SPICE Model for Effective and Accurate Time Domain Simulation of Power Transformers

Lucian MANDACHE, Dumitru TOPAN Dept. of Electrical Engineering

University of Craiova Craiova, Romania

lmandache@elth.ucv.ro; dtopan@central.ucv.ro

Mihai IORDACHE, Ioana Gabriela SIRBU Dept. of Electrical Engineering

University Politehnica of Bucharest Bucharest, Romania

iordache@elth.pub.ro; osirbu@elth.ucv.ro

Abstract—The paper proposes an effective time-domain

modeling and simulation strategy of power transformers, using the SPICE circuit simulator. The nonlinear phenomena of the iron core are carefully considered, including saturation, static hysteresis and eddy currents. The principle of magnetic circuit modeling is based on analog lumped equivalent circuits and the SPICE implementation uses the principle of modularity. Such a module includes a transformer winding and its core leg together with the ferromagnetic phenomena. The method allows simulating normal operation modes, as well as critical transients and faulty conditions, the simulation result containing all electromagnetic quantities as time-domain functions. The method is remarkable through its extremely short computation time and reasonable accuracy, it being conceived firstly as useful tool for design purposes.

Keywords—time-domain modeling and simulation; SPICE implementation; power transformer; iron core; eddy current

I. INTRODUCTION The ferromagnetic cores are basic constructive elements

of transformers and inductors for wide application area, ranging from analog and digital microelectronics toward power converters and power systems. It is well known that the rigorous study and design optimization of such electromagnetic devices is difficult because of nonlinearity, electromagnetic inertial behavior and other related phenomena, as saturation, anisotropy, magnetic hysteresis and induced eddy currents.

The optimal design of transformers requires considering the whole electromagnetic system, including the power supply and the load, in rated operation mode as well as in malfunction modes. This is only possible by means of a computer aided design, based on an adequate modeling and simulation using software tools capable to provide the analysis of the entire equipment with adequate accuracy. Even though cosimulation solutions involving a FEM-based simulator for the electromagnetic and thermal field analysis and a time-domain circuit simulator could offer the most accurate results [1,2], they are obviously the most costly regarding hardware and software requirements, computation effort and simulation time and, not least, the high level qualification required to the design engineer.

In this context, in order to achieve reasonable design costs with reasonable accurate results, the paper proposes an extremely efficient modeling and simulation method of any transformer. It can be supplied by harmonic or distorted voltages, the load being linear, nonlinear and/or time-

dependent. Thus, the procedure allows treating transformers in switching modes as in switched-mode power supplies.

The procedure uses the concept of modeling the whole electromagnetic system through equivalent diagrams with lumped circuits, so that only one simulation software is necessary. The FEM-based analysis is therefore avoided. The modeling solution is SPICE-compatible, so that the mathematical model is reduced to a nonlinear differential-algebraic equation system. Although this modeling and simulation principle has been promoted previously by some authors [3-8], our systematic study brings a significant improvement through the degree of generality and the ease of use of the conceived procedure.

We paid special attention to conceive the modeling and simulation method so that it could become a really useful tool for researchers and designers. Event though, in principle, almost any circuit simulator can be used as software environment, the implementation in SPICE is described here, due to its accessibility and widespread within the scientific world.

The modeling procedure follows the specific phenomena related to the wired ferromagnetic pieces, explained by the Maxwell's equations. A concept of modularity has been exploited, in order to confer flexibility and ease of use. Three basic modules have been developed in order to build any magnetic circuit, as follows.

The main module is conceived as a subcircuit that combines: a ferromagnetic piece as magnetic field path, a winding spooled on this ferromagnetic piece and the corresponding flux leakage path. Depending on the extern diagram, the winding of the model can play either the role of primary or secondary coil. The ferromagnetic piece model involves the nonlinearity with saturation, magnetic hysteresis and induced eddy currents. The second module contains only a nonlinear, hysteretic and eddy current carrying iron core piece with flux leakage path, while the third module corresponds to a generic airgap. Associated graphic symbols have been created for each module.

The modeling procedure has a wide range of generality, being suitable to be extended to inductors, linear actuators and rotating motors.

The section II describes briefly the modeling principles, and the SPICE implementation is shown in the section III. A common example of a single-phase transformer has been chosen to prove the benefits of the developed procedure (section IV).

II. MODELING PRINCIPLES The simplest model of a ferromagnetic piece of cross

section S and length l flowed by a magnetic flux assumed as uniform within the cross section (with the flux density B) consists in a voltage controlled nonlinear resistance flowed by a current numerically equal to the magnetic flux, the voltage across it being numerically equal to the magnetic force [8]. Such a model does not consider the hysteresis effect, nor eddy currents. The nonlinear characteristic of the model resistance reproduces the anhysteretic curve.

For design purposes, the initial magnetization curve is commonly specified by the manufacturer of the ferromagnetic material as a nonlinear dependence )(HB . Some modeling strategies deal with the anhysteretic magnetization curve given by points, as a lookup table according to the manufacturer specifications. Alternatively, it can be given as analytic functions, the most preferred being based on the Langevin approximation of the anhysteretic magnetization given by the implicit equation [9]:

⎟⎠⎞

⎜⎝⎛

α+−α+=

MHa

aMHMHM satan coth)( , (1)

where the saturation magnetization satM and the parameters a,α can be established enough accurately starting from the characteristic specified in the datasheet.

Starting from the magnetic force lHum ⋅= , as consequence of a given ampere-turns, the block diagram of Fig. 1 shows the computation chain based on the expression (1) to obtain the corresponding magnetic flux.

Langevin function l

1×

α×

S×0μ× ϕ+

mu + H

M B

Fig. 1. Computation chain of the anhysteretic magnetization based on an analytic model.

In order to achieve an accurate model one does not neglect the induced eddy currents and their magnetic field, even if the transformer works at the grid frequency. Since the eddy current losses can be easily assessed for common power transformers working at the grid frequency (starting from the specific power losses specified in the datasheet), the problem of eddy currents becomes more complicated in distorting operation modes, as under weak power quality conditions, at medium frequencies or in switching mode. We developed previously an original model for eddy current modeling, in order to asses their instantaneous and mean power losses, as well as their parasitic magnetic field [8,11]. The principle is based on the computation of an equivalent eddy current whatever the shape or the iron core piece, and an equivalent resistance that dissipates the same power loss as in the real case. Massive ferromagnetic pieces, as well as silicon steel sheets have been treated. The calculus is based on the quantitative evaluation of the phenomena explained

by the Maxwell's theory, according to the law of Faraday (to compute the electromotive forces induced in the iron core by the time-variable magnetic field), the law of Ohm (to compute the induced eddy current), the law of Joule (to compute the power loss) and the law of Ampere (to compute the parasitic magnetic field).

The equivalent eddy current in the time domain was obtained as:

t

KtBf

lpKti eddyeddy dd

dd

2)( 2

02

020 ϕ⋅=⎟

⎠⎞

⎜⎝⎛ ϕ⋅

πγ⋅= , (2)

where 0p is the specific power loss [W/kg] given in the datasheet for the reference frequency 0f and the reference flux density 0B in harmonic behavior. For silicon steel laminated sheets the reference conditions are usually 50 Hz and 1 T. K is a shape factor and γ is the mass density in kg/m3. If the eddy current mean power loss is computed in reference conditions, the equivalent eddy current resistance results:

SlpBf

KReddy γ

π⋅=

0

20

20

2

221 . (3)

It dissipates accurately the actual power loss when it is flowed by the equivalent eddy current computed before, whatever the time-domain variation of the magnetic flux.

Therefore, the anhysteretic or hysteretic magnetization model has to be completed with additional elements in order to consider the eddy current, its power loss and its magnetic field, as the computation chain shown in Fig. 2. It must be noticed that the input quantity is the external ampere-turn, as primary source of the magnetic field. The eddy current ampere-turn opposes to the external one.

Expanding the concept of modeling magnetic circuits through electric circuits, each of the models described above can be likened to subcircuits involving all corresponding phenomena.

Ampere-turn

eddyR×

ϕmU+

eddyK× eddyI

dtd

( )2

Hysteretic or anhysteretic

model –

eddyp

Fig. 2. Computation chain of eddy currents and their effect in ferromagnetic core pieces.

A core piece carrying a coil is now considered, like a leg together with the primary winding of a transformer (Fig. 3a). All other windings are assumed in open-circuit and both the main flux ϕ and the leakage flux lϕ have been considered. Obviously, they are fascicular magnetic fluxes.

ϕ

1i

1u lϕ

ϕ

1lϕ

dtd

N l )( 11

ϕ+ϕ 1mlR

11iN

1i

1u

1R

(a) (b)

Fig. 3. Primary winding with its core leg.

The voltage across the winding of 1N turns and resistance 1R contains the voltage drop across the resistance and the electromotive force induced by the total magnetic flux:

t

NiRu ld

)d( 11111

ϕ+ϕ+= . (4)

Therefore, an equivalent diagram of the structure of Fig. 3a is conceived as in Fig. 3b, where the core piece is represented as a nonlinear resistance flowed by the main magnetic flux ϕ . The magnetic reluctance corresponding to the leakage flux 1lϕ is 1mlR . The ampere-turn of the winding is the current controlled voltage source with the control resistance equal to its turn number 1N . The second term of (4) is modeled as the current controlled voltage source controlled by the time-derivative of the input current, numerically equal to the total magnetic flux 1lϕ+ϕ .

If a core piece carrying the secondary winding of 2N turns and resistance 2R is considered (Fig. 4a), using a similar diagram and similar notations, the voltage balance according to the voltage Kirchhoff's law is (Fig. 4b):

t

NiRu ld

)d( 22222

ϕ−ϕ+−= . (5)

ϕ

2i

2ulϕ

ϕ

2lϕ

dtd

N l )( 22

ϕ−ϕ 2mlR

22iN

2i

2u

2R

(a) (b)

Fig. 4. Secondary winding with its core leg.

III. SPICE IMPLEMENTATION According to the principles exposed in the previous

section, a complex model of a wired ferromagnetic piece has been developed and implemented as SPICE subcircuit (version ICAP4 from Intusoft [12]). The quasi-similarity between of expressions (4) and (5), as well as the equivalent diagrams of Fig. 3 and Fig. 4, allow conceiving a common model for both structures. It is based on an equivalent lumped circuit diagram. A suggestive graphical symbol was associated to the new model (Fig. 5) which allows a modular construction of any transformer.

Fig. 5. Graphical SPICE symbol associated to the winding-leg model.

The model contains four terminals, two terminals of the electric circuit (winding terminals) and two terminals of the magnetic circuit. The rule of terminal marking is related to primary windings: if the current flows toward the marked terminal of the electric circuit, then the corresponding magnetic flux flows toward the marked terminal of the magnetic circuit. This model can play either the role of primary or secondary. Depending on the transformer construction, the model allows placing the primary and secondary on the same core leg or on different legs. The number of secondary windings is practically unlimited.

A subcircuit netlist built according to the model above is detailed here (it was called LEDDY). It implements the computation chains of Fig. 1 and Fig. 2:

*SYM=LEDDY .SUBCKT LEDDY 11 14 15 17 {N=100, RCC=1.5, SFE=10e-4, + LFE=0.1, K=20, BSAT=1.8, B0=1, F0=50, P0=3.2, GAMMA=7650} R1 11 10 {RCC} H1 10 3 V3 {N} V1 3 14 H2 4 15 V1 {N} V2 4 5 H3 6 0 V2 1 V3 6 7 C1 7 0 1 VA 1 0 {BSAT/(3*MUMAX*3.14*4e-7)} B1 2 0 V=ABS(I(V4)/V(1))>1e-3 ? {BSAT}*(1/TANH(I(V4)/V(1))- +V(1)/I(V4)) : {MUMAX}*3.14*4e-7*I(V4) G1 5 16 2 0 {SFE} G2 8 2 5 16 {1/LFE} V4 0 8 F1 0 12 V3 {K/2*P0*GAMMA*LFE/(3.14*F0*B0)^2} B2 13 0 V=I(V3)/{K*SFE} V5 12 13 H5 16 17 V5 1 R2 15 17 {LFE/(3.14*4e-7*3*SFE)} .ENDS

The subcircuit parameters are: the winding number of turns, the DC resistance of the winding, the cross section, the length and the shape factor of the ferromagnetic core leg, the saturation flux density, the initial relative permeability, the specific power loss and the its reference conditions and, finally, the mass density of the ferromagnetic material.

Important quantities were passed outside the subcircuit in order to be computed and judged: the equivalent eddy current and the instantaneous eddy current loss, according to the considerations above.

The detailed diagram of the subcircuit is shown in Fig. 6, extracted directly from the SPICE-ICAP4 interface. Some explanations on it refer to the significance of the most important elements: the subcircuit terminals 11 and 14 correspond to the electric circuit, while the terminals 15 and 17 correspond to the magnetic circuit; the current controlled voltage source H1 provides the electromotive voltage given by the second term of the right hand of (4); V1 plays the role of an ammeter for the winding current; the source H2 provides the ampere-turns of the winding; V2 is an ammeter for the current numerically equal to the magnetic flux; B1 is a user-defined voltage source that provides the anhysteretic magnetization starting from the magnetic strength given by the voltage-controlled current source G2; G1 becomes the flux density, multiplies it with the core cross-section and provides the magnetic flux; the pair H3 – C1 assures the time-derivative of the flux; F1 becomes the flux derivative and provides the equivalent eddy current; B2 is a user-defined resistance that dissipates the eddy current power loss; the resistance R2 (on the last line of the subcircuit netlist, not shown on Fig. 6) is the leakage magnetic reluctance.

H1V3

V1

H2V1

V2

H3V2 V3 C1

1

I(V2)FLUX

B1V=1

G110E-4

G210

V4

I(V3)DFDT

F1V3

R215

I(V5)IE

H5V5

V(5)UM

V(16)UMR

V(11)Z1

V(14)Z2

11

3

14

4

15

5

6 7

2

16

8

12

13 17

Fig. 6. Detailed diagram of the subcircuit LEDDY.

If the winding terminals are not connected (if no winding is placed on the iron piece), the model works properly. However, two more simplified models have been derived for simplicity reasons: a model for the core piece only (without winding), and other model for airgap magnetic reluctances. All models were included in the SPICE model library, in order to be called in future applications.

IV. EXAMPLE A simple low voltage two-leg, two-winding transformer

has been simulated using the SPICE models detailed in the section III. The circuit diagram (Fig. 7) contains all three types of subcircuits: the winding-leg subcircuits (X1, X2), two core pieces without winding corresponding to the yokes (X3, X4) and two subcircuits corresponding to the linear magnetic reluctances of the technological airgaps (X5, X6).

X1LEDDY2

V2SIN

I(V3)PRIM

V(1)VPRIM

I(V1)FLUX

X2LEDDY2

V(10)VSEC

I(V4)ISEC

RS5K

X3LEDDY0

X5AIR1

X6AIR1X4

LEDDY0

RNETW0.5

LNETW40E-6

1

13

3

2 6

7

4 10

8

9

5

Fig. 7. Example SPICE diagram.

The primary winding is supplied by the sinusoidal source V2 and the network longitudinal impedance parameters were considered.

The simulations were focused on possible faulty conditions (as the grid voltage increasing and load short-circuit) and transient behaviors (as the network connection in no-load condition). Some results are presented below.

In Fig. 8 is depicted the influence of the grid voltage on the primary no-load (magnetization) current, where the curve 1 represents the rated value (slightly distorted), the curve 2 correspond to a 20% lower voltage and the curve 3 corresponds to a 20% higher voltage, this latter being strongly distorted due to the core saturation. The RMS values of the currents are also given on the label attached to the figure.

2

31

930M 940M 950M 960M 970MTIME [Secs]

1.60

800M

0

-800M

-1.60

IPR

IM [A

mps

]

AIAIAI

88.022.037.0

%120

%80

%100

===

12

3

Fig. 8. Influence of the grid voltage on the primary no-load current.

Fig. 9 represents the magnetization characteristics flux-current for the three situations considered above, revealing the level of saturation and the core loss.

2

-1.60 -800M 800M 1.60IPRIM [Amps]

3.20M

1.60M

0

-1.60M

-3.20M

FLU

X [W

b]

1

4

1

2 3

Fig. 9. Influence of the grid voltage on the flux-current curve.

If the no-load transformer is connected to the grid at the moment of zero-crossing voltage (see curve 1 in Fig. 10), the transient behavior is critical, the peak value of the current reaching about 90 times the rated value (curve 2). The corresponding magnetization curve is shown (curve 3) to explain the phenomenon.

3

2

8.00M 24.0M 40.0M 56.0M 72.0MTIME [Secs]

60.0

40.0

20.0

0

-20.0

IPR

IM [A

mps

] 1

2

3

Fig. 10. No-load transformer connected to the grid at the moment of zero-

crossing voltage.

Contrarily, if it is connected at the peak value of the voltage (see curve 1 in Fig. 11), the transient behavior is extremely slight.

2

8.00M 24.0M 40.0M 56.0M 72.0MTIME [Secs]

800M

400M

0

-400M

-800M

IPR

IM [A

mps

]

1

2

3

Fig. 11. No-load transformer connected to the grid at the peak value of the

voltage.

It must be emphasized that the simulation time for each studied case was about 10 seconds, on a common PC.

V. CONCLUSION The proposed modeling and simulation method is capable

to offer complete information on conventional power transformers, as well as on those conceived for special applications, as in power electronics, due to the time-domain modeling. The complex nonlinear phenomena in ferromagnetic cores are also considered. The simplifying assumptions (see the uniformity of the flux density) are reasonable for the envisaged applications.

The flexibility and simplicity of implementation are obvious due to the modular structure. The method is remarkable through the extremely short computation time, comparing to other modeling and simulation strategies. The method is also robust and reliable of the point of view of the computation stability.

The effectiveness and accuracy recommend this modeling and simulation method for research purposes and design optimization.

The proposed modeling and simulation strategy can be easily extended for almost any electromagnetic device with soft iron core.

ACKNOWLEDGMENT This work was supported by The Romanian Ministry of

Education, Research, Youth and Sport-UEFISCDI, project number 678/2009 PNII - IDEI code 539/2008.

REFERENCES [1] M.C. Costa, S.I. Nabeta, J.R. Cardoso, Modified Nodal Analysis

Applied to Electric Circuits Coupled with FEM in the Simulation of a Universal Motor, IEEE Transactions on Magnetics, vol. 36, no. 4, July 2000, pp. 1431-1434.

[2] R.Escarela-Perez, E. Melgoza, J.A.-Ramirez, Systematic Coupling of Multiple Magnetic Field Systems and Circuits Using Finite Element and Modified Nodal Analyses, IEEE Transactions on Magnetics, vol. 47, no. 1, January 2011, pp. 207-213.

[3] L.O. Chua, K.. Stromsmoe, Lumped-Circuit Models for Nonlinear Inductors Exhibiting Hysteresis Loops, IEEE Transactions on Circuit Theory, vol. CT-17, no. 4, November 1970, pp. 564-574.

[4] J.H. Chan, A. Vladimirescu, X.C. Gao, P. Liebmann, J. Valainis, Nonlinear Transformer Model for Circuit Simulation, IEEE Transactions on Computer-Aided Design, vol. 10, no. 4, April 1991, pp. 476-482.

[5] J.T. Hsu, K.D.T. Ngo, Subcircuit Modeling of Magnetic Cores with Hysteresis in PSpice, IEEE Transactions on Aerospace and Electronic Systems, vol. 38, no. 4 October 2002, pp. 1425-1434.

[6] P.R. Wilson, J.N. Ross, A.D. Brown, Simulation of Magnetic Component Models in Electric Circuits Including Dynamic Thermal Effects, IEEE Transactions on Power Electronics, vol. 17, no. 1, January 2002, pp. 55-65.

[7] D.W.P. Thomas, J. Paul, O. Ozgonenel, C. Christopoulos, Time-Domain Simulation of Nonlinear Transformers Displaying Hysteresis, IEEE Transactions on Magnetics, vol. 42, no. 7, July 2006, pp. 1820-1827.

[8] L. Mandache, D. Topan, Modeling and Time-Domain Simulation of Wired Ferromagnetic Cores for Distorted Regimes, Buletinul Institutului Politehnic din Iaşi, Univ. Tehnică “Gh. Asachi”; Tomul LIV (LVIII), fasc. 3, 2008, Electrotehnică, energetică, electronică, pag. 303-310; ISSN 1223-8139.

[9] D.C. Jiles, J.B. Thoelke, M.K.Devine, Numerical Determination of Hysteresis Parameters the Modeling of Magnetic Properties Using the Theory of Ferromagnetic Hysteresis, IEEE Transactions on Magnetics, vol. 28, no. 1, January 1992, pp. 27-35.

[10] D.C. Jiles, D.L. Atherton, Ferromagnetic Hysteresis, IEEE Transactions on Magnetics, vol. MAG-19, No. 5, September 1983, pp. 2183-2185.

[11] D. Topan, L. Mandache, Chestiuni speciale de analiza circuitelor electrice, Ed. Universitaria, Craiova, 2007.

[12] IsSpice4 User’s Guide, Intusoft co., San Pedro, Ca., 1995. [13] M. Iordache, L. Mandache, M. Perpelea, Analyse numérique des

circuits analogiques non linéaires, Ed. Groupe Horizon, Marseille, 2006.

[14] D.W.P. Thomas, J. Paul, O. Ozgonenel, C. Christopoulos, Time-Domain Simulation of Nonlinear Transformers Displaying Hysteresis, IEEE Transactions on Magnetics, vol. 42, no. 7, July 2006, pp. 1820-1827.