Distributed Models for Transmission Lines

Distributed models for transmission lines (for example, the W element model in Nexxim and HSPICE) are based on Telegrapher 's equations without any approximations to the partial derivatives.

where:

• V(z,t) = [v₁, v₂, v₃,..., v_n] are vectors of the signal voltages.
  • v_n is the voltage between the n^th conductor and ground.
• t is time, in seconds.
• L is the nxn matrix of self- and mutual- inductances between lines.
• R is the nxn matrix of losses resulting from series resistance.

and:

where:

• I(z,t) = [v₁, v₂, v₃,..., v_n] are vectors of the signal voltages.
  • i_n is the current flowing in the n^th conductor
• z is the position along the length of the line.
• C is the nxn matrix of self- and mutual- capacitances between lines.
• G is the nxn matrix of losses resulting from conductance between lines.

The equations are used to derive frequency domain wave equations for lossy and lossless transmission lines. These equations and the wave equations are fully coupled. To make them easier to solve in the time domain, the software uses modal analysis to decouple the equations. This allows a single, coupled, n-conductor transmission line to be modeled using a set of n uncoupled single transmission lines.

Typically, ports are defined at the ends of the lines and the Telegrapher's equations are then solved analytically to compute the port responses.

If you solve for a frequency sweep, 2D Extractor can produce a frequency dependent distributed model. Otherwise, the distributed model is independent of frequency.