Magnetic Measurements

Introduction and Context

Magnetic measurements determine the magnitude, distribution, direction, and time variation of magnetic quantities in electrical materials and devices. They are essential in the design and validation of transformers, rotating machines, inductors, magnetic sensors, permanent magnets, magnetic shielding systems, relays, actuators, and power-electronic components.

The principal quantities are magnetic flux Φ, flux density B, magnetic field strength H, magnetomotive force F, permeability μ, reluctance ℜ, and magnetic moment. Although these quantities are related, they are not interchangeable. A reliable measurement system must distinguish field quantities from material properties, static from dynamic behaviour, and local measurements from measurements averaged over a specimen.

At postgraduate level, magnetic measurement is not merely the reading of a gaussmeter. It involves selecting a physical measurement principle, controlling the magnetic circuit, accounting for demagnetization and leakage, correcting instrument nonlinearity, and estimating uncertainty.

Learning Objectives

After studying this chapter, the reader should be able to:

1. distinguish B, H, Φ, permeability, and magnetic moment;
2. explain fluxmeter, ballistic galvanometer, search-coil, Hall-effect, magnetoresistive, and vibrating-sample methods;
3. derive and apply key measurement equations with appropriate assumptions;
4. measure hysteresis, permeability, loss, and permanent-magnet characteristics;
5. identify major error sources and construct an uncertainty budget;
6. select an appropriate method for laboratory, industrial, and research applications;
7. evaluate contemporary digital and automated magnetic-measurement systems.

1. Fundamental Magnetic Quantities

1.1 Magnetic flux and flux density

Magnetic flux is the surface integral of magnetic flux density:

Here Φ is flux in weber (Wb), B is flux density in tesla (T), dA is an oriented elemental area in square metres, and S is the chosen surface. If the field is uniform and normal to a plane surface,

where A is area in m². This special form is not valid when the field is nonuniform or oblique unless the appropriate component is used.

1.2 Field strength and magnetomotive force

For a magnetic path, Ampère’s law gives

where H is magnetic field strength in A/m, dl is a path element, N is the number of turns, and I is current in A. For a uniform path of length l,

This approximation assumes a uniform field and negligible fringing. The product NI is magnetomotive force (A-turns).

In a linear isotropic medium,

where μ is permeability in H/m, μ₀ = 4π × 10⁻⁷ H/m is the permeability of free space, and μᵣ is relative permeability. Ferromagnetic materials are generally nonlinear and hysteretic, so B/H is an apparent or secant permeability rather than a universal material constant.

1.3 Reluctance model

For a simple magnetic circuit,

where reluctance is

The circuit analogy is useful for design, but it can be inaccurate when leakage flux, fringing, saturation, anisotropy, or distributed air gaps are significant. A magnetic circuit may contain several sections, in which case reluctances in series add, while parallel branches divide flux according to their reluctances.

1.4 Energy and co-energy

For a linear inductor,

where W is stored magnetic energy in joules, L is inductance in henries, I is current, and V is the magnetic-field volume. The integral expression is more general, but the simple quadratic expression assumes linearity and a fixed geometry.

2. Magnetic Material Behaviour

2.1 Initial magnetization and saturation

When an initially demagnetized ferromagnetic specimen is subjected to increasing H, its flux density rises through domain-wall motion and domain rotation. At high field, most domains align and the material approaches saturation. The initial magnetization curve is therefore nonlinear.

The differential permeability is

It describes the local response to a small field variation. The secant permeability is

for a specified operating point. These quantities should not be confused with the permeability of free space or with a manufacturer’s nominal relative permeability.

2.2 Hysteresis

When the field is cycled, B does not return along the original magnetization path. The resulting hysteresis loop has several important parameters:

• saturation flux density B_s;
• remanent flux density B_r, the residual flux density when H = 0;
• coercive field H_c, the reverse field required to reduce B to zero;
• loop area, proportional to energy dissipated per cycle and per unit volume.

For a quasistatic cycle, energy loss density is

with units J/m³ per cycle. The sign and orientation depend on the direction of traversal; the physical loss is the positive area enclosed by the loop. At frequency f, the average hysteresis power for specimen volume V is approximately

provided the loop corresponds to the actual waveform, frequency, and temperature. This relation does not by itself include eddy-current or excess dynamic losses.

2.3 AC loss and dynamic effects

Under alternating excitation, total core loss commonly includes hysteresis, classical eddy-current, and excess or anomalous components. A frequently used empirical representation is the Steinmetz form

where P_v is loss per unit volume, k, α, and β are experimentally fitted constants, f is frequency, and B_m is peak flux density. The constants depend on material, waveform, temperature, and frequency range; therefore the equation must not be extrapolated outside its calibration range.

3. Measurement Principles and Instruments

3.1 Search-coil method

A search coil measures changing flux through Faraday’s law:

where e(t) is induced voltage in V and N is the number of turns. Integration gives

A search coil is therefore fundamentally a flux-change instrument. It cannot measure a perfectly steady flux without a change, although a controlled reversal or extraction from the field can create the required transient.

If the coil has area A and is fully linked by a uniform normal field,

A search coil is especially useful for mapping magnetic fields, calibrating electromagnets, measuring transformer flux, and obtaining hysteresis loops.

3.2 Ballistic galvanometer and fluxmeter

A ballistic galvanometer responds to the total charge produced by a brief induced current. If the coil resistance is R and the flux changes by ∆Φ, the charge is approximately

under the assumptions that the pulse duration is short compared with the mechanical period, the electrical time constant is appropriate, and the induced pulse is not distorted by saturation or excessive damping.

A fluxmeter is a modern integrating instrument. Its output is proportional to the time integral of input voltage and is commonly used with search coils. Integration drift, amplifier offset, leakage, and polarity reversal are important practical issues.

3.3 Hall-effect gaussmeter

A Hall element develops a transverse voltage when current flows through it in a magnetic field. A simplified relation is

where V_H is Hall voltage, K_H is the calibrated Hall coefficient or sensitivity, I_c is control current, and B_n is the component normal to the active Hall surface.

Hall probes measure DC and AC fields over a broad range and are convenient for field mapping. Their limitations include temperature coefficient, transverse sensitivity, probe-position errors, electronic offset, and sensitivity to mechanical alignment. A three-axis probe can measure vector field components, but its axes require orthogonality and sensitivity calibration.

3.4 Magnetoresistive sensors

Anisotropic magnetoresistance, giant magnetoresistance, and tunnel magnetoresistance sensors measure field-dependent resistance. They offer high sensitivity and small size, but may exhibit hysteresis, saturation, temperature dependence, and directional ambiguity. They are widely used in current sensing, position measurement, compassing, and magnetic encoders.

3.5 Faraday and magneto-optical methods

In Faraday rotation, the polarization plane of light rotates in a magneto-optic medium:

where θ_F is rotation angle in radians, V is the Verdet constant, B is field, and L is optical path length. The Verdet constant depends strongly on wavelength and temperature. Optical methods provide electrical isolation and can be valuable near high voltage or in strong electromagnetic interference.

3.6 Vibrating-sample magnetometer

A vibrating-sample magnetometer (VSM) mechanically vibrates a specimen in a nearly uniform applied field. The changing magnetic dipole moment induces a signal in pickup coils. The measured quantity is usually magnetic moment m in A·m². Dividing by specimen volume gives magnetization M in A/m:

where V_s is specimen volume. VSM measurements require careful background subtraction, field calibration, centering, and demagnetization correction.

4. Measuring Hysteresis Loops

A practical hysteresis measurement requires two independent quantities. The applied field is often inferred from magnetizing current:

where l_m is effective magnetic path length. Flux density is obtained by integrating the voltage from a secondary search winding:

Here v_s is secondary voltage, A is effective cross-sectional area, and B(t₀) is the initial reference value. The reference condition must be known or the integration constant must be determined by a reset or flux-balancing procedure.

A digital system normally samples magnetizing current and search-coil voltage simultaneously. The signal chain should include anti-alias filtering, calibrated gain, synchronized sampling, numerical integration, and baseline correction. The field waveform should be checked because a sinusoidal voltage does not necessarily produce a sinusoidal flux waveform when the circuit is nonlinear.

For toroidal specimens, demagnetization is small when the magnetic path is closed. For open samples, the internal field is approximately

where H_int is internal field, H_appl is applied field, N_d is the dimensionless demagnetizing factor for the relevant direction, and M is magnetization. The factor depends on specimen geometry and equals different values along different axes of an ellipsoid. Neglecting this correction can substantially distort coercivity and permeability.

5. Measurement of Permanent Magnets

Permanent magnets are commonly characterized by a second-quadrant demagnetization curve. Important quantities include B_r, H_c, and the maximum energy product (BH)_max. The energy product is evaluated from the magnitude of the product BH in the operating quadrant:

Its unit is J/m³ because T·A/m is equivalent to J/m³. A hysteresisgraph or permeameter can obtain the demagnetization curve. The measurement must control temperature, because remanence and coercivity are temperature dependent. Mechanical stress, previous magnetic history, and field uniformity also affect results.

6. Measurement of AC Core Loss

The two-wattmeter or electrical-input method measures the real power supplied to a transformer core under specified excitation. For a two-winding test, copper loss, winding resistance, instrument phase error, and waveform distortion must be separated from core loss.

A direct method uses the measured H and B waveforms and computes loop area:

The numerical integration must preserve the phase relationship between the two channels. Channel delay can create a false loop area, particularly at high frequency. Sensor bandwidth, sampling rate, anti-alias filters, and probe compensation must therefore be documented.

The Epstein frame and single-sheet tester are standardized approaches for electrical steels. A toroidal specimen is convenient because the magnetic path is closed, but winding geometry, joint effects, stress, and packing factor influence the result. Measurements should state frequency, peak flux density, waveform, temperature, specimen mass or volume, and conditioning history.

7. Worked Example: Search-Coil Flux Measurement

A search coil has 200 turns and an effective area of 4.0 × 10⁻⁴ m². During a controlled field reversal, the measured voltage integral is ∫v dt = 0.160 V·s. Determine the change in uniform normal flux density.

Using Faraday’s law,

Therefore,

The result is a change of 2.0 T. If the reversal is from -1.0 T to +1.0 T, the value is consistent. The calculation assumes complete flux linkage, uniform field over the coil, negligible parasitic induced voltage, and accurate integration.

8. Uncertainty, Calibration, and Error Control

Magnetic measurements are often dominated by systematic rather than random error. Important sources include:

• probe misalignment and finite sensing area;
• calibration drift and temperature dependence;
• uncertainty in coil turns, area, and effective magnetic length;
• fringing, leakage, and demagnetization;
• integrator offset and baseline drift;
• current-sensor phase error;
• specimen geometry, stress, packing, and magnetic history;
• electromagnetic interference and ground loops;
• insufficient bandwidth or aliasing.

For a result y = f(x₁, x₂, ..., x_n), a first-order combined standard uncertainty is

where u(xᵢ) is the standard uncertainty of input xᵢ and u(xᵢ,xⱼ) is covariance. If inputs are independent, covariance terms vanish. For the special result B = Φ/A, small independent relative uncertainties give

Calibration should be traceable to recognized standards. A Hall probe should be calibrated against a reference field or precision current-and-coil arrangement. Search coils require characterization of turns, effective area, and integration electronics. A complete report should include the calibration date, uncertainty, environmental conditions, measurement orientation, and correction model.

9. Illustrative Case: Qualification of a Transformer Core Material

Background

A manufacturer is evaluating two batches of grain-oriented electrical steel for a 50 Hz distribution transformer. The target is low specific core loss at a peak flux density of 1.6 T.

Problem

Initial electrical-input measurements show that Batch B has lower apparent power than Batch A, but the result is questioned because the excitation waveform is distorted and the winding resistance was not corrected.

Analysis

The laboratory mounts representative strips in a standardized frame, measures primary current and secondary voltage synchronously, and calculates H and B. The voltage is integrated to obtain flux, while the current is corrected for the magnetizing-winding resistance. The loop-area method is then compared with the corrected real-input-power method.

The team also controls specimen clamping pressure and temperature. It discovers that Batch B was tested at a slightly lower peak flux density, which explains much of its apparent advantage.

Approach and solution

Both batches are conditioned by a defined demagnetization cycle. The test is repeated at exactly 50 Hz and 1.6 T with the same waveform. The acquisition system is checked for channel delay using a common reference signal. Core loss is reported in W/kg with uncertainty and test conditions.

Outcome

After correction, Batch B still performs better, but its advantage is smaller than the initial result suggested. The manufacturer selects Batch B only for transformer designs whose clamping and annealing conditions reproduce the laboratory result.

Learning points

The case demonstrates that magnetic measurements are inseparable from test conditions. A numerical result without waveform, temperature, specimen preparation, calibration, and uncertainty information is not an adequately defined material property.

10. Contemporary Developments and Professional Practice

Modern instruments combine precision current sources, FPGA-based acquisition, digital lock-in detection, automated field ramps, and temperature control. Finite-element models are increasingly coupled with measured hysteresis data to predict local saturation, stray fields, and losses. Machine-learning methods can assist in parameter identification and anomaly detection, but they do not remove the need for traceable measurements and physically valid training data.

In high-frequency power electronics, broadband current and voltage sensors, nanocrystalline or amorphous cores, and nonsinusoidal excitation require measurement bandwidth substantially beyond the fundamental frequency. In research laboratories, automated VSM, alternating-gradient magnetometers, and magneto-optical imaging provide spatially resolved or nanoscale-sensitive information.

Best practice requires: defining the measurand before selecting the sensor; controlling magnetic history; documenting geometry and orientation; calibrating the full signal chain; checking polarity and phase; validating numerical integration; repeating measurements; and reporting uncertainty and environmental conditions. Safety procedures are essential around strong magnets, stored magnetic energy, high currents, cryogenic systems, and implanted medical devices.

Chapter Summary

Magnetic measurement concerns both field quantities and material response. Flux density is measured in tesla, field strength in A/m, flux in weber, and magnetic moment in A·m². Ampère’s law relates current and field, while Faraday’s law provides the basis of search-coil and fluxmeter methods. Hall and magnetoresistive sensors are convenient for local DC or AC field measurements, whereas VSM and hysteresisgraph methods characterize magnetic materials.

Permeability may be differential or secant, and ferromagnetic materials must be described through magnetization history and hysteresis. Loop area gives energy loss per cycle and, with frequency and volume, allows loss estimation. Accurate measurement requires attention to demagnetization, leakage, fringing, waveform, phase error, calibration, temperature, and uncertainty. Contemporary systems improve automation and bandwidth, but physical modelling and traceability remain essential.

Key Terms

Magnetic flux: Surface integral of magnetic flux density, measured in weber.

Flux density: Magnetic flux per unit area, measured in tesla.

Field strength: Magnetizing field, measured in A/m.

Magnetomotive force: Product of turns and current, NI.

Permeability: Ratio describing the magnetic response of a material.

Differential permeability: Local slope dB/dH of a magnetization curve.

Secant permeability: Ratio B/H at a specified operating point.

Reluctance: Opposition to magnetic flux in a magnetic circuit.

Saturation: Region in which increasing field produces relatively little additional flux density.

Hysteresis: Dependence of magnetic state on previous field history.

Remanence: Residual flux density after removal of applied field.

Coercivity: Reverse field required to reduce flux density or magnetization to a specified zero condition.

Search coil: Coil used to infer flux change from induced voltage.

Fluxmeter: Integrating instrument for measuring flux change.

Hall effect: Transverse voltage generated by current in a magnetic field.

Demagnetizing factor: Geometry-dependent factor used to obtain internal field.

Vibrating-sample magnetometer: Instrument that measures magnetic moment from vibration-induced signal.

Core loss: Energy dissipated in a magnetic core, usually through hysteresis and dynamic effects.

Traceability: Property of a measurement linked through documented calibrations to a recognized standard.

Review Questions

Short Answer Questions

1. Distinguish magnetic flux density B from magnetic field strength H.
2. State Faraday’s law as applied to a search coil.
3. Define remanent flux density and coercive field.
4. Why is demagnetization important for an open magnetic specimen?
5. Name two principal error sources in Hall-probe measurements.

Long Answer Questions

1. Explain the construction and operation of a digital hysteresis measurement system.
2. Compare search-coil, Hall-effect, magnetoresistive, and VSM methods.
3. Derive the relationship between hysteresis-loop area and energy loss per unit volume.
4. Discuss the measurement of permanent-magnet demagnetization curves and (BH)_max.
5. Explain how waveform, phase error, temperature, and specimen preparation affect AC core-loss measurements.

Analytical / Application Questions

1. A 500-turn search coil of area 2.5 × 10⁻⁴ m² produces a voltage-time integral of 0.075 V·s during a field reversal. Calculate the change in uniform normal flux density and state the assumptions.
2. Design an uncertainty budget for a flux-density measurement obtained from B = Φ/A, including at least four input quantities or effects.
3. A hysteresis loop is measured on an open rectangular specimen. Propose a procedure for correcting the applied field to internal field and discuss the information required to determine the demagnetizing factor.

Multiple Choice Questions with Answers

Question 1

Which quantity is directly related to the surface integral of magnetic flux density?

A. Magnetic flux

B. Magnetic reluctance

C. Coercive field

D. Magnetic moment

Correct Answer: A. Magnetic flux

Question 2

A 100-turn search coil has an effective area of 2 × 10⁻⁴ m². If the measured voltage-time integral is 0.020 V·s, what is the flux-density change?

A. 0.1 T

B. 1.0 T

C. 10 T

D. 100 T

Correct Answer: B. 1.0 T

Question 3

The Hall voltage of a Hall probe is primarily proportional to which field component?

A. The field component normal to the active Hall surface

B. Only the field component parallel to the probe cable

C. The time derivative of current in the probe

D. Magnetic reluctance

Correct Answer: A. The field component normal to the active Hall surface

Question 4

The area enclosed by a quasistatic B–H loop represents energy loss per unit volume in which unit?

A. J/m³ per cycle

B. W/m²

C. Wb/s

D. A·m²

Correct Answer: A. J/m³ per cycle

Question 5

Which expression defines differential permeability?

A. B/H

B. dB/dH

C. H/B

D. NI/l

Correct Answer: B. dB/dH

Question 6

Why is a search coil unable to measure an unchanging flux directly?

A. Its resistance is always zero

B. The induced voltage depends on flux change

C. It responds only to temperature

D. Its turns cannot be calibrated

Correct Answer: B. The induced voltage depends on flux change

Question 7

For an open specimen, the internal field is reduced relative to the applied field mainly because of which effect?

A. Hall rotation

B. Demagnetization

C. Skin depth in the probe cable

D. Fluxmeter integration gain

Correct Answer: B. Demagnetization

Question 8

A vibrating-sample magnetometer primarily measures which quantity?

A. Magnetic moment

B. Reluctance

C. Coil resistance

D. Electric permittivity

Correct Answer: A. Magnetic moment

Question 9

Which condition is necessary when using the simple relation P_h = fVw_h?

A. The loop area must represent the actual operating waveform and frequency

B. The material must be nonmagnetic

C. The applied field must be zero

D. The specimen volume must be unknown

Correct Answer: A. The loop area must represent the actual operating waveform and frequency

Question 10

In a magnetic circuit with uniform field and negligible fringing, which relation gives field strength?

A. H = NI/l

B. H = l/(NIA)

C. H = BA

D. H = μ/B

Correct Answer: A. H = NI/l