Series connection of transmission lines described by telegrapher's equations

Question

My issue concern the definition of series connection of multiple transmission lines described by the so called constant parameter model, to be next simulated in EMTP-RV. Let us take as an example two transmission lines of length respectively $$\Delta x_1,\Delta x_2$$ described by the telegapher's equations

$$ L_i\frac{\partial I_i(x,t)}{\partial t}=-R_iI_i(x,t)+\frac{\partial V_i(x,t)}{\partial x}\\ C_i\frac{\partial V_i(x,t)}{\partial t}=-G_iV_i(x,t)+\frac{\partial I_i(x,t)}{\partial x}. $$

with i=1,2. Is it possible to obtain an equivalent aggregated model in the form of telegrapher's equation via linear combination of the two systems? By setting for example $$\Delta x_{eq}=\Delta x_1+\Delta x_2$$ which are the corresponding values of $$L_{eq},C_{eq},R_{eq},G_{eq}$$ ?

Answer 1

Is it possible to obtain an equivalent aggregated model in the form of telegrapher's equation via linear combination of the two systems?

No, it's not possible.

Reason: at the junction of two real but different transmission lines, there will be a reflection coming back to the source due to the disparity of the two characteristic impedances and, this cannot be adequately modelled by a single "new and modified" transmission line because it will not naturally produce reflections at some distance along its length.

Answer 2

As Andy suggests, the discontinuity must be modeled. This upset of 1_D energy movement needs to keep track of bi_directional energy, which is not an assumption or intention of the standard Telegrapher maths.