# Three phase Bergeron transmission line

This section describes Bergeron three phase transmission line based on a distributed parameter traveling wave model with lumped resistance.

Properties for this model are length of the line in kilometers, per length resistance and ground resistance in ohms/km, per length core and shield capacitance in farads/km, per length inductance and mutual inductance between phases in henries/km and execution rate in seconds.

## Description

In case of balanced line $(\mathrm{L}\mathrm{a}\mathrm{a}=\mathrm{L}\mathrm{b}\mathrm{b}=\mathrm{L}\mathrm{c}\mathrm{c}=\mathrm{L}\mathrm{s};\mathrm{}\mathrm{M}\mathrm{a}\mathrm{b}=\mathrm{M}\mathrm{b}\mathrm{c}=\mathrm{M}\mathrm{c}\mathrm{a}=\mathrm{M};\mathrm{}\mathrm{C}\mathrm{a}\mathrm{b}=\mathrm{C}\mathrm{b}\mathrm{c}=\mathrm{C}\mathrm{c}\mathrm{a}=\mathrm{C}\mathrm{c};\mathrm{}\mathrm{C}\mathrm{e}\mathrm{a}=\mathrm{C}\mathrm{e}\mathrm{b}=\mathrm{C}\mathrm{e}\mathrm{c}=\mathrm{C}\mathrm{e})$

mathematical model of the circuit can be written in the matrix form:

$-\left(\frac{\partial}{\partial x}\right)\left[e\right]=\left[L\right]\left(\frac{\partial}{\partial t}\right)\left[i\right]$

$-\left(\frac{\partial}{\partial x}\right)\left[i\right]=\left[C\right]\left(\frac{\partial}{\partial t}\right)\left[e\right]$

where matrices [ L ] and [ C ] are: $L\left[\begin{array}{ccc}Ls& M& M\\ M& Ls& M\\ M& M& Ls\end{array}\right]$

$C\left[\begin{array}{ccc}2{C}_{c}+{C}_{e}& -{C}_{e}& -{C}_{e}\\ -{C}_{e}& 2{C}_{c}+{C}_{e}& -{C}_{e}\\ -{C}_{e}& -{C}_{e}& 2{C}_{c}+{C}_{e}\end{array}\right]$

Model of the three phase system is transformed into three decoupled single phase line models, so called "modal" lines, where transform matrix is: $T\left[\begin{array}{ccc}1& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right]$ and inverse transform matrix is ${T}^{-1}\left[\begin{array}{ccc}\frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& -\frac{1}{3}& 0\\ \frac{1}{3}& 0& -\frac{1}{3}\end{array}\right]$

To transform matrices [ L ] and [ C ] into modal domain we use transform matrix:

$\left[{L}_{mode}\right]=\left[{T}^{-1}\right]\left[L\right]\left[T\right]$

$\left[{C}_{mode}\right]=\left[{T}^{-1}\right]\left[C\right]\left[T\right]$

For each of single phase modal lines, travel time can be found and simulation
time step,* T* , selected according to them (Note: in discrete system
simulation adjusted time steps can only be integer multiples of *T*).

The calculations previously done in Single phase Bergeron line are made in the modal domain before being converted back to phase values .

Representation of the lossless three phase power line can be expanded to model losses by adding series conductors' resistances and ground resistance.

$R\left[\begin{array}{ccc}R+{R}_{g}& {R}_{g}& {R}_{g}\\ {R}_{g}& R+{R}_{g}& {R}_{g}\\ {R}_{g}& {R}_{g}& R+{R}_{g}\end{array}\right]$

To transform [ R ] into modal domain:

$\left[{R}_{mode}\right]=\left[{T}^{-1}\right]\left[R\right]\left[T\right]$

## References

[1] Dommel, H., “Digital Computer Solution of Electromagnetic Transients in
Single and Multiple Networks,”* IEEE**®** Transactions on Power
Apparatus and Systems*, Vol. PAS-88, No. 4, April, 1969.