hysteresis in ferromagnetic materials

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Hysteresis in Ferromagnetic Materials

Bruno Murta, [email protected], Churchill College, Cambridge, UK (Practical partner Songran Shi, Fitzwilliam College, Cambridge, UK)

Experiment performed Friday 8th November 2013

ABSTRACT The properties of ferromagnetic materials were investigated using mild steel and transformer iron samples. Such samples were introduced in the core of a secondary solenoid. A 6 V power supply generated an alternating current across a primary coil, inside which was the secondary coil. The resulting magnetic field generated an electromotive force across the terminals of the secondary coil. The relative permeabilities (µr) of mild steel and transformer iron were determined to be 104.2 ± 2.4 and 221 ± 29 in the low-field region, and 11.7 ± 1.2 and 14.3 ± 1.0 in the high-field region, respectively. The energy lost per unit volume per cycle of hysteresis was of (38 ± 3) kJ m-3 cycle-1 for mild steel and (10.9 ± 1.4) kJ m-3 cycle-1 for transformer iron. To study the temperature dependence of ferromagnetic properties, the secondary coil and a CuNi alloy core were immersed in water, which was heated and cooled to vary the temperature of the sample. CuNi was found to be paramagnetic at 48ºC and the relative permeability was 1.28 ± 0.06. At 5ºC CuNi exhibited ferromagnetic behaviour. In the low-field region µr = 2840 ± 50, whereas in the high-field region µr = 197 ± 3. The energy lost per unit volume per cycle at this temperature was (490 ± 50) J m-3 cycle-1. The Curie point of CuNi alloy was confirmed to be between 5ºC and 48ºC.

I. INTRODUCTION Ferromagnetic materials play a large and growing

role in today’s technology. Many common devices, such as transformer cores and memory devices, explore such magnetic properties. These technological advances are only possible if the underlying physical phenomena are understood.

The objectives of this experiment are therefore: • to study the properties of ferromagnetic

materials, namely: o how the induced magnetic field flux

density B and the relative permeability µr vary with the applied field H;

o the energy loss per unit volume per cycle of hysteresis;

• to investigate the temperature dependence of ferromagnetic properties.

An electrical circuit with a secondary solenoid inside a primary solenoid was set up. A 6 V power supply was used to generate an alternating current across the primary coil. The resulting magnetic field induced an electromotive force across the terminals of the secondary coil. To study the properties of

ferromagnetic materials, mild steel and transformer iron samples were introduced in the core of the secondary coil. To investigate the temperature dependence, the secondary coil and the CuNi alloy core were immersed in water, which was heated and cooled to vary the temperature of the sample.

A more detailed explanation of the theory is disclosed in the next section. Section III provides with a careful description of the method and includes the results of the experiment. In section IV such results are discussed and improvements to the experiment are mentioned. The overall conclusions are presented in section V.

II. THEORETICAL BACKGROUND Magnetic moments are produced by spinning

electrons orbiting the atomic nucleus. Some atoms and ions behave as magnetic dipoles.

The magnetisation of a material is a function of how strong and how well aligned the separate moments are. The relative orientation may be influenced by the neighbouring moments and by an external field. Depending upon the material considered, these individual dipole moments may be

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perfectly aligned, randomly oriented or somewhere in between.

In some materials, such as air, there are some unpaired electrons in the molecular orbitals. Because such electrons are isolated and non-interacting, the magnetic moments are randomly aligned and the net moment is zero. If, however, an external field H is applied, there is partial alignment along the direction of H, which leads to a low magnetization. This phenomenon is known as paramagnetism.

However, other materials have many unpaired electrons in partially filled shells, and therefore there is a strong interaction between the individual moments. In fact, cooperative alignment tends to be favoured to minimise the energy of the system (a quantum mechanical effect known as exchange interaction [1]). Moments align with the applied field and such orientation may become permanent, i.e. moments may remain aligned even without an external field. The phenomenon associated with this non-linear behaviour is known as ferromagnetism.

At high temperatures thermal agitation competes with exchange interaction. As a result, the magnetic properties of ferromagnetic materials are temperature-dependent: there is a temperature above which the material is no longer ferromagnetic but paramagnetic. Such temperature is known as the Curie point [2].

A current I passing through a long solenoid (primary coil) of nP turns and length LP generates a magnetic field B = µ0 I nP / LP inside it. Such current I can be measured by passing it through a resistance R* and measuring the voltage VX across it:

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Provided that I is varying, if there is a second

solenoid (secondary coil) of nS turns and cross-sectional area AS inside the primary coil, according to Faraday’s Induction Law an electromotive force (EMF) will be induced across the terminals of the secondary coil:

234#$#05#65#7)-78######################(9,#

Integrating the EMF with respect to time yields:

!"# !!" ! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Dividing by RC, where R and C are associated

with the (ideal) integrator circuit, gives a quantity VY with dimensions of voltage:

!! !!!" !"# !!" ! !!!!!!!!!" !!!!!!!!!!!!!!

When the material that fills the secondary coil is

paramagnetic, VY is expected to be proportional to VX. Conversely, for ferromagnetic materials, this linear behaviour is not observed. For the latter case, the expression for B has to be replaced by B = µr µ0 I nP / LP, where µr is a dimensionless parameter whose value is not constant for the particular material, as it depends on the externally applied magnetic field H. Hence, equation (1) becomes:

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III. METHOD AND RESULTS

A. Set Up The electrical circuit used in all parts of the

experiment is shown in the figure below:

The current that passes through the primary

solenoid was measured by passing it through a

Figure 1: Schematic diagram of electric circuit. The capacitance C was chosen so as to approximate the integrator to an ideal integrator. PicoScope Oscilloscope Software was used instead of a conventional analog oscilloscope. The 2.2 ! resistor R* limits the current across the primary coil and hence prevents overheating.

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resistance R* and measuring the voltage across it (VX). This resistor also limits the current and prevents the coil from overheating.

A suitable integrator was designed in order to measure the output voltage (VY) by integrating the EMF generated at the secondary coil. Given that the integrator was assumed to be approximately ideal in section II, the capacitor impedance was set to be much smaller than the resistance of the shunt resistor R1 at a frequency of 50 Hz (the frequency of the current across the primary coil). Using a 1 M! shunt resistor, the capacitance was required to be much greater than 3.2 nF. Hence, a 1 µF capacitor was used. A TL071 amplifier was used to make the integrator.

The gain, A, of the ideal integrator [3] is given by:

! ! !!!!!!!! !! ! !!

!!!! ! !! ! !

!!!" !!!!!!!!!!!! The performance of the integrator was tested in

two ways: 1) Since an integrator with the same properties (i.e. same shunt resistor and capacitor) had been used in a previous experiment, the gain at 50 Hz was measured and compared with the value from the previous experiment. The gain was determined to be A = 0.34 ± 0.02. Hence, the gain previously determined, A = 0.32, is within the error range. 2) A square wave was set as the input signal. The observed output signal in the oscilloscope was a triangular wave, as expected.

The dimensions of the core samples and the secondary coil were measured with callipers. The properties of the electrical components used in the circuit (e.g. resistance, capacitance) were measured using a bridge.

B. Calibration The experiment was calibrated by measuring the

permeability of air, which is constant and tabulated [4]. Combining equations (2) and (3) from section II gives:

!!!!! !!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

The experiment was performed with air occupying

the whole volume inside the secondary coil. The value of µr was determined from the slope of the VY versus VX graph (figure 2).

Sample Cross-Sectional Area / m2 Secondary Coil (2.290 ± 0.004) x 10-5

Mild Steel (8.35 ± 0.20) x 10-6 Transformer Iron (2.91 ± 0.08) x 10-6

CuNi Alloy (2.0 ± 0.3) x 10-5

Quantity Measurement nS 500 nP 400 LP (4.2 ± 0.2) x 10-2 m R* 2.22 ± 0.01 ! R (9.87 ± 0.01) x 103 ! R1 (9.79 ± 0.01) x 105 ! C (1.01 ± 0.01) x 10-6 F

µ0 (definition) 4" x 10-7 H / m

Table 1: Summary of cross-sectional areas of core samples and secondary coil. The diameters of the samples were measured with a calliper.

Table 2: Summary of measurements. The resistances and capacitances were measured using a bridge.

6 4 2 0 2 4 60.08

0.06

0.04

0.02

0

0.02

0.04

VX / V

V Y / V

VY versus VX (Air)

data 1 lineary = 0.00724*x 0.0201

Figure 2: VY versus VX when air fills the whole volume inside the secondary coil. Using the slope from the linear regression and equation (7) the calibrated permeability of air was determined to be 1.16 ± 0.06

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The experimental value of the relative permeability of air was determined to be µr = 1.16 ± 0.06. Hence, the tabulated value, µr = 1.00, is not within the error range.

The main cause for the difference between the measurement and the lecture value is the paramagnetic background, likely due to the material around which the secondary coil is wrapped. Such material is probably some steel or possibly aluminium and its presence in the experimental apparatus leads to this linear behaviour typical of paramagnets.

This paramagnetic background was accounted for in the following parts of the experiment.

C. The Experiment

a) Mild Steel A mild steel core was introduced in the secondary

solenoid. Although the sample did not fill the inner volume of the solenoid, the effective cross-sectional area AS was approximated to that of the mild steel sample, since its relative permeability (~102) is much greater than that of air (~1). As such, the effect due to air is negligible when compared with that due to mild steel.

In order to plot the hysteresis loop, the x-axis was calibrated in units of B/(µr µ0) (i.e. A/m) and the y-axis in units of B (i.e. T). Using equations (4) and (5), this implies multiplying VX by a factor of nP / (LP R*) and VY by RC / (nS AS), respectively.

To account for the correction due to calibration, the background effect was subtracted from the B vs. B/(µr µ0) plot in the high field range (for |H| > 2 x 104 A/m). Such correction was not required in the low field range, because the background effect is negligible: the relative permeability of mild steel is still much greater than the corrected permeability of air.

The energy loss per unit volume of material per cycle round the loop, Udissipated, is given by the area enclosed by the hysteresis loop, since:

!!"##"$%&'! ! !!!!!!

!" !!!!!!!!!!!!!!!!!!!!!!!!!!!

This area was determined using approximations with triangles and parallelograms. The energy lost per unit volume per cycle of hysteresis was estimated to be (38 ± 3) kJ m-3 cycle-1.

Figure 4: Estimation of area of hysteresis loop for mild steel. The area of the loop was estimated by summing the areas of the five simple polygons drawn in the figure. The energy lost per unit volume per cycle was estimated to be (38 ± 3) kJ m-3 cycle-1.

In order to estimate the range of relative permeability µr for mild steel, two cases were considered:

3 2 1 0 1 2 3x 104

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0.5

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1.5

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B /(µ0 µr) / A/m

B / T

B versus H (Mild Steel)

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1

3

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3 2 1 0 1 2 3x 104

2

1.5

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0.5

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0.5

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B /(µ0 µr) / A/m

B / T


Figure 3: Hysteresis loop of mild steel. At high-field range (|H| > 2 x 104 A/m) the paramagnetic background determined in the calibration was subtracted from the original experimental data.

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1) Low Field Range: the maximum value of µr is observed since this corresponds to the steepest sections of the hysteresis loop (the rate of change of B is greatest);

2) High Field Range: the minimum value of µr is observed, since the rate of change of B is lowest.

The upper and lower values of µr were determined directly from the slope of the hysteresis loop in the appropriate ranges, since the slope is µr µ0. The two trendlines used to estimate the upper value of µr are shown in figure 5 in red, whereas the two lines used to estimate the lower value are in green. The final estimate of each value is the average of the two respective estimates.

In the low-field region, µr = 104.2 ± 2.4. In the high-field region, µr = 11.7 ± 1.2.

Figure 5: Estimation of range of relative permeability µr of mild steel. The red lines were used to estimate the upper limit of µr in the low-field range, whereas the lower limit in the high-field range was estimated using the green lines. The final estimate of each value is the average of the values from each line. In the low-field region, µr = 104.2 ± 2.4. In the high-field region, µr = 11.7 ± 1.2.

b) Iron Transformer A similar method was followed when the

transformer iron sample was used as the core inside the secondary solenoid. The assumptions regarding the effective cross-sectional area AS and the correction due to calibration are still valid.

Figure 7: Estimation of area of hysteresis loop for transformer iron. The area of the loop was estimated by summing the areas of the three simple polygons drawn in the figure. The energy lost per unit volume per cycle was estimated to be (10.9 ± 1.4) kJ m-3 cycle-1.

In the low-field region, µr = 221 ± 29. For high fields, µr = 14.3 ± 1.0. The energy lost per unit volume per cycle was (10.9 ± 1.4) kJ m-3 cycle-1.

3 2 1 0 1 2 3x 104

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B /(µ0 µr) / A/m

B / T


3 2 1 0 1 2 3x 104

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B /(µ0 µr) / A/m

B / T

B versus H (Transformer Iron)

Figure 6: Hysteresis loop of iron transformer. At the high field range (|H| > 1.2 x 104 A/m) the paramagnetic background determined in the calibration was subtracted from the original experimental data.

3 2 1 0 1 2 3x 104

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1.5


B / T

B /(µ0 µr) / A/m

1

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3

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Figure 8: Estimation of range of relative permeability µr of transformer iron. In the low-field region, µr = 221 ± 29. In the high-field region, µr = 14.3 ± 1.0.

c) Cu/Ni Alloy In the case of the last sample, a Cu/Ni alloy, a

change in hysteresis properties occurs around 40ºC, the Curie point.

The secondary solenoid and the Cu/Ni alloy core inside it were immersed in hot water in a beaker with a thermometer. The water was heated before the coil was immersed, otherwise melting, fire or shorting could result. The temperature was allowed to stabilize and was confirmed to be above 40ºC.

The method described in a) was performed.

However, since Cu/Ni was found to be paramagnetic at 48ºC, the cross-sectional area considered was that of the secondary coil and not of the Cu/Ni core. In addition, since no hysteresis loop was observed, the energy loss per cycle was not determined.

The relative permeability, µr, for CuNi alloy at 48ºC was determined to be 1.28 ± 0.06.

Later, ice cubes were added to the water in the beaker in order to lower the temperature below 40ºC. The temperature was again allowed to stabilise and confirmed to be below 40ºC.

The method was repeated exactly as described in a). The assumptions regarding the effective cross-sectional area AS and the correction due to calibration are valid, as Cu/Ni was found to be ferromagnetic at this temperature.

In the low-field region, µr = (2.84 ± 0.05) x 103. In

the high-field region, µr = 197 ± 3. The energy lost per unit volume per cycle was (490 ± 50) J m-3 cycle-1.

3 2 1 0 1 2 3x 104

2

1.5

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0.5

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0.5

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B /(µ0 µr) / A/m

B / T


3 2 1 0 1 2 3x 104

0.08

0.06

0.04

0.02

0

0.02

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B /(µ0 µr) / A/m

B / T

B versus H (CuNi Alloy, T = 48ºC)

data 1 lineary = 1.64e 06*x 0.0224

Figure 9: B versus H for CuNi alloy sample at 48ºC. Using the slope from the linear regression and the fact that the slope equals µr µ0, the permeability of paramagnetic CuNi at this temperature was determined to be 1.28 ± 0.06

3000 2000 1000 0 1000 2000 30002

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B /(µ0 µr) / A/m

B / T


Figure 10: Hysteresis loop of iron transformer. At the high field range (|H| > 800 A/m) the paramagnetic background determined in the calibration was subtracted from the original experimental data.

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Figure 11: Estimation of area of hysteresis loop for ferromagnetic CuNi alloy at 5ºC. The area of the loop was estimated by calculating the area of the parallelogram drawn in the figure. The energy lost per unit volume per cycle was estimated to be (490 ± 50) J m-3 cycle-1.

Figure 12: Estimation of range of relative permeability µr of ferromagnetic CuNi alloy at 5ºC. In the low-field region, µr = 2840 ± 50. In the high-field region, µr = 197 ± 3.

IV. DISCUSSION A. Interpretation of Results The B versus B/(µr µ0) graph for the mild steel

core exhibits a clear hysteresis loop. Hence, this experiment confirmed the ferromagnetic behaviour of this material. This analysis also applies to the transformer iron.

The relative permeability of mild steel was found to vary from 104.2 ± 2.4 in the low-field range to 11.7 ± 1.2 in the high-field range. Likewise, for transformer iron, µr was found to be 221 ± 29 (low-field) and 14.3 ± 1.0 (high-field).

The energy released per unit volume per cycle was determined to be (38 ± 3) kJ m-3 cycle-1 for mild steel and (10.9 ± 1.4) kJ m-3 cycle-1 for transformer iron.

Despite the similar magnetic properties of these two materials, a relevant difference can be observed when comparing the two hysteresis loops: the coercive field of transformer iron is clearly lower than that of mild steel, which is clearly noticeable in terms of the width of the hysteresis loop.

As a result, transformer iron allows easy switching from positive to negative saturation magnetization (which explains why this material is used in transformer cores), whereas the higher coercive field of mild steel makes it a better material to be used in permanent magnets.

The study of the magnetic properties of the CuNi sample illustrated the temperature dependence of ferromagnetism. Indeed, at 48ºC the B vs. B/(µr µ0) graph was a straight line, and therefore the CuNi alloy is paramagnetic at this temperature. However, the same sample exhibited ferromagnetic behaviour at 5ºC, because a hysteresis loop was observed. This experiment therefore confirms that the Curie point of CuNi is between 5ºC and 48ºC.

The relative permeability at 48ºC was found to be 1.28 ± 0.06, which is a typically low value associated with paramagnetic materials. At 5ºC µr = (2.84 ± 0.05) x 103 at low field, which is one order of magnitude greater than the analogous values of mild steel or transformer iron. Although the coercive field of CuNi at 5ºC is low (~100 Am-1), the remanent magnetization is probably not high enough for this material to be used in transformer cores.

B. Improvements to the Experiment The apparatus used to measure the temperature of

the CuNi sample could be improved. In fact, the thermometer did not measure the actual temperature of the sample, but rather the temperature of the water. Alternatively, using a thermometer attached to the CuNi alloy sample would be recommended.

In addition, the CuNi sample was immersed in water even at a temperature above 40ºC, since this

3000 2000 1000 0 1000 2000 30002

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B / T


3000 2000 1000 0 1000 2000 30002

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B / T


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was the only way to heat the sample. Given that the relative permeability of CuNi at this temperature is low, water may have contributed with a non-negligible systematic error that could not be evaluated during the practical session. As a result, developing a different method for heating the sample should be considered.

It is possible to verify that both the B vs. H graphs for the transformer iron and the CuNi alloy are not centred at the origin, even though the PicoScope was operating in the AC mode. This systematic error might have to be further investigated, but a possible solution would be adding a high-pass filter to the circuit.

The exact composition of the samples used in this experiment should have been determined in order to allow the comparison of these experimental values with those of future experiments. This could not be accomplished with the apparatus available in the laboratory.

V. CONCLUSIONS The magnetic properties of mild steel and

transformer iron were investigated. Both materials were confirmed to be ferromagnetic. The relative permeabilities (µr) of mild steel and transformer iron were determined to be 104.2 ± 2.4 and 221 ± 29 in the low-field region, and 11.7 ± 1.2 and 14.3 ± 1.0 in the high-field region, respectively. The energy lost per unit volume per cycle of hysteresis was of (38 ± 3) kJ m-3 cycle-1 for mild steel and (10.9 ± 1.4) kJ m-3 cycle-1 for transformer iron.

CuNi alloy was used to study the effect of temperature in ferromagnetic properties. CuNi was found to be paramagnetic at 48ºC and the relative permeability was 1.28 ± 0.06. At 5ºC CuNi exhibited ferromagnetic behaviour. In the low-field region µr = 2840 ± 50, whereas in the high-field region µr = 197 ± 3. The energy lost per unit volume per cycle at this temperature was (490 ± 50) J m-3 cycle-1. The Curie temperature of CuNi alloy was therefore confirmed to be between 5ºC and 48ºC.

VI. REFERENCES [1] Z. Barber, Materials Science Part IA Course B

Handout: Materials for Devices, pp. 36-38, Department of Materials Science and Metallurgy, University of Cambridge, 2012.

[2] D. J. Griffiths, Introduction to Electrodynamics, p 281, New Jersey, 1999

[3] Systems and Measurements – IB Physics A & Physics B Practicals – Michaelmas 2013, pp. 23-24 Cavendish Laboratory, University of Cambridge, 2013.

[4] Richard A. Clarke, “Clarke, R. “Magnetic Properties of Materials”, surrey.ac.uk”.

hysteresis in ferromagnetic materials

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