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PROFESSOR: Measurement of the B-H characteristic
of a magnetizable core provides the constitutive law
that relates the magnetization density, m, or equivalently,
the magnetic flux density, B, to the macroscopic magnetic field
intensity, H, within a material.
The objective here is to observe the establishment of H
by an imposed current in accordance with Ampere's law
and to deduce B from the voltage it induces in accordance
with Faraday's law.
A toroidal coil is tightly wound onto a donut-shaped
magnetizable core.
Each turn carries a current i.
For the toroidal geometry, Ampere's integral law
is enough to relate the current i
to the H field in the magnetizable core material.
Symmetry about the toroidal axis suggests
that H is phi-directed.
Ampere's integral law is written for a contour, C, circulating
about the toroidal axis within the core.
Because the major radius, R, of the torus
is large compared to the minus radius,
the field is essentially uniform over the cross-section
of the torus.
We approximate the average radius by the radius R. Thus,
the line integral of H is essentially 2 pi R times H.
To evaluate the right-hand side of Ampere's law,
we see that the surface, S, spanned by this contour, C,
is pierced by the current i as many times
as there are turns in 1.
So the surface integral of j is in N1i.
Thus, Ampere's law reduces to the product of H phi
and the contours circumference 2 pi R equally N1i.
Here is the excitation winding.
The current is supplied by this Variac.
The voltage drop across the small resistance
in series with the driving winding
is used to record the driving current i.
This voltage drives the horizontal amplifier
of the oscilloscope.
Because the H field is proportional to current,
the horizontal scope deflection is therefore
proportional to the magnetic field intensity, H,
in the core material.
The magnetic flux density is measured
by means of a second sensing or secondary winding,
also wound on the donut-shaped core.
This coil is terminated in a high enough impedance
that it carries a negligible current.
Thus, the H field established by the current i of the excitation
winding is unchanged.
The flux linked by each turn of the sensing coil
is essentially the flux density, B,
multiplied by the cross-sectional area
of the core.
The flux linked by the N2 turned coil is then N2 times the flux
linked by each turn.
An RC integrating network on the sensing of secondary coil
integrates the terminal voltage so that the flux linking
the coil is recorded.
The circuit acts as an integrator
because the capacitor has a reactant
at the frequency of operation, 60 Hertz, that
is much smaller than the series resistance.
With the capacitor essentially a short circuit,
the current is approximately v2 over R2.
From Faraday's law, the terminal voltage of the sensing coil
is the time rate of change of the magnetic flux linking
the coil.
This current then charges the capacitor.
By comparing these last two
Equations, we can see that the voltage across the capacitor
is proportional to the magnetic flux linking the sensing coil.
Thus, by connecting the terminals of the sensing coil
through the integrator to the vertical trace
of the oscilloscope, we make the vertical axis proportional
to B. So we should see a display of B versus H
on the oscilloscope.
Here is the experiment.
The magnetizable material is in this donut-shaped toroid.
The excitation or primary coil connected to these terminals
is driven by the current at 60 Hertz from this Variac.
The voltage across this series, 1-ohm resistance.
Then gives a horizontal deflection
of the oscilloscope proportional to i and, thus, H.
The terminals of the sensing coil
are connected through the integrating network
with resistance 100K ohms.
And capacitance is 1 microfarad to the vertical deflection
terminals of the oscilloscope.
Thus, the vertical deflection is proportional
to the integral of the terminal voltage, the flux lambda,
and, hence, to B.
We turn up the Variac current.
The scope shows the B-H hysteresis loop
typical of a magnetically soft material.
For our primary coil of 83 turns,
the major radius is about 3.2 centimeters.
The current viewing resistor on the primary coil is 1 ohm.
Thus, the scope-scaling factor relating H to V1 equal IR1
is 413.
With a horizontal scope scale of a 1/2 a volt per centimeter,
each scope horizontal deflection of 1 centimeter
is about 210 amps per meter.
The minor diameter of the secondary coil
is 1.2 centimeters.
There are 537 turns.
The integrating circuit has resistance
100K ohms and capacitance 1 microfarad.
The scaling factor relating integrating capacitor voltage V
sub C to magnetic flux density B is 1.65.
With a vertical scale a 1/2 a volt per centimeter,
each vertical deflection of 1 centimeter is 0.83 Teslas.
Thus, for this core, saturation is
reached at B, approximately 2 Teslas,
for an H field of about 500 amps per meter.
Polycrystalline ferromagnetic material at the domain level
has randomly oriented magnetic moments
that tend to cancel in the absence of an applied field.
As a magnetic field is applied, the domain walls shift.
Those domains with magnetic dipole moments
align with the applied field, grow at the expense of domains
having oppositely directed magnetic dipole moments.
The result is a net magnetization
that tends to be in the direction of H.
In ideal materials at high magnetic fields, saturation
results when all of the domains combine into one.
In real materials used for transformers,
imperfections and boundaries result
in the domain walls encountering a resistance to motion.
A phase leg then develops between the magnetization
and the magnetic field intensity, H.
The B-H trajectory makes this evident.
It shows that even when the applied field, H, is
instantaneously 0, there remains a remnant flux density, B sub
R. And the field must be reversed to minus H sub
C, the coercive field intensity, to reduce the flux
density to 0.
Hysteresis can be both useful and harmful.
Permanent magnetization at the remnants point
is the basis for permanent magnets
and for the storage of information on magnetic tapes
and disks.
However, periodic excitation of the hysteretic material,
such as in a transformer or rotating machine,
dissipates energy and generates heat.
The area enclosed within the B-H hysteresis loop
is the energy per unit volume dissipated in one cycle.
The dissipation arises from the resistance
to propagation of the domain walls.