3.4: Modeling of Transmission Lines

Describing the signal on a line in terms of \(E\) and \(H\) requires a description of the \(E\) and \(H\) fields in the transverse plane. This can be quite difficult. It is fortunate that current and voltage descriptions can be successfully used to describe the state of a transmission line at a particular position along the line. This is an approximation and the designer needs to be aware of situations where this breaks down. Such extraordinary effects are left to the next chapter. Once the transmission line descriptions have been simplified to current and voltage, \(R\), \(L\), \(G\), and \(C\) models of the line can be developed. A range of models are used for transmission lines depending on the accuracy required and the frequency of operation.

Uniform interconnects (with regular cross section) can most accurately be modeled using EM modeling software. Most commonly a specialized type of software called \(2\frac{1}{2}\)D EM is used, which only considers current flowing in the horizontal plane or in the vertical direction. A consequence is that planar interconnects are modeled as having zero thickness, as shown in Figure 3.3.2. This is reasonable for microwave interconnects, as the thickness of a planar strip is usually much less than the width of the interconnect. Many analytic formulas have also been derived for the characteristics of uniform interconnects. These formulas are important in arriving at synthesis formulas that can be used in design (i.e., arriving at the physical dimensions of an interconnect structure from its required electrical specifications). Just as importantly, the formulas provide insight into the effects of materials and geometry.

Simplification of the geometry of the type illustrated in Figure 3.3.2 for microstrip can lead to appreciable errors in some situations. More elaborate computer programs that capture the true geometry must still simplify the real situation. An example is that it is not possible to account for density variations of the dielectric. Consequently characterization of many RF and microwave structures requires measurements to “calibrate” simulations. Unfortunately it is also difficult to make measurements at microwave frequencies. Thus one of the paradigms in RF circuit engineering is requiring intuition, measurements, and simulations to develop self-consistent models of transmission lines and distributed elements.