1. Transformer

2. Basic transformer model
L11 and L22 are the self-inductance of winding 1 and 2 respectively, and L12 and L21 are the mutual inductance between the windings.

3. Basic transformer model
Example: Consider a transformer with a 10% leakage reactance equally divided between the two windings and a magnetising current of 0.01 p.u.

4. Basic transformer model

5. Numerical implementation

6. Numerical implementation

7. Numerical implementation
Transformer equivalent after discretisation

8. Modelling of non-linearities
Typical studies requiring the modelling of saturation are: Inrush current on energising a transformer, steady-state overvoltage studies, core-saturation instabilities and ferro-resonance.

9. Modelling of non-linearities
to impose a decay time on the inrush currents, as would occur on energisation or fault recovery:

10. Three-Phase Transformer Inductance Matrix Type (Two Windings)
The phase windings of the transformer are numbered as follows:
1 and 4 on phase A
2 and 5 on phase B
3 and 6 on phase C
This core geometry implies that phase winding 1 is coupled to all (other phase windings (2 to 6)

11. Three-Phase Transformer Inductance Matrix Type (Two Windings)
Transformer Model
The Three-Phase Transformer Inductance Matrix Type:

12. Three-Phase Transformer Inductance Matrix Type (Two Windings)
R1 to R6 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 6 diagonal terms and 15 off-diagonal terms of the symmetrical inductance matrix.

13. Three-Phase Transformer Inductance Matrix Type (Two Windings)
The self and mutual terms of the (6x6) L matrix are obtained from excitation currents (one three-phase winding is excited and the other three-phase winding is left open) and from positive- and zero-sequence short-circuit reactances X112 and X012 measured with three-phase winding 1 excited and three-phase winding 2 short-circuited.

14. Three-Phase Transformer Inductance Matrix Type (Two Windings)
Q11= Three-phase reactive power absorbed by winding 1 at no load when winding 1 is excited by a positive-sequence voltage Vnom1 with winding 2 open
Q12= Three-phase reactive power absorbed by winding 2 at no load when winding 2 is excited by a positive-sequence voltage Vnom2 with winding 1 open
X112= Positive-sequence short-circuit reactance seen from winding 1 when winding 2 is short-circuited
Vnom1, Vnom2= Nominal line-line voltages of windings 1 and 2

15. Three-Phase Transformer Inductance Matrix Type (Two Windings)

16. Three-Phase Transformer Inductance Matrix Type (Two Windings)
Extension from the following two (2x2) reactance matrices in positive-sequence and in zero-sequence

17. Three-Phase Transformer Inductance Matrix Type (Two Windings)
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 windings. If winding 1 is selected, the resistances are computed as:

18. Three-Phase Transformer Inductance Matrix Type (Two Windings)

19. Three-Phase Transformer Inductance Matrix Type (Two Windings)

20. Three-Phase Transformer Inductance Matrix Type (Two Windings)

21. UMEC (Unified Magnetic Equivalent Circuit) model
Single-phase UMEC model

22. UMEC (Unified Magnetic Equivalent Circuit) model
Single-phase UMEC model

23. UMEC (Unified Magnetic Equivalent Circuit) model
Three-limb three-phase UMEC

24. UMEC (Unified Magnetic Equivalent Circuit) model
Three-limb three-phase UMEC