7.7.1 The microstrip line and why it is not TEM

A microstrip line (Figure 7.7.1a) is a conducting strip of width $w$ on a dielectric substrate of relative permittivity ${\epsilon }_r$ and height $h$, which covers a conducting ground plane. The strip is thin, $t\ll h$, and the line is uniform along $z$.

Why not TEM?

Suppose the wave were TEM, $E_z=H_z=0$, with the factor ${\mathrm{e}}^{-\mathrm{j}\tau z}$. The $x$-component of $\nabla \times E=-\mathrm{j}\omega {\mu }_{0}H$ and the $y$-component of $\nabla \times H=\mathrm{j}\omega \epsilon E$ then read, in each region,

wherever the field is not zero. In the air this demands $\tau =k_0$, in the substrate $\tau =k_0\sqrt{{\epsilon }_r}$. The field lives in both regions and has one $\tau$, so a TEM wave is impossible. The true modes are hybrid, with small longitudinal components $E_z$ and $H_z$ (Orfanidis, Section 11.3, shows the same from the boundary conditions at the air–substrate interface).

Quasi-TEM

As long as the substrate is thin compared with the wavelength, $h\ll \lambda$, the longitudinal fields stay tiny and the transverse field is, to a very good approximation, the static field of the two conductors. The wave then behaves as a TEM wave in a uniform medium of an effective permittivity ${\epsilon }_{\mathrm{eff}}$, with $1<{\epsilon }_{\mathrm{eff}}<{\epsilon }_r$. This quasi-TEM picture is the basis of all microstrip design.