Substrate-integrated waveguide

A SIW is composed of a thin dielectric substrate covered on both faces by a metallic layer. The substrate embeds two parallel rows of metallic via holes delimiting the wave propagation area. The organization of the vias and the geometric parameters are described in the attached figure.

The width of a SIW is the distance ${\displaystyle a}$ between its two vias rows, which is defined from center to center. An effective width ${\displaystyle a_{e}}$ may be used to characterize more precisely the wave propagation. The distance between two successive vias of the same row is ${\displaystyle s}$, and the vias diameter is denoted by ${\displaystyle d}$.

In classical solid-walled rectangular waveguide, the general formulation of propagation involves a superposition of transverse electric (TE) and transverse magnetic (TM) modes. Each of these is associated with particular fields and currents. In the case of TM modes, the current in the vertical walls is longitudinal, i.e. parallel to the propagation axis, usually denoted as ${\displaystyle z}$. Then, given the vertical geometry of the vias, it is impossible for such modes to appear in SIWs: the electrical current cannot propagate from via to via. Only TE modes are able to propagate through SIW.

Each mode appears above a precise cut-off frequency determined by the waveguide dimensions and the filling medium. For TM modes, decreasing the waveguide thickness (usually denoted as ${\displaystyle b}$) increases the cut-off frequency with ${\displaystyle 1/b}$. In the case of SIW, the thickness is so low that the cut-off frequency of TM modes is much higher than the dominant mode.

One of the objectives of the SIW geometry is to reproduce the characteristic propagation modes of rectangular waveguides inside a thin template. The width ${\displaystyle a}$ of the waveguide is an essential parameter of those modes. In the typical SIW geometry, ${\displaystyle a}$ is the distance between the two vias rows from center to center (see figure). Due to the vias geometry, this distance cannot be used directly; because of the space between successive vias and their circular shape, the signal inside the guide does not behave exactly as it would in a perfectly rectangular waveguide of the same width.

To apply waveguide theory to SIWs, an effective width ${\displaystyle a_{\text{eff}}}$ can be used. Its takes into account the shape of the vias and the space in-between. Its value lies between ${\displaystyle a+d}$ and ${\displaystyle a-d}$.

A common simple definition is[3][4]

${\displaystyle a_{\text{eff}}=a-{\frac {d^{2}}{0.95s}},}$

and a more refined definition used for large values of ${\displaystyle d/a}$ is[5]

${\displaystyle a_{\text{eff}}=a-1.08{\frac {d^{2}}{s}}+0.1{\frac {d^{2}}{a}}.}$

With this effective width, the propagation constant of a SIW is similar to that of a classical rectangular waveguide whose width is ${\displaystyle a_{\text{eff}}}$. The formulas given above are empirical: they were established comparing the dispersion characteristics of different SIWs to those of rectangular waveguide filled with the same dielectric material.[4]

Part of the signal attenuation is due to the surface current density flowing through the metallic walls of the waveguide. These currents are induced by the propagating electromagnetic fields. These losses may also be named ohmic losses for obvious reasons. They are linked to the finite conductivity of the metals: the better the conduction, the lower the losses. The power lost per unit length ${\displaystyle P_{l}}$ can be calculated by integrating the current densities ${\displaystyle J_{s}}$ on a path ${\displaystyle C}$ enclosing the waveguide walls:[11]

$${\displaystyle P_{l}={\frac {1}{2}}R_{s}\int _{C}|J_{s}|^{2}\,dl.}$$

It can be shown that in a classical rectangular waveguide, the attenuation of the dominant mode ${\displaystyle {\text{TE}}_{10}}$ due to conduction currents is given, in nepers per meter, by

$${\displaystyle \alpha _{C}={\frac {R_{s}}{a^{3}b\beta k\eta }}(2b\pi ^{2}+a^{3}k^{2})={\frac {R_{s}}{\beta k\eta }}\left({\frac {2\pi ^{2}}{a^{3}}}+{\frac {k^{2}}{b}}\right),}$$

where

${\displaystyle a}$ is the width of the waveguide,

${\displaystyle b}$ its height,

${\displaystyle \eta ={\sqrt {\mu /\epsilon }}}$ the wave impedance,

${\displaystyle k=\omega {\sqrt {\mu \epsilon }}}$ the wave vector,

${\displaystyle \delta =1/{\sqrt {\pi f\mu \sigma }}}$ the skin depth in the conductor,

${\displaystyle R_{s}={\frac {1}{\delta \sigma }}={\sqrt {\frac {\omega \mu }{2\sigma }}}}$ is the sheet resistance (of surface impedance).

It is noticeable that ${\displaystyle \alpha _{C}}$ is directly linked to the substrate thickness ${\displaystyle b}$: the thinner the substrate, the higher the conduction losses. This can be explained keeping in mind that this ohmic losses are determined by integrating the current density on a path enclosing the waveguide walls.

On the top and bottom horizontal metallic plates, the current is scaled with ${\displaystyle 1/b}$, due to the modification of the field intensity on these plates: when ${\displaystyle b}$ increases, the field intensity decreases, as well as the currents. In the vertical walls, this variation of ${\displaystyle J_{s}^{2}}$ is compensated by the lengthening of the integration path ${\displaystyle C}$. As a result, the contribution of the vertical vias to the conductor losses is unchanged with ${\displaystyle b}$.[14] This is why there is two terms in the expression of ${\displaystyle \alpha _{C}}$: the first is independent of ${\displaystyle b}$, while the second one varies with ${\displaystyle 1/b}$.

Another key point of the conduction losses experienced by the SIWs is linked to the roughness of the surfaces that may appear due to the synthesis processes. This roughness decreases the effective conductivity of the metallic walls and subsequently increases the losses. This observation is of crucial importance for the design of SIWs, as they are integrated on very thin substrates. In this case, the contribution of the conduction losses on the global attenuation is predominant.[3][15][13]

The attenuation due to the dielectric behavior of the filling medium can be determined directly from the propagation constant.[11] Indeed, it can be proven that, making use of a Taylor expansion of the function ${\displaystyle {\sqrt {a^{2}+x^{2}}}}$ for ${\displaystyle x\ll a}$, the propagation constant is

$${\displaystyle \gamma =\alpha _{D}+j\beta \simeq {\frac {k^{2}\tan \delta }{2\beta }}+j\beta ,}$$

where ${\displaystyle \tan \delta }$ is the loss tangent of the dielectric substrate. This approximation is correct if ${\displaystyle \tan \delta \ll 1}$, which is usually the case in microwave electronics (at 10 GHz, ${\displaystyle \tan \delta =0}$ in air, ${\displaystyle 1.5\times 10^{-4}}$ in Teflon, and ${\displaystyle 1\times 10^{-4}}$ in bulk alumina). Then the following identification can be made:

$${\displaystyle \alpha _{D}={\frac {k^{2}\tan \delta }{2\beta }}.}$$

This relation is correct for both electrical and magnetic transverse modes.

The dielectric losses ${\displaystyle \alpha _{D}}$ depend only on the substrate and not on the geometry: unlike the conduction losses, ${\displaystyle \alpha _{D}}$ is not influenced by the substrate thickness. It transpires that the only way to reduce ${\displaystyle \alpha _{D}}$ consists in choosing a template with better dielectric properties: the lower the loss tangent ${\displaystyle \tan \delta }$, the lower the attenuation.

Because the vertical walls of the SIW are not continuous, radiation leakages may flow between the vias. These leakages can significantly affect the global transmission quality if the vias geometry is not chosen carefully. Some studies have been conducted to describe, predict and reduce the radiation losses. They have resulted in some simple geometric rules that have to be satisfied in order to reduce the radiation losses.[1][5][14][16][17]

The geometric parameters of interest are the diameter ${\displaystyle d}$, the width of the SIW ${\displaystyle a}$ and the center-to-center distance between the vias ${\displaystyle s}$. They must be tuned in such a way to approximate the behavior of a continuous metallic wall: the spacing of the vias has to remain small compared to their diameter, while the diameter must be small compared to the waveguide guided wavelength (${\displaystyle \lambda _{g}}$). To keep the radiation losses reasonably small, the recommended values are

$${\displaystyle s\leq 2d\quad {\text{and}}\quad d\leq {\frac {\lambda _{g}}{5}}.}$$

For a specific traveling mode, the leakages decrease with the increasing frequency and are maximal at the cut-off frequency of the mode. The radiation leakage factor ${\displaystyle \alpha _{R}}$ is independent of the substrate properties and independent of the height of the guide.