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Model delay-based or lumped parameter transmission line
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
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_{2}( t – τ ) Z_{0}
v_{2}( t ) – i_{2}( t ) Z_{0} = v_{1}( t – τ ) + i_{1}( t – τ ) Z_{0}
where:
v_{1} is the voltage across the left-hand end of the transmission line.
i_{1} is the current into the left-hand end of the transmission line.
v_{2} is the voltage across the right-hand end of the transmission line.
i_{2} is the current into the right-hand end of the transmission line.
τ is the transmission line delay.
Z_{0} is the line characteristic impedance.
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}
The block lets you specify either L or Z_{0} when using the lumped parameter model.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.