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Transmission Line

Model delay-based or lumped parameter transmission line

Passive Devices

Description

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

1. Delay-based and lossless

2. Delay-based and lossy

3. Lumped parameter L-section

4. Lumped parameter pi-section

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

Delay-Based and Lossless

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:

v1( t ) – i1( t ) Z0 = v2( tτ ) + i2( tτ ) Z0

v2( t ) – i2( t ) Z0 = v1( tτ ) + i1( tτ ) Z0

where:

• v1 is the voltage across the left-hand end of the transmission line.

• i1 is the current into the left-hand end of the transmission line.

• v2 is the voltage across the right-hand end of the transmission line.

• i2 is the current into the right-hand end of the transmission line.

• τ is the transmission line delay.

• Z0 is the line characteristic impedance.

Delay-Based and Lossy

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.

Lumped Parameter L-Section

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.

Lumped Parameter Pi-Section

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.

Lumped Parameter Line Model Parameterization

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 · Z02

The block lets you specify either L or Z0 when using the lumped parameter model.

Basic Assumptions and Limitations

The Transmission Line model has the following limitations:

• 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.

Parameters

Model type

Select one of the following transmission line models:

• `Delay-based and lossless` — Model the transmission line as a fixed impedance, irrespective of frequency, plus a delay term, as described in Delay-Based and Lossless. This is the default method. It provides the best simulation performance.

• `Delay-based and lossy` — Model the transmission line as a number of delay-based components, connected in series via a set of resistors, as described in Delay-Based and Lossy.

• `Lumped parameter L-section` — Model the transmission line as a number of L-line segments, connected in series, as described in Lumped Parameter L-Section.

• `Lumped parameter pi-section` — Model the transmission line as a number of pi-line segments, connected in series, as described in Lumped Parameter Pi-Section.

Transmission delay

The total transmission line delay. This parameter appears for delay-based models only. The parameter value must be greater than zero. The default value is `5` ns, which is a typical value for a one-meter coaxial cable.

Characteristic impedance

The characteristic impedance of the transmission line. This parameter appears for delay-based models, and for lumped parameter models where Parameterization is ```By characteristic impedance and capacitance```. The parameter value must be greater than zero. The default value is `50` Ω.

Parameterization

This parameter appears for lumped parameter models only. Select the model parameterization method, as described in Lumped Parameter Line Model Parameterization:

• `By characteristic impedance and capacitance` — Specify values for the Characteristic impedance and Capacitance per unit length parameters. This is the default method.

• `By inductance and capacitance` — Specify values for the Inductance per unit length and Capacitance per unit length parameters.

Inductance per unit length

The effective inductance of the transmission line per unit length. For lumped parameter models where Parameterization is ```By inductance and capacitance```, this parameter appears instead of the Characteristic impedance parameter. The parameter value must be greater than zero. The default value is `220` μH/m.

Capacitance per unit length

The transmission line capacitance per unit length. This parameter appears for lumped parameter models only. The parameter value must be greater than zero. The default value is `90` pF/m.

Resistance per unit length

The total transmission line resistance (that is, the sum of the resistance for the two conducting paths) per unit length. This parameter appears for `Delay-based and lossy` and for lumped parameter models. The parameter value must be greater than zero. The default value is `0.3` Ω/m.

Insulation conductance per unit length

The conductance between the two transmission line conductors per unit length. This parameter appears for lumped parameter models only. The parameter value must be greater than, or equal to, zero. The default value is `5e-6` S/m.

Line length

The total transmission line length. This parameter appears for ```Delay-based and lossy``` and for lumped parameter models. The parameter value must be greater than zero. The default value is `1` m.

Number of segments

The number of model segments used to represent the transmission line. This parameter appears for `Delay-based and lossy` and for lumped parameter models. The parameter value must be an integer greater than, or equal to, 1. The default value is `1`.

Ports

The block has four conserving electrical ports. For a coaxial cable, the two top ports correspond to the inner conductor, and the two lower ports to the external shielding conductor.

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.