#### 4.1. Plane Wave Reflection From an Infinite Wall of Cylinders

[17] 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:

[18] 1. The dependence of Γ with frequency arises from (6). The surface impedance *η*_{st} in (6) is much smaller than the free space impedance *η*_{0}. 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 *η*_{st}.

[19] 2. For *R*/*W*, the surface impedance *η*_{st} as given in (6) is related to the sum of the reciprocals of the free space impedance *η*_{0} and the surface impedance *η*_{s0} 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.

[20] 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.

[21] A SIW with cylinder walls then becomes identical to a rectangular waveguide, with no leakage, when 4*R*/*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

[22] 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 ɛ_{r} = 2.33, the SIW width *a*′ = 0.5*λ*_{0} (in ɛ_{r}), while the frequency *f*_{0} = 10 GHz.

[23] 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* = 0.047*λ*_{0} = 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* = 4*R* = 1.4 mm (*R* = 0.0117*λ*_{0}), 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.

[24] To verify the validity of the theoretical analysis, one SIW prototype in *X*-band has been fabricated. The width is *a*′ = 0.708*λ*_{0} in air at *f*_{0} = 10 GHz, the top and bottom metal plates are supported by two rows of copper cylinders with radius *R* = 0.0167*λ*_{0}, *W* = 0.133*λ*_{0}, the length of the SIW *L* = 4.274*λ*_{0}. 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.

[25] 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.

[26] 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 *S*_{11}, 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 *S*_{11} 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.

[27] 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.

[28] 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 *α*_{c (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.

[29] 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

[30] 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.

[31] 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.