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.

Figure 1: Properties window of Bergeron single phase model

Figure 2: Symbol

Description

Figure 3: Model of the two-point lossless line

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 = 1 l c and the characteristic impedance Z o = l c .

Transport delay is τ = d v , where d is the length of the line and v is the propagation speed.

The model equations are:

i s t = 1 Z o e s t + I s h ( t )

math i r t = 1 Z o e r t + I r h ( t )

And two current sources are:

I s h t - τ = - 1 Z o e r t - τ - i r ( t - τ )

I r h t - τ = - 1 Z o e s t - τ - i s ( t - τ )

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 = r d is lumped, 1 4 at each end and 1 2 in the middle of the line. When losses are taken into account, equations are obtained as:

i s t = 1 Z e s t + I s h ' ( t - τ )

I s h ' t - τ = R 4 Z - 1 Z e s t - τ - Z 0 - R 4 Z i s ( t - τ ) + Z 0 Z - 1 Z e r t - τ - Z 0 - R 4 Z i r ( t - τ )

i r t = 1 Z e r t + I r h ' ( t - τ )

I r h ' t - τ = R 4 Z - 1 Z e r t - τ - Z 0 - R 4 Z i r ( t - τ ) + Z 0 Z - 1 Z e s t - τ - Z 0 - R 4 Z i s ( t - τ )

where:

Z=Zo+rd4

Figure 4: Schematic diagram of single phase Bergeron transmission line

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.