1

ln x = x + 1

The solution of this equation is x = ro /r i = 3.591. By substituting this in (3.79), we obtain Z 0 = 76.7V.

3.8 Microstrip Line

A microstrip line consists of a metal strip on one side and a ground plane on the other side of a substrate, as shown in Figure 3.13. The substrate is made of a low-loss dielectric material such as polytetrafluoroethylene (Teflon), aluminum oxide (alumina), or quartz.

A pure TEM wave mode can propagate in a microstrip line only if all fields are in the same medium. Then the solution for the field can be derived from Laplace’s equation. In a case where the nonstatic fields are in two different media, the field has also longitudinal components. At low frequencies, or more precisely when l >> h, the fields are nearly the same as those in a static case, and we call them quasi-TEM. However, the analytical solution

Figure 3.13 The cross section of a microstrip line and the characteristic impedance Z 0 as a function of the ratio of strip width to substrate height w /h for different substrate materials.

The width of the strip corresponding to a known Z 0 is obtained for w /h ≤ 2 from

When w /h ≥ 2,

wh ≈ p2 HB − 1 − ln (2B − 1) + er2−er 1 Fln (B − 1) + 0.39 − 0e.61r GJ (3.91)

where

In the preceding equations, it has been assumed that the thickness t of the strip is very small. In practice, the capacitance of the line per unit length will be slightly larger due to the finite thickness t compared to the case t = 0. This will lower the characteristic impedance. Due to the extra capacitance, the width of the strip seems to increase by

where D = 2h /t , when w /h ≥ 1/(2p ), and D = 4pw /t, when w /h ≤ 1/(2p). Equation (3.93) is valid for t < h and t < w /2. Equations (3.86), (3.88), (3.89), and (3.91) can be used for a strip having a finite thickness by replacing w by we = w + Dwe .

64 Radio Engineering for Wireless Communication and Sensor Applications

The sources of losses in a microstrip line are:

•Conductor loss in the strip and ground plane;

•Dielectric and conduction losses in the substrate;

•Radiation loss;

•Surface wave loss.

Assuming a constant current distribution over the strip width, the

The accuracy of this equation is best for a wide strip. In practice, the value of the surface resistance R s is larger than the theoretical one. For example, the surface roughness of the substrate increases R s . The thickness of the conductors should be at least four times the skin depth ds = √2/(vmsm ). A metallization thinner than twice the skin depth would yield excessive attenuation.

Dielectric loss is usually much lower than conductor loss. The attenua-

where l0 is the wavelength in free vacuum.

Discontinuities of the microstrip line produce radiation to free space. For a given line, radiation loss increases rapidly as the frequency increases. To avoid leakage and interference, microstrip circuits are usually shielded within a metal case. In microstrip antennas, leaking radiation is harnessed into use.

Surface waves are waves that are trapped by total reflection within the substrate. They may produce unwanted radiation from the edges of the substrate and spurious coupling between circuit elements.

Example 3.5

Find the width of the strip for a 50-V microstrip line. The substrate has a thickness of h = 0.254 mm and a relative permittivity of er = 9.7. The