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.
The circuit diagram
Tolerances are only significant on R2, R6 and C1 (which is polyester
or polycarbonate). C2 and C3 are ceramic.
Winding the core
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 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 R_{2}, R_{6} and C_{1} 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
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, I_{p},
changes by 146mA. From this you can find the change in field strength in
Am^{-1} as:
- Equation PMA
- H = N_{p}×I_{p} / l_{e}
Where N_{p} 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
- V_{s} = N_{s}×dΦ/dt
Where V_{s} is the voltage on the secondary winding and
N_{s} 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
- dV_{o}/dt = -V_{s}/(C_{1}R_{6})
Where V_{o} is the voltage on pin 1. Substituting the previous
result for V_{s} we get:
- Equation PMD
- dV_{o}/dt = -N_{s}(dΦ/dt)/(C_{1}R_{6})
We have time rates of change on both sides of this equation so we
can integrate wrt time and get:
- Equation PME
- V_{o} = -N_{s}Φ/(C_{1}R_{6})
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 = -V_{o}(C_{1}R_{6})/N_{s}
Putting in the known values:
- Equation PMG
- Φ = 0.239(1×10^{-6}×1×10^{3})/25 = 9.56×10^{-6} webers
We now get the flux density from:
- Equation PMH
- B = Φ/A_{e} tesla
The core has a roughly rectangular cross section of 6.3 × 3.2 =
20.2 mm^{2}. 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 = μ_{0} μ_{r} = B/H H m^{-1}
- Equation PMK
- 4×10^{-7}π μ_{r} = 0.473/116 H m^{-1}
Giving μ_{r} = 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:
- Equation PML
- W_{R} = 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
- W_{A} = 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 × 25×10^{3} = 140 kW m^{-3}
The mean core diameter is 9.5 mm so the toroid volume is
- Equation PMO
- V_{T} = 20.2×10^{-6} × 9.5×10^{-3}π = 6.03×10^{-7} m^{3}
So the total core hysteresis loss is
- ^{ }Equation PMP
- P = 140×10^{3} × 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:
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 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.
Troubleshooting
Insufficient signal on the output of the integrator: try reducing C_{1}. You could also reduce R_{6} 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.