THEORETICAL FOUNDATIONS OF EDDY-CURRENT TESTING OF FERROMAGNETIC MATERIALS

V. E. DREIZIN

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

In the theoretical consideration of the eddy-current method, one has to investigate the electromagnetic processes in the system sensor—tested article. The mathematical description of these electromagnetic processes is based on the classical theory of the electromagnetic field. In the general case, the basic equations of a time-varying electromagnetic field have the form:

\[ \operatorname{rot}\,\overline{H}=\frac{\partial \overline{D}}{\partial t}+\delta, \tag{1} \]

\[ \operatorname{rot}\,\overline{E}=-\frac{\partial \overline{B}}{\partial t}, \tag{2} \]

\[ \operatorname{div}\,\overline{H}=0, \tag{3} \]

\[ \operatorname{div}\,\overline{E}=0. \tag{4} \]

To these are added the expressions characterizing the relation of the field vectors in a certain material medium:

\[ \overline{B}=\mu \overline{H}, \tag{5} \]

\[ \overline{D}=\varepsilon \overline{E}, \tag{6} \]

\[ \delta=\sigma \overline{E}. \tag{7} \]

Here \(\mu\), \(\varepsilon\), \(\sigma\) are parameters expressing the macroscopic electromagnetic properties of the medium. In the case where they are constant and independent of the magnitudes of the vectors, Maxwell’s equations turn out to be linear and, accordingly, such media are called “linear.” However, there also exist, and are of technical importance, media distinguished by a noticeable dependence of the macroscopic parameters on the field vectors. In this case Maxwell’s equations turn out to be nonlinear, and such media are called “nonlinear.” A striking example of such media is ferromagnets—substances whose magnetic permeability depends significantly and in a complex manner on the magnetic field. In addition to nonlinearity, ambiguity is also observed here.

values of magnetic permeability at different values of the magnetic field, as determined by the phenomenon of hysteresis. The ferromagnetic metals of interest to us, without exception, possess magnetic hysteresis, even if the field changes are infinitely slow (quasistatic). Moreover, as the rate of field variation increases, the lag of the magnetic induction increases still further and is due not only to “static” hysteresis but also to the finite rate of change of the field \(H\) inside the ferromagnet. This additional increase in the lag is commonly called magnetic viscosity.

In order to be able to solve Maxwell’s equations while taking into account the nonlinearity of a ferromagnetic medium, it is necessary to express analytically the magnetization loop of a ferromagnet in alternating fields and to solve Maxwell’s equations as nonlinear differential equations. However, it is not possible to express the hysteresis loop in general form analytically. Consequently, it is necessary to characterize the magnetic properties of ferromagnets by certain approximate characteristics. Therefore, when investigating ferromagnets in alternating fields, it is necessary in each particular case to determine carefully in what way, and by what parameters, its electromagnetic properties can be characterized most completely and most accurately.

In the domestic and foreign literature on nondestructive testing of metal products by the eddy-current method, the problem of inspecting ferromagnetic alloys is either not addressed at all, or due importance is not attached to the question of the electromagnetic parameters of the material. In all works known to the author, when passing from the study of nonmagnetic materials to magnetic ones, \(\mu_0\) in Maxwell’s equations was simply multiplied by \(\mu_r\), and nowhere was it specified what physical meaning should be assigned to this characteristic, how it could be measured; that is, it remained completely unclear what it represented. At the same time, of course, all the specific phenomena occurring in ferromagnetic materials during magnetization reversal turned out to be completely unaccounted for.

At the same time, Professor V. K. Arkad’ev, in his works on magnetodynamics and magnetic spectroscopy, proposed a characteristic that reflects the magnetic properties of a ferromagnetic medium in weak alternating electromagnetic fields with sufficient completeness and accuracy. This characteristic is the complex magnetic permeability of the substance. However, to date this characteristic has been used only in magnetic spectroscopy, probably because it could not be measured directly. It is found indirectly from calculated curves, which were compiled by Arkad’ev and his school only for two cases: for a straight round wire through which a high-frequency current is passed directly, and for a thin sheet rolled into a triode, onto which an exciting winding is uniformly wound [2, 3, 4]. In the present work, the legitimacy of applying the complex magnetic permeability of a substance in all cases where a ferromagnet is in an alternating electromagnetic field is substantiated; the physical essence of this characteristic is revealed; and a solution of Maxwell’s equations is given using this characteristic for the case of incidence of a cylindrical electromagnetic wave on the surface of a solid isotropic ferromagnetic cylinder. Thus, the problem of the theory of a feed-through transducer for the inspection of solid cylindrical ferromagnetic bodies is solved, taking into account all

specific phenomena occurring in a ferromagnet during magnetization reversal.

In order to take a well-founded approach to revealing the physical essence of the complex magnetic permeability of a substance, it is necessary to consider briefly the behavior of a ferromagnet in alternating fields.

When a ferromagnetic body is magnetized in an alternating sinusoidal field, the magnetic state of the body changes along a symmetrical loop analogous to the static (or quasistatic) hysteresis loop. However, unlike the static loop, the area of the dynamic magnetization-reversal loop is proportional not only to hysteresis losses, but also to eddy-current losses, and at sufficiently high frequencies also to energy losses due to the magnetic viscosity of the material. At low induction values (up to 100 G), the shape of the dynamic loop is an ellipse. This indicates that, in a sinusoidal field, the induction is also sinusoidal, but will lag in phase behind the magnetic-field intensity. Consequently, the dynamic magnetization-reversal loop of a body in weak fields can be expressed in analytical form and the magnetic properties of the given body in the given field can be completely characterized by the complex magnetic permeability of the body, which is the ratio, expressed in complex form, of the induction averaged over the cross section of the body to the magnetic-field intensity at the surface of the body. Indeed, writing the magnetic-field intensity at the surface of the body as

\[ H_0 = H_{0m}\sin \omega t, \]

and the induction averaged over the cross section of the body as

\[ B_{\mathrm{cp}} = B_{\mathrm{cp},m}\sin(\omega t-\delta), \]

we obtain a parametric representation of the equation of an ellipse. Expressing the induction and the magnetic-field intensity in the form of complexes and taking their ratio, we obtain

\[ \dot{\mu}_{\partial} = \frac{\dot{B}_{\mathrm{cp}}}{\dot{H}_0} = \frac{B_{\mathrm{cp},m} e^{j(\omega t-\delta)}}{H_{0m} e^{j\omega t}} = \mu_n e^{-j\delta} = \mu_1 - j\mu_2 . \tag{8} \]

Thus, the process of magnetization reversal of a ferromagnetic body in an alternating electromagnetic field is fully characterized by the dynamic loop, which, in weak fields, in turn is fully characterized by the complex dynamic magnetic permeability of the body. It should be emphasized, however, that the dynamic loop characterizes not the substance, but the body. That is, with a change in the shape and dimensions of the body, as well as in the frequency of the electromagnetic field and the electrical conductivity of the material, the dynamic loop, and consequently also the dynamic complex magnetic permeability, will be different. Therefore, the magnetic properties of a ferromagnet as a substance cannot be characterized in alternating fields by this quantity, since it expresses, as it were, the total electromagnetic properties of some quite definite body when it is located in a quite definite electromagnetic field.

In order to obtain a magnetic permeability that characterizes the substance itself, one must know the magnetic-field intensity and induction at every point of the given ferromagnetic body; that is, it is necessary to take into account the additional field of the eddy currents arising in the ferromagnet when it is placed in an alternating electromagnetic field. Obviously, in this case the magnetization-reversal loop of the substance will be determined-

is characterized only by “static hysteresis” and magnetic viscosity. It is quite reasonable to assume that if the dynamic loop of the body has the form of an ellipse, then the remagnetization loop of the substance, determined by the true values of the field strength and induction at the given point, will also have the form of an ellipse. In this case we can fully characterize it by the complex magnetic permeability of the substance, defined as the ratio of the induction at the given point of the field to the field strength at this same point, expressed as complex quantities,

\[ \mu'=\frac{\dot{B}}{\dot{H}}=|\mu'|e^{-j\delta}=\mu-j\rho . \tag{9} \]

The real component \(\mu\) is called the conservative, or “elastic,” permeability and is associated with elastic, that is, reversible, processes occurring during magnetization, which are not accompanied by energy losses. The imaginary part \(\rho\) is called the consumptive, or viscous, permeability and is associated with irreversible, that is, viscous, processes determined by hysteresis and magnetic viscosity. Thus, the complex magnetic permeability of a substance \(\mu'\) completely determines the macroscopic magnetic properties of a ferromagnet in weak alternating fields. However, it should be remembered that this quantity determines the magnetic properties not at any instant of time, but over a complete remagnetization cycle, that is, over a period. It is precisely because of this that we are able to characterize the true magnetic properties of a nonlinear medium by a constant complex quantity. It should be noted, however, that the magnetic properties of ferromagnets determined by the quantity \(\mu'\) are themselves a function of the frequency of the electromagnetic field. These dependences are studied by the special branch of magnetodynamics founded by V. K. Arkad’ev—magnetic spectroscopy. The change in magnetic permeability (dispersion and absorption) as a function of frequency is due here not to some secondary phenomena, such as the shielding action of eddy currents, but to an actual change in the magnetic properties of substances, determined by the processes occurring in the crystals and atoms of the ferromagnetic substance. However, for most materials in the frequency range up to \(1000\) Hz the magnetic permeability \(\mu'\) is constant; then, in the range from \(10^3\) to \(10^9\) Hz, a slow decrease in it is observed, which is less noticeable the smaller the absolute value of the magnetic permeability of the material; the principal changes occur in the region of shorter waves, and, finally, at a wavelength of about \(1\) cm magnetic materials completely lose their magnetic properties. All this should be borne in mind; however, we shall be interested only in the frequency range up to \(1\) MHz, where changes in the magnetic permeability of the substance are insignificant for most materials.

Let us now return to Maxwell’s equations. For metals the displacement current

\[ \frac{\partial D}{\partial t} \]

is vanishingly small in comparison with the conduction current \(\vec{\delta}\) and, consequently, it may be neglected. Then, taking into account \(\vec{\delta}=\sigma \vec{E}\) and \(\vec{B}=\mu_0\mu'\vec{H}\), Maxwell’s equations are obtained in the following form:

\[ \operatorname{rot}\vec{H}=\sigma\vec{E}, \]

\[ \operatorname{rot}\vec{E}=-\mu_0\mu'\frac{\partial\vec{H}}{\partial t}. \]

Thus, by introducing \(\mu'\), we have obtained formally linear equations. For sinusoidal fields, instead of \(\vec H\) and \(\vec E\) there should appear the symbols \(\dot{\vec H}\) and \(\dot{\vec E}\), denoting complex quantities, i.e., quantities varying sinusoidally in time. Then

\[ \operatorname{rot}\dot{\vec H}=\sigma \dot{\vec E}, \tag{10} \]

\[ \operatorname{rot}\dot{\vec E}=-j\omega \mu_0\mu' \dot{\vec H}. \tag{11} \]

In the case of testing cylindrical bodies in a feed-through transducer, a cylindrical electromagnetic wave will be incident on this body, exciting eddy currents in it. The latter, by forming their own electromagnetic field, will lead to a change in the resultant field in the cavity of the transducer, which determines its total complex impedance (in the case of a single-winding transducer) and the e.m.f. in the measuring winding (in the case of a two-winding transducer). Thus, we need to obtain expressions for the magnetic- and electric-field intensities at any point of the ferromagnetic cylinder being tested. Finding from the first equation

\[ \dot{\vec E}=\frac{1}{\sigma}\operatorname{rot}\dot{\vec H} \tag{12} \]

and substituting it into the second, we obtain

\[ \frac{1}{\sigma}\operatorname{rot}\operatorname{rot}\dot{\vec H} = -j\omega\mu_0\mu'\dot{\vec H}. \tag{13} \]

In view of the fact that

\[ \operatorname{rot}\operatorname{rot}\dot{\vec H} = \operatorname{grad}\operatorname{div}\dot{\vec H} - \Delta \dot{\vec H} = -\Delta \dot{\vec H}, \]

where \(\Delta\) is the Laplace operator, we obtain

\[ \Delta \dot{\vec H}=j\omega\sigma\mu_0\mu'\dot{\vec H}. \tag{14} \]

Using a cylindrical coordinate system and taking into account that in our case

\[ \dot{\vec H}=\vec I_z \dot H_z; \]

\[ \dot{\vec E}=\vec I_\varphi \dot E_\varphi, \]

and also that \(\dfrac{\partial \dot{\vec H}}{\partial \varphi}=0\) as a consequence of the circular symmetry of the system, and \(\dfrac{\partial \dot{\vec H}}{\partial z}=0\), since we shall neglect edge effects, we obtain:

\[ \Delta \dot{\vec H} = \frac{\partial^2 \dot{\vec H}}{\partial r^2} + \frac{1}{r}\frac{\partial \dot{\vec H}}{\partial r}, \tag{15} \]

\[ \operatorname{rot}\dot{\vec H} = -\vec I_\varphi \frac{\partial \dot H}{\partial r}. \tag{16} \]

Substituting these expressions into (12) and (14), we have:

\[ \dot{\vec E} = -\vec I_\varphi \frac{1}{\sigma}\frac{\partial \dot H}{\partial r}, \tag{17} \]

\[ \frac{\partial^2 \dot{\vec H}}{\partial r^2} + \frac{1}{r}\frac{\partial \dot{\vec H}}{\partial r} = j\omega\sigma\mu_0\mu'\dot{\vec H}. \tag{18} \]

Equation (18) can be reduced to a modified Bessel equation of zero order with complex argument

\[ r\sqrt{j\omega\sigma\mu_0\mu'}=r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}: \]

\[ \frac{\partial^2 \dot{\vec H}} {\partial \left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right]^2} + \frac{1}{r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}}\cdot \frac{\partial \dot{\vec H}} {\partial \left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right]} -\dot{\vec H}=0 . \tag{19} \]

Its general solution is found in the form

\[ \dot{\vec H} = \dot{\vec A}\, I_0\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] + \dot{\vec B}\, K_0\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right], \tag{20} \]

where \(I_0\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right]\) and \(K_0\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right]\) are the modified Bessel functions, respectively of the first and second kind, of zero order;

\(\dot{\vec A}\) and \(\dot{\vec B}\) are integration constants determined from the boundary conditions.

The first term of equation (20) describes the incident wave propagating from the surface of the cylinder to its axis, and the second term the reflected wave propagating from the axis of the cylinder to the surface. However, in a solid isotropic cylinder there are no reflecting surfaces, and there will be no reflected wave, i.e. \(\dot{\vec B}=0\). Taking the magnetic-field intensity at the surface of the cylinder (for \(r=r_0\)) to be equal to \(\dot{\vec H}_0\) and substituting \(r=r_0\) into equation (20), we obtain

\[ \dot{\vec H}_0 = \dot{\vec A}\, I_0\!\left[r_0\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right], \]

whence

\[ \dot{\vec A} = \dot{\vec H}_0 \frac{1} {I_0\!\left[r_0\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right]} . \]

Thus we obtain

\[ \dot{\vec H} = \dot{\vec H}_0 \frac{ I_0\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] }{ I_0\!\left[r_0\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] } = \bar{\vec i}_z\,\dot H_0 \frac{ I_0\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] }{ I_0\!\left[r_0\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] }. \tag{21} \]

Substituting the expression found for \(\dot{\vec H}\) into (17), we obtain

\[ \dot{\vec E} = -\bar{\vec i}_{\varphi}\frac{1}{\sigma}\frac{\partial}{\partial r} \left\{ H_0 \frac{ I_0\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] }{ I_0\!\left[r_0\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] } \right\} = \]

\[ = -\bar{\vec i}_{\varphi}\frac{\dot H_0}{\sigma} \sqrt{\omega\sigma\mu_0(\mu'+j\mu)} \frac{ I_1\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] }{ I_0\!\left[r_0\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right] }, \tag{22} \]

where \(I_1\!\left[r\sqrt{\omega\sigma\mu_0(\mu'+j\mu)}\right]\) is the Bessel function of the first kind of first order.

Thus, we have obtained expressions (21) and (22), which determine the values of the vectors of electric- and magnetic-field intensity at any point of a cylindrical ferromagnetic body placed coaxially in the cavity of a solenoid carrying alternating current.

Further, by the usual procedure, one can obtain expressions for the increments of resistance (for a single-winding transducer) and of the emf in the measuring winding (for a double-winding transducer). It is obvious that

the concept of the complex magnetic permeability of a substance should be used also in the analysis of a surface probe and, in general, in any case where it is required to describe the behavior of a ferromagnet in alternating sinusoidal electromagnetic fields, since this characteristic most fully and accurately reflects the true magnetic properties of a ferromagnetic medium in an alternating electromagnetic field.

It should be noted that complex magnetic permeability can be used not only in weak fields, when the induction is sinusoidal and the dynamic loop has the form of an ellipse, but also in sufficiently strong fields, when the dynamic loop differs in shape from an ellipse. In this case we, as it were, replace the actual dynamic loop by an ellipse equal to it in area, owing to which the numerical value of the losses remains the same.

REFERENCES

1. V. K. Arkad’ev. Theory of the electromagnetic field in a ferromagnetic metal. ZhRFKhO, Phys. Dept., 45, 312, 1913.

2. V. K. Arkad’ev. Electromagnetic Processes in Metals. Part II, Energoizdat, Moscow, 1936.

3. V. K. Arkad’ev, K. A. Volkova. Calculation of the magnetic permeability of ferromagnetic wires and plates in an alternating field. Collection “Problems of Ferromagnetism and Magnetodynamics,” Publishing House of the USSR Academy of Sciences, 1946.

4. K. M. Polivanov. Determination of the complex magnetic permeability of a substance from the effective complex magnetic permeability of a body by the method of V. K. Arkad’ev. Collection “Problems of Ferromagnetism and Magnetodynamics,” Publishing House of the USSR Academy of Sciences, 1946.