# Three-Phase Transformer Inductance Matrix Type (Three Windings)

Implement three-phase three-winding transformer with configurable winding connections and core geometry

## Library

Fundamental Blocks/Elements

## Description

The Three-Phase Transformer Inductance Matrix Type (Three Windings) block is a three-phase transformer with a three-limb core and three windings per phase. Unlike the Three-Phase Transformer (Three Windings) block, which is modeled by three separate single-phase transformers, this block takes into account the couplings between windings of different phases. The transformer core and windings are shown in the following illustration.

The phase windings of the transformer are numbered as follows:

• 1 , 4, 7 on phase A

• 2, 5, 8 on phase B

• 3, 6, 9 on phase C

This core geometry implies that phase winding 1 is coupled to all other phase windings (2 to 9), whereas in Three-Phase Transformer (Three Windings) block (a three-phase transformer using three independent cores) winding 1 is coupled only with windings 4 and 7.

 Note   The phase winding numbers 1, 2, and 3 should not be confused with the numbers used to identify the three-phase windings of the transformer. Three-phase winding 1 consists of phase windings 1,2,3, three-phase winding 2 consists of phase windings 4,5,6, and three-phase winding 3 consists of phase windings 7,8,9.

### Transformer Model

The Three-Phase Transformer Inductance Matrix Type (Three-Windings) block implements the following matrix relationship:

$\left[\begin{array}{c}{V}_{1}\\ {V}_{2}\\ ⋮\\ {V}_{9}\end{array}\right]=\left[\begin{array}{cccc}{R}_{1}& 0& \dots & 0\\ 0& {R}_{2}& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {R}_{9}\end{array}\right]\cdot \left[\begin{array}{c}{I}_{1}\\ {I}_{2}\\ ⋮\\ {I}_{9}\end{array}\right]+\left[\begin{array}{cccc}{L}_{11}& {L}_{12}& \dots & {L}_{19}\\ {L}_{21}& {L}_{22}& \dots & {L}_{29}\\ ⋮& ⋮& \ddots & ⋮\\ {L}_{91}& {L}_{92}& \dots & {L}_{99}\end{array}\right]\cdot \frac{d}{dt}\left[\begin{array}{c}{I}_{1}\\ {I}_{2}\\ ⋮\\ {I}_{9}\end{array}\right].$

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.

The self and mutual terms of the (9x9) L matrix are obtained from excitation currents (one three-phase winding is excited and the other two three-phase windings are left open) and from short-circuit reactances.

The following short-circuit reactances are specified in the mask parameters:

X112, X012 — positive- and zero-sequence reactances measured with three-phase winding 1 excited and three-phase winding 2 short-circuited

X113, X013 — positive- and zero-sequence reactances measured with three-phase winding 1 excited and three-phase winding 3 short-circuited

X123, X023 — positive- and zero-sequence reactances measured with three-phase winding 2 excited and three-phase winding 3 short-circuited

Assuming the following positive-sequence parameters for three-phase windings i and j (where i=1,2,or 3 and j=1,2,or 3):

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:

$\begin{array}{c}{X}_{1}\left(i,i\right)=\frac{{V}_{{\text{nom}}_{i}}^{2}}{Q{1}_{i}}\\ {X}_{1}\left(j,j\right)=\frac{{V}_{{\text{nom}}_{j}}^{2}}{Q{1}_{j}}\\ {X}_{1}\left(i,j\right)={X}_{1}\left(j,i\right)=\sqrt{{X}_{1}\left(j,j\right)\cdot \left({X}_{1}\left(i,i\right)-X{1}_{ij}\right)}.\end{array}$

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 following two (3x3) reactance matrices in positive-sequence and in zero-sequence

$\begin{array}{l}\left[\begin{array}{ccc}{X}_{1}\left(1,1\right)& {X}_{1}\left(1,2\right)& {X}_{1}\left(1,3\right)\\ {X}_{1}\left(2,1\right)& {X}_{1}\left(2,2\right)& {X}_{1}\left(2,3\right)\\ {X}_{1}\left(3,1\right)& {X}_{1}\left(3,2\right)& {X}_{1}\left(3,3\right)\end{array}\right]\\ \left[\begin{array}{ccc}{X}_{0}\left(1,1\right)& {X}_{0}\left(1,2\right)& {X}_{0}\left(1,3\right)\\ {X}_{0}\left(2,1\right)& {X}_{0}\left(2,2\right)& {X}_{0}\left(2,3\right)\\ {X}_{0}\left(3,1\right)& {X}_{0}\left(3,2\right)& {X}_{0}\left(3,3\right)\end{array}\right]\end{array}$

to a (9x9) matrix, is performed by replacing each of the nine [X1 X0] pairs by a (3x3) submatrix of the form:

$\left[\begin{array}{ccc}{X}_{s}& {X}_{m}& {X}_{m}\\ {X}_{m}& {X}_{s}& {X}_{m}\\ {X}_{m}& {X}_{m}& {X}_{s}\end{array}\right]$

where the self and mutual terms are given by:

Xs = (X0 + 2X1)/3
Xm = (X0X1)/3

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:

$R{1}_{i}=\frac{{V}_{{\text{nom}}_{i}}^{2}}{P{1}_{i}}\text{ }R{0}_{i}=\frac{{V}_{{\text{nom}}_{i}}^{2}}{P{0}_{i}}.$

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 three-phase winding 2 or 3, an extra outport port labeled `n2` or `n3` is generated.

#### Excitation Current in Zero Sequence

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 following figure shows a three-limb core with a single three-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

$\begin{array}{c}{V}_{B}={Z}_{s}{I}_{B}\\ {V}_{A}={Z}_{m}{I}_{B}=-{V}_{B}/2\\ {V}_{C}={Z}_{m}{I}_{B}=-{V}_{B}/2\\ {Z}_{s}=\frac{2{Z}_{1}+{Z}_{0}}{3}\\ {Z}_{m}=\frac{{Z}_{0}-{Z}_{1}}{3}\\ {V}_{A}={V}_{C}=\frac{{Z}_{m}}{{Z}_{s}}{V}_{B}=-\frac{\frac{{Z}_{1}}{{Z}_{0}}-1}{2\frac{{Z}_{1}}{{Z}_{0}}+1}{V}_{B}=-\frac{\frac{{I}_{0}}{{I}_{1}}-1}{2\frac{{I}_{0}}{{I}_{1}}+1}{V}_{B}=-k{V}_{B},\end{array}$

where k= ratio of induced voltage (with k slightly lower than 0.5)

Therefore, the I0/I1 ratio can be deduced from k:

$\frac{{I}_{0}}{{I}_{1}}=\frac{1+k}{1-2k}.$

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.

#### Winding Connections

The three-phase windings can be configured 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).

## Dialog Box and Parameters

### Configuration Tab

Core type

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.

Winding 1 connection

The winding connection for three-phase winding 1.

Winding 2 connection

The winding connection for three-phase winding 2.

Winding 3 connection

The winding connection for three-phase winding 3.

Connect windings 1 and 2 in autotransformer

Check to connect the three-phase windings 1 and 2 in autotransformer (three-phase 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).

In autotransformer mode you must specify the same winding connections for the three-phase windings 1 and 2. If you select `Yn` connection for both winding 1 and winding 2, the common neutral N connector is displayed on the left side.

The following figure illustrates winding connections for one phase of an autotransformer when the three-phase windings are connected respectively in Yg,Yg, and Delta..

If V1 > V2:

If V2 > V1:

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

• Phase A: W1=1, W2=4, W3=7

• Phase B: W1=2, W2=5, W3=8

• Phase C: W1=3, W2=6, W3=9

Measurements

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 three-phase windings 2 and 3, except that `1` is replaced by `2` or `3` in the labels.

### Parameters Tab

Nominal power and frequency

The nominal power rating, in volt-amperes (VA), and nominal frequency, in hertz (Hz), of the transformer.

Nominal line-line voltages [V1 V2 V3]

The phase-to-phase nominal voltages of windings 1, 2, 3 in volts RMS.

Winding resistances [R1 R2 R3]

The resistances in pu for windings 1, 2, and 3.

Positive-sequence no-load excitation current

The no-load excitation current in percent of the nominal current when positive-sequence nominal voltage is applied at any three-phase 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 three-phase winding terminals (ABC, abc2, or abc3).

Positive-sequence short-circuit reactances

The positive-sequence short-circuit reactances X12, X23, and X13 in pu. 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 selected, the short-circuit reactances are labeled XHL, XHT, and XLT. 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).

Zero-sequence no-load excitation current with Delta windings opened

The no-load excitation current in percent of the nominal current when zero-sequence nominal voltage is applied at any three-phase winding terminals (ABC, abc2, or abc3) connected in Yg or Yn.

 Note:   If your transformer contains delta-connected windings (D1 or D11), the zero-sequence current flowing into the Yg or Yn winding connected to the zero-sequence voltage source does not represent the net excitation current because zero-sequence currents are also flowing in the delta winding. Therefore, you must specify the no-load zero-sequence circulation current obtained with the delta windings open.

If you want to measure this excitation current, you must temporarily change the delta windings connections from D1 or D11 to Y, Yg, or Yn, and connect the excited winding in Yg or Yn to provide a return path for the source zero-sequence currents.

Zero-sequence no-load losses with Delta windings opened

The core losses plus winding losses at no-load, in watts (W), when zero-sequence nominal voltage is applied at any three-phase winding terminals (ABC, abc2, or abc3) connected in Yg or Yn. The Delta windings must be temporarily open to measure these losses.

 Note:   Note: If your transformer contains delta-connected windings (D1 or D11), the zero-sequence current flowing into the Yg or Yn winding connected to the zero-sequence voltage source does not represent the net excitation current because zero-sequence currents are also flowing in the delta winding. Therefore, you must specify the no-load zero-sequence circulation current obtained with the delta windings open.
Zero-sequence short-circuit reactances

The zero-sequence short-circuit reactances X12, X23, and X13 in pu. 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. 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).

Zero-sequence X12 measured with winding 3 Delta connected

Select this check box if the available zero-sequence short circuit tests are obtained with tertiary winding (winding 3) connected in Delta.

## Limitations

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 the same connection (Yg, D1 or D11) and same winding resistance for the two windings connected in parallel. Specify the Y or Yg connection for the secondary winding and leave it open. Specify appropriate voltage, power ratings, and the saturation characteristics that you want. The saturation characteristic is obtained when the 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, the saturation model also gives acceptable results even if zero-sequence flux circulates outside of the core and returns through the air and the transformer tank surrounding the iron core. 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 of the iron core. This model will give acceptable results whether the three-limb transformer has a delta or not.

See the `power_Transfo3phCoreType``power_Transfo3phCoreType` example showing how saturation is modeled in an inductance matrix type two-winding transformer.