A Rigorous Theory of an Inductance Coil in a Radio-Frequency Alternating-Current Circuit and Some Engineering Methods for the Approximate Calculation of the Parameters of a Single-Layer Coil

R. M. Kessenikh

(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)

The design of radio apparatus in many cases requires the use of two-terminal and multi-terminal networks with small energy losses and with resonant and antiresonant frequencies that are stable with respect to external influences.

In accordance with the general theory of circuits, the impedance of a two-terminal network is determined by expression 1 through \(P\)—the average power absorbed by the circuit over a period—and the period-averaged values of the energy \(W_\mu\) of the magnetic field and \(W_\э\) of the electric field created by currents and charges in the elements of the circuit.

\[ z=\frac{P}{I^2}+\frac{i2\omega\left(W_\mu-W_\э\right)}{I^2}=R(\omega)+ix(\omega). \tag{1} \]

As is known, the resonant and antiresonant frequencies for a two-terminal network are the frequencies at which \(|z|\) has an extremum and \(x(\omega)=0\). In this case the corresponding frequency \(\omega_k\) will be resonant if \(|z(\omega_k)|\) has a minimum at the point \(\omega=\omega_r\), and antiresonance if this point corresponds to a maximum of the modulus of the total resistance.

It follows from the foregoing that, for small losses

\[ \left(\frac{P}{W_\mu+W_\э}\ll 1\right) \]

the resonant properties are determined by the frequency dependence of the quantity

\[ \frac{1}{\omega}x(\omega)=\frac{2\overline{W}_\mu-\overline{W}_\э}{I^2}. \tag{2} \]

The constructive methods for implementing circuits with the properties of a two-terminal network are at present very diverse; however, most types of two-terminal networks characterized by sharp resonance and antiresonance curves may be assigned to one of three groups:

a) an oscillatory circuit with separated inductance and capacitance;

b) a volume resonator;

c) a long line or waveguide resonator.

Systems of type “c” may be implemented in the form of a combination of two-terminal networks of type “a” (artificial lines). The possibility of conce—

concentration of relatively large reserves of magnetic-field energy in a small volume with a small amount of metal is of special value for the design of small-size apparatus. If one leaves aside the type of systems with iron, ferrodielectric, and ferrite cores that lies outside the scope of our consideration, then, for the frequency range from \(1\text{--}3\) to \(50\text{--}100\) MHz, such a possibility is created by the use of single-layer solenoidal conductors in the form of an inductance coil, with or without a shield.

For this frequency range, resonant dipoles in the form of cavity resonators and long lines are applicable only in rare cases, when the dimensions and weight of the device play a secondary role. For small-size portable communications equipment intended for operation in the short- and ultrashort-wave ranges, resonant systems with separated inductance and capacitance have long been used and retain their primary importance.

The inductance in such systems is realized by means of single-layer self-inductance coils. Multilayer coils have appreciable losses and are used in the medium- and long-wave ranges.

In accordance with the main task of the present work, we shall consider here systems that most closely approximate, in their properties, a pure inductance, i.e., systems for which, in the operating frequency range,

\[ \frac{\overline{W}_{\mathrm{e}}}{\overline{W}_{\mu}} \ll 1. \]

The selection of parts for constructing oscillatory circuits of radio equipment must be made on the basis of specifying definite characteristics of these parts; moreover, of course, the number of characteristics and parameters of the parts must be minimal, and their physical meaning must be as clear as possible to the designer.

The quasistationary model of an alternating-current electric circuit is very attractive for constructing the equivalent circuit of an oscillatory circuit.

The simplest case, to which the consideration of oscillatory elements of radio-engineering devices very often has to be reduced, is a circuit with lumped inductance, capacitance, and resistance.

The transition from an idealized circuit to a laboratory mock-up of a radio-engineering device, and then to a specimen and to standard parts, introduces substantial changes into the idealized circuit. In the case of the simplest oscillatory circuit, such changes reduce to taking account of what would seem to be secondary corrections—such as distributed capacitance and distributed losses of all kinds in the inductance, distributed inductance and losses of all kinds in the capacitance, and the capacitance and inductance of resistances.

In some cases the “parasitic phenomena” that necessitate the introduction of such corrections become so sharply expressed that the original simplified equivalent circuit loses its meaning (a choke with a large capacitance at high frequencies, a bifilar resistance at high frequencies, etc.). In such cases it becomes necessary to change the construction of the part substantially in order to restore its qualitative features. In other cases, parasitic influences of the second order, without violating the fundamental aspect of the circuit operation, have a substantial effect on such properties as stability. The aim of the present work, as already indicated, is the consideration of secondary

phenomena in such components of high-radio-frequency radio circuits (3–150 MHz), which are due to the influence of the dielectric properties of the material of inductor coil forms on the electric field of the distributed charges of the coil.

As follows from the very formulation of the problem, a complete and sufficiently rigorous treatment of the question of the influence of the electric field in the form material of an inductor on the oscillatory parameters of the coil under operating conditions requires consideration of very complex systems, both geometrically and electrically, as complex electrodynamic problems. This approach to the processes in a single-layer coil has long attracted the attention of many theoretical physicists and experimentalists. It is sufficient to mention that the problem of the single-layer coil in its most simplified form has been considered theoretically and experimentally [2, 3, 4, 5]. An attempt at a more rigorous solution of the problem of free and forced electric oscillations of a single-layer coil was made in [6]. Works in which this problem is considered in a more simplified engineering formulation do not take into account the theoretical results obtained earlier [7, 8, 9].

Naturally, the general electrodynamic formulation of the problem in all its completeness is of no value for engineering unless at least limiting cases are established in which the rigorous formulation confirms the simplified engineering treatment of the problem, in agreement with experiment and with the practice of designing radio components.

Field equations for a complex system of conductors and the transition to the equations of quasi-stationary alternating currents

The equations of quasi-stationary alternating currents are obtained from the equations of the electromagnetic field by passing from partial differential equations for the field \(\vec E\) in the volume occupied by the system under consideration to a system of ordinary differential equations obtained after integrating the electric-field vectors over contours corresponding to the circuits into which the system is divided. In the general case the starting equation is

\[ \operatorname{rot}\vec E=-\frac{\partial \vec B}{\partial t}. \tag{1} \]

Since \(\operatorname{div}\vec B=0\) and \(\vec B=\operatorname{rot}\vec A\), where \(\vec A\) is the vector potential, it follows, on the basis of Stokes’ theorem applied to the equation

\[ \operatorname{rot}\left(\vec E+\frac{\partial \vec A}{\partial t}\right)=0, \]

that for any closed contour \(\Gamma_i\),

\[ \oint_{\Gamma_i}\left\{\vec E+\frac{\partial \vec A}{\partial t}\right\}\vec{ds}=0 \tag{2} \]

The condition for passing from this contour integral to the differential equation of a quasi-stationary alternating-current circuit reduces to the possibility of representing:

\[ \oint \vec E\,\vec{ds}=\sum U_{ik}+\sum R_{ik}I_k \tag{3} \]

\[ \oint \frac{\partial \vec A}{\partial t}\,\vec{ds} = \frac{\partial}{\partial t}\sum L_{ik}I_k + \int \frac{\partial A_0}{\partial t}\,\vec{ds}, \tag{4} \]

where \(U_{ik}\) are the potentials induced in the sections of the \(i\)-th loop by the \(k\)-th charge;
\(R_{ik} I_k\) is the voltage drop across the resistances of the \(i\)-th loop, due to the current

\[ I_k=\frac{\partial q_k}{\partial t} \tag{5} \]

and \(l_{ik}\dfrac{\partial I_k}{\partial t}\) is the electromotive force of mutual induction induced by the current \(I_k\) in the \(i\)-th loop;
\(\vec A_0\) is the vector potential of the primary field, producing an electromotive force in the loop.

Let us consider how this scheme for matching equations can be applied to calculating the distribution of charges and currents in a coil (wound on an insulating former made of a material possessing dielectric losses) forming an oscillatory circuit in series with a capacitor. Such a system is shown schematically in Fig. 1.

The distribution of the insulating material and the conductor is shown in cross section in Figs. 2 and 3. This scheme may correspond to the commonly used construction of self-inductance coils with a ceramic former and the coil winding metal applied by firing.

Fig. 1. Inductance coil with capacitance

Figs. 2–3. Distribution of insulating material and conductor

In the general case, the transition from a differential equation in partial derivatives to equations of electric circuits corresponds to the choice of a minimum number of loops \(\Gamma_i\), the traversal of which is, with accuracy sufficient for the stated purpose, equivalent to the complete specification of the boundary conditions of the problem. The zeroth approximation will correspond to the introduction of one traversal loop, shown schematically in Fig. 1.

In this figure the section \(P_1-P_2\) of the loop is a spiral passing along the surface of the conductor of the coil, \(P_2-P_3\) and \(P_4-P_1\) are along the surface of the connecting conductors, and \(P_3-P_4\) is through the dielectric of the capacitor \(C_0\). In this case, under the assumption that the conductivity of the metal \(\sigma \to \infty\),

\[ \int_{P_1}^{P_2} \vec E\, d\vec s+ \int_{P_2}^{P_3} \vec E\, d\vec s+ \int_{P_4}^{P_1} \vec E\, d\vec s=0 \tag{6} \]

As a consequence of the equality to zero of the tangential component of \(E\) on the surface of an ideal conductor. Further,

\[ \int_{P_3}^{P_4} \vec E\,d\vec s = -\int_{P_3}^{P_4} \operatorname{grad}\psi\,d\vec s - \frac{\partial}{\partial t}\int_{P_3}^{P_4}\vec A\,d\vec s . \tag{7} \]

The second simplifying assumption consists in the fact that the potential

\[ \varphi = \frac{1}{4\pi\varepsilon_0} \int_{v} \frac{\rho(\xi,\eta,\zeta),\,t-\dfrac{r}{c}}{r}\,d\tau \tag{8} \]

is replaced by an expression without allowance for retardation (one of the conditions of quasistationarity), in accordance with the usual calculation of the field and charge in a plane capacitor.

Under this assumption

\[ -\int_{P_3}^{P_4}\operatorname{grad}\psi\,d\vec s = \psi(P_3)-\psi(P_4)=U . \tag{9} \]

The potential difference \(U\) will be equal to

\[ U=\frac{q}{C_0}, \tag{10} \]

where \(q\) is the charge on a plate;
\(C_0\) is the capacitance of the circuit capacitor.

The third simplification reduces to the assumption that at all points of the conductor of the coil and of the connecting wires

\[ \operatorname{div}\vec j=0. \tag{11} \]

This assumption is equivalent to the assertion that the surface charge density on the conductor of the coil and on the connecting wires is equal to zero and that, consequently, outside the capacitor the electric field has no component of intensity vector normal to the surface of the conductor. From this assumption it follows that

\[ \int_s j\,ds=J \tag{12} \]

the current through a cross section of the conductor is the same at all its points, i.e., also a condition of quasistationarity.

Applying (6), (7), (10), and (12) to (2) in the given idealized case, we obtain

\[ \oint \left( \vec E+\frac{\partial \vec A}{\partial t} \right) d\vec s \simeq \frac{q}{C_0} - \frac{\partial}{\partial t}\int_{P_3}^{P_4}\vec A\,d\vec s + \frac{\partial}{\partial t} \int \left\{ \int_{(P_4,P_1,P_2,P_3)} \frac{j\left(s_1,t-\dfrac{r}{c}\right)}{r}\,d\tau \right\} d\vec s + \]

\[ + \frac{\partial}{\partial t}\int_{P_3}^{P_4}\vec A\,d\vec s + \frac{\partial}{\partial t}\int \vec A_0\,d\vec s . \tag{13} \]

And, finally, after an approximate replacement, also using the condition of quasistationarity, of the integral on the right-hand side of (13) over ob-

of the conductors \((P_4, P_1, P_2, P_3)\), assuming that equality (2) remains valid, we obtain

\[ -\Sigma+\frac{q}{C_0}+\frac{\partial J}{\partial t}\iint \frac{\overrightarrow{ds'}\,\overrightarrow{ds}}{r}=0 \tag{14} \]

where

\[ \Sigma=-\frac{\partial}{\partial t}\int \vec A_0 \overrightarrow{ds} \]

is the electromotive force of induction of the external field.

The quantity

\[ \iint \frac{\overrightarrow{ds'}\,\overrightarrow{ds}}{r}=L_{\parallel} \]

is the self-inductance of the coil and of the connecting wires.

Taking also into account that in the region of the capacitor equation (11) becomes

\[ \operatorname{div}\vec j=-\frac{\partial \rho}{\partial t}, \qquad \int j_{\parallel}\,ds=I; \qquad I=-\frac{\partial}{\partial t}\int \rho\,d\tau=\frac{\partial q}{\partial t} \]

and replacing \(\partial/\partial t\) by \(d/dt\), we obtain the ordinary differential equation of a lossless oscillatory circuit:

\[ L_{\parallel}\frac{d^2q}{dt^2}+\frac{q}{C_0}=\Sigma(t). \tag{15} \]

This well-known in principle path is reproduced here in detail in order to indicate more distinctly the admissible simplifications.

Without dwelling here on the simplification of the integral

\[ \int\left\{\int \frac{\vec j\left(s',\,t-\frac{r}{c}\right)}{r}\,d\tau \right\}\overrightarrow{ds}, \]

which, under rigorous consideration, even in the case of ideal conductivity, contains, in addition to the self-inductance, also the radiation resistance of the coil and of the connecting wires, and also depends on the effect of the proximity of neighboring turns, one should, in accordance with the principal task of the work, direct primary attention to assumption (11)

\[ \operatorname{div}\vec j=0. \]

This assumption asserts that the instantaneous value of the total current of the coil remains constant and, consequently, that there is no distribution of capacitance of the coil. The first approximation in the solution of the problem, obtained when condition (11) is abandoned while the assumption of ideal conductivity of the coil conductor is retained, immediately leads to the necessity of taking into account the electric field of the charge distributed over the surface of the coil conductor and of the connecting conductors of the oscillatory circuit.

All further approximations to the rigorous formulation of the problem may be regarded as the replacement of the ordinary differential equation of the zero approximation (15) by a system of equations with first time derivatives of the charges \(q_i\), or of the corresponding currents

\[ I_i=\frac{dq_i}{dt}. \]

It seems that such a system of differential equations corresponds to the representation of the coil in the form of an electrical network with a finite number of elementary cells.

In practice, this first approximation means adding to the capacitance of the circuit \((C_0)\) the self-capacitance of the coil, which we shall denote by \(C'_0\). This approximation is not sufficient for a coil with a large number of turns. A more complex equivalent circuit may be required, in which the gradual branching of the total current between the turns of the coil and the distributed capacitance of the turns will be taken into account.

Integral equation of a closed oscillatory circuit under free and forced oscillations and its application to the problem of the self-capacitance and quality factor of a single-layer coil

Abandoning equation (11) restores the rigorous relation

\[ \operatorname{div}\left(\vec{j}-\frac{\partial \vec{P}}{\partial t}\right)=-\frac{\partial \rho}{\partial t} \tag{16} \]

where \(\vec{P}\) is the polarization vector of the dielectric of the frame material and of the conductor insulation of the coil winding;

\(\rho\) is the volume density of free charges.

The scalar potential of the field in this case will be determined by the expression

\[ \psi=\frac{1}{4\pi\varepsilon_0}\left\{ \int \frac{\rho\left(t-\frac{r}{c}\right)}{r}\,d\tau - \int \frac{\operatorname{div}\vec{P}\left(t-\frac{r}{c}\right)}{r}\,d\tau \right\} \tag{17} \]

or

\[ \frac{\partial \psi}{\partial t} = -\frac{1}{4\pi\varepsilon_0} \int \frac{\operatorname{div}\vec{j}\left(t-\frac{r}{c}\right)}{r}\,d\tau . \tag{18} \]

The electric-field intensity \(\vec{E}\) is determined through the scalar potential (17) and the vector potential \(\vec{A}\)

\[ \vec{A} = \frac{\mu_0}{4\pi} \int \frac{\vec{j}\left(t-\frac{r}{c}\right)}{r}\,d\tau + \vec{A}_0, \tag{19} \]

where \(\vec{A}_0\) is the vector potential of the external field by means of the expression

\[ \vec{E} = -\operatorname{grad}\psi - \frac{\mu_0}{4\pi} \int \frac{\dfrac{\partial \vec{j}\left(t-\frac{r}{c}\right)}{\partial t}}{r}\,d\tau - \frac{\partial \vec{A}_0}{\partial t}. \tag{20} \]

Substitution into (20)—(18), after differentiation with respect to time, gives

\[ \frac{\partial \vec{E}}{\partial t} = \frac{1}{4\pi\varepsilon_0}\operatorname{grad} \int \frac{\operatorname{div}\vec{j}\left(t-\frac{r}{c}\right)}{r}\,d\tau - \frac{\mu_0}{4\pi} \frac{\partial^2}{\partial t^2} \int \frac{\vec{j}\left(t-\frac{r}{c}\right)}{r}\,d\tau - \frac{\partial^2 \vec{A}_0}{\partial t^2}. \tag{21} \]

The projection of (21) onto the tangent \(\vec{s}\) to the axial line passing through the central part of the wire gives, under the assumption of ideal conductivity of the medium,

\[ \frac{1}{4\pi\varepsilon_0} \frac{\partial}{\partial s_i} \int \frac{\operatorname{div}\vec{j}\left(t-\frac{r}{c}\right)}{r}\,d\tau - \frac{\mu_0}{4\pi \partial t^2}\,\vec{s}_i \int \frac{\vec{j}\left(t-\frac{r}{c}\right)}{r}\,d\tau - \frac{\partial^2 A_0(t)}{\partial t^2}. \tag{22} \]

If one assumes a purely harmonic dependence of $\vec A_0(t)$ on time in the form $\vec A_0(t)=Re\vec A_0 e^{-i\omega t}$ and discards, in equation (22), the solution expressing the natural oscillations, then one can obtain an integro-differential equation for the current density $\vec j$ in the conductor of the coil. This question is considered in [6].

For forced oscillations corresponding to a specified external field with complex amplitude of the vector potential $\vec A_0$, the determination of the self-capacitance reduces to the following: the coil is represented in the form of an equivalent circuit (Fig. 4).

The total current $I$ entering the two-terminal coil is decomposed into two parts: the current $I_L$ in the coil-current bundle and the equivalent current $I_{C_0}$ corresponding to the capacitive branch of the equivalent circuit.

Fig. 4. Equivalent circuit of the coil

Both rigorously and approximately, it is possible to determine the total current $I$ and the current $I_L$. Hence $I_{C_0}$ is also determined as

$$ I_{C_0}=I-I_L. \tag{23} $$

Under the condition that the equivalent loss resistance in the self-capacitance is small, $I_{C_0}=i\omega C_0U$, where $U$ is the voltage drop across the coil.

The aim of our study is to estimate the influence of the electric field in the space between the turns on the quality factor of a coil wound on a former. For an engineering solution of this problem it is necessary to determine the self-capacitance of the coil, and then the influence of the dielectric properties of the former material on the quality factor.

The problem posed can be solved with the aid of equivalent circuits and comparison of them with experiment. Figure 5 gives circuits for several variants of a rough calculation of $C'_0$ for the case approximating our problem. It can be seen from Fig. 5 that, for practical calculations, it is expedient to use a linear dependence of the self-capacitance $C'_0$, having the form (Fig. 5, a)

$$ C'_0=C_1+C_2\varepsilon. \tag{24} $$

Figure 6 gives a circuit representing the coil as a two-terminal network. The calculation shows that, for the quality factor of the coil $Q$, the formula [10] is valid

$$ Q=Q_1(1-Q_{10}^{2}C_gL\,\operatorname{tg}\delta), \tag{25} $$

Fig. 5. Diagrams of several variants for calculating the self-capacitance of a coil

Fig. 6. Coil as a two-terminal network

Fig. 7. Dependence of the self-capacitance on the dielectric permittivity of the former material

Fig. 8. Dependence of the quality factor on the tangent of the angle of dielectric losses of the former material

where \(Q_1\) is the quality factor, under the condition \(Q_1\omega^2 C_g \tan\delta \ll 1\), of a form-less coil;
\(C_g\) is the capacitance through the dielectric of the form;
\(L\) is the inductance;
\(\tan\delta\) is the tangent of the angle of dielectric losses of the form material.

The formulas (24) and (25) proposed by us give good agreement with experimental results. To illustrate the foregoing, Figs. 7 and 8 present experimental data obtained by us in checking (24) and (25).

REFERENCES

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