Lecture 1 - EE743

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Transformer
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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
This core geometry implies that phase winding 1
is coupled to all (other phase 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 zero-sequence short-circuit
  reactances X112 and X012 measured with
  three-phase 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 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
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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