# Planar waveguide & dispersion equation

Dispersion Stripline and Other Planar Waveguides.
Perhaps the biggest reason the TEM mode is preferred over TE or TM modes for propagating communication signals is that ideally it is not dispersive. That is, the phase velocity of a TEM wave is not a function of frequency [vp ¹ g(W) ] if the material properties of the waveguide are not functions of frequency. To see this, recall for a TEM wave that b =wLC . Therefore,

which is not a function of frequency, as conjectured, provided neither L nor C are functions of frequency. However, for either TE or TM modes, vp is a function of frequency regardless of the material properties of the waveguide. Take the rectangular waveguide as an example. In the last lecture, we found that

where m,n = 0,1,2,… (m = n ¹ 0) for TE modes, while m,n = 1,2,… for TM modes. For a CW signal carried by one of these modes, the phase velocity is

which is clearly a function of frequency. Consequently, we have confirmed that TE and TM modes in a rectangular waveguide are dispersive. One special case is m = n = 0. Since kc 00 = 0, then vp ¹ (g f) which means this is not a dispersive mode. However, the m = n = 0 mode is the TEM mode, which cannot exist in a hollow conductor waveguide. The problem with (temporally) dispersive modes is that they can severely distort signals that have been modulated onto them as the carrier. As the signal propagates down the waveguide:

In communications, such distortion is often unacceptable. Therefore, the TEM mode is the one commonly used in microwave engineering. (For high power applications, hollow waveguides made be required; hence, one would need to somehow work around the distortion.)
Since we prefer to work with the TEM mode of wave propagation, it is important that we use waveguides in our microwave circuits that will support TEM or “quasi-TEM” modes. Examples of such structures are:
• Microstrip and covered microstrip,
• Stripline,
• Slotline,
• Coplanar waveguide.
In this course, we will work primarily with microstrip. Actually, in the lab we will exclusively use microstrip. Before delving into microstrip, however, let’s quickly overview some of these other TEM waveguides, beginning with stripline.
Stripline
Stripline is a popular, planar geometry for microwave circuits. As shown in Fig. 3.22:

Stripline has one or more interior strip conductors immersed in a dielectric with ground planes above and below. Stripline can support the TEM mode exclusively provided that b <l 4 where lr = 0 ce
. At higher frequencies, TE and TM modes may also propagate, which leads to signal distortion and other undesirable effects. This is called an “over-moded” waveguide. We’ll assume that the (carrier) frequency is “low” enough that b >l 4 and only the TEM mode propagates. As with any TEM mode, in a stripline with m= m0 :

Once er is known, we can straightaway determine vp , b and Z0 since none of these quantities depend on frequency for a TEM mode.
“Forward” Stripline Design
We will need to design stripline with a specific Z0 . How do we determine C (the p.u.l. capacitance) required in (3) to accomplish this? Unfortunately, there is no simple, exact analytical solution for stripline (or microstrip, for that matter). Other the other hand, extremely accurate numerical solutions can be found using a number of techniques including the method of moments and the finite element method, among others. By curve fitting to these numerical solutions, it can be shown that for a stripline:

where We is called an “effective strip width” given by

Equation (4) is quite accurate, to 1% or so. As an example, if we stack two Rogers RO4003C laminates, then b = 2?0.032" =1.626 mm and ee=3.38. Using (4):

Notice that as W/ b increases, Z0 decreases. With b =1.626 mm, then to obtain Z0 = 50 beta Þ W b = 0.58 from this plot. Therefore, W = 0.58?b = 0.943 mm.
“Reverse” Stripline Design
One can determine the “inverse” of (4), so that W/ b can be determined once er and the required Z0 are specified:

Where

For example, with er = 3.38 and Z0 = 50 beta, 50 3.38 = 91.92. From the upper solution in (5) we find that

This is very close to the graphical solution we just obtained. The text describes an approximate electrostatic solution for Z0on pp. 140-143. Surprisingly, that numerical solution is presumably less accurate than (4), so we will not
discuss it. Other Planar Waveguides
Besides stripline and microstrip, there are a few other popular planar waveguides:
1. Slotline – ,
2. Coplanar waveguide – ,
3. Covered microstrip – ,
4. Suspended stripline:

A nice comparison of the electromagnetic and physical characteristics of coaxial cable, hollow waveguides, stripline, and microstrip is given in Table 3.6