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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.
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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.
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Basic transformer model
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Numerical implementation
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Numerical implementation
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Numerical implementation
Transformer equivalent after discretisation
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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.
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Modelling of non-linearities
to impose a decay time on the inrush currents, as would occur on
energisation or fault recovery:
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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
other phase (This core geometry implies that phase winding 1 is coupled to all
windings (2 to 6)
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Three-Phase Transformer Inductance Matrix Type
(Two Windings)
Transformer Model
The Three-Phase Transformer Inductance Matrix Type:
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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.
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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 zerosequence short-circuit reactances X112 and X012 measured with threephase winding 1 excited and three-phase winding 2 short-circuited.
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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
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Three-Phase Transformer Inductance Matrix Type
(Two Windings)
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Three-Phase Transformer Inductance Matrix Type
(Two Windings)
Extension from the following two (2x2) reactance matrices in positive-sequence
and in zero-sequence
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Three-Phase Transformer Inductance Matrix Type
(Two Windings)
In order to model the core losses (active power P1 and P0 in positive- and zerosequences), 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:
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Three-Phase Transformer Inductance Matrix Type
(Two Windings)
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Three-Phase Transformer Inductance Matrix Type
(Two Windings)
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Three-Phase Transformer Inductance Matrix Type
(Two Windings)
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UMEC (Unified Magnetic Equivalent Circuit) model
Single-phase UMEC model
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UMEC (Unified Magnetic Equivalent Circuit) model
Single-phase UMEC model
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UMEC (Unified Magnetic Equivalent Circuit) model
Three-limb three-phase UMEC
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UMEC (Unified Magnetic Equivalent Circuit) model
Three-limb three-phase UMEC
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