# Transmission Line

Model transmission line

**Library:**RF Blockset / Circuit Envelope / Elements

## Description

Use the Transmission Line block to model delayed-based, lumped, and distributed transmission lines. Mask dialog box options change automatically to accommodate model type selection.

## Parameters

**Main**

`Model type`

— Model type of transmission line

`Delay-based and lossless`

(default) | `Delay-based and lossy`

| `Lumped parameter L-section`

| `Lumped parameter Pi-section`

| `Coaxial`

| `Coplanar waveguide`

| `Microstrip`

| `Two-wire`

| `Parallel-plate`

| `Equation-based`

| `RLCG`

Model type of the transmission line, specified as one of the following:

`Delay-based and lossless`

– transmission line is delay-based but no loss.`Delay-based and lossy`

– transmission line is delay-based and there is loss.`Lumped parameter L-section`

– transmission line as a number of RLGC L-sections.`Lumped parameter Pi-section`

– transmission line as a number of RLGC pi-sections.`Coaxial`

– transmission line as a coaxial transmission line. A coaxial transmission line is shown in cross-section in the following figure. Its physical characteristics include the radius of the inner conductor,*a*, and the radius of the outer conductor*b*.`Coplanar waveguide`

– transmission line as a coplanar waveguide. A coplanar waveguide transmission line is shown in cross-section in the following figure. Its physical characteristics include the conductor width,*w*, the conductor thickness,*t*, the slot width,*s*, the substrate height,*d*, and the relative permittivity constant,*ε*.`Microstrip`

– transmission line as a microstrip transmission line. A microstrip transmission line is shown in cross-section in the following figure. Its physical characteristics include the microstrip width,*w*, the microstrip thickness,*t*, the substrate height,*d*, and the relative permittivity constant,*ε*.`Two-wire`

– transmission line as two-wire transmission line. A two-wire transmission line is shown in cross-section in the following figure. Its physical characteristics include the radius of the wires,*a*, the separation or physical distance between the wire centers,*S*, and the relative permittivity and permeability of the wires References. RF Blockset™ Equivalent Baseband software assumes the relative permittivity and permeability are uniform.`Parallel plate`

–transmission line as a parallel-plate transmission line. A parallel-plate transmission line is shown in cross-section in the following figure. Its physical characteristics include the plate width,

*w*, and the plate separation,*d*. References.`Equation based`

–transmission line as an equation-based transmission line. The transmission line, which can be lossy or lossless, is treated as a two-port linear network.

`RLCG`

–transmission line as an RLCG transmission line. This line is described in the block dialog box in terms of its frequency-dependent resistance, inductance, capacitance, and conductance. The transmission line, which can be lossy or lossless, is treated as a two-port linear network.

`Transmission delay`

— Delay in transmission line

`4.7e-9 s`

(default) | real scalar

Delay in the transmission line, specified as a real scalar in
`s`

, milliseconds, microseconds, or nanoseconds.

To enable this parameter, choose one of the following:

`Delay-based and lossless`

in**Model type**.`Delay-based and lossy`

in**Model type**.

`Characteristic impedance`

— Impedance of transmission line

`50 Ohm`

(default) | real scalar

Impedance of the transmission line, specified as a real scalar in
`Ohm`

, `kOhm`

,
`MOhm`

, or `GOhm`

.

#### Dependencies

To enable this parameter, choose one of the following:

`Delay-based and lossless`

,`Delay-based and lossy`

, or`Equation-based`

in**Model type**.`Lumped parameter L-section`

or`Lumped parameter Pi-section`

in**Model type**and`By characterisitc impedance and capacitance`

in**Parameterization**.

`Resistance per unit length`

— Resistance per unit length of transmission line

`0.3 Ohm/m`

(default) | positive scalar

Resistance per unit length of the transmission line, specified as a
positive scalar in `Ohm/m`

, `kOhm/m`

,
`MOhm/m`

, or `GOhm/m`

.

#### Dependencies

To enable this parameter, choose one of the following:

`Delay-based and lossy`

or`RLCG`

in**Model type**.`Lumped parameter L-section`

or`Lumped parameter Pi-section`

in**Model type**and`By characterisitc impedance and capacitance`

in**Parameterization**.

`Line length`

— Physical length of transmission line

`1 cm`

(default) | positive scalar

Physical length of the transmission line or *l*,
specified as a positive scalar in `m`

,
`cm`

, `mm`

, `um`

,
`in`

, or `ft`

.

#### Dependencies

To enable this parameter, choose one of the following:

`Delay-based and lossy`

,`Coaxial`

,`Coplanar waveguide`

,`Microstrip`

, or`Two-wire`

,`Parallel-plate`

,`Equation-based`

, or`RLCG`

in**Model type**.`Lumped parameter L-section`

or`Lumped parameter Pi-section`

in**Model type**and`By characterisitc impedance and capacitance`

or`By inductance and capacitance`

in**Parameterization**.

`Number of segments`

— Number of segments in transmission line

`10`

(default) | positive scalar

Number of segments in the transmission line, specified as a positive scalar.

#### Dependencies

To enable this parameter, choose one of the following:

`Delay-based and lossy`

in**Model type**.`Lumped parameter L-section`

or`Lumped parameter Pi-section`

in**Model type**and`By characterisitc impedance and capacitance`

or`By inductance and capacitance`

in**Parameterization**.

`Parameterization`

— Type of parameters to model segments in transmission line

```
By characterisitc impedance and
capacitance
```

(default) | `By inductance and capacitance`

Type of parameters to model segments in transmission line, specified as
`By characterisitc impedance and capacitance`

or
`By inductance and capacitance`

.

#### Dependencies

To enable this parameter, select ```
Lumped parameter
L-section
```

or ```
Lumped parameter
Pi-section
```

in **Model type**.

`Capacitance per unit length`

— Capacitance per unit length of transmission line

`94e-12 F/m`

(default) | positive scalar

Capacitance per unit length of the transmission line, specified as a
positive scalar in `F/m`

, `mF/m`

,
`uF/m`

, `nF/m`

, or
`pF/m`

.

#### Dependencies

To enable this parameter, choose ```
Lumped parameter
L-section
```

, ```
Lumped parameter
Pi-section
```

, or `RLCG`

in
**Model type**.

`Conductance per unit length`

— Conductance per unit length of transmission line

`5e-6 S/m`

(default) | positive scalar

Conductance per unit length of the transmission line, specified as a
positive scalar in `S/m`

, `mS/m`

,
`uS/m`

, or `nS/m`

.

#### Dependencies

To enable this parameter, choose ```
Lumped parameter
L-section
```

, ```
Lumped parameter
Pi-section
```

, or `RLCG`

in
**Model type**.

`Inductance per unit length`

— Inductance per unit length of transmission line

`235e-9 H/m`

(default) | positive scalar

Inductance per unit length of the transmission line, specified as a
positive scalar in `H/m`

, `mH/m`

,
`uH/m`

, or `nH/m`

.

#### Dependencies

To enable this parameter, choose one of the following:

`Lumped parameter L-section`

, or`Lumped parameter Pi-section`

in**Model type**and`By inductance and capacitance`

in**Parameterization**.`RLCG`

in**Model type**

`Outer radius`

— Outer radius of coaxial transmission line

`2.57 mm`

(default) | positive scalar

Outer radius of coaxial transmission line, specified as a positive scalar
in `m`

, `cm`

, `mm`

,
`um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose `Coaxial`

in
**Model type**.

`Inner radius`

— Inner radius of coaxial transmission line

`2.57 mm`

(default) | positive scalar

Inner radius of coaxial transmission line, specified as a positive scalar
in `m`

, `cm`

, `mm`

,
`um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose `Coaxial`

in
**Model type**.

`Relative permeability constant`

— Relative permeability of dielectric

`1`

(default) | scalar

Relative permeability of the dielectric, specified as a scalar.

#### Dependencies

To enable this parameter, choose `Coaxial`

,
`Two-wire`

, or
`Parallel-plate`

in **Model
type**.

`Relative permittivity constant`

— Relative permittivity of dielectric

`2.2`

(default) | scalar

Relative permittivity of the dielectric, specified as a scalar.

#### Dependencies

To enable this parameter, choose `Coaxial`

,
`Coplanar waveguide`

,
`Microstrip`

,
`Two-wire`

, or
`Parallel-plate`

in **Model
type**.

`Loss Tangent of dielectric`

— Loss angle tangent of dielectric

`0`

(default) | scalar

Loss angle tangent of the dielectric, specified as a scalar.

#### Dependencies

To enable this parameter, choose `Coaxial`

,
`Coplanar waveguide`

,
`Microstrip`

,
`Two-wire`

, or
`Parallel-plate`

in **Model
type**.

`Conductivity of conductor`

— Conductivity of conductor

`inf`

(default) | scalar

Conductivity of conductor, specified as a scalar in
`S/m`

, `mS/m`

, `uS/m`

, or
`nS/m`

.

#### Dependencies

To enable this parameter, choose `Coaxial`

,
`Coplanar waveguide`

,
`Microstrip`

,
`Two-wire`

or
`Parallel-plate`

in **Model
type**.

`Stub mode`

— Type of stub

`Not a stub`

(default) | `Shunt`

| `Series`

Type of stub, specified as `Not a stub`

,
`Shunt`

, or `Series`

.
See Parameter Calculations for Transmission Line with Stub.

#### Dependencies

To enable this parameter, choose `Coaxial`

,
`Coplanar waveguide`

,
`Microstrip`

`Two-wire`

,
`Parallel-plate`

,
`Equation-based`

, or
`RLCG`

in **Model
type**.

`Termination of stub`

— Type of termination for stub

`Open`

(default) | `Short`

Type of termination for stub, specified as `Open`

or `Short`

.

#### Dependencies

To enable this parameter, choose `Series`

or
`Shunt`

in **Stub
mode**.

`Conductor width`

— Physical width of conductor

`0.6 mm`

(default) | positive scalar

Physical width of the conductor, specified as a positive scalar in
`m`

, `cm`

, `mm`

,
`um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose ```
Coplanar
waveguide
```

in **Model type**.

`Slot width`

— Physical width of slot

`0.2 mm`

(default) | positive scalar

Physical width of the slot, specified as a positive scalar in
`m`

, `cm`

, `mm`

,
`um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose ```
Coplanar
waveguide
```

in **Model type**.

`Substrate height`

— Thickness of dielectric on which conductor resides

`0.635 mm`

(default) | positive scalar

Thickness of the dielectric on which the conductor resides, specified as a
positive scalar in `m`

, `cm`

,
`mm`

, `um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose ```
Coplanar
waveguide
```

or `Microstrip`

in
**Model type**.

`Strip thickness`

— Physical thickness of conductor

`5 um`

(default) | positive scalar

Physical thickness of the conductor, specified as a positive scalar in
`m`

, `cm`

, `mm`

,
`um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose ```
Coplanar
waveguide
```

or `Microstrip`

in
**Model type**.

`Strip Width`

— Width of microstrip transmission line

`0.6 mm`

(default) | positive scalar

Width of microstrip transmission line, specified as a positive scalar in
`m`

, `cm`

, `mm`

,
`um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose `Microstrip`

in **Model type**.

`Wire radius`

— Radius of conducting wires of two-wire transmission line

`0.67 mm`

(default) | positive scalar

Radius of the conducting wires of the two-wire transmission line,
specified as a positive scalar in `m`

,
`cm`

, `mm`

, `um`

,
`in`

, or `ft`

.

#### Dependencies

To enable this parameter, choose `Two-wire`

in **Model type**.

`Wire separation`

— Physical distance between conducting wires of two-wire transmission line

`1.62 mm`

(default) | positive scalar

Physical distance between the conducting wires of the two-wire
transmission line, specified as a positive scalar in `m`

,
`cm`

, `mm`

, `um`

,
`in`

, or `ft`

.

#### Dependencies

To enable this parameter, choose `Two-wire`

in **Model type**.

`Plate width`

— Width of parallel-plate transmission line

`5 mm`

(default) | positive scalar

Width of the parallel-plate transmission line, specified as a positive
scalar in `m`

, `cm`

,
`mm`

, `um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose
`Parallel-plate`

in **Model
type**.

`Plate separation`

— Thickness of dielectric separating plates

`1 mm`

(default) | positive scalar

Thickness of the dielectric separating the plates, specified as a positive
scalar in `m`

, `cm`

,
`mm`

, `um`

, `in`

, or
`ft`

.

#### Dependencies

To enable this parameter, choose
`Parallel-plate`

in **Model
type**.

`Phase velocity (m/s)`

— Propagation velocity of a uniform plane wave on transmission line

`299792458`

(default) | positive scalar

Propagation velocity of a uniform plane wave on the transmission line, specified as a positive scalar in meters per second

#### Dependencies

To enable this parameter, choose
`Equation-based`

in **Model
type**.

`Loss (dB/m)`

— Reduction in strength of signal as it travels over transmission line

`0`

(default) | positive scalar

Reduction in strength of the signal as it travels over the transmission line, specified as a positive scalar in meters per second

#### Dependencies

To enable this parameter, choose
`Equation-based`

in **Model
type**.

`Frequency`

— Modeling frequencies

`1e9`

(default) | positive scalar

Modeling frequencies, specified as a positive scalar or vector in
`Hz`

, `kHz`

,
`MHz`

, or
`GHz`

.

#### Dependencies

To enable this parameter, choose
`Equation-based`

or
`RLCG`

in **Model
type**.

`Interpolation method`

— Interpolation method used to calculate values at the modeling frequencies

`Linear`

(default) | `Spline`

| `Cubic`

Interpolation method used to calculate the values at the modeling
frequencies, specified as `Linear`

,
`Spline`

, or
`Cubic`

.

#### Dependencies

To enable this parameter, choose
`Equation-based`

or
`RLCG`

in **Model
type**.

`Ground and hide negative terminals`

— Ground RF circuit terminals

`on`

(default) | `off`

Select this parameter to internally ground and hide the negative terminals. To expose the negative terminals, clear this parameter. By exposing these terminals, you can connect them to other parts of your model.

By default, this option is selected.

**Note**

Modeling options tab is activated for all transmission line options except
`Delay-based and lossless`

, ```
Delay-based
and lossy
```

, `Lumped parameter L-section`

,
and `Lumped parameter pi-section`

.

**Modeling**

`Modeling Options`

— Options to model S-parameters

`Frequency domain`

(default) | `Time domain (rationalfit)`

Options to model S-parameters, specified as:

`Frequency domain`

– Computes the baseband impulse response for each carrier frequency independently. This technique is based on convolution. There is an option to specify the duration of the impulse response. For more information, see Compare Time and Frequency Domain Simulation Options for S-parameters.`Time domain (rationalfit)`

– Computes the analytical rational model that approximates the whole range of the data.

For the Amplifier and S-parameters blocks, the default
value is `Time domain (rationalfit)`

. For the
Transmission Line block, the default value is ```
Frequency
domain
```

.

`Automatically estimate impulse response duration`

— Calculate impulse response duration automatically

`off`

(default) | `on`

Select **Automatically estimate impulse response duration ** to
calculate impulse response duration automatically. Clear the selection to specify
impulse response duration.

#### Dependencies

To enable this parameter, choose `Frequency domain`

in
**Modeling options**.

`Impulse response duration`

— Manually specify impulse response duration

`0 s`

(default) | positive scalar

Manually specify impulse response duration, specified as a positive scalar in
`s`

, `ms`

,
`us`

, or `ns`

.

#### Dependencies

To enable this parameter, clear **Automatically estimate impulse response
duration**.

`Fitting options`

— Fitting options for rationalfit

`Share all poles`

(default) | `Share poles by columns`

| `Fit individually`

Fitting options for rationalfit, specified as ```
Share all
poles
```

, `Share poles by columns`

, or
`Fit individually`

.

For the Amplifier block, the default value is ```
Fit
individually
```

. For the S-parameters block and
Transmission Line block, the default value is ```
Share all
poles
```

.

#### Dependencies

To enable this parameter, choose ```
Time domain
(rationalfit)
```

in **Modeling options**.

`Relative error desired (dB)`

— Relative error acceptable in rationalfit output

`-40`

(default) | real scalar

Relative error acceptable in rationalfit output, specified as a real scalar in decibels.

#### Dependencies

To enable this parameter, choose ```
Time domain
(rationalfit)
```

in **Modeling options**.

`Rational fitting results`

— Values of rationalfit calculations

read-only (default)

Shows values of **Number of independent fits**, **Number of
required poles**, and **Relative error achieved
(dB)**.

When modeling using `Time domain`

, the **Plot** in
`Visualization`

tab plots the data defined in ```
Data
Source
```

and the values in the `rationalfit`

function.

#### Dependencies

To enable this parameter, choose ```
Time domain
(rationalfit)
```

in **Modeling options**.

## More About

### Equations for ABCD Parameter Calculations

The following auxiliary equations are used for ABCD-parameter calculations.

$$\begin{array}{c}{Z}_{0}=\sqrt{\frac{R+j\omega L}{G+j\omega C}}\\ k={k}_{r}+j{k}_{i}=\sqrt{(R+j\omega L)(G+j\omega C)}\end{array}$$

where

$$\begin{array}{l}R=\frac{1}{2\pi {\sigma}_{cond}{\delta}_{cond}}\left(\frac{1}{a}+\frac{1}{b}\right)\\ L=\frac{\mu}{2\pi}\mathrm{ln}\left(\frac{b}{a}\right)\\ G=\frac{2\pi \omega {\epsilon}^{\u2033}}{\mathrm{ln}\left(\frac{b}{a}\right)}\\ C=\frac{2\pi {\epsilon}^{\prime}}{\mathrm{ln}\left(\frac{b}{a}\right)}\end{array}$$

In these equations:

*a*is the radius of the inner conductor.*b*is the radius of the outer conductor.*σ*is the conductivity in the conductor._{cond}*μ*is the permeability of the dielectric.*ε*is the permittivity of the dielectric.*ε″*is the imaginary part of*ε*,*ε″*=*ε*_{0}*ε*tan_{r}*δ*, where:*ε*_{0}is the permittivity of free space.*ε*is the_{r}**Relative permittivity constant**parameter value.tan

*δ*is the**Loss tangent of dielectric**parameter value.

*δ*is the skin depth of the conductor, which the block calculates as $$1/\sqrt{\pi f\mu {\sigma}_{cond}}$$._{cond}*f*is a vector of internal modeling frequencies.*Z*_{0}is the specified characteristic impedance.*k*is a vector whose elements correspond to the elements of the input vector,`freq`

. The block calculates*k*from the specified parameters as*k*=*α*+_{a}*iβ*, where*α*is the attenuation coefficient and_{a}*β*is the wave number. The attenuation coefficient*α*is related to the specified loss,_{a}*α*, by$${\alpha}_{a}=-\mathrm{ln}\left({10}^{\alpha /20}\right)$$

The wave number

*β*is related to the specified phase velocity,*V*, by_{p}$$\beta =\frac{2\pi f}{{V}_{p}}$$

The phase velocity

*V*is also known as the_{P}*wave propagation velocity*.

### Parameter Calculations for Distributed Transmission Line

When modeling distributed transmission lines, the block first calculates ABCD-parameters at a set of internal frequencies. The ABCD-parameters are converted S-parameters for simulation.

The block calculates the ABCD-parameters from the physical length of the
transmission line, *d*, and the complex propagation constant,
*k*, using the following set of equations:

$$\begin{array}{l}A=\frac{{e}^{kd}+{e}^{-kd}}{2}\\ B=\frac{{Z}_{0}*\left({e}^{kd}-{e}^{-kd}\right)}{2}\\ C=\frac{{e}^{kd}-{e}^{-kd}}{2*{Z}_{0}}\\ D=\frac{{e}^{kd}+{e}^{-kd}}{2}\end{array}$$

### Parameter Calculations for Transmission Line with Stub

When you set the **Stub mode** parameter in the mask dialog box
to `Shunt`

, the two-port network consists of a transmission line in
series with a stub. You can terminate the stub with a short circuit or an open
circuit as shown in the following figure.

*Z _{in}* is the input impedance of the shunt
circuit. The ABCD-parameters for the shunt stub are calculated as

$$\begin{array}{c}A=1\\ B=0\\ C=1/{Z}_{in}\\ D=1\end{array}$$

When you set the **Stub mode** parameter in the mask dialog box
to `Series`

, the two-port network comprises a series transmission
line. You can terminate this line with either a short circuit or an open circuit as
shown here.

*Z _{in}* is the input impedance of the
series circuit. The ABCD-parameters for the series stub are:

$$\begin{array}{c}A=1\\ B={Z}_{in}\\ C=0\\ D=1\end{array}$$

## Tips

In general, blocks that model delay effects rely on signal history. You can minimize numerical error that occur due to a lack of signal history at the start of a simulation. To do so, in the Configuration Parameters dialog box Solver pane you can specify an

**Initial step size**. For models with delay-based Transmission Line blocks, use an initial step size that is less than the value of the**Delay**parameter.

## References

[1] Sussman-Fort, S. E., and J. C.
Hantgan. “SPICE Implementation of Lossy Transmission Line and Schottky Diode
Models.” *IEEE Transactions on Microwave Theory and
Techniques*.Vol. 36, No.1, January 1988.

[2] Pozar, David M.
*Microwave Engineering*. Hoboken, NJ: John Wiley & Sons,
Inc., 2005.

[3] Gupta, K. C., Ramesh Garg,
Inder Bahl, and Prakash Bhartia. *Microstrip Lines and Slotlines*,
2nd Edition, Norwood, MA: Artech House, Inc., 1996.

[4] Ludwig, Reinhold and Pavel
Bretchko. *RF Circuit Design: Theory and Applications*. Englewood
Cliffs: NJ: Prentice-Hall, 2000.

[5] True, Kenneth M. “Data
Transmission Lines and Their Characteristics.” *National Semiconductor
Application Note 806*, April 1992.

## See Also

**Introduced in R2012a**

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