# Single phase Bergeron transmission line

This section describes Bergeron single 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 in ohms/km, per length core and shield capacitance in farads/km, per length inductance in henries/km and execution rate in seconds.

## Description

This model represents the L and C elements in a distributed manner while resistance R is lumped. In the case when line is lossless( r = 0), model is characterized by two values: the wave propagation speed $v=\frac{1}{\sqrt{lc}}$ and the characteristic impedance ${Z}_{o}=\sqrt{\frac{l}{c}}$ .

Transport delay is $\tau =\frac{d}{v}$ , where d is the length of the line and v is the propagation speed.

The model equations are:

${i}_{s}\left(t\right)=\frac{1}{{Z}_{o}}{e}_{s}\left(t\right)+{I}_{sh}\left(t\right)$

math ${i}_{r}\left(t\right)=\frac{1}{{Z}_{o}}{e}_{r}\left(t\right)+{I}_{rh}\left(t\right)$

And two current sources are:

${I}_{sh}\left(t-\tau \right)=-\frac{1}{{Z}_{o}}{e}_{r}\left(t-\tau \right)-{i}_{r}\left(t-\tau \right)$

${I}_{rh}\left(t-\tau \right)=-\frac{1}{{Z}_{o}}{e}_{s}\left(t-\tau \right)-{i}_{s}\left(t-\tau \right)$

This set of equations can be implemented for time discrete modelling of the line if the transport delay, τ, is integer multiple of the simulation step, T , or can be approximated as such: τ = NT.

For lossy lines, total resistance $R=rd$ is lumped, $\frac{1}{4}$ at each end and $\frac{1}{2}$ in the middle of the line. When losses are taken into account, equations are obtained as:

${i}_{s}\left(t\right)=\frac{1}{Z}{e}_{s}\left(t\right)+{I}_{sh}^{\text{'}}\left(t-\tau \right)$

${I}_{sh}^{\text{'}}\left(t-\tau \right)=\frac{\frac{R}{4}}{Z}\left[-\frac{1}{Z}{e}_{s}\left(t-\tau \right)-\frac{{Z}_{0}-\frac{R}{4}}{Z}{i}_{s}\left(t-\tau \right)\right]+\frac{{Z}_{0}}{Z}\left[-\frac{1}{Z}{e}_{r}\left(t-\tau \right)-\frac{{Z}_{0}-\frac{R}{4}}{Z}{i}_{r}\left(t-\tau \right)\right]$

${i}_{r}\left(t\right)=\frac{1}{Z}{e}_{r}\left(t\right)+{I}_{rh}^{\text{'}}\left(t-\tau \right)$

${I}_{rh}^{\text{'}}\left(t-\tau \right)=\frac{\frac{R}{4}}{Z}\left[-\frac{1}{Z}{e}_{r}\left(t-\tau \right)-\frac{{Z}_{0}-\frac{R}{4}}{Z}{i}_{r}\left(t-\tau \right)\right]+\frac{{Z}_{0}}{Z}\left[-\frac{1}{Z}{e}_{s}\left(t-\tau \right)-\frac{{Z}_{0}-\frac{R}{4}}{Z}{i}_{s}\left(t-\tau \right)\right]$

where:

$Z={Z}_{o}+\frac{rd}{4}$

## 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.