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A NEW METHOD FOR REDUCING LOSSES DUE TO THE SKIN EFFECT
Reducing the losses caused by the phenomenon of the skin effect is one of the cardinal problems of electrical engineering and radio engineering. Recently a fundamentally new method for solving this problem was proposed,^1,2 and was then verified experimentally.^3
Let us explain the idea of the method. Consider the propagation of an electromagnetic wave in an anisotropic medium whose dielectric constant and conductivity in the direction of the \(OX\) axis are equal to \(\varepsilon_x\) and \(\sigma_x\), and in the direction of the \(OY\) axis to \(\varepsilon_y\) and \(\sigma_y\). An example of such a medium is the layered structure shown in Fig. 1. The shaded strips are metallic sheets (electrical conductivity \(\sigma\), thickness \(W\)); between them are layers of dielectric (dielectric constant \(\varepsilon\), thickness \(t\)). The effective dielectric constant and conductivity of such a layered structure are equal to^1,4:
\[ \begin{aligned} \varepsilon_x&=\varepsilon\left(\frac{t}{t+W}\right);& \sigma_x&=\sigma\left(\frac{W}{W+t}\right),\\ \varepsilon_y&=\varepsilon\left(1+\frac{W}{t}\right);& \sigma_y&=\sigma\left(\frac{\omega\varepsilon}{\sigma}\right)^2 \frac{W}{t}\left(\frac{W}{t}+1\right). \end{aligned} \tag{1} \]
Metallic sheets possess a certain conductivity \(\sigma\), and therefore the component of the field \(E_x\) is different from zero. Then from the relation \(\operatorname{div} D=0\) it follows that
\[ \varepsilon_y\frac{\partial E_y}{\partial y} + \varepsilon_z\frac{\partial E_z}{\partial z} \ne 0, \]
i.e. the wave propagating in such a medium is inhomogeneous: \(E=E(y,z)e^{i\omega t-ikx}\). We shall seek a solution of the simplest type: independent of \(z\), and, moreover, we shall assume \(E_z=0\). In this case Maxwell’s equations reduce to a system of three equations:
\[ \left. \begin{aligned} \frac{\partial H_z}{\partial y}&=(i\omega\varepsilon_x+\sigma_x)E_x;\\ -\frac{\partial H_z}{\partial x}&=(i\omega\varepsilon_y+\sigma_y)E_y;\\ \frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}&=-i\omega\mu_0 H_x^{*}). \end{aligned} \right\} \tag{2} \]
We seek the solution of equations (2) in the form of an exponential factor \(e^{-\alpha y-ikx+i\omega t}\), multiplied by the corresponding amplitude: \(E_x\), \(E_y\), \(H_x\). Substituting these expressions into (2), we find:
\[ \left. \begin{aligned} E_x&=-\frac{\alpha}{i\omega\varepsilon_x+\sigma_x}H_z,\\ E_y&=\frac{ik}{i\omega\varepsilon_y+\sigma_y}H_z, \end{aligned} \right\} \tag{3} \]
\[ \text{*) } B_x=\mu_0H_x,\ \text{here and everywhere below the system of units MKS is used.} \]
where \(H_z=e^{-\alpha y-ikx}\), with exponent
\[ \alpha=\pm\sqrt{\frac{i\omega\varepsilon_x+\sigma_x}{i\omega\varepsilon_y+\sigma_y} \left[i\omega\mu_0\sigma_y-\omega^2\mu_0\varepsilon_y+k^2\right]} . \tag{4} \]
In continuous metallic conductors \(\sigma_x=\sigma_y,\ \varepsilon_x=\varepsilon_y\), and (4) takes the form:
\[ \alpha=\pm\sqrt{i\omega\mu_0\sigma-\omega^2\mu_0\varepsilon+k^2}. \]
Usually \(\sigma\gg\omega\varepsilon\); the last two terms in (4) are small in comparison with the first, and thus we obtain from (4) the usual expression for the thickness of the skin layer \(\delta\):
\[ \delta=\sqrt{\frac{2}{\omega\mu_0\sigma}}. \]
The anisotropic layered medium shown in Fig. 1 behaves differently.
Let us assume that the main part of the cross section of the transmission line is an isotropic medium with dielectric constant \(\varepsilon_1\), while the layered conductor bounds this medium only at the edges. Then the propagation constant \(k\) will be determined chiefly by the isotropic dielectric, and therefore one may take \(k^2=\omega^2\mu_0\varepsilon_1\). Taking (1) into account, we find from (4):
\[ \alpha_0=\pm\left\{ \frac{1}{i}\,\sigma_x i\omega\mu_0 \left[ \left(\frac{\varepsilon_1}{\varepsilon_x}-1\right) +i\,\frac{\omega\varepsilon W}{\sigma t} \right] \right\}^{1/2}. \tag{5} \]
For \(\varepsilon_1=\varepsilon_x\), i.e. when the effective dielectric constant of the “layered structure” at the edges and of the principal dielectric filling the line are equal, \(\alpha\) attains its minimum value
\[ \alpha_0=\frac{Wk}{W+t}=\frac{W}{W+t}\cdot\frac{2\pi}{\lambda}, \]
where \(\lambda\) is the wavelength, \(\lambda=2\pi/k\), i.e. the field penetrates into the layered medium to a depth
\[ \delta_0=\frac{\lambda}{2\pi}\left(1+\frac{t}{W}\right), \]
many times exceeding the thickness of the skin layer of a solid metal. Let us explain the physical meaning of this result. From (2) it is not difficult to find the change in \(E_x\) over the thickness of one metallic layer,
\[ (\Delta E_x)_{\mathrm{met}}=i\omega\mu_0 H_z W. \]
The change in \(E_x\) over the thickness of one dielectric layer is
\[ (\Delta E_x)_{\mathrm{diel}}=i\omega\mu_0 H_z\left(1-\frac{\varepsilon_1}{\varepsilon}\right). \]
The essence of the effect is that, for \(\varepsilon=\varepsilon_1\), both these increments are equal: the decrease of \(E_x\) in the metallic layer (the ordinary skin effect) is compensated by the increase of \(E_x\) in the neighboring dielectric layer. It is precisely for this reason that the electromagnetic field penetrates into layered structures to a considerable depth.
Above, the plane case was considered; in the case of cylindrical symmetry—in a coaxial line—the physical picture is the same as in plane layers, but the calculation is somewhat more complicated\(^5\).
In all the preceding discussion, a very substantial circumstance was not taken into account: the finite thickness of the metal plates and of the dielectric layers. In reality the layered medium is inhomogeneous. By replacing this really inhomogeneous medium by a homogeneous but anisotropic one with effective mean values \(\varepsilon_x,\sigma_x,\varepsilon_y,\sigma_y\), determined from (1), we introduce an error which is the smaller the smaller \(W\) and \(t\) are. Unfortunately,
From Current Literature
the characteristic length in the skin-effect phenomenon—the thickness of the skin layer \(\delta\)—is usually of the same order as \(W\) and \(t\). This circumstance greatly complicates the calculation. Relations (1)—(5) should therefore be regarded as a qualitative explanation of the idea of the method, but not as computational formulas.
A more exact treatment,¹ ² taking into account the finite thickness of the metal and dielectric layers, shows that the wave penetrates into the layered medium to a depth
\[ \delta_W=\sqrt{3\left(1+\frac{t}{W}\right)\frac{\delta^2}{W}} \quad (\text{for } W\ll \delta), \]
which is much smaller than
\[ \delta_0=\frac{\lambda}{2\pi}\left(1+\frac{t}{W}\right), \]
but nevertheless many times exceeds the thickness of the skin layer of a solid conductor
\[ \delta=\frac{1}{\sqrt{\pi f\mu\sigma}}. \]
For example, for a copper conductor with \(W=0.005\) mm at a frequency of 18.6 MHz, \(\delta\sim 0.17\) mm. For \(t=0.02\) mm, \(\delta_W\sim 20\delta\), i.e. the use of a layered conductor increases the skin-layer thickness by a factor of 20 and correspondingly reduces the losses.
For an experimental verification of the theory,³ a coaxial quarter-wave line 232 cm long was constructed (Fig. 2). The outer conductor of the line was a copper tube with an inner diameter of 10.16 cm. The inner conductor, 6.9 mm in diameter, was made layered, analogous in structure to that shown in Fig. 1. It consisted of 50 layers of copper foil 0.005 mm thick, separated by insulation layers 0.027 mm thick and with \(\varepsilon=2.4\) (loss tangent \(\operatorname{tg}\delta=10^{-3}\)). This inner conductor was wound on a solid dielectric rod 3.7 mm in diameter. Titanium dioxide disks and tubes (\(\varepsilon=99\)) were placed in the space between the inner and outer conductors of the coaxial line. Depending on the degree to which the line was filled with these disks and tubes, the average dielectric constant of the medium between the inner and outer conductors of the line, \(\varepsilon_1\), took values from 1 to 4.5. In this way, by gradually inserting more and more TiO\(_2\) tubes and disks, it was possible to obtain a curve of the dependence of the attenuation in the line on the dielectric constant \(\varepsilon_1\) (Fig. 3; the experimental points are indicated by circles).
Fig. 2.
Fig. 3.
Graph labels: vertical axis, “losses in decibels”; horizontal axis, \(\varepsilon_1\). Legend: “theory”; “experiment”; dashed curve: “solid conductor.”
As follows from the theory (see (5)), at \(\varepsilon_1=\varepsilon_x\) the losses in the line reach their minimum (since in this case the thickness of the skin layer of the inner conductor is maximal). For comparison, Fig. 3 shows the theoretical curve (solid line), as well as the loss curve in the same line but with a solid inner conductor (dashed curve).
As can be seen from Fig. 1, the theory agrees well with experiment. A certain discrepancy between experiment and theory is explained by inhomogeneities that were present in the line, and first of all in the inner conductor, which was made by hand and not by a machine method.
M. P.
References
- A. M. Clogstone, Proc. Inst. Radio Eng. (PIRE), 39, 767 (1951).
- A. M. Clogstone, Bell. Syst. Techn. Journ., 30, 491 (1951).
- H. S. Black, C. O. Mallinckrodt, S. P. Morgan, PIRE, 40, 902 (1952).
- Sakurai, Journ. Phys. Soc. (Japan), 5, No. 6, 394 (1950).
- S. Morgan, Bell. Syst. Techn. Journ. 31, 883 (1952).