On Determining the Shape of Guiding Surfaces in Lossless Lines with Inhomogeneous Insulation and Constant Characteristic Impedance
Abstract Generated abstract
This paper considers how to choose the shape of guiding surfaces in lossless TEM transmission-line sections with inhomogeneous insulation so that the characteristic impedance remains constant and reflections are avoided. It argues that, because replacing the inhomogeneous dielectric by a homogeneous one does not change the elemental inductance and because phase velocity is set by the medium rather than guide geometry, constancy of wave impedance can be reduced to the condition that the product of the capacitance distributions for the homogeneous and inhomogeneous dielectric cases remain constant. The paper proposes electrical modeling, such as an electrolytic bath, as a practical way to determine the required guide shape, while noting that the analysis excludes possible excitation of other wave types.
Full Text
ON DETERMINING THE SHAPE OF GUIDING SURFACES IN LOSSLESS LINES WITH INHOMOGENEOUS INSULATION AND CONSTANT WAVE IMPEDANCE
V. D. Es’kov
(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)
Transmission of energy from a source to a receiver by means of electromagnetic waves of the TEM type is possible only when there are at least two guiding surfaces forming a transmission line (for example, two wires, a coaxial cable, or a conical transition) [1]. To avoid the appearance of reflected waves, which carry away part of the energy and distort the form of the transmitted signal, the line must be matched to the load, i.e., the input impedance of the latter must be equal to the wave impedance of the line.
The wave impedance of a homogeneous lossless line is determined by the formula
\[ z_{\mathrm{в}}=\sqrt{\frac{L_0}{C_0}}, \tag{1} \]
where \(L_0\) and \(C_0\) are the inductance and capacitance per unit length of the line and are uniquely related to each other, since the phase velocity
\[ v=\frac{1}{\sqrt{L_0C_0}} \tag{2} \]
is a constant quantity for the given medium [2].
Individual sections of lines (for example, the junction with the receiver) may have inhomogeneous insulation, which is due to technological or design considerations. If the length of such a section is comparable with the wavelength, then it is necessary to take care in choosing the shape of the guiding surfaces, ensuring such a distribution of the specific (differential) parameters \(L_{\mathrm{н}}\) and \(C_{\mathrm{н}}\) along the length of the section that the wave impedance of each element
\[ z_{\mathrm{в}}=\sqrt{\frac{L_{\mathrm{н}}}{C_{\mathrm{н}}}} \tag{3} \]
would remain unchanged.
Taking into account that, when an inhomogeneous dielectric is replaced by a homogeneous one, the inductance of the line elements does not change, i.e.,
\[ L_{\text{n}}=L_0, \tag{4} \]
from equations (2) and (3) it is not difficult to obtain
\[ z_{\text{v}}=\frac{v}{\sqrt{C_0 C_{\text{n}}}} . \tag{5} \]
Despite the fact that in the present case, even for a homogeneous medium, \(L_0\) and \(C_0\) are distributed nonuniformly and the wave impedance determined by formula (1) does not remain constant, the phase velocity depends only on the properties of the medium and does not depend on the geometry of the guides. Therefore, for the constancy of the wave impedance on the section with inhomogeneity, it is sufficient for condition (5) to be satisfied:
\[ C_0 C_{\text{n}}=\mathrm{const}. \tag{6} \]
For a practical determination of the shape of the guides in accordance with this condition, one may recommend electrical modeling (for example, in an electrolytic bath) [3]. In the model, the distribution of currents along the length of the electrodes will be analogous to the distribution of charges in the original system and, consequently, to the distribution of capacitances along the length of the line. It should be emphasized that the determination of \(C_0\) and \(C_{\text{n}}\) must be carried out with the same shape of guides with a homogeneous and an inhomogeneous dielectric, respectively.
In conclusion, it is necessary to note that the arguments presented do not take into account the possibility of the occurrence of waves of other types.
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