Transmission Line (3ph)
Purpose
3-phase transmission line
Library
Electrical / Passive Components
Description
This component implements a three-phase transmission line. A transmission line is characterized by a uniform distribution of inductances, resistances and neutral capacitances along the line. In multi-wire lines, there are also uniformly distributed mutual inductances and coupling capacitances.
The user has the choice between two different implementations: one with series-connected pi sections of lumped elements and another one with distributed parameters based on traveling wave theory. A stiff solver is recommended for simulating models containing this component.
Pi-Section Line
In many cases, the uniformly distributed parameters of a transmission line can be approximated by a series of pi sections consisting of lumped inductors, capacitors and resistors. The figure below illustrates a single pi section exemplified for a 2-phase line. Depending on the desired fidelity at higher frequencies, the number of series-connected pi sections can be configured.
Let be the length of the line and
the number of pi sections representing the
line. The inductance
, the resistance
, the neutral capacitance
as well as the coupling capacitances
and mutual inductances
of the discrete elements can then be calculated from their per-unit-length
counterparts
,
,
,
and
using the following equations:
It is possible to specify the parameters for each phase individually in order to model asymmetric lines. In this case, the parameters must be provided in vector format. Otherwise, the parameter can be a scalar assigning the same value to all phases.
Distributed Parameter Line
The implementation of a distributed parameter line is based on the traveling wave theory, which describes the time delay phenomenon. This approach is numerically more efficient due to the absence of numerous state variables and should be used in large models.
Modeling asymmetric lines is not supported, therefore all parameters need to be scalar.
Single-Phase Lossless Line




Since a wave entering the sending end "s" of the line must remain unchanged when it arrives at the receiving end "r" (and vice versa), the following expression is derived:
where
with surge impedance and travel time
. This model can be represented by a two-port equivalent circuit, where the electrical
conditions at port "s" are transferred after a time delay
to port "r' via the controlled current source
.
Approximation of Series Resistance
Since the shunt conductance is usually negligible, the series resistance is
responsible for the major part of the power losses. Such series resistance can
be approximated by three lumped resistors, two of which with the value
are placed at both ends of the line while one with the value
is placed in the middle.
is the total series resistance of the line. After aggregation and substitution, the
original expression of the two equivalent current source becomes:
with and
.
Three-Phase Line
The differential equations of a 3-phase system with vector variables ,
can be expressed as:
Under the assumption of symmetrical phase parameters, the per unit length inductance, capacitance and resistance can be written in matrix form:
The presence of off-diagonal elements (mutual inductance and coupling
capacitance) in the matrix make it difficult to solve the equation system.
However, this can be overcome with the help of modal transformations. If the
differential equations are multiplied by a transformation matrix on the left
side
with
the off-diagonal elements of the inductance, capacitance and resistance matrix can be eliminated:
Thus the original system, in which the three phases are coupled, has
been converted to three decoupled systems in the modal domain (denoted as
,
,
). They can be treated separately in the same way as the single-phase system.
The simulation output in the modal domain should be eventually transformed
back into the phase domain via the inverse of the matrix .
Parameters
- Self inductance per unit length
- The series self inductance
per unit length. If the length
is specified in meters (m) the unit of
is henries per meter (H/m).
For a pi-section line, the self inductance can be specified individually per phase by providing a 3-element vector of the form
.
- Mutual inductance per unit length
- The series mutual inductance
per unit length. If the length
is specified in meters (m) the unit of
is henries per meter (H/m).
In a pi-section line, the mutual inductances
between the i-th and j-th phase can be specified individually by providing a 3-element vector
containing the upper triangular coupling matrix.
- Resistance per unit length
- The series resistance
per unit length. If the length
is specified in meters (m) the unit of
is ohms per meter (
/m).
For a pi-section line, the parameter can be a vector of the form
.
- Neutral capacitance per unit length
- The line-to-neutral capacitance
per unit length. If the length
is specified in meters (m) the unit of
is farads per meter (F/m).
For a pi-section line, this parameter can be a vector of the form
.
- Coupling capacitance per unit length
- The line-to-line capacitance
per unit length. If the length
is specified in meters (m) the unit of
is farads per meter (F/m).
In a pi-section line, the coupling capacitance
between the i-th and j-th phase can be specified individually by providing a 3-element vector
containing the upper triangular coupling matrix.
- Length
- The length
of the line. The unit of
must match the units
,
,
,
and
are based on.
- Number of pi sections
- Number of sections used to model the transmission line.
The default is
. This parameter only affects the pi-section implementation.
Reference
- H. Dommel: "Digital Computer Solution of Electromagnetic Transients in Single and Multiple Networks", IEEE Transactions on Power Apparatus and Systems, Vol. PAS88, No. 4, April, 1969