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Plotting Magnetization Curves

The curve tracer

Here is a circuit you can use to measure the B-H characteristics of a ferromagnetic component.

It works best with ring cores (toroids) but may be usable with other shapes having a closed magnetic path. The circuit, as shown, will plot the hysteresis loop for a half-inch diameter, high permeability ferrite ring; adaptations for other components are also given.

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The circuit diagram

Schematic of BH-test circuit.


Figure 1: Schematic of circuit for plotting the B-H characteristics of an inductor.

Tolerances are only significant on R2, R6 and C1 (which is polyester or polycarbonate). C2 and C3 are ceramic.
 

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Winding the core

The test toroid showing 22 turns of thick wire for the primary.
Photo. 1: A toroidal ferrite core custom wound to serve as a test inductor.

I chose to use two windings. Although a more complicated circuit could be devised (which only required one winding) this circuit is cheaper, easier to understand and more flexible. There is nothing special about the number of turns used - just as long as you know how many you have.

The secondary can be made from wire that is as thin as you like, while the primary need only be sufficiently thick not to get hot enough to heat the core much (the saturation level falls fairly rapidly with temperature). I used 0.2mm and 0.5mm respectively.
 

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The equipment

You will need a source of AC current of about 0.3 amp. If you're feeling lazy and don't want to wind as many turns on the primary then you'll need a higher current. I used a lab supply, which gave up to 25V at 50Hz, together with R1 to limit the current. You can improvise other solutions. A mains variac followed by a step-down transformer should work well.

Note: if you wish to measure very small rings with low permeability (such as those used in radio receivers) then you may need a source running at a few kilohertz in order to get sufficient secondary voltage. If you do this then you should also decrease C1.

The oscilloscope must be a dual channel model able to operate in an 'X-Y mode' (with the horizontal deflection controlled by a signal input rather than the timebase). Although you can use AC coupled inputs on the oscilloscope during initial tests make sure that they are set to DC coupling for the highest fidelity plot. I used an HP 54600 digital storage 'scope. A DSO is handy if you wish to plot initial magnetization curves.

Component tolerances for R2, R6 and C1 will affect the accuracy of your results.

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Adjusting the circuit

The op-amp is used as a voltage integrator. A common problem with this circuit is drift due to voltage and current offsets. R7 helps keep drift under control but you will still need to adjust R5 so that, with no signal in or out of the integrator, the output on pin 1 remains steady.

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Interpreting the curves

A narrow B-H loop showing hysteresis but not saturation.

Figure 2: Screnshot of B-H curve for the test toroid under moderate MMF.

X-axis = voltage on R2. Y-axis = Vo (voltage on U1 pin 1)


Figure 2 was obtained using a low current: This shows the characteristic hysteresis effect. Looking at the horizontal axis you see that the limits of the curve span a change in voltage of 146mV. Because R2 is 1 ohm you know that the primary current, Ip, changes by 146mA. From this you can find the change in field strength in Am-1 as:

Equation PMA
H = Np×Ip / le


Where Np is the number of turns on the primary. For the core I used this gives H = 22 × 0.146 / 0.0276 = 116Am-1.

OK, that's the field strength found. Flux density is a bit trickier. Faraday's Law tells us:

Equation PMB
Vs = Ns×dΦ/dt


Where Vs is the voltage on the secondary winding and Ns is the number of turns on the secondary and Φ is the the magnetic flux in the core. Now, all text books on op-amps give the result:

Equation PMC
dVo/dt = -Vs/(C1R6)


Where Vo is the voltage on pin 1. Substituting the previous result for Vs we get:

Equation PMD
dVo/dt = -Ns(dΦ/dt)/(C1R6)


We have time rates of change on both sides of this equation so we can integrate wrt time and get:

Equation PME
Vo = -NsΦ/(C1R6)


This is a good result because it establishes that the op-amp voltage is proportional to the core flux. You can understand why an integrator circuit is required because Faraday's Law demands that, at any instant, the coil voltage represents the rate of change or differential of core flux. By carrying out an integration (which is the reverse of differentiation) we 'get back' to a signal representing the actual flux in webers.

Equation PMF
Φ = -Vo(C1R6)/Ns


Putting in the known values:

Equation PMG
Φ = 0.239(1×10-6×1×103)/25 = 9.56×10-6   webers


We now get the flux density from:

Equation PMH
B = Φ/Ae   tesla


The core has a roughly rectangular cross section of 6.3 × 3.2 = 20.2 mm2. So

Equation PMI
B = 9.56×10-6/20.2×10-6 = 0.473   tesla


Now we can work out the permeability (at this level of field strength) from:

Equation PMJ
μ = μ0 μr = B/H   H m-1
Equation PMK
4×10-7π μr = 0.473/116   H m-1


Giving μr = 3240.

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Finding the hysteresis losses

Save the image above to disk and then open it with an image editing program such as Photoshop. Draw a rectangular selection marquee around the limits of the curve and choose Image:Histogram (in Photoshop CS select Window:Histogram then in the histogram options Expanded_View and Show_Statistics). At the bottom of the dialog box is a value for the total number of pixels selected: 45122. Now, using the polygon lasso tool trace the outline of the hysteresis loop. This encloses 4605 pixels. If our loop had the completely rectangular shape then the energy contained would be:

Equation PML
WR = H × B = 116 × 0.473 = 54.9   J m-3


However, the actual area of our loop is smaller by the fraction 4605/45122 giving an actual energy value of

Equation PMM
WA = 54.9 × 4605/45122 = 5.60   J m-3


If we ran the core at 25 kHz this would mean a hysteresis loss rate of

Equation PMN
P = 5.60 × 25×103 = 140   kW m-3


The mean core diameter is 9.5 mm so the toroid volume is

Equation PMO
VT = 20.2×10-6 × 9.5×10-3π = 6.03×10-7   m3


So the total core hysteresis loss is

  Equation PMP
P = 140×103 × 6.03×10-7 = 84.4   mW


Now, the above calculation isn't to be taken too seriously - there are several shaky assumptions, but as an indication then it should be worthwhile.

When the primary current is increased you will see a curve something like figure 3:

A B-H loop showing saturation.

Figure 3: Screnshot of B-H curve for the test toroid under higher MMF.

X-axis = voltage on R2. Y-axis = Vo (voltage on U1 pin 1)

Note the change of scale on the X-axis.


The difference in shape is due to the onset of saturation.

A magnetization curve showing flattening at large H.

Figure 4: The normal magnetization curve as the locus of the tip for each hysteresis loop.

If you repeat this measurement at different values of primary current then you can get a curve like figure 4:

As you raise the primary current the tip of the hysteresis loop traces out a normal magnetization curve. It is similar in shape to the initial magnetization curve, and is a convenient way of describing the material behavior.

Measured mur curve for toroid.

Figure 5: Relative permeability for toroidal ferrite core derived from the normal magnetization curve.

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Troubleshooting

Insufficient signal on the output of the integrator: try reducing C1. You could also reduce R6 but there's a risk that the secondary current will start to affect the H field.

You see extraneous loops at the tips of the B-H curve: use DC coupling on the 'scope inputs.


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