Plotting Magnetization Curves

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 should 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.

About your browser: if this character '×' does not look like a multiplication sign, or you see lots of question marks '?' or symbols like 'Rectangular box symbol' or sequences like '&cannot;' then please accept my apologies.

See also ...
[ Producing wound components] [A guide to the terminology used in the science of magnetism] [ Power loss in wound components] [Further example results with this circuit for other types of components] [ Faraday's law] [ Professional B-H plotting]

The circuit diagram

Circuit to produce B-H curves

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

Winding the core

Photo of ferrite toroid with primary and secondary windings 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.

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 best accuracy. 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.

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.

Interpreting the curves

The following curve was obtained using a low current:

Screen shot of low H curve
X-axis = voltage on R2. Y-axis = Vo (voltage on U1 pin 1)

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:

H = Np×Ip / le
Equation PMA

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:

Vs = Ns×dΦ/dt
Equation PMB

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:

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

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

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

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

Vo = -NsΦ/(C1R6)
Equation PME

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.

Φ = -Vo(C1R6)/Ns
Equation PMF

Putting in the known values:

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

We now get the flux density from:

B = Φ/Ae   tesla
Equation PMH

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

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

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

μ = μ0 μr = B/H   H m-1
Equation PMJ


4×10-7πμr = 0.473/116   H m-1
Equation PMK

Giving μr = 3240.

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:

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

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

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

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

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

The mean core diameter is 9.5 mm so the volume is

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

So the total core hysteresis loss is

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

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 this: Screen shot of curve at higher H
X-axis = voltage on R2. Y-axis = 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.

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

Measured mu-r curve for toroid
X-axis = Field strength (Am-1). Y-axis = relative permeability

Incidentally, 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 useful way of describing the material behavior.

The normal magnetization curve


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

Last modified: 2008 May 10th.