Summary: Part 1 and Part 2 described the perfect and ideal transformer models, and how to add lossy elements to transformer models created with ideal transformer primitives. This part discusses the methods that can be used to create core elements which can include core loss, magnetization and hysteresis in these models.

Creating Core Models:

SPICE is NOT a design tool, but rather an analysis tool. While it may be true that an overall simulation can reveal performance information which can be used to modify some model parameters, as an example in the feedback compensation of a switching mode power supply, its primary use is to validate an existing design.

It is assumed herein that one knows what primary inductance one needs, what the approximate voltage and current levels are, the frequency (s) of operation, the modes of operation (continuous/discontinuous, and the overall network topology. These and other considerations will drive the core selection process. It is assumed one has selected an appropriate core material and core topology for the intended final design.

Searching the net, one finds there are about six methods to arrive at core models. The first way is to find a core suitably characterized already that one could use immediately. Generally this is rather unlikely to occur. This will not characterize parasitic elements such as winding resistances, stray capacitances and leakage inductances, but it may include hysteresis and eddy current effects.

The second way is to create your own core model using the Jiles Atherson method.

The third method is to hire a shop to arrive at an inductor/transformer model for you. The model itself has to be tailored to the specific version of SPICE one is using, as extensions and additions in one version of SPICE may not be present in the others. Most commonly the models may be tailored to one of the more popular SPICE products, containing non-standard extensions/features from time to time. However, unless they create a model geared to a specific production method of the transformer as part of their effort (using method 5 or some alternate which will estimate them), the problem of parasitics remains.

The fourth way is to empirically measure the characteristics of an existing transformer or inductor, and use that that as a basis for a core/transformer model. This is not a bad method, however as magnetic materials can vary widely in many parameters, several devices may be required for test to ensure that the sample is not at one extreme or the other.

The fifth method is to use software that will enable you to create core models including in some cases construction details of the finished inductor/transformer. This can be quite good IF the core library supports the selected core material and mechanical configuration, and the mechanical configuration is properly optimized. These methods do not in general include hysteresis and eddy current effects.

The sixth way is to use vendor data, entered into a specialized SPICE subcircuit model of a core. Again, the parasitics remain a problem but the hysteresis and eddy current effects may be modeled.

Methods one and two are often fruitless. The third method will work, but one must find the right shop, and the cost can be prohibitive. Method four can be done by you or by an outside shop, but methods three and four often presume the existence of a working finished transformer/inductor to characterize. This is due to construction methods of the physical end product affecting the model.

Method five will work, and is often the best choice. These products do not produce models exhibiting hysteresis in general. Still, the BEST model is the simplest one that works. This paper will discuss the sixth method, namely behavioral modeling of a core.

Behavioral modeling of a core (or any other device) consists of the arrangement of devices connected to provide, at the external terminals, a model that behaves like the 'real' device, although the internal device configuration may be very dissimilar to that of the 'real' device. This is commonly done with such devices as opamps. This still leaves such questions as leakage inductance and other construction related parameters open,

However, in accordance with the maxim that one should ALWAYS use the simplest model which provides suitable simulation results, progressing to more complex models only when required, one can add these missing elements later as required. One may always use this model when simulations are complete, as a basis for a design in method 5, or in conjunction with it.

Refer to Reference 1. In this interesting paper, which appears in many places on the WEB, both as shown and as a reference, a behavioral model of a magnetic core is presented. This is shown in Figure 1. This basic circuit can be modified slightly to account for the low frequency loss effects of a core, as contained in Reference 2, which presents the same model with some modifications. Reference 2, the Magnetics Designers Handbook, is available for free download from Intusoft at their website. This is a MUST item to acquire and read.

There are other behavioral models that can be used to model a core. In most cases hysteresis does not need to be included and is not. Indeed, this is an effect to be avoided in the transformer design in many if not most cases, as an undesirable second order effect.

The specific model, as described in Reference 1, is shown in Figure 1.

Figure 1
Saturable Core Model

The netlist for this model can be found in the reference. As this is not in B2 SPICE format, it is easily conformable. Before doing this, it might be well to examine the model a bit, although it is covered in the reference materials. The 'G' element integrates the core voltage and produces a voltage across capacitor CB proportional to the core flux. The 'E' element applies this voltage (flux) to shaping elements. VM measures this current.

RB represents the magnetic impedance, with the VP, VN, D1 and D2 elements representing saturation effects as the voltage (flux) levels increase, decreasing the impedance seen by the 'E1' source. The current sensed through VM is an input to the 'F' element that is fed back to the input terminals creating an incremental inductance value at the 'core' model input. (Note: Reference 2 shows how to create low frequency loss effects in the equivalent core, and they will be included, with the correction of one slight error.)

An equivalent B2 SPICE realization of this circuit with some modifications is shown in Figure 2.

Figure 2
B2 SPICE core model

The terminals of this equivalent inductor are N2 and N1. An auxiliary output 'Flux' representing the core flux, the voltage across capacitor C1, is provided. This is in this realization 'buffered' by E2, with resistor 'Rx' providing a return path. There is no compelling reason NOT to ground the internal circuitry save for the input terminals, as the input terminals, when the circuit is imbedded in a larger circuit, will find a ground path, and neither input need be connected directly to ground. Resistor 'R3' is provided to ease simulations in the same manner as is 'Rx', both being arbitrarily large.

Some other changes were made in the model, one being that the passed parameters were modified to make the turns a subcircuit parameter. In general this is not a major change, but it enables the basic core turns to be modified easily without recalculating several subcircuit input parameters.

These changes are subjective, and the original definitions may be retained if desired. The following are the parameters to be passed, their units and definitions:

SVSEC = BSAT * AE Core capacity, in Volt-seconds/turn
IVSEC = B * Ae Initial condition, in Volt-seconds/turn
LMAG = (µ0 * µr * AE)/LM Magnetizing Inductance, in Henries/turn2
LSAT = (µ0 * AE)/LM Saturation Inductance, in Henries/turn2
IHYST = H0 * LM Magnetizing current at zero flux, in Amperes/turn
N = turns
FEDDY = Frequency where the permeability vs frequency is 3 dB down, in Hz.

Where: BM is the maximum flux density in Gauss
H0 is the magnetic field strength in Oersteds
AE is the effective area of the core in cm2
LM is the effective magnetic path length in cm

There may be occasion to use the following conversions, among others, when using manufacturer's data sheets.

It is assumed that most if not all of the above can be determined from manufacturer's data sheets. It might be useful to create a parameterized subcircuit model in which ALL of the data is passed for the maximum flexibility. This is a matter of preference. Some calculations may be required to prepare the inputs, however it would seem the terms are most meaningful to engineers as shown. A netlist for this parameterized subcircuit (assumed to be created) is as follows:

The problem now is to obtain the required values. Examination of a core material data will provide this, with most of the data coming from a material B-H curve. This is done in the following manner, using an example B-H curve. Refer to Figure 3 following:

Figure 3
Tape wound core material typical hysteresis loop

Figure 3 came from Magnetics Incorporated, in this case being specific to a Square Permalloy tape. Refer to Reference 3, and be sure to look at many of the helpful articles Magnetics Incorporated provides. Typical brochures for their cores will show a B-H loop such as this, where hysteresis is a major contributor to core losses and/or a necessary design consideration. Other manufacturers will do likewise.

In this case, using the Figure 3 curve as a representative sample, several core elements may be determined. Assuming our frequency is 6kHz, corresponding to the dashed curve, the BM and BR points on the curve correspond the BM and B values in the equations for SVSEC and IVSEC respectively. For square loop cores the manufacturer will usually provide a value or a range for BM as well.

In this event, use the mean value. If this is done, then one can arrive at a value of (µ0 * µr) using the curve and the relation that:

(µ0 * µr) = B/H

Alternately, one can choose a point on the midrange value of BM from the curve, use the corresponding H value, and arrive at the permeability value if it is not specified. Also,

(µ0 * µr) = ?B/?H

So one may use an average value from selected points. Or, the manufacturer may provide a µ vs H curve(s) from which the values can be determined. If the manufacturer provides an 'AL' value, inductance per thousand turns, this may be used with the turns to compute LMAG. Knowing the permeability over this range from the B-H curve or however, with AE and LM known or to be determined, one can find the 'missing' parameters.

AE and LM are usually specified, but they can be estimated if not. AE is usually the hardest to estimate; especially when a core has a single air gap, as flux fringing effects will modify the physical gap area. Usually in a toroidal core with a distributed air gap (powdered core material) a value for the AE is given. ML is often given as well, but it is usually close to the mean circumference of the core magnetic element in a toroidal core. It the coil is NOT a toroid, the manufacturer will usually provide a value or a formula which can be used to determine this. The manufacturer can assist you with these values if they are not shown the data sheet. Now we have enough information to determine LMAG. LSAT can be determined in a similar manner, and it will be essentially that of an air-core inductor.

H0 may be found as the x-axis value of the dashed curve when the flux is zero.

The hardest parameter to find is usually FEDDY. If one is fortunate, one can find a curve of permeability vs frequency for the core material and select the 3dB down frequency. Usually this is provided for ferrite cores, or where the core is essentially linear. I could not find a value for FEDDY readily available for the specific tape core material

Recognizing that FEDDY is an input parameter, and that one can construct a test circuit to 'tweak' the parameters to fit the B-H curve properly to the manufacturers curves, an estimate may be found from curves of core loss versus frequency. Select a value of frequency where the core will be operating. A lower value is more conservative.

It should be noted that if one does not have a bipolar drive across the core model, and where the core is reset as in a flyback transformer mode, the core flux will describe a 'minor loop' wherein the 'working' B-H full loop is not traversed. These details are out of scope here, but the model will nonetheless work, although the Flux output may seem unusual at first glance. Figure 3 of reference 4 shows this effect. It is expected this will be different for non tape-wound-cores, and will differ where there are significant material differences.

XMag2 empirical Core Modeling:

The basic core model of Figure 2 is also capable of being modified slightly to account for changes in core permeability using empirical equations. Figure 3 of reference 2 provides a schematic and a netlist for such a circuit, which unfortunately do not agree with each other.

References 5 and 6 discuss this model also, and the model of Reference 5 was chosen to be generated in B2 SPICE format for this circuit, resolving the documentation problems. This is shown in Figure 4:

Figure 4
XMag2 model

This circuit is very similar to that of Figure 2, and it is drawn in a manner to show the similarities. This circuit is best suited for cores with little hysteresis. The model used user defined equations as shown in the following netlist. Because of the user defined equations, it is best NOT to convert this into a subcircuit until it has been tested and verified. Then, it will represent a specific core model ('N' will be a parameter to be passed, along with IC) or if a specific coil is described, 'N' will be imbedded in the model and only 'IC; will remain as a parameter to be passed.

The model as shown does not have a 'flux' output, as does that of Figure 3. It can be provided as a source replacing B4, or as an additional VCVS generator E1 with controlling input nodes of N8 to N2. I would prefer a flux output myself, but it is a matter of preference. In any event, the netlist for the model as shown follows:

More details can be found in Reference 5. It also provides a test circuit and values for testing the model. However, an application of the model in Figure 2 is present in Reference 6 provides details of a simulation of an actual core which may be used to validate the Figure 3 model. Note that is some implementations that R2 is shown in parallel with C1, and in other cases it is in series with it.

In this case, Reference 5 provides the following parameters to be passed to the XMag2 model of Figure 4, to simulate a Kool Mu 77140 core. Specifically,

RDCR = .01
AL = 26 mH
Lm = 0.817
FEDDY = 7.07E6
µ = 125
N = 20

The test circuit becomes as shown in Figure 5:

Figure 5
XMag2 Test circuit

The netlist for Figure 5 is:

In using the test circuit, it should be noted that if the inductor is driven into saturation in the first half cycle, or close thereto, then the operation on the coil B-H loop will 'walk' up the B-H loop to the point where a net DC current will be drawn from the input source. As the core becomes lossier, the effect will be minimized. It is interesting to adjust the parameters of the core model to see the effects of the various parameters.

Practically, the only way to really test the coil is to imbed it within a circuit whose input stimulus to the coil is dynamically adjusted by the circuit to produce no net-DC to the coil, or to 'ramp up' the coil stimulus, or to provide an initial condition for capacitor 'C1' such that the operation is centered about the origin.

One way would be to use the core model in a Royer saturating core circuit model. Another would be to use the core with an ideal transformer in a 'flyback' configuration where the stored energy in the core was allowed to totally reset the core each cycle. Either of these is out of scope for this paper, however the references should provide information to do this if it is desired. Still, in any real core, walking up the B-H loop will occur to some extent although it will be small if saturation is never reached.

I have never had the need to model a saturating transformer circuit using SPICE. But, if it is desired, it can be done using B2 Spice or indeed any version of SPICE that one has using one of the two models shown herein, and the ideal transformer models previously created.

Caveats:

One item should be noted. It is assumed that the transformer coupling from any single winding to any other is independent of which winding is being driven. In the case of a transformer such as a toroid, or a cup-core or E-I core where all the windings are on a single bobbin this will be true. However, were one to wind a transformer on an E-I core with a normally primary winding on the center leg, and two output windings, each on an outer leg of the 'E', then the effects of either of the output windings on the other would NOT be as expected. This can be seen by drawing the flux produced by the center leg core and its relation to the outer cores, and then the flux of either outer winding and the resultant flux in the remaining windings. Fortunately this construction is seldom done now.

References:

1. Article Title: "SPICE Models For Power Electronics"
Author: L.G. Meares and Charles E. Hymowitz
http://www.eettaiwan.com/ARTICLES/2002APR/PDF/2002APR22_POW_EDA_DA_AN215.PDF

2. Intusoft Power Specialist's App Notebook.
http://www.intusoft.com/psbook.htm

3. TWC-S2, How to Select the Proper Core for Saturating Transformers, Magnetics Inc.
http://www.mag-inc.com/pdf/twc-s3.pdf

4. TWC-600, Tape Wound Cores Design Manual.
Magnetics Incorporated
http://www.mag-inc.com/pdf/TWC-600.pdf

5. Non-linear Saturable Kool Mu Core Model
Scott Frankel, Analytic Engineering
http://www.aeng.com/pdf/core2.pdf

6. PCB Café book exerpt
SMPS Simulation with SPICE3
Stephen M. Sandler
http://www.pcbcafe.com/BOOKS/SMPS/222coresb.php