Intusoft is excited to release its latest ICAP/4 8.x.11 software this summer. Over the past several months a dedicated technical team has put forth an unprecedented effort to provide valuable extensions to the existing ICAP/4 2450 build. The first of these capabilities stems from an intense modeling effort, assembled from the engineering community’s custom modeling requests. Many of the devices were very complex in nature. To the surprise of the requestors, the devices were successfully modeled, and with a very high level of accuracy. ICs such as sophisticated PWMs, regulators, and specialized operational amplifiers were amongst the list. Discrete devices included select Schottky diodes, an extensive collection of saturable magnetic cores to Magnetics Designer, and special power MOSFETs. Even a solar cell/panel was added to the ever-growing IsSpice4 model collection, which now totals more than 1,000 new full custom devices since the free service began a year ago. Several unique generic models have also been added, enabling the engineer to easily tweak inherent modeling parameters to create unique versions of ICAP/4’s newest op-amps, UJTs, SCRs, and power MOSFETs.

1.0 Where  is the magnetic flux N is the number of turns L is the inductance i is current t is time It’s convenient to define the magnetic field, And flux density as, , where A is the core cross sectional area and is the magnetic path length. Ampere’s law, derived from Maxwell’s equations states: 1.1 where F is the magneto-motive force and H is the magnetic field. The core consists or n series segments. For a simple core with an air-gap, there are 2 segments consisting of the core and the gap. We’ll get back to the reluctance model after deriving the equation for inductance. Two common system of units are used, CGS with H in oersteds and B in gauss. The Standard or SI system uses the MKS system and defines H in ampere-turns/meter, and B in teslas (volt-seconds/sq-meter/turn) , where 1 tesla = 1e4 gauss. Substituting and , you get the following equation for inductance in SI units: 2.0 where 2.1 is the relative permeability is the permeability of free space. In CGS units, ,   2.2 and in SI units, ,   Then, the B-H curves and -H curves are not dependant on core area or path length; they are just material properties. The CGS system makes because so there is still a following that prefers the older CGS system because the integral of vs. H gives B vs. H without adjusting for units. For a specific core we can let and . F becomes analogous to voltage in the magnetic circuit and acts like current. Then a new quantity, Reluctance, can be defined by . For a simple core, , on the other hand, if the core consists of several magnetic paths with different permeability and geometry, the reluctance model can be used like a network of resistors to solve the magnetic equation. The attractive thing about this model is that the network topology looks just like the physical core topology, making the reluctance model an engineering favorite. Here’s what it looks like for a series gap. 3.1 and using equation 1.0, the inductance is given by:   SI units 3.2 The model is shown in Figure 4.

Figure 4:

The reluctance model includes an air gap. Flux is calculated as I(Vmphi) using a reluctance model, and Faradays equation is implemented using L1 and Be such that

Magnetic cores are described by their B-H loop properties. Inductance is proportional to the permeability, which is proportional to the derivative or slope of B with respect to H. A typical B-H curve and permeability-H curve is shown in Figure 5. For most of our discussion, hysteresis will be ignored; it can be shown that the area enclosed by the hysteresis curve is proportional to core loss energy. Moreover, the loss increases significantly at higher frequencies. Placing a resistor, Rcore, across the nonlinear inductor accurately models the high-frequency loss. Hysteresis will appear in the B-H curve if the current used to compute H includes the current delivered to the combination of the resistor and the nonlinear inductor.

Accurate modeling of inductance requires a good estimate of the permeability-H characteristic. Then the function in the model, func(x) in Figure 4, is formed by integrating the permeability curve.

Inductors are made by using a magnetic material in combination with an air-gap. The gap can be introduced as a discrete gap or as a distributed gap. The most common way of making a distributed gap is by mixing non-magnetic filler with powdered magnetic material. Discrete gaps are made by inserting a dimensionally stable material, such a paper, between the core pieces.

The inductor’s core can be modeled using a table, or by finding continuous functions that approximate the B-H characteristics. Continuous functions are less susceptible to discontinuities, and as a result they should work better in a simulator. When using a table model, the most commonly available data is from a B-H curve. Inaccuracies in transcribing the data from a data sheet curve to a table will result in magnifying the error in the permeability function (that’s because permeability is the derivative of the B-H curve). The B-H curves must be carefully filtered and their derivatives tested to assure a stable simulation. On the other hand, continuous functions can be fit to permeability-H curves using optimization techniques. Errors in transcribing data are reduced statistically, and because the continuous functions tend to describe the device physics. Many data sheets give what appear to be valid continuous functions as the ratio of polynomials. The problem with this approach is that the polynomial ratio departs from physical reality outside the region where data was used to make the best fit. This shortcoming is illustrated in Figure 6. It also illustrates how easy it is to use the IntuScope waveform viewer to evaluate and plot equations.

Powdered cores exhibit a larger swing in H for the same percentage change in permeability. The reason for this is because the powdered particles will saturate first where they are almost touching each other, slowly increasing the air gap as the magnetic material saturates. Solid cores with a single gap tend to saturate uniformly so that the permeability drops more rapidly. The packing arrangements and shape of the powdered particles vary, based on the manufacturing process. The solution of Maxwell’s equations in a closed form function is unknown, even for simple packing geometry. We will get around this problem by examining several reluctance models based on four generic solutions.

Continuous functions for permeability are useful for defining the -H curves. Many Ferrite curves follow a 1/(1+(mH)^2) curve. This function is integrated as follows:

	4.0
	4.1
	4.2
Solving for m and substituting into 4.1 yields
	4.3
Finally, solving for and setting
	4.4

The (ATAN) function in IsSpice4 has no singularities in itself or its derivatives, so that it’s a safe function to use in behavioral models. The complete atan model is shown in Figure 7.

Notice that the permeability curves, for example Figure 5, look very much like magnitude vs. frequency plots. When the slope of log(permeability)/log(H) is an integer multiple of 20db/decade, then an even polynomial ratio can be used to describe the curve, and in theory an integral can be found. Note: the restriction on even values arises from the symmetry of permeability for negative MMF-. There are 3 more shapes commonly found. First, permeability can be described as:

, then

5.1

Notice that it retains the even symmetry, but its slope for goes as 1/H. This property tends to match the characteristics for Kool Mu and High Flux cores. The integral in CGS units yields:

5.2

and for H=0,

5.3

solving for a, we fine a=1, and k1 = uro
Finally

5.4

Unlike the atan function, this one grows to infinity. But the slope drops below 1, the permeability of free space. We can therefore define

as

5.5

and from 5.1 with

Notice that for all of the models, a Usat term is added in the B1 current (Flux analog) source. Usat is usually set to 1, the relative permeability of free space. It may become convenient to give it a larger value to fit data for very high magnetic fields.

The next shape is the sigmoid B-H curve. It’s extremely sharp and can be used for square-loop material such as Orthonal or Permalloy. Starting from the B-H curve, the sigmoid is defined as:

	6.1
	6.2
	6.3
	6.4

and finally

6.5

Figure 9 shows the competed model.

Notice that this model contains a scaling parameter, Kni. It’s used to reduce the voltage that represents the MMF at node NI to normalize the voltage range, so that the VSECTOL convergence option gives good results. It’s usually set to kh, conveniently making the voltage at node NI equal to H in Oersteds.

Yet another B-H loop comes from combining a linear relation below a field, , as:

Then an exponential function is used thereafter such that the inductance at . The parameters , , and define the core as follows.

where

This exponential model is used internally in Magnetics Designer for the reluctance model used to compute inductance verses current. It provides a reasonably accurate model for powdered cores; however, there is an abrupt change in the derivative of inductance where the two functions are joined, as seen in Figure 10. It predicts sharper cutoff characteristics for Ferrite cores.

Figure 11 illustrates the differences for a gaped core.

For cores that have distributed gaps, such as MPP or Kool Mu powered cores, a special reluctance model is used. In general, two cores are made in parallel, each with its own gap. The mix parameter tells how much the first core contributes to the H=0 permeability. Note: parallel cores correspond to series inductors.

Table 2 shows the combination of free parameters used to get the best fit to the published data. Free parameters are Model (Log or Atan), Bsat, Usat, uro, ur1, mix. Log cores have no explicit Bsat parameter, and if Mix=1, ur1=1.

Table 2: parameters for powdered cores

The RMS error is in general based on data for percent initial permeability = 80,50,20. When these permeability values can’t be achieved, three different values are substituted. The accuracy to which the charts can be read is larger than the error in the curve fit. Moreover, the initial permeability tolerance is much greater than the curve fit tolerance. Most inaccuracy occurs for percent permeability below 10%, or for deep saturation. Figures 12-16 show the percent permeability verses H for the families in the above table.

It is recommended that the atan model be used for other ferrite cores, and the sigmoid model selected for solid cores. Provisions are made for making your own user models, and connecting them to the Magnetics Designer SPICE model. All of the non-linear models use the IsSpice4 syntax and need to be translated if you use another simulator.

Documentation on how to convert to IsSpice4 syntax is located:
http://www.intusoft.com/convert2isspice4.htm