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CACD Group
updated 2000.03.30
Author Arpad Buermen

Modelling and Simulation of a Non-linear Transformer

Abstract: every device not found in the basic SPICE library must be modelled using the elements from the library. A transformer is no exception. Sure, there are some devices present in the basic SPICE library that can be used directly to model a transformer (coupled inductances combined with resistors provide a quite useful model). Unfortunately the simple coupled-inductances model is appropriate only for linear analysis since it models the characteristics of a linear transformer - a non-existent ideal of the real transformer that can be used only when the core is well below saturation. In many specific simulations of a circuit including a transformer the non-linear model must be used; e.g. transient responses to power surges, power-on transient response, etc.
When building a model the main issue is what to put in and what to neglect. In case the care for details is exaggerated, the model is too complex and the simulation takes long time even on state of the art hardware with results just slightly more accurate than those obtained with a less complex model. On the other hand excess simplification makes things even worse since the results are inaccurate and in some cases SPICE algorithms are unable to reach convergence.
A simple model of a non-linear transformer is presented below. The model is then used to obtain the power-on transient response of a circuit consisting of a transformer and various loads.

Basic info:

    • Core:
      • relative permitivity below the point of saturation ur = 3980
      • relative permitivity above the point of saturation ur' = 180
      • vacuum permitivity u0 = 4*PI*10-7 Vs/Am
      • point of transition between linear operation and saturation - Hk = 200A/m, Bk = 1T
      • average cross-section area of the core A = 2*10-4 m2
      • average length of a line of force lsr = 0,2m
    • Coils:
      • primary coil windings N1 = 4470, N1*A = 0,894m2
      • secondary coil windings N2 = 570, N2*A = 0,144m2
      • primary coil resistance R1 = 300ohm
      • secondary coil resistance R2 = 5ohm
      • primary coil to ground resistance 1Gohm
      • secondary coil to ground resistance 1Gohm
    The latter two resistances are necessary to define the potential of the primary and secondary coil since SPICE doesn't allow floating circuits.
Transformator
The transformer: windings produce magnetic fluxes that support each other.

Modelling the core
    Magnetilna karakteristika jedra
    The B(H) characteristic of the core. Units on x-axis (magnetic field, H) are A/m and on y-axis (magnetic flux density, B) are T.
     
    Since SPICE works with voltages and currents, equivalents must be chosen for magnetic flux density and magnetic field. Let the current be the equivalent of the magnetic flux density and the voltage the equivalent of the magnetic field. The basic idea is to build a circuit excited with voltage which is the equivalent of magnetic field (H). The response is the current which is the equivalent of magnetic flux density (B). Since magnetic field in the core depends on primary and secundary current, two linear current controlled voltage sources connected in series can model the relation between the coil currents and the magnetic field in the core:
      H = (N1 * i1 + N2 * i2) / lsr
    Since SPICE provides a generic non-linear current/voltage controlled current/voltage source, it will be used to represent the latter equation in the model. The primary and secondary coil currents are obtained using two 0V independent voltage sources VTR1 and VTR2 (see the schematics below).
    To model the non-linear B(H) response a non-linear voltage controlled current source BTR2 controlled with the TR1 node voltage is used.. The non-linear characteristic used in the model consists of three linear segments - one for linear operation (-Hk to Hk) and two for saturation (below -Hk and above Hk).
      B = u0 ( ur * H + (ur' - ur) * ((H - Hk) * u(H - Hk) + (H + Hk) * u( -(H + Hk)))
    u(x) is the unit step function (0 for x<0, 1 otherwise).
    As mentioned before the loop current is the equivalent of the magnetic flux density (B). To model the interaction between the magnetic field in the core and the voltage induced in the primary and the secondary coil the first order derivative of magnetic flux density (dB/dt) must be available. It can be obtained by running the current generated by BTR2 through a 1H inductance. Thus node TR3 provides us with voltage equivalent of dB/dt.
    Independent voltage source VTR0 is used to obtain the current which is the equivalent of magnetic flux density (B). Magnetic field can be obtained as the TR1 node voltage.

    The above discussion gives the following model of the core: 

      BTR1 TR1 0 V=(4470*i(VTR1)+570*i(VTR2))/.20
VTR0 TR1 TR2 0V
BTR2 TR2 TR3
  + I=(3980*V(TR1)+
  + (180-3980)*((V(TR1)-200)*u( V(TR1)-200 )+
  + (V(TR1)+200)*u(-(V(TR1)+200))))
  + *1.256637e-6 
LTR TR3 0 1H
Model jedra
Model of the transformer core.
The current flowing through the independent voltage source VTR0 is the equivalent of the magnetic flux density (B). TR1 node voltage is the equivalent of the magnetic field (H).
The TR3 node voltage is the equivalent of the first order derivative of B (dB/dt).
The non-linear voltage source BTR1 (magnetic field (H)) is controlled with primary and secondary coil current.
The non-linear current source BTR2 (magnetic flux density (B)) is controlled with the TR1 node voltage.
 
Modelling the coils
    Primary coil is modelled with a coil resistance RTR1, an independent 0V voltage source  VTR1 used to obtain the primary coil current and a voltage controlled voltage source ETR1 modelling induced voltage. The latter is obtained using the following equation:
      Ui1 = N1 * A * dB/dt
    A resistance is added to define the potential of the primary coil (RITR).

    The same goes for the secondary coil. The following equation is used to determine the induced voltage:

      Ui2 = N2 * A * dB/dt
    Note the negative sign is missing in the latter two equations since it was accounted for when the ETR1 and ETR2 were put in the circuit (polarization).

    Primary coil model: 

      RTR1 1 TR10 300ohm
VTR1 TR10 TR11 0V
ETR1 TR11 2 TR3 0 0.894
RITR 2 0 1Gohm
    Secondary coil model: 
      RTR2 6 TR20 5ohm
VTR2 TR20 TR21 0V
ETR2 TR21 7 TR3 0 0.114
ROTR 7 0 1Gohm
 
Model primarnega in sekundarnega navitja
Primary and secondary coil model.
RTR1 and RTR2 model the primary and secondary coil resistance.
VTR1 and VTR2 are 0V independent voltage sources used to obtain the primary and secondary coil current used to calculate the magnetic filed (H) in the core.
ETR1 and ETR2 are linear voltage controlled voltage sources controlled with the TR3 node voltage (dB/dt) modelling the induced voltages.
RITR and ROTR are 1Gohm resistances used to define the potential of the primary and secondary coil.
In case a coil is grounded one of these two resistances can be removed (RITR for the primary and ROTR for the secondary).

Examples
Conclusion
A non-linear transformer can be efficiently modelled using a simple circuit. Since the model developed above  is so simple, the analysis is quick. It poses no problem to incorporate a more complex non-linear core B(H) characteristic since a non-linear voltage controlled current source was used providing a wide variety of functions (exp, sin, cos, th, etc.) for describing a curve.
In the next stage of the development modelling of hysteresis, imperfect magnetic coupling of coils and eddy current effects in the core should be considered.
 

The author is aware of his imperfect English. All suggestions concerning grammar and spelling are welcome.

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