| SimPowerSystems™ | |
Elements
The standard Three-Phase Transformer (Three Windings) block uses three single-phase transformers to implement a three-phase model. When a three-phase transformer is built with a three-limb core or a five-limb core this model does not represent the couplings between windings of different phases.
A three-phase transformer using a three-limb core and three windings per phase is shown on the figure below.Windings are numbered as follows: 1, 2, 3 for phase A, 4,5,6 for phase B and 7, 8, 9 for phase C.
This core geometry implies that winding 1 is coupled to all other windings (2 to 9), whereas in a three-phase transformer using three independent cores (as in Three-Phase Transformer (Three Windings) block) winding 1 is coupled only with windings 2 and 3.
The Three-Phase Transformer Inductance Matrix Type (Three-Windings) block implements the following matrix relationship:
R1 to R9 represent the winding resistances. The self inductance terms Lii and the mutual inductance terms Lij are computed from the voltage ratios, the inductive component of the no load excitation currents and the short-circuit reactances at nominal frequency. Two sets of values in positive-sequence and in zero-sequence allow calculation of the 9 diagonal terms and 36 off-diagonal terms of the symmetrical inductance matrix.
When the parameter Core type is set to Three single-phase cores, the model uses three independent circuits with (3x3) R and L matrices. In this condition, the positive-sequence and zero-sequence parameters are identical and you need only specifying positive-sequence values.
Calculation of the inductance matrix is performed as follows.
Assuming, the following positive-sequence parameters for two sets of three-phase windings i and j:
Q1i= Three-phase
reactive power absorbed by winding i at no load when winding i is
excited by a positive-sequence voltage Vnomi with
winding j open
Q1j= Three-phase
reactive power absorbed by winding j at no load when winding j is
excited by a positive-sequence voltage Vnomj with
winding i open
X1ij = positive-sequence
short-circuit reactance seen from winding i
when winding j is short-circuited
Vnomi, Vnomj = nominal line-line voltages of windings i and j
The positive-sequence self and mutual reactances are given by:
The zero-sequence self reactances X0(i,i), X0(j,j) and mutual reactance X0(i,j) = X0(j,i) are also computed using similar equations.
Extension from the two 2x2 reactance matrices in positive-sequence and in zero-sequence
to a 6x6 matrix, is performed by replacing each of the four scalar terms by a 3x3 sub-matrix of the form:
where the self and mutual terms are given by:
In order to model the core losses (active power P1 and P0 in positive- and zero-sequences), additional shunt resistances are also connected to terminals of one of the three-phase winding. If winding i is selected, the resistances are computed as:
The block takes into account the connection type you select, and the icon of the block is automatically updated. An input port labeled N is added to the block if you select the Y connection with accessible neutral for winding 1. If you ask for an accessible neutral on winding 2 or 3, an extra outport port labeled n2 or n3 is generated.
Often, the zero-sequence excitation current of a transformer with a 3-limb core is not provided by the manufacturer. In such a case a reasonable value can be guessed as explained below.
The figure below shows a 3-limb core with a 3-phase winding. Only phase B is excited and voltage is measured on phase A and phase C. The flux Φ produced by phase B shares equally between phase A and phase C so that Φ/2 is flowing in limb A and in limb C. Therefore, in this particular case, if leakage inductance of winding B would be zero, voltage induced on phases A an C would be -k.VB=-VB/2. In fact, because of the leakage inductance of the three windings, the average value of induced voltage ratio k when windings A, B and C are successively excited must be slightly lower than 0.5
Assume:
Zs = average value
of the three self impedances
Zm =average
value of mutual impedance between phases
Z1 =
positive-sequence impedance of three-phase winding
Z0 =
zero-sequence impedance of three-phase winding
I1 =
positive-sequence excitation current
I0 =
zero-sequence excitation current
where k= ratio of induced voltage (with k slightly lower than 0.5)
Therefore, the I0/I1 ratio can be deduced from k:
Obviously k cannot be exactly 0.5 because this would lead to an infinite zero-sequence current. Also, when the three windings are excited with a zero-sequence voltage the flux path should return through the air and tank surrounding the iron core. The high reluctance of the zero-sequence flux path results in a high zero-sequence current.
Let us assume I1= 0.5%. A reasonable value for I0 could be 100%. Therefore I0/I1=200. According to the equation for I0/I1 given above, one can deduce the value of k. k=(200-1)/(2*200+1)= 199/401= 0.496.
Zero-sequence losses should be also higher than the positive-sequence losses because of the additional eddy current losses in the tank.
Finally, it should be mentioned that neither the value of the zero-sequence excitation current nor the value of the zero-sequence losses are critical if the transformer has a winding connected in Delta because this winding acts as a short circuit for zero-sequence.
The three windings of the transformer can be connected in the following manner:
Y
Y with accessible neutral
Grounded Y
Delta (D1), delta lagging Y by 30 degrees
Delta (D11), delta leading Y by 30 degrees
Note The D1 and D11 notations refer to the following clock convention. It assumes that the reference Y voltage phasor is at noon (12) on a clock display. D1 and D11 refer respectively to 1 PM (delta voltages lagging Y voltages by 30 degrees) and 11 AM (delta voltages leading Y voltages by 30 degrees). |
Select the core geometry: Three single-phase cores or Three-limb or five-limb core. If you select the first option only the positive-sequence parameters are used to compute the inductance matrix. If you select the second option, both the positive- and zero-sequence parameters are used.
The winding connection for winding 1.
The winding connection for winding 2.
The winding connection for winding 3.
Check this parameter to connect windings 1 and 2 in autotransformer.
Specifying this connection connects windings 1 and 2 in series with additive voltage. If the first voltage specified in the Nominal line-line voltages parameter is higher than the second voltage, the low voltage tap is connected on the right side (a2,b2,c2 terminals). Otherwise, the low voltage tap is connected on the left side (A,B,C terminals). If you select this option you must specify the same winding connections for windings 1 and 2. If you select Yn for both winding 1 and winding 2, the common neutral N connector is displayed on the left side.
Select Winding voltages to measure the voltage across the winding terminals of the Three-Phase Transformer block.
Select Winding currents to measure the current flowing through the windings of the Three-Phase Transformer block.
Select All measurements to measure the winding voltages and currents.
Place a Multimeter block in your model to display the selected measurements during the simulation. In the Available Measurements list box of the Multimeter block, the measurements are identified by a label followed by the block name.
If the Winding 1 connection parameter is set to Y, Yn, or Yg, the labels are as follows.
Measurement | Label |
|---|---|
Winding 1 voltages | Uan_w1:, Ubn_w1:, Ucn_w1: or Uag_w1:, Ubg_w1:, Ucg_w1: |
Winding 1 currents | Ian_w1:, Ibn_w1:, Icn_w1: or Iag_w1:, Ibg_w1:, Icg_w1: |
If the Winding 1 connection parameter is set to Delta (D11) or Delta (D1), the labels are as follows.
Measurement | Label |
|---|---|
Winding 1 voltages | Uab_w1:, Ubc_w1:, Uca_w1: |
Winding 1 currents | Iab_w1:, Ibc_w1:, Ica_w1: |
The same labels apply for windings 2 and 3, except that 1 is replaced by 2 or 3 in the labels.
The nominal power rating, in volt-amperes (VA), and nominal frequency, in hertz (Hz), of the transformer.
The phase-to-phase nominal voltages of windings 1, 2, 3 in volts RMS.
The resistances in pu for windings 1, 2 and 3.
The no-load excitation current in percent of the nominal current when positive-sequence nominal voltage is applied at any group of winding terminals (ABC, abc2 or abc3).
The core losses plus winding losses at no-load, in watts (W), when positive-sequence nominal voltage is applied at any group of winding terminals (ABC, abc2 or abc3).
The positive-sequence short-circuit reactances X12, X23 and X13 in pu, where Xij is the reactance measured from winding i when winding j is short-circuited.
When the Connect windings 1 and 2 in autotransformer parameter is checked, the short-circuit reactances are labeled XHL, XHT and XLT, where H, L and T indicate the following terminals: H= high voltage winding (either winding 1 or winding 2), L=low voltage winding (either winding 1 or winding 2) and T=tertiary (winding 3).
The no-load excitation current in percent of the nominal current when zero-sequence nominal voltage is applied at any group of winding terminals (ABC, abc2 or abc3).
If you want to measure this excitation current, you must connect the excited winding in Yg or Yn in order to provide a return path for the source zero-sequence currents. Also, the Delta connections of the other windings must be open; otherwise, these Delta connections would produce a short-circuit for zero sequence. You must therefore temporarily change the open winding connections from D1 or D11 to Y, Yg, Yn.
The core losses plus winding losses at no-load, in watts (W), when zero-sequence nominal voltage is applied at any group of winding terminals (ABC, abc2 or abc3). The Delta windings must be temporarily open in order to measure these losses.
The zero-sequence short-circuit reactances X12, X23 and X13 in pu, where Xij is the reactance measured from winding i when winding j is short-circuited. If the Zero-sequence X12 measured with winding 3 Delta connected check box is not selected, X12 represents the short-circuit reactance when winding 3 is not connected in Delta.
When the Connect windings 1 and 2 in autotransformer check box is selected, the short-circuit reactances are labeled XHL, XHT and XLT, where H, L and T indicate the following terminals: H= high voltage winding (either winding 1 or winding 2), L=low voltage winding (either winding 1 or winding 2) and T=tertiary (winding 3).
Select this check box if the available zero-sequence short circuit tests were obtained with tertiary winding (winding 3) connected in Delta.
This transformer model does not include saturation. If you need modeling saturation connect the primary winding of a saturable Three-Phase Transformer (Two Windings) in parallel with the primary winding of your model. Use same connection (Yg, D1 or D11) and same winding resistance for the two windings connected in parallel. Specify Y or Yg connection for the secondary winding and leave it open. Specify appropriate voltage, power ratings and desired saturation characteristics. The saturation characteristic is the characteristic obtained when then transformer is excited by a positive-sequence voltage.
If you are modeling a transformer with three single-phase cores or a five-limb core, this model will produce acceptable saturation currents because flux stays trapped inside the iron core.
For a three-limb core, it is less evident that this saturation model also gives acceptable results because zero-sequence flux circulates outside of the core and returns through the air and the transformer tank surrounding the iron core. However, as the zero-sequence flux circulates in the air, the magnetic circuit is mainly linear and its reluctance is high (high magnetizing currents). These high zero-sequence currents (100% and more of nominal current) required to magnetize the air path are already taken into account in the linear model. Connecting a saturable transformer outside the three-limb linear model with a flux-current characteristic obtained in positive sequence will produce currents required for magnetization the iron core. This model will give acceptable results whether the three-limb transformer has a delta or not.
See the power_Transfo3phCoreType demo showing how saturation is modeled in an inductance matrix type two-winding transformer.
Linear Transformer, Multimeter, Three-Phase Transformer (Two Windings), Three-Phase Transformer (Three Windings), Three-Phase Transformer Inductance Matrix Type (Two Windings)
| Three-Phase Transformer Inductance Matrix Type (Two Windings) | Three-Phase V-I Measurement | |
| © 1984-2008- The MathWorks, Inc. - Site Help - Patents - Trademarks - Privacy Policy - Preventing Piracy - RSS |