Single Phase Bergeron Model

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

Figure 1. A schematic block diagram of a Single Phase Bergeron Transmission Line
Figure 2. Schematic diagram of a Single Phase Bergeron Transmission Line
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.

It is important to note that transport delay value must be greater than execution rate.

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 = Z o + r d 4

Ports

  • A1
    • Bergeron transmission line A1 port
  • B1
    • Bergeron transmission line B1 port
  • A2
    • Bergeron transmission line A2 port
  • B2
    • Bergeron transmission line B2 port

Properties

  • Unit system
    • Visible if Model definition is set to R-L
    • Specifies if metric or imperial unit system is used for parameter definition
    • Available properties are metric and imperial
  • Length
    • Transmission line length
    • Units are in km if Unit system is set to metric, or in miles if Unit system is set to imperial
  • Resistance per unit length
    • Transmission line resistance per unit length
    • Units are in Ω/km if Unit system is set to metric, or in Ω/miles if Unit system is set to imperial
  • Core capacitance per unit length
    • Cable core capacitance per unit length
    • Units are in F/km if Unit system is set to metric, or in F/miles if Unit system is set to imperial
  • Shield capacitance per unit length
    • Cable shield capacitance per unit length
    • Units are in F/km if Unit system is set to metric, or in F/miles if Unit system is set to imperial
  • Inductance per unit length
    • Cable inductance per unit length
    • Units are in H/km if Unit system is set to metric, or in H/miles if Unit system is set to imperial
  • Execution rate
    • Execution rate of the component

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.