Frequency Independent Lumped Circuit Model

Lumped circuit models attempt to represent the behavior of a transmission line by approximating the Telegrapher's equations used to solve Distributed Models. A finite difference expression replaces the partial derivative with respect to the spatial coordinate. The resulting equations can be realized using lumped RLC circuit elements.

Lumped circuit models are an approximation to the Telegrapher's equations. In the frequency domain, the equations for a single line are:

The lumped approximation makes use of a finite difference formula to eliminate the partial derivative with respect to z. We approximate the partial derivative of the current at z as:

This can be rewritten as:

Inserting this into the first Telegrapher's equation yields:

This is equivalent to the following Kirchhoff's Voltage Law (KVL) equation:

This equation can be realized using the following lumped equivalent circuit:

The two circuit models can then be combined into an RLC "ladder" equivalent network to model the entire transmission line:

The ladder model above can be generalized to handle coupled systems of transmission lines. Each line will have its own ladder network, and in addition to the "self" RLC elements there will be coupling capacitances and mutual inductances between them.

To export a circuit model that is frequency independent and lumped:

1. Follow the general procedure for exporting a circuit equivalent.
2. Select Capacitance and/or Conductance.
3. Select Inductance and/or Resistance.
4. Choose Lumped to model the transmission line as a lumped SPICE equivalent. Increasing the number of cells allows you to more accurately model faster rise times using additional lumped elements.

The following figure shows the ladder network for a frequency independent lumped model: