A certain microstrip line has fused quartz (\varepsilon_{r}=3.8) as a substrate. If the ratio of line width to substrate thickness is w/h = 4.5, determine
(a) The effective relative permittivity of the substrate
(b) The characteristic impedance of the line
(c) The wavelength of the line at 10 GHz
(a) For w/h=4.5, we have a wide strip. From eq. (11.70):
\varepsilon_{eff}=\frac{(\varepsilon_{r}+1)}{2}+\frac{(\varepsilon_{r}-1)}{2\sqrt{1+12h/w}}
\varepsilon_{eff}=\frac{4.8}{2}+\frac{2.8}{2}\left[1+\frac{12}{4.5}\right]^{-1/2}=3.131
(b) From eq. (11.71):
Z_{o}=\begin{cases}\frac{60}{\sqrt{\varepsilon_{eff}}}\ln\left(\frac{8h}{w}+\frac{w}{4h}\right) & w/h\leq1 \\ \frac{1}{\sqrt{\varepsilon_{eff}}}\frac{120\pi}{[w/h+1.393+0.667\ln(w/h+1.444)]} & w/h\geq1\end{cases}
Z_{o}=\frac{120\pi}{\sqrt{3.131}[4.5+1.393+0.667\ln(4.5+1.444)]}=30.08\Omega
(c)
\lambda=\frac{u}{f}=\frac{c}{f\sqrt{\varepsilon_{eff}}}=\frac{3\times10^{8}}{10^{10}\sqrt{3.131}}=1.69\times10^{-2}m=16.9 mm