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Delay-based or lumped parameter transmission line

**Library:**Simscape / Electrical / Passive / Lines

The Transmission Line block lets you choose between the following models of a transmission line:

Delay-based and lossless

Delay-based and lossy

Lumped parameter L-section

Lumped parameter pi-section

Distributed parameter line

The first option provides the best simulation performance, with options 2, 3 and 4 requiring progressively more computing power.

This first option, `Delay-based and lossless`

, models the
transmission line as a fixed impedance, irrespective of frequency, plus a delay
term. The defining equations are:

v( _{1}t ) –
i( _{1}t )
Z =
_{0}v( _{2}t –
τ ) + i( t –
τ )
Z_{0} | (1) |

_{2}

v_{2}( t ) –
i( _{2}t )
Z_{0} =
v( _{1}t –
τ ) + i(
_{1}t – τ )
Z_{0} | (2) |

where:

*v*is the voltage across the left-hand end of the transmission line._{1}*i*is the current into the left-hand end of the transmission line._{1}*v*is the voltage across the right-hand end of the transmission line._{2}*i*is the current into the right-hand end of the transmission line._{2}*τ*is the transmission line delay.*Z*is the line characteristic impedance._{0}

To introduce losses, the second option, ```
Delay-based and
lossy
```

, connects *N* delay-based components, each
defined by the above equations, in series via a set of resistors, as shown in the
following illustration.

*N* is an integer greater than or equal to 1. *r* = *R* ·
*LEN* / *N*, where *R* is the line resistance per unit length
and *LEN* is the line length.

The following block diagram shows the model of one L-line segment.

The lumped parameter parameterization uses *N* copies of the
above segment model connected in series.

Parameters are as follows:

*R*is line resistance per unit length.*L*is the line inductance per unit length.*C*is the line capacitance per unit length.*G*is the line conductance per unit length.*LEN*is the length of the line.*N*is the number of series segments.

The following block diagram shows the model of one pi-line segment.

The lumped parameter parameterization uses *N* copies of the
above segment model connected in series. The parameters are as defined for the
L-section transmission line model. Unlike the L-section model, the pi-section model
is symmetric.

The lumped-parameter models (L-section or pi-section) are the most challenging to simulate, typically needing many more segments (greater N) than for the delay-based and lossy model [1].

Cable manufacturers do not typically quote an inductance value per unit length, but instead give the characteristic impedance. The inductance, capacitance, and characteristic impedance are related by:

L = C ·
Z_{0}^{2} | (3) |

The block lets you specify either *L* or
*Z*_{0} when using the lumped parameter
model.

A distributed parameter line gives more accurate simulation at particular
frequency compared to the other options provided by this block. However, this is not
a frequency-dependent model. The accuracy of the model will drop outside the
frequency point specified in the **Frequency used for rlcg
specification** parameter.

For a frequency-dependent transmission line model, see Frequency-Dependent Overhead Line (Three-Phase).

The electromagnetic behavior of a multiconductor transmission line is described by the telegrapher's equation.

${I}_{2}-{Y}_{c}{V}_{2}=-H({I}_{1}+{Y}_{c}{V}_{1})$ | (4) |

${I}_{1}-{Y}_{c}{V}_{1}=-H({I}_{2}+{Y}_{c}{V}_{2})$ | (5) |

Define:

${I}_{sh,1}={Y}_{c}{V}_{1}$ — Shunt current vector produced at terminal 1 by injected voltages

*V*_{1}${I}_{sh,2}={Y}_{c}{V}_{2}$ — Shunt current vector produced at terminal 2 by injected voltages

*V*_{2}${I}_{rfl,1}=\frac{1}{2}({I}_{1}+{Y}_{c}{V}_{1})$ — Reflected currents of terminal 1

${I}_{rfl,2}=\frac{1}{2}({I}_{2}+{Y}_{c}{V}_{2})$ — Reflected currents of terminal 2

You can then rewrite and solve equations 4 and 5:

${I}_{1}={I}_{sh,1}-2H{I}_{rfl,2}$

${I}_{2}={I}_{sh,2}-2H{I}_{rfl,1}$

The following block diagram shows the equivalent circuit for a distributed parameter line.

For the lumped parameter options, MathWorks recommends that you use a trapezoidal solver such as

`ode23t`

. This is because lumped parameter transmission models have very lightly damped internal dynamics, which are best suited to trapezoidal solvers for numerical accuracy.The lumped parameter pi-section model has a parallel capacitor at both ends. This means that you should not connect it directly to an ideal voltage source, that is, a source with no internal resistance. The lumped parameter L-section model, however, has a series input resistor, and therefore you can connect it directly to an ideal voltage source.

Refer to the figure for port locations.

[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.