Three-Winding Autotransformer

The Two-Winding Transformer Inductance Matrix Type is a three-phase transformer with two windings per phase. Unlike other three-phase transformer models, which are represented by three separate single-phase transformers, this model takes into account the connections between the windings of the different phases. The core and windings of the transformer are shown in the following figure.

The phase windings of the transformer are numbered as follows:

This type of core implies that phase winding 1 is connected to all other phase windings (2 to 6), whereas in a unit Three-Phase Transformer (Two Windings) (three-phase transformer with three independent cores), winding 1 is connected only to winding 4.

The Two-Winding Transformer Inductance Matrix Type model realises the following matrix relationship:

,

where

Eigen and mutual terms of the inductance matrix are calculated from the ratio of voltages, inductive component of no-load currents and short-circuit reactances at rated frequency. Two sets of values in the forward and zero sequence allow the calculation of 6 diagonal terms and 15 non-diagonal terms of the symmetrical inductance matrix.

The inductance matrix terms (6x6) are calculated from the no-load currents (one three-phase winding excited and the other three-phase winding open) and the forward and zero-sequence short-circuit reactances and , measured with the excited three-phase primary winding and the short-circuit of the three-phase secondary winding.

Let’s assume the following parameters of the direct sequence:

The intrinsic and mutual reactive resistances of the direct sequence are determined by the expressions:

,

,

.

The intrinsic reactive resistances , and mutual reactive resistances of the zero sequence are determined by usage of similar expressions.

Expansion of the following two (2x2) matrices of direct and zero-sequence reactive resistances

to matrix (6x6) is done by replacing each of the four pairs ] by a submatrix (3x3) of the form:

,

where the eigen and reciprocal terms are defined as

,

.

In order to simulate core losses (active power and forward and zero sequence), additional shunt active resistors are also connected to the leads of one of the three-phase windings.

If the primary winding is selected, the resistors are calculated as:

,

.

The unit takes into account the selected connection type and the unit icon is automatically updated. The input port n1 is added to the block if connection Y with available primary neutral (`Wye with neutral port') is selected. If an available secondary neutral is selected, an additional n2 port will be created.

If the Core type parameters are set to Three single-phase cores, then the model uses three independent circuits with and matrices (3x3). In this case parameters of direct and zero sequence are identical and only direct sequence values are specified.

,

,

.

For an autotransformer, you must specify the same connections for the three-phase primary and secondary windings. If you select the `Wye with neutral port' connection for the primary and secondary windings, the common neutral port `n1' is shown on the left. The figure shows the winding connection of one phase of an autotransformer when both three-phase windings are star-connected (with earthed neutral).

If

If

W1, W2 windings correspond to the following phase winding numbers:

The Three-Winding Transformer Inductance Matrix Type model is a three-phase transformer with three windings per phase. Unlike other three-phase transformer models, which are represented by three separate single-phase transformers, this model takes into account the connections between the windings of the different phases. The core and windings of the transformer are shown in the following figure.

The phase windings of the transformer are numbered as follows:

This type of core implies that phase winding 1 is connected to all other phase windings (2 to 9).

The Three-Winding Transformer Inductance Matrix Type model realises the following matrix relationship:

,

where

Eigen and mutual terms of the inductance matrix are calculated from the ratio of voltages, inductive component of no-load currents and short-circuit reactances at rated frequency. Two sets of forward and zero-sequence values allow the calculation of 9 diagonal terms and 36 non-diagonal terms of the symmetric inductance matrix.

The inductance matrix terms (9x9) are calculated from the no-load currents (one three-phase winding excited and the other two three-phase windings open) and short-circuit reactances.

Short-circuit reactances:

and are the reactive resistances of the direct and zero sequences when the three-phase primary winding is excited and the three-phase secondary winding is short-circuited.

and - reactive resistances of direct and zero sequences at excited three-phase primary winding and short circuit of three-phase tertiary winding.

and - reactive resistances of direct and zero sequences at excited three-phase secondary winding and short circuit of three-phase tertiary winding.

Let’s assume the following parameters of direct sequence for three-phase windings and ( and have values 1, 2 or 3):

The own and mutual reactive resistances of the direct sequence are defined by the expressions:

,

,

.

The intrinsic reactive resistances , and mutual reactive resistances of the zero sequence are determined by usage of similar expressions.

Expansion of the following two (3x3) forward and zero-sequence reactive impedance matrices:

,

to the matrix (9x9) is done by replacing each of the four pairs ] by a submatrix (3x3) of the form:

,

where the eigen and reciprocal terms are defined as

,

.

In order to simulate core losses (active power and forward and zero sequence), additional shunt active resistors are also connected to the leads of one of the three-phase windings.

If winding is selected, the resistors are calculated as:

,

.

The block takes into account the selected connection type and the block icon is automatically updated. The n1 port is added to the block if connection Y with available primary neutral (Wye with neutral port) is selected. If an available secondary or tertiary neutral is selected, an additional n2 or n3 port will be created.

If the Core type parameters are set to Three single-phase cores, then the model uses three independent circuits with and matrices (3x3). In this case the parameters of the direct and zero sequence are identical and only the values of the direct sequence are specified.

,

,

.

For an autotransformer, you must specify the same connections for the three-phase primary and secondary windings. If you select the Wye with neutral port connection for the primary and secondary windings, the common neutral port n1 is displayed on the left. The figure shows the single phase winding connection of an autotransformer when the primary and secondary three-phase windings are star-connected (with earthed neutral) and the tertiary three-phase winding is delta-connected.

If

If

W1, W2, W3 windings correspond to the following phase winding numbers:

Often the zero-sequence no-load current of a transformer with a three-bar core is not provided by the manufacturer. If this is the case, the value can be roughly calculated as described below.

The following figure shows a three-core with one three-phase winding. Only phase B is excited, the voltage is measured at phase A and phase C. The flux , generated by phase B, is equally distributed between phase A and phase C, so that the flux /2 flows into rod A and into rod C. Hence, in this particular case, if the dissipation inductance of winding B is zero, the voltage induced on phases A and C will be equal to . In fact, due to the dissipation inductance of the three windings, the average value of the induced voltage coefficient when windings A, B and C are excited in series should be slightly lower than 0.5.

Let’s assume:

- the average value of the three eigenresistances.

- average value of mutual resistance between phases.

- direct sequence resistance of a three-phase winding.

- zero sequence resistance of a three-phase winding.

- no-load current of the direct sequence.

- zero-sequence no-load current.

,

,

,

,

,

,

where is the induced voltage coefficient (just below 0.5).

Hence, the ratio can be obtained relative to :

.

Obviously, cannot be exactly equal to 0.5 because this would result in an infinite zero-sequence current. In addition, when the three windings are excited by zero-sequence voltage, the magnetic flux must short-circuit through the air and tank surrounding the iron core. The high magnetic resistance of the zero-sequence flux path results in high zero-sequence current.

Assume that . A reasonable value of may be 100%. Thus . According to the above equation for we can determine the value of :

.

The zero sequence losses must also be higher than the direct sequence losses due to additional eddy current losses in the tank.

Finally, the magnitude of the zero-sequence excitation current and the magnitude of the zero-sequence losses are not relevant if the transformer has a delta winding, since this winding acts as a short circuit for the zero sequence.

The three-phase windings of a transformer can be connected as follows:

For more information on the operating modes of a three-phase system connected in a Delta pattern, see Problem solving for delta connection of transformer windings.

These models are not designed for saturation. To simulate saturation, connect the primary winding of a Two-Winding Transformer (Three-Phase) in parallel with the primary winding. Use the same connection (Yg, D1 or D11) and the same resistance for the two windings connected in parallel. Specify the Y or Yg connection for the secondary winding and leave it open-circuited. State the required voltage, power rating and saturation characteristic. The saturation characteristic is obtained when the transformer is excited by direct sequence voltage. If a transformer with three single-phase cores or a five-bar core is modelled, this model produces acceptable saturation currents because the flux remains inside the steel core. For a three rod core, this saturation model still produces acceptable results even though the zero-sequence flow circulates outside the core and returns through the air and transformer tank surrounding the steel core. Because the zero-sequence flux circulates in the air, the magnetic circuit is essentially linear and its magnetic resistance is high (high magnetising currents). These high zero-sequence currents (100% or more of rated current) required to magnetise the air path are already accounted for in the linear model. Connecting the saturating transformer outside the three-bar linear model with the magnetic-current characteristic obtained for the direct sequence produces the currents required to magnetise the steel core. This model gives acceptable results whether the three-bar transformer has a delta-connected winding or not.