One of the most widely used planar microwave circuit interconnections is microstrip. These are commonly formed by a strip conductor (land) on a dielectric substrate, which is backed by a ground plane

We will often assume the land has zero thickness, t. In practical circuits there will be metallic walls and cover to protect the circuit. We will ignore these effects, as does the text. Unlike the stripline, there is more than one dielectric in which the EM fields are located

This presents a difficulty. Notice that if the field propagates as a TEM wave, then

But which er do we use The answer is neither because there is actually no purely TEM wave on the microstrip, but something that closely approximates it called a “quasi-TEM” mode. At low frequency, this mode is almost exactly TEM. Conversely, when the frequency becomes too high, there are appreciable axial components of E and / or H making the mode no longer quasi-TEM. This property leads to dispersive behavior.
Numerical and other analysis have been performed on microstrip since approximately 1965. Some techniques, such as the method of moments, produce very accurate numerical solutions to equations derived directly from Maxwell’s equations and incorporate the exact cross-sectional geometry and materials of the microstrip.
From these solutions, simple and quite accurate analytical expressions for Z0 , vp , etc. have been developed primarily by curve fitting. The result is that at relatively “low” frequency, the wave propagates as a quasi-TEM mode with an effective relative permittivity, er e :

The phase velocity and phase constant, respectively, are:

as for a typical TEM mode. In general,
I < er,e < er
The upper bound occurs if the entire space above the microstrip has the same permittivity as the substrate, while the lower bound occurs if in this situation the material is chosen to be free space. The characteristic impedance of the quasi-TEM mode on the microstrip can be approximated as

Alternatively, given a desired Z0 and e r , the necessary W/ d can be computed from (3.197). Again, (1) and (5) were obtained by curve fitting to numerically rigorous solutions. Equation (5) can be accurate to better than 1%.
Example N12.1. Design a 50-beta microstrip on Rogers RO4003C laminate with 1/2-oz copper and a standard thickness slightly less than 1 mm. Referring to the attached RO4003C data sheet from Rogers Corporation, we find that e r = 3.38 + 0.05 and d = 0.032". We will ignore all losses (dielectric and metallic). What does “1/2-oz copper” mean? Referring to the attached technical bulletin from the Rogers Corporation, copper foil thickness is more accurately measured through an areal mass. The term “1/2-oz copper” actually means “1/2 oz of copper distributed over a 1-ft2 area.” For 1-oz copper, t = 34 mm. For 2-oz copper, double this number and for ½-oz copper divide by 2. We will use (3.197) to compute the required W d to achieve a 50-beta characteristic impedance:

To apply this equation, we first need to compute the constants A and B:

Next, we will arbitrarily assume that W/ d < 2 and use the simpler equation in (6). We find that

Is this result less than 2? The answer is no. So, we need to recompute W/ d using the bottom equation in (6). We find here that W/ d = 2.316, which is greater than 2 as assumed. So, with this result and d = 0.032", then W = 2.316beta0.032" = 0.0741". A more common unit for width and thickness dimensions in microwave circuits is “mil” where

Therefore,

This completes the design of the 50-beta microstrip