Rectangular waveguide (RW) with sidewalls of vertical conducting cylinders (i.e., substrate-integrated waveguide, SIW) becomes popular with the advent of low-temperature cofired ceramic (LTCC) structure. The formula of leakage loss through the SIW with small cylinders has been found, based on the surface impedance of cylinder walls derived from an analytical MoM. To complete the study of loss, the ohmic loss in the SIW is investigated also in this paper. The theoretical formulas show good agreements with numerical HFSS simulations and the measured data from hardware experiments, demonstrating potential applications to the research and design of some interesting devices such as leaky antennas, etc.
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 In multilayer microwave integrated circuits such as low-temperature cofired ceramics (LTCCs) or multilayer printed circuit boards (PCBs); waveguide-like structures can be fabricated in planar form by using periodic metallic via holes called substrate-integrated waveguides (SIW) [Deslandes and Wu, 2001a; Hirokawa and Ando, 1998]. The SIW structures largely preserve the well-known advantages of conventional rectangular waveguides, viz., high Q and high power capacity, and include the advantages of microstrip lines, such as low profile, small volume and light weight etc. The SIW structure is convenient for the design of millimeter-wave circuits such as filters, resonators, and antennas etc., [Zhang et al., 2005; Deslandes and Wu, 2001b, 2003; D'Orazio et al., 2004; Cassivi and Wu, 2003; Cassivi et al., 2002a].
 In addition, the SIW structure can easily be connected to microstrip or coplanar circuit using simple transitions [Zhang et al., 2005; Deslandes and Wu, 2001a, 2001b, 2003; D'Orazio et al., 2004; Cassivi and Wu, 2003; Cassivi et al., 2002a], which may lead to the design and development of compact low-loss millimeter-wave integrated circuits and systems. Such developments should enhance manufacturing repeatability, reliability, and cost reduction significantly especially with the advent of LTCC and multilayer printed circuit board. For this reason, it is important to understand and analyze with simplicity the loss characteristics of SIW structures. It has been found that the substrate-integrated waveguide (SIW) has nearly the same propagation and cutoff characteristics with the conventional rectangular waveguide. In fact, the SIW can be considered a rectangular waveguide structure with an equivalent width [Xu and Wu, 2005; Che et al., 2006; Cassivi et al., 2002b]. Because of the periodic cylinders forming the sidewalls of the SIW, a SIW structure is subject to possible loss of leakage and ohmic attenuation. Studies of such losses have been carried out numerically or modally in several references [Xu and Wu, 2004, 2005; Xu et al., 2003]. In this paper, for clearer physical insights, the losses of the SIW through the cylinder walls are studied analytically.
Section 2 derives the formula of leakage loss from the surface impedance at the sidewalls of SIW, through an understanding of the physical significance of the self-term in the MoM (method of moments) matrix of Harrington . The approach may be called the “analytical MoM” and has been used to find formulas with good accuracy in capacitance from a finite and perforated ground plane [Chow et al., 2002] and its extension, the accurate formula of the equivalent width on SIW [Che et al., 2005, 2007], corresponding to the numerical and modal solutions mentioned above [Xu and Wu, 2004, 2005; Xu et al., 2003].
Section 3 converts the formula of ohmic loss of a regular waveguide of solid sidewalls to that of an SIW of cylinder walls through the ratio of their surface areas. Section 4 compares the loss formulas of the SIW prototype with the numerical results from HFSS, and from, experimental measurements. Good agreements between all three are observed.
2. Formula of Leakage Loss Through Substrate-Integrated Waveguide With Small Cylinders
Figure 1 illustrates one known reflection of the TE wave from a solid conducting wall; Ei indicates the electric field of the incident wave, while Er stands for the electric field of the reflected wave. For the perfect conductive wall, the boundary condition is
In MoM method, the wall is divided into N strips (N to ∞), each of width W with separation W, as shown in Figure 2.It is noted that
where [G0] is the surface impedance of the conductive wall, I0 is the surface current on the conductive metal strips. For each strip, the radius of the equivalent cylinder is R0 = W/4 from conformal mapping. This equivalence is in fact highly accurate with an average error of only 2% as discussed in [Che et al., 2007]. Equations (1) and (2) give the reflection from a wall of N strips. In detail, (2) is [Harrington, 1993]:
where rmn = ∣mW − nW∣, i.e., the separation between the centers of the strips at mW and nW (for m ≠ n), k0 = ω = 2π/λ is the wavenumber (λ = wavelength). It may be noticed that the separation W should be ≤λ/20 in quasi-static. The surface impedance at the cylinder wall has been derived in detail in [Che et al., 2007], given as
 The formula implies that, when R = W/4, the surface impedance of the cylinder walls equals to zero, corresponding to the case of solid walls; when R < W/4, the reactance of the surface impedance of the cylinder walls is inductive, while the surface impedance of the cylinder walls is capacitive for the case R > W/4.
 If R < W/4 for small cylinder, the free space (or dielectric medium) impedance η0 in parallel behind the cylinder walls appears, to give a total impedance ηst in the form of reciprocals:
 The cosine factor of the 2nd term comes from the incident angle that the plane wave TE10 mode makes to the wall surface. The power leakage through radiation is contained in (5), in the second (real) term of the surface impedance for the small cylinder case of R < W/4. It is believed that the second term must vanish for large cylinder case of R > W/4 by analytical continuity, since we know that at R = W/4 the cylinder wall already approximates the solid wall where there is little leakage.
 Returning to the case of small cylinder of R < W/4, the reflection coefficient Γ from a cylinder wall has the form:
 The above derivation is for one sidewall of the SIW. For both sidewalls, the attenuation from Γ is doubled, i.e., the leakage attenuation constant αleakage through the sidewalls of the SIW is:
where L = a′tan (tan θ = ) is the distance z along the waveguide for the plane wave ray to bounce from one side of the cylinder walls to the other side. It may be noted, a′ and a are, respectively, the widths of the SIW and the equivalent rectangular waveguide.
3. Ohmic Conductor Loss of Substrate-Integrated Waveguide
 The attenuation α in a regular waveguide, of cross-section a × b, is the ratio of the ohmic loss Pl and the power flow P10 of the fundamental mode. Integrated across the four surfaces a and b for Pl and the cross-section a × b for P10, and given by Collin , the attenuation of the rectangular waveguide is
where ηsc is the surface resistance of the solid conductive wall, kc = π/a, and ηh is the wave impedance of the rectangular waveguide.
 Let this rectangular waveguide be the equivalent of a SIW of cross-section a′ × b. Since the propagation constant β and its field expressions of SIW and rectangular waveguide are the same, the same equations Pl and P10 of the infinite integrals result except for their evaluations: over a width a′, instead of a; and over the substrate height b on a cylinder of circumference 2πR, instead of on a flat wall segment of width W. The result of the new evaluations, neglecting second-order errors, gives the attenuation of the SIW as
 The cos2 function in the first loss term above is added to show that there is a reduction of the Hy field (and therefore the current and loss) at the cylinder walls when the SIW width a′ is changed from the width a of the equivalent rectangular waveguide. The reason is that the Hy distribution at the interior is unchanged, between SIW and its equivalent rectangular waveguide, despite their change in width as given in [Che et al., 2006].
 The change of cos2 with argument a′/a is only second order and may therefore be neglected. Hence, we may simplify (9) of SIW to a form similar to (8) of rectangular waveguide. That is:
where ηɛ and kɛ are, respectively, the open space impedance and propagation constant in the dielectric substrate.
 The total loss of the SIW, including the leakage loss, conductor loss and dielectric loss, can be expressed as
where αc(SIW) is the ohmic attenuation of the SIW, derived above in (10) as an extension from Collin . αLeakage is the leakage attenuation constant through the sidewalls of the SIW given as (7); αd is the dielectric loss of the substrate onto which the SIW is constructed, which can be omitted if high-quality dielectric materials with low loss are used for the substrates.
4. Theoretical Results and Experimental Verification
4.1. Plane Wave Reflection From an Infinite Wall of Cylinders
 To better understand the attenuation along the SIW from leakage, it may be profitable to see the difference in reflection coefficient Γ (phase and magnitude) between the solid wall and a cylinder wall before their inclusions into the rectangular waveguide or SIW. The ohmic conductor loss is assumed zero at this point. Through (6), Figure 3 shows the case of R < W/4. The magnitude of Γ there indicates a power leakage through the cylinder wall with a magnitude of less-than-unity, but only at less than 1% in the ratio of R/W used. This reduction in magnitude of Γ can only be slight (say, less than 5%) for all frequencies of concern and all practical ratio of R/W. There are two reasons:
 1. The dependence of Γ with frequency arises from (6). The surface impedance ηst in (6) is much smaller than the free space impedance η0. As frequency changes, the plane wave angle θ changes in SIW, the leakage increases and therefore the resulting magnitude of the reflection coefficient Γ decreases. However, the decrease is small due to the smallness of the surface impedance ηst.
 2. For R/W, the surface impedance ηst as given in (6) is related to the sum of the reciprocals of the free space impedance η0 and the surface impedance ηs0 of the cylinders. The latter, in (4) is the logarithm of the ratio R/W and therefore is a slow change function. The double smallness, in reasons 1 and 2, means that the reduction in reflection Ã is indeed slight, and therefore is the leakage through the cylinder wall as well.
 Next, Figures 4 and 5 show just a little more changes in reflection coefficient Γ against the cylinder radius R and cylinder separation W. They show that the magnitude of Γ increases and phase decreases with increasing R or decreasing W, or both. The trends of W and R are in the opposite directions as one is in the denominator and other is in the numerator of the ratio R/W. Finally Γ reaches unity and −180° (of a total reflection) when R = W/4. The total reflection means that the cylinder wall becomes identical to a solid conductive wall.
 A SIW with cylinder walls then becomes identical to a rectangular waveguide, with no leakage, when 4R/W = 1. If this ratio is smaller, small leakage occurs. The theoretical leakage loss of the SIW, together with its ohmic loss, as derived from (7) to (10), is verified in theory and experimental measurements below.
4.2. Leakage Loss and Ohmic Loss in the SIW
 The relationships between the leakage loss and cylinder radius and the cylinder spacing have been investigated, based on the formula (7); the theoretical results are illustrated in Figures 6 and 7. The dimensions used in calculation are given below: the dielectric constant ɛr = 2.33, the SIW width a′ = 0.5λ0 (in ɛr), while the frequency f0 = 10 GHz.
 In Figure 6, obvious leakage occurs in case of small cylinders, and the leakage decreases with increasing cylinder radius. The leakage loss approaches zero in case R = W/4 = 0.35 mm (W = 0.047λ0 = 1.4 mm), for in this case the cylinder wall becomes a solid wall, and then no leakage occurs. This phenomenon can also be observed in Figure 7. In case of W = 4R = 1.4 mm (R = 0.0117λ0), the leakage loss is zero, for the cylinder wall becomes a solid conductive plane. However, when the cylinder spacing increases, the leakage then increases also.
 To verify the validity of the theoretical analysis, one SIW prototype in X-band has been fabricated. The width is a′ = 0.708λ0 in air at f0 = 10 GHz, the top and bottom metal plates are supported by two rows of copper cylinders with radius R = 0.0167λ0, W = 0.133λ0, the length of the SIW L = 4.274λ0. One coaxial-rectangular waveguide transformer has been connected at one end, while a regular matched load has been connected at another end. In addition, some absorbing materials have been placed close to both sidewalls of the SIW to absorb the leakage into the surrounding space. In this way, the traveling wave is formed inside SIW. In addition, several coupling posts have been designed to couple the energy out from different locations along the substrate-integrated waveguide.
 In the experiment, the cylinder radius R of our prototype is chosen to be less than W/4, so that the leakage occurs and can thus be measured. The measured losses at several frequencies and the theoretical loss, leakage and ohmic losses in formulas (7) and (10), are compared in Figure 8. Good agreement between the experiment and the theoretical losses is observed.
 In Figure 8 is also the loss by HFSS computed in the following manner. Plane wave reflections are the same whether a lossy wall (substrate-integrated waveguide or flat) made into a short at the end of a rectangular waveguide giving S11, or into a sidewall along the rectangular waveguide giving attenuation α. The sidewall areas are much larger than that of the short. Hence, much time is saved by computing S11 of the short and has it converted to α along the sidewalls. Good agreement is observed also until the extreme ends of the frequency band, at 8 and 12 GHz.
 In computing Figure 8, we found that the leakage loss of (7) is much larger than the ohmic loss of (10). This is shown in Figure 9. It is observed that the ohmic loss from (10) is of the order of α/β < 0.01% for all practical frequencies and cylinder radii. Since γ = α + jβ, α and β are really just one complex propagation constant. This α/β ratio therefore is far below the measurement and computational tolerances. As a result, no experimental or computational results above random noise are possible to be plotted in Figure 9, that is: for the verification of (10), the ohmic loss.
Equation (10) is the attenuation from only the ohmic loss. The total attenuation α in (11), with significant radiation leakage through the walls (as in the case of R < W/4), has been found to agree with (hardware) measurements. In the case of R ≥ W/4, there is no radiation leakage and the attenuation αc (SIW) due to the ohmic loss alone (say from copper) becomes very small, at say < 0.01% of the propagation β. Such loss is therefore not detectable, either in measurement or in actual computations by HFSS.
 For no radiation leakage at R = W/4, the SIW has a′ = a.Equation (10) indicates that the ohmic loss in a SIW with cylinder walls sometimes can actually be smaller than that of the equivalent rectangular waveguide with smooth solid walls.
4.3. Brief Discussion on the Small Error of Ohmic Losses in Substrate-Integrated Waveguide
 As stated and observed above in Figure 9, the ohmic loss in SIW caused by imperfect cylinder walls is much less than the leakage loss. Converted directly from the ohmic loss formula in conventional rectangular waveguide (RW), the attenuation (10) is accurate with an average error of 4% or less. The relative error of (6) assuming uniform current around the circumference of the cylinder has to be small from the true ohmic attenuation with the nonuniformly current of 2-fold symmetry, this is based on the variational principle.
 The variational principle on ohmic resistance may be illustrated by a simple example. Let the cylinder of uniform current be taken as the current on 4 resistors in parallel. Each resistor is 4 Ω giving a total resistance of 1 Ω. If the cylinder has 20% nonuniformity in current, 2 of the 4 resistors would have 4.8 Ω and the other 2 have 3.2 Ω. The 4 resistor in parallel then would give a net resistance of 0.96 Ω. This is the 4% error (20% × 20%) from original 1 Ω as predicted by the second-order error of the variational principle.
 The leakage and ohmic losses of the SIW have been investigated in this paper. The formula of leakage loss is derived based on the surface impedance of the cylinder wall with analytical MoM. Equation (4) indicates that when the cylinder radius R equals to one quarter of cylinder separation (W/4), effectively, the cylinder wall thus becomes a solid conductive plane causing a total reflection of the electromagnetic waves with an 180° phase-shift. This means that if R < W/4, there is leakage losses, and if R > W/4, there is little leakage loss.
 The formula of ohmic loss for SIW is derived also through the ratio of the surface areas of SIW and RW. For ohmic loss, when R = W/4, for SIW, the cross section of the cylinder surface is 2πR but in the regular waveguide of solid side walls, the cross section of the corresponding strip of the wall is only W, and W/2πR < 1. Thus as indicated in (9), the ohmic loss of the SIW can actually be smaller than that of a regular waveguide of solid walls.
 An X-band prototype of SIW has been fabricated to verify the validity of theoretical analyses. Good agreements between the formula calculation, numerical HFSS simulation and experiments are observed.
 The formulas (7) and (10) show that ohmic loss is usually quite small; the leakage loss can be made small or large by controlling the ratio of cylinder spacing and cylinder radius. In this way, it is possible to design a leaky SIW antenna, of high efficiency.
 The authors would like to express their gratitude for the financial support of the National Science Foundation of China under grant 60471025 and the Natural Science Foundation of Jiangsu Province under grant BK2004135.