for a prediction of the functional operating frequency bandwidth and constituting a useful tool in the design process.

2 Modal fields and dispersion diagrams In [11], we have shown that structure depicted in Fig. 1 supports a modal field distribution that can be approximated by individual modal structures in the bed of nails region (|x| . w/2) and in the ridge region (|x| ≤ w/ 2). These modes are matched by continuity of tangential fields at the top wall. In this process, the bed of nails surface is approximated by a spatially homogeneous dispersive surface as in [5]. In the bed of nail regions, this approximation leads to a pair of TMy and TEy modes propagating along the longitudinal direction (z), attenuating along the lateral direction (x), and stationary along the vertical direction ( y). In the ridge region, a single mode is adopted to represent the field; this mode is obtained by a plane wave bouncing between the two region boundaries placed at |x| ¼ w/2 with angle of incidence close to grazing aspects. In its point-matching continuity, the three-mode representation constitutes an approximation of a global propagating quasi-TEMz mode that exhibits two evanescent tales in the bed of nail regions. For IET Microw. Antennas Propag., 2010, Vol. 4, Iss. 8, pp. 1073 – 1080 doi: 10.1049/iet-map.2009.0613

www.ietdl.org convenience, the analytical form of this global quasi-TEMz mode is summarised hereinafter.

where

2.1 Field structure of the dominant quasi-TEMz mode Inside the gap region (|x| , w/2) one has kx x)e−jkz z Ey = E0 cos(

(1a)

kz cos( kx x)e−jkz z jk

(1b)

Hz = −jE0

kx sin( kx x)e−jkz z jk

(1c)

where E0 is a constant related to the incident power at the input port, and kx is the propagation constant along x such 2 2 2 that k = kx + kz . In the bed of nails region (|x| . w/2) the global mode is represented by the summation of a TMy mode kz g˜ (x, z) cos[k˜ y (y − h)] Hx = jATM 2 k2 − k˜ y

(2a)

a˜ x Hz = −ATM g˜ (x, z) cos[k˜ y (y − h)] 2 k2 − k˜ y

(2b)

j 2 ˜2 k − ky g˜ (x, z) cos[k˜ y (y − h)] k j kz k˜ y Ez = ATM g˜ (x, z) sin[k˜ y (y − h)] k 2 ˜2 k − ky

Ey = −jATM

(2c)

(2d) (2e)

and a TEy mode as kz g˜˜ (x, z) sin[k˜˜ y (y − h)] Ex = jATE 2 k2 − k˜˜ y

a˜˜ x Ez = −ATE g˜˜ (x, z) sin[k˜˜ y (y − h)] 2 k2 − k˜˜ y ˜ 1 a˜˜ x k˜ y ˜ Hx = −jATE g˜ (x, z) cos[k˜˜ y (y − h)] jk 2 k2 − k˜˜ y 1 Hy = −jATE jk

2 k2 − k˜˜ y g˜˜ (x, z) sin[k˜˜ y (y − h)]

˜ 1 kz k˜ y Hz = ATE g˜˜ (x, z) cos[k˜˜ y (y − h)] jk 2 k2 − k˜˜ y

(4a)

˜ g˜˜ (x, z) = e−jkz z e−a˜ x (|x|−w/2)

(4b)

For the above equations, the dispersion relations k2z = k2 −

Hx = −E0

j a˜ x k˜ y Ex = −jATM g˜ (x, z) sin[k˜ y (y − h)] k 2 ˜2 k − ky

g˜ (x, z) = e−jkz z e−a˜ x (|x|−w/2)

(3a)

2 2 2 k˜ y + a˜ 2x and k2z = k2 − k˜˜ y + a˜˜ x hold, where a˜ x and a˜˜ x are TM and TE lateral attenuation constants, respectively. In the modal field expressions, six unknowns appear, such as field coefficients and lateral and longitudinal wavenumbers. By imposing the matching of the z-component of the wave vector as well as a field continuity at the top wall of the different regions (x ¼ +w/2, y ¼ h), the six unknowns are eliminated by solving a linear equation system; thus, leading to the following closed form expression of the dispersion equation that relates kz to k

w 2 ˜2 (k − ky ) k2 − k2z tan k2 − k2z 2 2 2 2 ˜2 2 kz ky − k k2z − k2 + k˜ y k2z − k2 + k˜˜ y + =0 2 ˜ 2 2 kz − k + k˜ y

The above equation can be solved numerically through standard routines for searching complex zeros of complex functions. Since k˜ y can be purely imaginary or purely real [11], the solution kz will show two different behaviours as well, as it will be clear from the numerical results shown further in the next. In (5), k˜ y and k˜˜ y are the transverse dominant TM and TE eigenvalues of a parallel plate waveguide composed by a metallic wall on top and a homogenised surface (representing the bed of nails) on bottom [11]. Their expressions are found through the solution of the implicit dispersion equation shown in the Appendix. Under the assumption of nails with infinitesimal period, it can be found that k˜˜ y = p/(h + d ) (see the Appendix). Furthermore, the unknown coefficients can be explicitly calculated as

(3b)

(3c)

(3d) (3e)

IET Microw. Antennas Propag., 2010, Vol. 4, Iss. 8, pp. 1073 – 1080 doi: 10.1049/iet-map.2009.0613

(5)

ATM

ATE

k 1 w 2 k − k2z = jE0 cos 2 j 2 ˜ 2 k − ky

(6a)

2 k2 − k˜˜ y k˜ y kz w 2 k − k2z = jE0 cos 2 2 2 ˜ 2 k − k y k˜˜ y k2z + k˜˜ y − k2 (6b)

The dispersion equation in (5) identifies two different modes. A first mode occurs for f , fd¼l/4 where k˜ y is purely imaginary. This case corresponds to a slow wave that propagates inside the bed of nails region, and it is not of interest for practical purposes (in this case kx becomes 1075


www.ietdl.org purely imaginary and the cosine functions in (1a) and (1b) become imaginary). The second mode is indeed the quasiTEMz mode, where the energy is confined in the ridge.

2.2 Field structure of the TE higher order mode A TEz higher order mode exhibits a odd parity; in the gap region it can be represented as Ey =

E0′

Hx = −E0′ Hz = jE0′

′ sin( kx x)e−jkz z

(7a)

′ kz sin( kx x)e−jkz z jk

(7b)

′ k′x cos( kx x)e−jkz z jk

(7c)

′ kx where the transverse wavenumber kx plays the same role as in (1). In the bed of nails region, the modal fields are still ′ and those of (2) and (3), with new field coefficients ATM ′ ATE By enforcing the same razor blade matching of transverse field as that applied to the dominant mode, a new dispersion equation is obtained

w 2 ˜2 (k − ky ) k2 − k2z cot k2 − k2z 2 2 2 2 ˜2 2 kz ky − k k2z − k2 + k˜ y k2z − k2 + k˜˜ y − =0 2 ˜ 2 2 ˜ kz − k + ky

(8)

′ ′ and ATE are while the new unknown coefficients ATM identified as

′ ATM

′ ATE

k 1 w 2 ˜2 k − ky = jE0 sin j 2 ˜2 2 k − ky

(9a)

2 k2 − k˜˜ y k˜ y kz w 2 ˜2 k − ky (9b) = jE0 sin 2 2 2 ˜ ˜k˜ k2 + k˜˜ 2 − k2 k − ky y z y

From the dispersion equation in (8) it is seen that the cut-off frequency of the higher order mode is characterised by the equation

w kc cot kc = 2 2 k˜ y − k2c

(10)

where kc is the cut-off wavenumber associated with the cutoff frequency. Equation (10) is obtained by setting kz ¼ 0 and k ¼ kc (cut-off condition of the mode) in (8). 1076 & The Institution of Engineering and Technology 2010

Equation (10) can be solved in terms of k˜ y , yielding k˜ y =

kc cos(kc (w/2))

(11)

Since k˜ y depends on k (see implicit expression (29) in the Appendix), and k ¼ kc under the cut-off condition, one can substitute (11) into (29) to obtain an explicit transcendental equation in terms of kc , that is 1 kc h tan + tan(kc d ) = 0 cos(kc (w/2)) cos(kc w/2)

(12)

2.3 Dispersion diagrams Let us consider the geometry depicted in Fig. 1, where d ¼ 7.5 mm, h ¼ 1 mm, w ¼ 5 mm, a ¼ 2 mm and the radius of pins b ¼ 0.5 mm. For the above geometry, Fig. 2 presents the dispersion diagrams obtained by solving (5) (black dots) and (8) (grey dots). While the dominant mode dispersion was already validated in [11], the odd mode dispersion diagram derived in this paper is a new feature, successfully tested here against a full wave analysis. The full wave analysis has been performed by CST MWS (diamond lines). The dispersion equation (5) and (8) have been solved in Matlab, by means of the standard fsolve routine. The CST MWS results employ the eigenvalue solver to the basic cell shown in the inset of Fig. 2. We note that (5) serves to characterise not only the quasiTEMz modal dispersion, but also the slow-wave quasi-TM modal dispersion below 10 GHz. As underlined in [11] the present approximation is valid whenever the pin structure is densely packed, that is, when d/a ≫ 1 [5]. This assumption allows studying the local dispersion of the bed of nails as done in [5], namely by homogenising the surface. In that work, the authors investigated the validity of the models in terms of superposition of a TEM mode and a TM (with respect to the axis of the pins) mode. Beyond that, in practical application, flexibility in the fabrication process must be admitted. Thus, the influence of varying the geometrical parameters has been object of investigation by some research groups [1, 2, 4, 6] who have demonstrated through parametric studies how varying the geometrical parameters mainly affects the operational bandwidth of the waveguide. The stop band of the bi-dimensional bed of nails structure, is denned by the condition for which k˜ y is real and greater than k (see the Appendix), implying attenuation along any direction along the pins surface. The stop band for the 2D structure obtained as a parallel plate with PEC top wall and homogenised bed of nail bottom stop ¼ wall is calculated as the range of frequency between flow stop f (d¼l/4) and fup ¼ f(d+h ¼ l/2) , which in the present case is 10–17.5 GHz. For the ridge gap waveguide, even beyond this upper limit a real value of kz can still be found when k˜ y stop is real but less than k, which occurs when f (d¼l/2) ¼ 2flow . stop However, in practical packaging, fup constitutes the actual IET Microw. Antennas Propag., 2010, Vol. 4, Iss. 8, pp. 1073 – 1080 doi: 10.1049/iet-map.2009.0613

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Figure 3 Solution w0(d, h) of dispersion equation (12) in which kc ¼ p/(d + h), which is the condition maximising the unimodal bandwidth Figure 2 Dispersion diagram for the ridge waveguide (dotted line) Black dots identify the dispersion equation in (5), the grey dots the equation (8). The geometry is the one in Fig. 1, with d ¼ 7.5 mm, h ¼ 1 mm, w ¼ 5 mm, a ¼ 2 mm and the ray of pins b ¼ 0.5 mm. The dispersion diagrams are compared with the one obtained through a CST-MWS simulation (diamond line) of the reference structure depicted in the inset. The light line (kz ¼ k) is shown by a continuous line

upper limit of unimodal bandwidth, since spurious cavity resonances can be observed in absence of the beneficial action of the bed of nail band-gap. Actually, the upper frequency usability of the dominant mode for the present geometry is dictated by the cut-off frequency of the higher order mode found in Section 2.2. It is intuitive that the cut off of this higher order mode is associated with the ridge width, but not in a simple way as in ordinary waveguides. The effort here is concentrated in determining, using the previous analysis, the maximum values of ridge width w0(d, h) that allows for the maximisation of the bandwidth. To this end one should bring the cut-off frequency to coincide with fupstop. This means to substitute lc ¼ 2(d + h) (kc ¼ p/(d + h)) into (12). Thus, one obtains the following limitation to the maximum strip width to have the largest unimodal bandwidth w ≤ w0 (d , h)

(13)

w0(d, h) has a linear dependence on d, with a weak dependence on h. To conclude this section, we note that the dispersion curve of the dominant mode and of the higher order mode both cross the light line at the same point. This property is rigorously obtained by setting in (5) and (8) kz ¼ k. For both dispersion equations it is indeed obtained k˜ y

k˜ y + k˜˜ y =0 k˜˜

(15)

y

which is satisfied for k˜ y = 0 (d ¼ l/2, see the Appendix).

3

Characteristic impedance

In order to give useful criteria to design the present gap waveguide, the approximate modal field expansion is used to define the dominant mode characteristic impedance. If the quasi-TEM mode is approximated as an ideal TEM mode, the characteristic impedance approaches ZTEM = j

h w

(16)

A better approximation, used in [12], suggests the use of a value equal to twice the characteristic impedance of a strip line, that is

where w0(d, h) is solution of 1 p h tan cos pw/2(d + h) d + h cos pw/2(d + h) pd =0 + tan d +h

Solution is plotted for different values of the gap height h, ranging between 0.5 and 1.5 mm, with step of 0.25 mm

ZqTEM = 2Zstrip = xZTEM (14)

Fig. 3 shows the value of w0(d, h) for different values of the gap height h, ranging between 0.5 and 1.5 mm. It is seen that IET Microw. Antennas Propag., 2010, Vol. 4, Iss. 8, pp. 1073 – 1080 doi: 10.1049/iet-map.2009.0613

(17)

where ZTEM ¼ jh/w and x is found to be as [13]

x=

1 a + 0.882h/w

(18)

1077


www.ietdl.org where

a=

⎧ ⎪ ⎪ ⎨ 1,

if

2h w 2 ⎪ ⎪ 0.35 − , 1 − ⎩ w 2h

w > 0.35 2h

(19)

otherwise

Actually [12], there are different ways to calculate the characteristic impedance since the structure is not doubly connected. In particular, being the quasi-TEM mode TEz , the voltage is not uniquely defined, while the current is. A first definition simply arises as Z(0) c ¼ Veq/I, where, for any z, the definition of the equivalent voltage is reasonably taken as the average of the line integral of the electric field from the two metals of the gap, that is 1 Veq = w

w/2 h −w/2

w Ey (x, y) dy dx = E0 hsinc kx 2 0

(20)

We noticed that Z(0) c shows the term ZTEM multiplied by the kx w/2)/FI (kz ); the first term correction factors k/kz and sinc( k/kz accounts for the displacement of the quasi-TEM mode propagation constant from the non-dispersive wavenumber k. The ratio k/kz is shown in Fig. 4a over the stop band frequency range. kx w/2)/FI (kz ) is The second correction term g (V /I ) = sinc( shown in Fig. 4b. A second definition of the characteristic impedance can be 2 given as Zc(2P/I ) = 2P/I 2 , where P is the average power obtained integrating the Poynting vector across the surface of the gap region; namely 1 P= 2

h w/2 0

−w/2

[−Ey (x, y)H ∗ (x, y)] dx dy

(24)

which yields

and 1 I =−

−1

Hx (x, y = h) dx = E0 w

kz F (k ) jk I z

(21)

The integral in (21) is performed from 21 to 1, namely to include the lateral evanescent part of the quasi-TEM mode, and it can be evaluated in closed form with straightforward algebraic manipulations; thus, obtaining ⎞ ⎛ 2 kx w kx w ⎝ k2 k˜ y ⎠ 2 cos + − FI (kz ) = sinc 2 a˜ x a˜˜ x 2 2 w(k2 − k˜ y ) (22) Therefore the characteristic impedance is found to be Zc(V /I ) =

Veq I

= ZTEM

k sinc( kx w/2) kz FI (kz )

1 wh kz

1 + sinc( kx w) P = |E0 |2 2 2 jk

(25)

The flux of power density in (24) is integrated in the ridge region only. Although the x-evanescent power contribution within the bed of nails region can be calculated in closed form, its additional contribution does not give significant change in the final results and complicated the formula without any practical utility. From (25) it is obtained 2

Zc(2P/I ) =

2P k 1 + sinc( kx w) = ZTEM 2 2 I kz 2FI (kz )

(26)

2

(23)

Again, Zc(2P/I ) shows the factor ZTEM multiplied by the correction factor k/kz , and the geometrical factor 2 g (2P/I ) = [1 + sinc( kx w)]/[2FI2 (kz )] whose behaviour is shown in Fig. 4b.

Figure 4 Factors appearing in the analytic expressions of the characteristic impedance in equations (23), (26) and (27) a Correction factor k/kz over the stop band frequency range of the same ridge gap waveguide of Fig. 2 b Geometrical terms for the three different impedances detailed in the text

1078 & The Institution of Engineering and Technology 2010


www.ietdl.org A2 third definition of the characteristic impedance is given Zc(V /2P) = Veq2 /(2P), where P is the power in (25), and Veq is the equivalent voltage in (20). In this case Zc(V

2

/2P)

=

Veq2 2P

= ZTEM

k 2sinc2 ( kx w/2) kz 1 + sinc( kx w)

(27)

that shows ZTEM as multiplied by the correction factor k/kz , 2 kx w/2)/(1 + and by the geometrical factor g (V /2P) = 2sinc2 ( sinc( kx w)) which is shown in Fig. 4b. Fig. 5 compares the diagram of the three different 2 definitions of characteristic impedances Zc(V /I ) , Zc(2P/I ) and (V 2 /2P) . Results are shown over the stop band frequency Zc range for the gap waveguide with dimensions w ¼ 5 mm, h ¼ 1 mm, a ¼ 2 mm, b ¼ 0.5 mm (see inset of Fig. 5). The characteristic impedances are compared with the TEM value ZTEM ¼ jh/w (dotted line) and with ZqTEM ¼ xZTEM in (17) (dashed line). This comparison shows that Zc(V /I ) is around 15 V above Zc(2P/I

2

)

and 25 V

2

Zc(V /2P)

for any frequency in the unimodal above bandwidth. However, being the definition of Zc(V /I ) and 2

Zc(V /2P) affected by ambiguity in the voltage, it should be reasonable to consider as more reliable the definition 2

Zc(2P/I ) that only involves power and current; both these quantities are uniquely defined. The latter is, by the way, the closest to the ZqTEM , which is the value adopted and tested in [12] and usually employed in microwave theory [13]. In Fig. 5 we also compare the analytical value of

2

Zc(2P/I ) (considered by us as the more reliable definition of the characteristic impedance), with the results obtained through a CST simulation for some frequency points (black dots). To obtain these results, the x component of the magnetic field and Poynting vector have been calculated by numerical integration on the transverse cross-section as in (21) and (24), respectively. The agreement is found satisfactory all over the frequency range of monomodal propagation, thus suggesting that the approximation given in (26) is reliable to be used for design purpose.

4

In this paper, we have provided an analytical expression of the modal dispersion equation of the first two modes and of the characteristic impedance of the dominant mode for a metal gap-waveguide surrounded by a bed of nails. The gap waveguide dominant (quasi-TEM) mode can propagate in the stop band imposed by the bed of nails. It possesses a cut-off wavelength around two times the pin lengths. Moving up from the cut-off frequency, the dominant mode dispersion curve very rapidly approaches the light line and goes parallel to it for an octave of band; thus, by further increasing the frequency, it crossed the light-line and goes in the slow wave regime, where the energy is confined mostly at the interface between the ridge and the metamaterial region. The dispersion equation associated to the higher order quasi-TEz mode starts from a cut-off frequency, which has been identified here in a closed form, and proceed with almost linear behaviour from the fast wave region to the slow wave region, crossing the light line close to the point where the dominant mode dispersion curve does the same. In order to maximise the unimodal bandwidth, the width of the ridge should be less than a specific value evaluated in an analytical form. Concerning the characteristic impedance of the dominant mode, three different closed form expressions, based on alternative definitions, are derived and compared with an approximate (non-dispersive) expression used in literature [12]. The three definitions exhibit similar global behaviour as a function of frequency, but they are displaced relative to each other by approximately 10 V. The most reliable of the three expressions is the one that does not involve directly any definition of equivalent voltage, since a rigorous voltage is undefined for the dominant mode.

5 Figure 5 Curves relevant to the three different definitions of characteristic impedance for the ridge gap waveguide 2

2

/2P ) ) Z(V/I from (23), Zc(2p/I ) from (26) and Z (V from (27) are c c compared with the TEM ideal value ZTEM ¼ jh/w (dotted line) 2 and with ZqTEM ¼ xZTEM from (18) (dashed line). Zc(2p/I ) is also compared with the values obtained through a CST simulation (black dots), where the x-component of the magnetic field and Poynting vector have been integrated as in (21) and (24), respectively


Concluding remarks

References

[1] KILDAL P.-S., ALFONSO E., VALERO-NOGUEIRA A., RAJO-IGLESIAS E.: ‘Local metamaterial-based waveguides in gaps between parallel metal plates’, Antennas Wirel. Propag. Lett., 2009, 8, pp. 84– 87 [2] KILDAL P.-S.: ‘Waveguides and transmission lines in gaps between parallel conducting surfaces’. European Patent EP08159791.6, 7 July 2008 1079


www.ietdl.org [3] HIROKAWA J. , ANDO M.: ‘Single-layer feed waveguide consisting of posts for plane TEM wave excitation in parallel plates’, IEEE Trans. Antennas Propag., 1998, 46, (5), pp. 625– 630 [4] VALERO-NOGUEIRA A. , ALFONSO E. , HERRANZ J.I. , KILDAL P.-S.: ‘Experimental demonstration of local quasi-TEM gap modes in single-hard-wall waveguides’, IEEE Microw. Wirel. Compon. Lett., 2009, 19, (9), pp. 536 – 538 [5] SILVEIRINHA M.G., FERNANDES C.A., COSTA J.R.: ‘Electromagnetic characterization of textured surfaces formed by metallic pins’, IEEE Trans. Antennas Propag., 2008, 56, (2), pp. 405– 415 [6] RAJO-IGLESIAS E., UZ ZAMAN A., KILDAL P.-S.: ‘Parallel plate cavity mode suppression in microstrip circuit packages using a lid of nails’, IEEE Microw. Wirel. Compon. Lett., 2009, 20, (1), pp. 31– 33 [7] YANG F.-R., MA K.-P., QIAN Y., ITOH T.: ‘A novel TEM waveguide using uniplanar compact photonic-bandgap (UC-PBG) structure’, IEEE Trans. Microw. Theory Technol., 1999, 47, (11), pp. 2092 – 2098 [8] NG MOU KEHN M., NANNETTI M., CUCINI A., MACI S., KILDAL P.-S.: ‘Analysis of dispersion in dipole-FSS loaded hard rectangular waveguide’, IEEE Trans. Antennas Propag., 2006, 54, (8), pp. 2275– 2282 [9] NG MOU KEHN M., KILDAL P.-S.: ‘Miniaturized rectangular hard waveguides for use in multi-frequency phased arrays’, IEEE Trans. Antennas Propag., 2005, 53, (1), pp. 100– 109 [10] BOSILJEVAC M., SIPUS Z., KILDAL P.-S.: ‘Construction of Greens functions of parallel plates with periodic texture with application to gap waveguides – a plane wave spectral domain approach’, IET Microw. Antennas Propag., Accepted for publication [11] POLEMI A., MACI S., KILDAL P.S.: ‘Dispersion characteristics of metamaterial-based parallel plate ridge waveguides’. Third European Conf. on Antennas and Propagation, (EUCAP 2009), 2009, pp. 1675– 1678 [12]

ALFONSO E., BAQUERO M., KILDAL P.-S., VALERO-NOGUEIRA A., RAJO-

IGLESIAS E., HERRANZ J.I.:

‘Design of microwave circuits in ridge

1080 & The Institution of Engineering and Technology 2010

gap waveguide technology’. IEEE MTT 2010 Int. Microwave Symp. (IMS2010), Anaheim, CA, 23 – 28 May 2010 [13] POZAR D.M.: ‘Microwave engineering’ (John Wiley and Sons, 2005)

6

Appendix

The TM eigenvalues k˜ y and TE k˜˜ y have been calculated in [11]. They are eigenvalue solutions of the problem constituted by a homogenised bed of nails surface covered by a metal plate. The guiding phenomenon is characterised by a wave bouncing in the parallel plate where the top wall is represented by homogeneous reflection coefficients GTM(ky) and GTE(ky), as in [5]. Under the assumption of nails with infinitesimal period, the TE reflection coefficient can be approximated by the one of the PEC ground of the nails, thus leading to k˜˜ y = p/(h + d ). By imposing the vanishing of the total tangential component of the electric field at the top wall, from (1) – (3), a transcendental equation in terms of k˜ y is obtained. After some algebraic manipulations, the eigenvalue k˜ y of the TM modes can be obtained by solving the following dispersion equation k˜ y k

⎡

tan k˜ y h + ⎣1 −

2

k2 − k˜ y

⎤

⎦ tan(kd ) 2 k2p + k2 − k˜ y 2 2 k2p − k˜ y k2 − k˜ y 2 + − tan k2p − k˜ y d = 0 2 k 2 2 ˜ k +k −k p

(28)

y

where k2p = 1/a2 (2p/ ln(a/2pb) + 0.5275) is the plasma wavenumber [5]. In doing this by an iterative process, a guess point ky ¼ k can be conveniently used. We note that k˜ y exhibits a different behaviour as the frequency changes. In particular, k˜ y is imaginary for d , l/4 and real and larger than k for l/4 , d , l/2 2 h; the latter range identify the band-gap of the structure. For very small periods, (28) can be rewritten as k˜ y k

tan k˜ y h + tan(kd ) = 0

(29)

which corresponds to assume a TEMy propagation inside the pin region.
