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FairRite Type 43
Material  BH Curve
16 March 2008. I've been looking at the BH curves for
toroidial core made with FairRite type 43 material and I thought the work might
be of general interest.
First of all, "what's a BH curve and why should I care
about it"?
Rather than reinvent the wheel, we'll start with the ARRL
Handbook's summary of magnetic components. 
An inductor creates a magnetic field when current flows through its
conductors. (ARRL Handbook, 2006, Fig. 4.41, page 4.25.)


The magnetomotive force (mmf) produced is proportional to the current
flowing in the conductor and the number of turns.
Of more immediate interest is the magnetic field
strength, H, measured (in the CGS system) in Oersteds, given by:
(ARRL Handbook, 2006, Equation 55, page 4.25)
Certain materials concentrate the magnetic flux lines,
increasing their density. The ratio of the density of the flux lines with the
material in place to that of the same form in a vacuum is defined as the
permeability of the material.
The magnetic flux density symbol is B and the
relationship between B and H is:
B=µH
B is the magnetic flux density in (CGS units) in Gauss
H is the magnetic field strength in Oersted
µ is the relative
permeability with vacuum = 1.0.
Air has a permeability essentially identical with that of
a vacuum.
Plot B versus H, shows a relatively complicated
relationship for most practical magnetic materials. 
Relationship between B and H for a typical magnetic
material. (ARRL Handbook, 2006, Fig 4.44, page 4.26)


The retentivity of magnetic core materials creates another
potential set of losses caused by hysteresis. [The above figure] illustrates the
change of flux density (B) with a changing magnetizing force (H). From starting
point A, with no residual flux, the flux reaches point B at the maximum
magnetizing force. As the force decreases, so too does the flux, but it does not
reach zero simultaneously with the force at point D. As the force continues in
the opposite direction, it brings the flux density to point C. As the force
decreases to zero, the flux once more lags behind. In effect, a reverse force is
necessary to overcome the residual magnetism retained by the core material, a
coercive force. The result is a power loss to the magnetic circuit, which
appears as heat in the core material. (ARRL Handbook, 2006, Page 4.27)
How do we measure B and H?
Measuring H is relatively simple. From the equation above
(ARRL Handbook 55), H is a function of the current, number of turns and the mean
inductance path. All of these parameters are known or measurable.
Measuring B is a bit more difficult. I'm not going to go
through the math here, but it can be shown that if the core is wound as a
transformer, the voltage on the secondary (if lightly loaded so that no
appreciable current is drawn) is proportional to the rate of change (derivative)
of B versus time. Thus, by integrating B, with, for example, an RC integrator,
it is possible to determine B. The relationship is given (See
Steward's
catalog 11D at page 33 for example) by:


The photo below shows the test transformer and RC integrator I used. Instead of
using R1 to sense the current, I used a Tektronix 6022 current probe.


The core is a FairRite P/N 5943001001, type 43 material, with dimensions of:
OD: 1.142", ID: 0.748", thickness: 0.295"
le = 7.3 cm (given just as l in equation 55 in the ARRL handbook)
Ae = 0.37 cm^{2}The primary and secondary
are wound with 30 turns no. 22 AWG magnet wire.
Current is measured with a Tektronix 6022 clipon current
probe. The signal generator is an HP200CD, and the power amplifier is an
Acoustic Research AR1. To protect the AR1 and to limit the current, a 4 ohm,
100 watt resistor is in series with the primary winding.
Data is taken at 10 KHz. The RC integrator uses a 10K/1%
resistor, and a 0.22uF polyester film capacitor, measured and selected to be
within 1% of nominal value. A Tektronix 2246 oscilloscope is used to display the
data, operated in XY mode. The X axis has the drive current (H) and the Y axis
displays the integrated voltage output (B).
The impedance of the integrating capacitor should be
around 1% of the resistance of R, at the test frequency. R should be chosen so
that the secondary current is negligible in comparison with the primary current.
At 10 KHz, the 0.22µF
integration capacitor has an impedance of about 72 ohms, so the 100:1 ratio is
satisfied. 
For ease of computation, I've substituted the actual values for R2, C, N, Ae and
le and restructured the equations as:
 H = 5.16I, where H is in oersted and I is amperes
 B = 19.8V, in gauss, where V is the integrator output
in millivolts.
Of course, these numerical relationships hold only
for this particular core size, number of turns and RC integration values.
FairRite provides the following data for Type 43
material. 


Let's look at the measured data.
The first illustration is at low drive. The horizontal axis is 2 mV/division,
and the 6022 current probe is in 1mV = 1mA mode. Hence the furthest upper right
point on the BH ellipse corresponds to a drive current of 7.6 mA and an
integrator output voltage of 3.2 mV. The corresponding B and H values are:
H = 5.16*0.0076 = 0.04 oe.
B = 3.2*19.8 = 64 gauss
The computed permeability is thus 64/.04 = 1600.
Note that Type 43 material has a nominal "initial permeability" of 800, or about
half the measured value. Why is this?
The answer is that what we have measured is the "amplitude
permeability" not the "initial permeability." The initial permeability is
measured at very low levels of drive, corresponding to < 10 gauss. The image
below is taken at 64 gauss, more than 6 times the level used in
incremental permeability specifications.
FairRite defines initial permeability as:
Permeability, initial  µi The
permeability obtained from the ratio of the flux density, kept at <10 gauss,
and the required applied field strength. Material initially in a specified
neutralized state.
Ferronics defines amplitude permeability
as:
Amplitude Permeability μa � The quotient
of the peak value of flux density and peak value of applied field strength
at a stated amplitude of either, with no static field present.
At the limit, as the applied H field is small
such that B is <10, the initial permeability and amplitude permeability will be
the same. My test setup does not permit accurate measurements at very low B
levels, however.


As we increase the drive level, we start to see signs of saturation.
In the image below, at the right top peak, B=2930 gauss and
H=3.25 oe. (Current = 630 mA peak) FairRite's data sheet suggests that for 3.25
oe, B should be about 2500 gauss, which is reasonable agreement considering
FairRite's data is taken for a smaller core.


In the final image, the peak current is 2 amperes, corresponding to H=10.3 oe
B is about 3800 gauss, as into gross saturation where B increases only slightly
with increasing H. FairRite data says that B at this drive level should be
around 2900 gauss, but again the data sheet is is for a smaller core.



The area inside the BH curve represents lost power. It takes
energy to move around the BH curve and the core loss due to hysteresis is
proportional to the area inside the BH curve.
Finally, we can note that when H = 0, i.e., when there is zero current applied,
there still is a B field. In the above figure, it's around 2600 gauss. 






