## Linear Models for Transmission Line Systems

This section provides information about the various types of linear models currently available in the Elements Browser to use in modeling distributed transmission line systems. It addresses basic methods and limitations of each general model type, providing you with the information required to choose between two models of different types.

In general, linear models of a given distributed transmission line system fit into the following categories:

• Discontinuity Models: These models are meant to predict electrical effects of the connections between transmission line sections. These models include bends, dimensional changes and the union of up to four transmission lines at a single junction.

• Transmission Line Models: These models are meant to model the lengths of lines between connections. These models include single lines and coupled lines.

These model types are described in the next two sections. Each section addresses the general types of models available and the limitations and advantages of models developed in that manner. Finally, a suggestion for the preferred approach to the design process is addressed in regard to the type of model used in various locations throughout the design process.

### Discontinuity Models

Quasi-Static or Empirical Models from Measured Data

The base set of linear models for discontinuities available in Cadence® AWR® Microwave Office® software are based on closed-form approximations of the electrical characteristics based on a quasi-static solution of an approximate problem, or upon fitting an equation to a limited amount of measured data. These models were developed throughout the years (late 1940's to present) by esteemed members of our profession and are typically published in technical literature. From a computer-aided design point of view, this closed-form implementation of these models is well suited to the vast number of evaluations required when optimizing or tuning over frequency. However, the accuracy of the approximation of electrical parameters varies from model to model and typically it also varies over frequency and the input parameter ranges.

• Models Based Upon Measured Data: The accuracy of models based on equations fit to a set of measured data can be far ranging and highly dependent upon the skill of the originator of the model. In each model of this type, Cadence has evaluated all of the models available in the public domain literature and has selected those which give the most accurate results. Considering the complexity of the problem, the quality of the models constructed is amazing. Consider for example, a lossless two-port device, such as a microstrip symmetric step in width, that has been parameterized by a limited set of five inputs (εr, h, freq, w1 and w2). For a lossless two-port, the electrical characteristics of the device can be uniquely described by deriving an equation for three parameters as a function of these five inputs. Now the selection of the model parameters is part of the art-in-model development: they might be S-parameters, inductances, capacitances or any other similar parameter. Once determined, you must then measure these three parameters as a function of the five input parameters. Thus you have three output parameters, which are mapped to five input parameters at discrete points (you have sampled the three output parameters in five spaces). You have to design an equation for each of these output parameters, which minimizes the error throughout this five-dimensional space. The amazing part about the development of these models is the time frame for their development (40's and on) -- a majority of this was done with a slotted line and slide rule. Having shown the daunting task associated with development of such models, you should look at their limitations. Models based on equations fit to a limited set of measured data tend to have good accuracy over the range of the parameter over which they are fit. However, these models are typically extended to approximate the performance over a wider range of input parameter ranges than the measured data set. For example, it is typical to have such a model developed on one dielectric constant (typically alumina) and then have the model extended to other dielectric constants via a scaling or other transformation of the original equations. Once we depart from the original data set on which these models are based, the associated error in the model increases. Finally, a model based on measured results tends to duplicate the original measurement error. As you can imagine, the quality and range of measurements has improved over the years. The implementation of Automatic Network Analyzers (ANA) and multi-term calibration methods has allowed the accuracy and frequency of measurement to extend to levels unheard of during the time of the development of some of these models. Thus, as you would expect, the accuracy and frequency range of the original data set on which the models were based results in degraded accuracy. This is especially true as you extend the model to frequencies beyond the original data set.

• Models Based on Closed-Form Quasi-Static Solutions: Models based on closed-form quasi-static evaluations of discontinuities have excellent performance at low frequencies. However, as the dimensions of the modeled component become large when compared to a wavelength, the accuracy of the model degrades. This results from the modeling of the stored energy of the discontinuity as equivalent capacitances or inductances where the actual reactance results from stored energy in higher-order evanescent (below cut-off) modes of the guiding structure. As in metallic waveguides, the frequency-dependence of these higher order modes do not have the same dependence as that of the modeled reactive components which cause the model to degrade over frequency. Further, the closed-form solution of Maxwell's equations typically require that the guiding structure and resulting fields conform to a standard coordinate system or a coordinate system which can be mapped to one of these systems. There is an extremely limited number of useful discontinuities in which this occurs, and most lie in the coaxial or metallic waveguide class of transmission lines. To further extend this technique to include some of the standard transmission line systems such as microstrip and stripline, an approximation of the transmission line system is used which fits into one of these coordinate systems. For example, multiple microstrip and stripline discontinuities are based on a parallel plate waveguide model, which consists of a rectangular waveguide with magnetic walls on the two sides and conductive walls on the top and the bottom. A closed-form solution of the quasi-static fields can be determined for this discontinuity and thus a lumped element equivalent circuit can be determined. However, in reaching this model, you have assumed that the approximate transmission line system has all of the traits of the original system. This difference in the actual and assumed transmission line systems results in simulation errors within the model. In some instances, the model developed using a quasi-static solution of an approximate model is augmented with measured data in order to minimize the associated error. While this is an improved model, accuracy will still suffer as frequency increases and as you extend the model out of the range of measurements taken.

#### Mode-Matching Models

A mode-matching model is very similar to a closed-form quasi-static model as discussed previously. The only difference in this method is that a full-wave solution of the model is developed using the higher order evanescent (below cut-off) modes. The term "mode-matching" originates from the fact that the fields at the discontinuity interface are expanded as a summation of the modes in each of the waveguide systems. The values of these modes are then determined such that the E and H fields at the interface are continuous (match). Due to the high number of calculations required in the mode-matching technique, you will discover that these models are not as well suited for tuning and optimization as the closed-form expressions shown in the previous section.

This form of model has excellent simulation performance for transmission line systems, which can be expressed or mapped to a regular coordinate system where expressions for all modes of the structure can be determined. However, this limits you to the use of this technique for coaxial and metallic waveguides. As with the closed-form quasi-static models, an approximate parallel plate waveguide structure can be used to simulate microstrip and stripline discontinuities and a full-wave closed-form solution can be determined. An error in the simulation results from using the approximate parallel plate waveguide to model the fields of a microstrip or stripline discontinuity.

AWR Microwave Office software currently implements four mode-matching models, which model a step and an offset step in the conductor width of a microstrip and stripline. Cadence determined that the computational expense required in these models does not result in drastic reductions in the modeling error when compared to a full EM simulation of the discontinuity and the quasi-static model. However, the microstrip and stripline offset step models available have no counterpart in a quasi-static solution and prove useful when you encounter this situation.

#### EM-based Models (X-models)

The X-models are a group of discontinuity models that use the results of full-wave electromagnetic solutions of the parameterized discontinuity in order to estimate the electrical performance of the discontinuity. These models are a result of ongoing internal research and development at Cadence and are intended to give you the most accurate discontinuity models at a computational speed adequate for tuning, optimization and yield analysis. See “X-models” for details on using X-models.

Although full-wave electromagnetic solutions are extremely time-consuming, the X-models only perform these simulations at predetermined sets of input parameters and the results are stored in a database. This database is then used to interpolate the electrical response from the stored data. Further, these models allow you to automatically generate the full database for a given discontinuity on a given substrate without a user presence. This allows you to generate the entire table for all discontinuities on a given substrate overnight or on the weekend. Once the database is filled, interpolation of the resulting electrical characteristics proceeds rather quickly.

If an X-model is available for the discontinuity you are using, this is the most accurate of all models available. As this internal research and development project at Cadence continues, you can expect to find more EM-based models available in AWR Microwave Office software.

#### Suggestions in the Use of Discontinuity Models

With a general knowledge of the advantages and limitations of the discontinuity models offered in AWR Microwave Office software, the following procedure for selecting the appropriate model is suggested:

1. In the initial conceptual stage of the design, the use of the closed-form discontinuities or no discontinuity model at all is suggested. In this stage, rapid evaluation and optimization of the circuit performance can be obtained, and several topologies investigated.

2. After a topology decision and an initial estimate of the input parameters is obtained, the discontinuity models can be added to increase the accuracy of the simulation. In this stage, the X-models should be used if available. A great time savings and reduction of data entry errors can be obtained by use of the intelligent discontinuity models which sense needed information from the surrounding parts.

3. The problem should then be re-optimized, correcting widths and lengths of the transmission line sections to negate the effects of the discontinues. If warnings occur for the discontinuity models, you should address them by changes in the circuit to eliminate the warning, or the discontinuity should be simulated using the EM simulator.

Discontinuity models function most accurately when attached to lines that match their corresponding edges. Directly connecting discontinuity models to one another reduces their accuracy.

The following table summarizes the discontinuity models sorted by system and model type:

Transmission Line System Closed-Form Transmission Line Models EM-based Models and Mode Matching Models
General GND_STRAP, VIA, WIRE, RIBBON
Interconnects TVIA
Lumped Element TFC
Microstrip MBENDA, MCROSS, MCURVE, MLEF, MLSC, MSTEP, MTAPER, MTEE, TFC2, TFR MBEND90X, MCROSSX, MLEFX, MOPENX, MRINDSBR, MSTEPX, MTEEX
Stripline SBEND, SCROSS, SCURVE, SGAP, SHOLE, SLEF, SLSC, SMITER, SSTEP, STAPER, STEE SSTEPO, SSTEP2

### Transmission Line Models

#### Closed-Form Transmission Line Models

Closed-form equations for the equivalent parameters of a given transmission line have been developed throughout history in the same manner as the discontinuity models previously mentioned. When compared to discontinuities, a greater variety of transmission lines either fit into a regular coordinate system or can be mapped into one of these systems. This is due to reduction of the problem to a 2-dimensional one, since the transmission line is assumed to be infinitely long with no changes in the transverse dimensions of the structure. Closed-form and quasi-static solutions of the characteristic parameters of a transmission line system have been determined for microstrip, stripline and co-planar waveguide among others. Typically, these systems make approximations about the problem being solved in order to reach these closed-form solutions. Such approximations may come in the form of one of the following: perfect electrical conductors, conductor thickness equal to zero, or transverse electro-magnetic (TEM) wave. Beginning with these base forms of the equations, corrections based on variational methods, measurements results, or full-wave electromagnetic simulation are used to estimate the effects of losses, metal thickness and dispersion. Again, the accuracy of these corrections varies considerably from one application to the next and must be viewed on an individual basis.

Although transmission line models are more easily obtained than discontinuity models for the same system, not all desired transmission line models can be obtained in this manner. Typically, these include coupled line models, which do not have a plane of symmetry. Often, these models are obtained by fitting equations to samples of measured data or data obtained via numerical electromagnetic simulations. The range of the sampled data and the skill of author at fitting the equations once again limits the accuracy of such models.

#### 2-Dimensional Method of Moments Based Models (Advanced Numerical Models)

LINPAR is a commercially available stand-alone software package from Artech House Publishers. The primary function of this software is to determine the electrical characteristics of transmission line models such as the characteristic impedance, propagation and attenuation constants for single lines. Further, this same analysis can be used to determine the characteristics of coupled- or multiple-coupled lines. This analysis is performed via a quasi-static electromagnetic simulation of the cross-section of the transmission line section. The method of analysis used is the Method of Moments (MoM) implementing the Galerkin Method of basis and testing functions. AWR Microwave Office software has integrated this simulation engine into a variety of transmission line topologies available to you.

The major benefit of these quasi-static MoM transmission models is the ability to model many transmission line systems which do not have adequate closed-form solutions. Examples of the types of transmission line systems that can be modeled using this method include: multiple coupled line of all types, co-planar waveguide (CPW), coupled CPW, suspended stripline, covered microstrip, and inverted microstrip. Further, this method also allows you to model the effects of parameters sometimes neglected or approximated in the closed-form solutions. An example of this is the ability of this method to model thick conductors.

The limitations of transmission line models incorporating this analysis are primarily due to two factors. First, calculation of the electrical parameters of the model requires an electromagnetic simulation, which can be computationally expensive. You should expect the simulation speed to decrease when using one of these models. Secondly, the performed electromagnetic simulation is only quasi-static, so you should expect the results to degrade when the cross-sectional dimensions of the transmission line approach a fraction of a wavelength. In transmission lines, this deviation from the quasi-static or DC characteristics is termed "dispersion". Importantly, you should realize that transmission line models constructed via a closed-form solution can model dispersion.

When choosing between a transmission line model developed using a closed-form solution and a quasi-static MoM, you must determine if the effects of dispersion will significantly influence the results. Importantly, dispersion only occurs for mixed dielectric transmission line structures such as microstrip. Single dielectric transmission lines such as coax and stripline only have dispersion as second order effects due to the frequency-dependent materials of which it is constructed.

#### Suggestions in the Use of Distributed Transmission Line Models

With a general knowledge of the advantages and limitations of the transmission line models offered in AWR Microwave Office software, the following procedure for selecting the appropriate model is suggested.

1. Before entering the initial design stage, compare the difference in performance of all available models over the design frequency range and at a frequency where dispersion can be ignored. This allows you to evaluate the electrical effects of dispersion and of the additional parameters available with the numerical models.

2. Once a familiarity with the model limitations is determined for your range of input parameters, an engineering judgment can be made on which model you should use for the initial design phase.

3. Finally, further accuracy can be obtained from this initial investigation by adding correction factors to accommodate the effects of dispersion or other input values. For example, you might notice that the addition of conductor thickness changes the even and odd mode impedance of a pair of coupled lines by ~2ohms. Thus, an effective change in the dispersion model parameters can then be obtained to realize this same change.

The following table summarizes distributed transmission line models sorted by system.

Transmission Line System Closed-Form Transmission Line Models 2D MoM-Based Models (Advanced Numerical Models)
Coplanar Waveguide   CPW1LIN, CPW2LIN, CPW3LIN
Microstrip MLIN, MACLIN, MCLIN, MCFIL M2CLIN, M3CLIN, M4CLIN, M5CLIN, M6CLIN, M7CLIN, M8CLIN,MEMLI
Stripline SLIN, SCLIN S1LIN, S2CLIN, S3CLIN, SEMLIN