Coplanar waveguide gap

According to W.J.Getsinger [56] a coplanar series gap (see fig. 12.5) is supposed to be the dual problem of the inductance of a connecting strip between twin strip lines.

Figure 12.5: coplanar waveguide series gap

The inductance of such a thin strip with a width $g$ and the length $W$ is given to a good approximation by

$$L = \dfrac{\mu_0\cdot W}{2\pi}\cdot\left(p - \sqrt{1 + p^2} + \ln{\left(\dfrac{1 + \sqrt{1 + p^2}}{p}\right)}\right)$$ (12.32)

where $p = g/4W$ and $g,W \ll \lambda$. Substituting this inductance by its equivalent capacitance of the gap in CPW yields

$$\begin{split}C &= L \cdot \dfrac{4\cdot \varepsilon_{r,eff}}{... ...{\left(\dfrac{1 + \sqrt{1 + p^2}}{p}\right)}\right) \end{split}$$ (12.33)