## Model Lossy Transmission Lines in Serial Link Designer

The Serial Link Designer app uses a frequency dependent RLGC model to represent a lossy transmission line model. The app uses a 2-D field solver with a transmission line editor. Using the transmission line editor, you can enter the cross-section of different model types. You can also define a table-driven loss model that you can import to model the transmission line.

### Types of Lossy Transmission Line Models

#### Single Conductor and Differential Lossy Transmission Line Models

You can create single conductor and differential lossy transmission line models for these model types:

• Simple lossy transmission line

• Microstrip

• Buried microstrip

• Stripline

The parameters in each model type define the relationship of the physical geometries for the transmission line model. You can also define etch shape and enable tabbed routing.

If you need to combine multiple dielectrics in the actual physical stackup in a single dielectric in the editor, combine them by their proportional thickness. For example, for a stripline with three dielectrics, the single εr is: `${\epsilon }_{r}=\frac{{\epsilon }_{r1}{\text{H}}_{\text{1}}+{\epsilon }_{r2}{\text{H}}_{\text{2}}+{\epsilon }_{r3}{\text{H}}_{\text{3}}}{{\text{H}}_{\text{1}}{\text{+H}}_{\text{2}}{\text{+H}}_{\text{3}}}$`

#### Coupled Lossy Transmission Line Model Types

You can create single-ended and differential coupled lossy transmission line models for these model types:

• Microstrip

• Buried microstrip

• Stripline

To create a coupled model, you have to enable the parameter Coupled and set the parameter Aggressor to an integer value in the Lossy Transmission Line Editor dialog box. For single conductors, set the aggressor value between 1 and 20. For differential conductors, set the aggressor value between 1 and 9.

### Analytical RLGC Model

#### Formulating Single-Line Models

For single-line case, the parameters R, L, G, and C are scalar.

From Telegrapher's equation:

`$\begin{array}{l}\frac{\partial V}{\partial x}=-RI-2\pi fLI\\ \frac{\partial I}{\partial x}=-GV-2\pi fCV\end{array}$`

The solution to the coupled partial differential equations results in two waves travelling in opposite directions along the x-axis. The solution can be written as:

`$\begin{array}{l}{Y}_{0}V={W}_{+}{e}^{-\gamma x}+{W}_{-}{e}^{-\gamma \left(l-x\right)}\\ I={W}_{+}{e}^{-\gamma x}-{W}_{-}{e}^{-\gamma \left(l-x\right)}\end{array}$`

where ${Y}_{0}\equiv \sqrt{\frac{G+2\pi fC}{R+2\pi fL}}$, $\gamma \equiv \sqrt{\left(G+2\pi fC\right)\left(R+2\pi fL\right)}$ and l is the length of the transmission line.

Evaluate these equations at boundary conditions x=`0` and x=l and set ${X}_{d}\equiv {e}^{-\gamma l}$:

`$\begin{array}{l}{Y}_{0}V\left(0\right)={W}_{+}+{X}_{d}{W}_{-}\\ {Y}_{0}V\left(l\right)={W}_{-}+{X}_{d}{W}_{+}\\ I\left(0\right)={W}_{+}-{X}_{d}{W}_{-}\\ I\left(l\right)={W}_{-}-{X}_{d}{W}_{+}\end{array}$`

Convert the frequency domain equations to time domain to perform transient analysis in SPICE:

`$\begin{array}{l}{Y}_{0}\ast V\left(0,t\right)={W}_{+}\left(t\right)+{X}_{d}\ast {W}_{-}\left(t\right)\\ {Y}_{0}\ast V\left(l,t\right)={W}_{-}\left(t\right)+{X}_{d}\ast {W}_{+}\left(t\right)\\ I\left(0,t\right)={W}_{+}\left(t\right)-{X}_{d}\ast {W}_{-}\left(t\right)\\ I\left(l,t\right)={W}_{-}\left(t\right)-{X}_{d}\ast {W}_{+}\left(t\right)\end{array}$`

Here, W+(t) is the wave that launches from the x=`0` end of the transmission line and travels towards the x=l end. Similarly, W-(t) is the wave that launches from the x=`l` end of the transmission line and travels towards the x=0 end.

Solve for W+(t) and W-(t):

`${W}_{+}\left(t\right)=\frac{1}{2}\left({Y}_{0}\ast V\left(0,t\right)+I\left(0,t\right)\right)$`
`${W}_{-}\left(t\right)=\frac{1}{2}\left({Y}_{0}\ast V\left(l,t\right)+I\left(l,t\right)\right)$`

#### Multi-Conductor Line

For multi-conductor line case, the parameters R, L, G, and C are appropriately sized symmetric positive-definite matrices. Using the `sqrtm` and `mldivide` functions to find the square root and division results, the equations for Y0 and γ becomes:

`γ=sqrtm(G+2πfC)(R+2πfL)`

and

`Y0=γ\(G+2πfC)`

And using the `expm` to find the exponential, the equation for Xd becomes:

`Xd=expm(-γl)`

These equations support the single line case, the symmetric two-line case, and the multiple line structure that comes from a uniform dielectric. But in general, the modes change with frequency. So removing the delay according to the high-frequency mode shape leads to ripples in the high-frequency response.

### Table-Driven Loss Model

A table-driven loss model is table of material loss parameters as a function of frequency. This table is used to scale the effect if the skin resistance (R) matrix, and to replace the behavior of the dielectric loss (G) matrix. Both the conductor losses and dielectric losses of a material are described in a table of loss parameters as a function of frequency. This table is expressed as a CSV formatted spreadsheet file.

#### File Format

Specification Frequency:  The table contains a row whose first cell contains the string `f_spec` and whose second cell contains a positive real value. This is the specification frequency (in Hz) at which the value of the real part of the dielectric constant is specified. If this row is not present, then the default value for the specification frequency is 1 GHz. The specification frequency is also the precise frequency at which the inductance and capacitance of any associated transmission line are defined.

Dielectric Loss:  The table contains a row whose first cell contains the string `er` and whose second cell contains a positive real value greater than or equal to 1.0. This is the real part of the dielectric constant at the specification frequency. If this row is not present, then the default value is 1.0.

Loss Data:  There must be a column header row before the beginning of the data. The first column must be labeled `Freq` and contain the frequencies (in Hz) for the data values. The subsequent columns can be:

• `R_factor` — The resistance factor used in both the Hammerstad and Huray models to scale the skin resistance of a perfectly smooth conductor to account for the effect of conductor roughness. The nominal value is 1.0, and the value is expected to increase to values somewhat larger than 1.0. A value less than 1.0 generates a warning message. A value less than zero results in an error message and aborts the calculation.

• `tande` — The dielectric loss tangent, using the definition ${\epsilon }_{r}={\epsilon }^{\text{'}}\left(1-j\mathrm{tan}\delta \right)={\epsilon }^{\text{'}}-j{\epsilon }^{\text{'}\text{'}}$.

• `er` — The real part of the complex dielectric constant, defined as ε'. If this optional column is present, it determines an appropriate value for the magnitude of the dielectric constant at the very high frequencies. A single value is chosen that results in the best fit to the data in this column.

Note

The contents of this column supersedes the value given for the dielectric constant at the specification frequency.

Each data row must be complete. A row with fewer entries than the number of column headers results in an error message and aborts the run.

#### Example Table with `er` Column

A loss model table with `er` defines a starting set of values for er in addition to values for the resistance factor and dielectric loss tangent. In this example, the specification frequency is the default 1 GHz.

```* Example materials properties table with er column Freq,R_factor,tande,er 1.0e6,1.01,0.003,4.05 1.0e7,1.03,0.007,4.02 1.0e8,1.06,0.009,4.00 1.0e9,1.12,0.01,3.97 2.0e9,1.15,0.01,3.97 5.0e9,1.23,0.011,3.96 1.0e10,1.31,0.012,3.95 2.0e10,1.4,0.013,3.94```

#### Example Table with Frequency

A loss model table with specification frequency uses the `f_spec` instead of the `er` column. In this example, the specification frequency is the default 1 GHz.

```* Example materials properties table with specification frequency f_spec,2e9 er,3.97 Freq,R_factor,tande 1.0e7,1.03,0.007 1.0e9,1.12,0.01 2.0e9,1.4,0.013```

### Editing Lossy Conductor Line Models

You can edit lossy transmission lines and coupled models for pre-layout analysis using the Lossy Transmission Line Editor dialog box. There are two modes for editing or creating models using the Lossy Transmission Line Editor dialog box:

• Standalone library mode — Quickly create and edit different types of lossy models and store them in a library location. You can create a common library directory and access the model from multiple projects. You can also reuse the models across different designs by copying them and pasting them in the new design. If you edit any model in the common library, the changes in the model appear in all the projects where the model is used.

To access the dialog box, select Tools > Lossy Transmission Line Editor from the app toolstrip.

• Interactive mode — Interactively use the Lossy Transmission Line Editor in the schematic to create and edit lossy models and coupled models on the fly for a selected transmission line. To access the editor, right click on the w-line element symbol in the schematic and select Edit T-line Properties.