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       Microwave circuit simulators have models for all the traditional transmission line cross-sections such as micro-strip, strip line and coplanar-wave guide (CPW). However, there are cases that the standard models do not cover. This is particularly true in multilayer environments such as MMIC''s, packages and printed circuit boards (PCB''s).
In all of these environments, we have the freedom to design transmission lines that may not fit any of the conventional definitions. We may need micro-strip or CPW with a dielectric overlay for an MMIC or package. In a multilayer PCB, we may have a buried strip line conductor with grounded isolations strips in close proximity and upper/lower ground planes that are not symmetrically located. This configuration is some kind of strip line/CPW hybrid depending on the strip widths and the various ground plane spacings. For any strip configuration, the effects of strip thickness and the actual
strip cross-section due to etching or other processing are also of interest. Fortunately, some inexpensive electromagnetic field-solvers can help us compute the impedance and effective dielectric constant of arbitrary transmission line cross-sections. For good engineering results, our goal is 1% to 2% accuracy. These tools also give us the freedom to ask many interesting "what if" questions regarding the configuration we have chosen.
II. Types of Field-Solvers
The use of electromagnetic field-solvers in microwave circuit design has increased dramatically in the last decade. We can divide these tools into three broad classes. We characterize each class not by the numerical method used but rather by the order of the geometry they can analyze. Within each class, any number of different numerical methods may be used. Model building time, numerical effort and solution time all increase dramatically as the geometry gets more complex.
The type of field-solver we will focus on in this work is the 2D cross-section solver, Fig. 1(a). This solver is suitable for strips or slots with uniform cross-section going into the page. For problems with one or two signal conductors, it is quite easy to compute the impedances and phase velocities of each mode. Numerically, we only have to consider a small, bounded 2D region, so solution time generally will not be an issue. Many of these field-solvers are stand-alone tools while some are integrated within a linear/non-linear simulator. Table I is a partial listing of 2D field-solvers.
If we want to solve more general planar circuits, we generally move to a 2.5D planar-solver, Fig. 1(b). These tools are also called 3D mostly planar solvers by some software vendors. With these tools, an arbitrary number of homogeneous dielectric layers are allowed. An arbitrary planar metal pattern can then be placed at the interface between any pair of dielectric layers. Via metal can also be used to connect metal layers. This is where the half

Finally, we come to the 3D field-solver, which allow us to analyze a truly arbitrary 3D structure, Fig. 1. the basic formulation for these solvers assumes a closed, metallic boundary around the solution region. However, an open environment can be approximated using various types of absorbing boundary. While these tools offer great flexibility, the penalty

Finally, we come to the 3D field-solvers, which allow us to analyze a truly arbitrary 3D structure, Fig. 1(c). The basic formulation for these solvers assumes a closed, metallic boundary around the solution region. However, an open environment can be approximated using various types of absorbing boundaries. While these tools offer great flexibility, the penalty is modeling time and solution time. Building a model in 3D is considerably more difficult than 2D or 2.5D modeling. Numerically, we are forced to solve for the fields in the entire 3D volume, which leads to a dramatic increase in solution time compared to the other two classes of solvers. For any given problem, one of the three general solver types will offer the most efficient solution. In the course of a design project, we might use all three types of solver at some point. However, in the very early stages of a project, we are typically choosing substrate materials and developing some intuition for the range of impedances that we can realize. We might also be experimenting with non-standard transmission line cross-sections. A fast computation of impedance using an inexpensive 2D cross-section solver is quite valuable at this stage.
III. Computing Zo and eff
The two most basic parameters that define a transmission line are characteristic impedance, Zo and the velocity of propagation, vp. For single strips and coupled pairs it is easy to compute these parameters using a stand-alone 2D field-solver. Characteristic impedance and velocity of propagation are defined in terms of inductance per unit length, L and capacitance per unit length, C.
L Zo (1)

These equations are for lossless lines and ignore skin depth effects [1]. At first glance, these equations also imply that we need two different 2D cross-sections solvers, an electrostatic solver to compute C and a magnetostatic solver to compute L. We can simplify the problem if we take advantage of one special case where we know the phase velocity in advance. This special case is any air dielectric transmission line where the velocity must be the speed of light,
m/s. 10 998 . 2 8 c (3)
Substituting into the equation for velocity of propagation,
we get
o C L c 1 (4)
where Co is the capacitance per unit length of the line
when all of the dielectrics are air ( r = 1). If all the materials are also non-magnetic ( r = 1) we can solve for L
o C c L 2
1 (5)
and substitute back into the equations for Zo and vp
o
o
C C c Z 1 (6)
C
C c v o p (7)
For mixed dielectric problems like microstrip, we can define an effective dielectric constant, eff that is related to the actual velocity of propagation in the medium by
2

p
c (8)
Substituting in the equation for vp as a function of C and
Co, we get
o
eff
C
C (9)