Influence of Dielectric Losses in the Frame Material on the Quality Factor of Single-Layer Cylindrical Coils

R. M. Kessenikh

(Presented by Professor Doctor A. A. Vorob’ev)

Introduction

A complete and sufficiently rigorous treatment of the question of the influence of the electric field in the frame material of a coil on its parameters under operating conditions is of considerable interest from the standpoint of designing stable components.

The corresponding electrodynamic problem, in its full formulation, is complex.

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 a simplified engineering treatment of the problem, one that is consistent with experiment and with the practice of designing radio components.

The problem of the single-layer cylindrical coil has attracted the attention of many theoretical physicists and experimentalists and has been considered both in its most simplified form and in a more rigorous formulation. However, problems connected with the consideration of secondary phenomena caused by the influence of the dielectric structural material of the frame on the electric field of the distributed charges of the coil have not been completely solved even with the aid of simplified engineering calculations. Thus, for example, matters stood this way with the separation of the component of the self-capacitance through the dielectric of the frame and of the dielectric losses in the frame material. One of the works [1] considers the distribution of current and voltage in a single-layer coil and gives a calculation formula for the self-capacitance of a frameless coil, without mentioning anything about the component through the dielectric of the frame.

Jackson [2] showed that the effective resistance of coils wound on a frame increases appreciably in comparison with the effective resistance of similar frameless coils.

In the same work Jackson proposed a formula for calculating the resistance \((R_{\partial.п.})\) introduced by the frame material of the coil.

The formula has the form:

\[ R_{\text{d.l.}}=L^2 C_0 \omega^3 \operatorname{tg}\delta, \tag{1} \]

where \(L\) is the inductance of the coil,
\(C_0\) is the self-capacitance of the coil,
\(\omega\) is the frequency,
\(\operatorname{tg}\delta\) is the tangent of the dielectric-loss angle of the frame material.

This formula (1) is often cited in radio-engineering textbooks. Thompson [3] obtained a formula relating the quality factor of coils to \(\operatorname{tg}\delta\) of the frame material, and on the basis of this formula calculation curves were given.

In experimentally checking his formula, Thompson found a discrepancy with the calculated data equal to 50%.

In the domestic specialized literature published over the last ten years there are the most contradictory statements concerning the influence of dielectric losses on the quality factor of HF and VHF inductance coils (see, for example, [4, 5, 6, 7, 8]).

In 1947 the author [9] obtained an expression relating the quality factor of a coil to \(\operatorname{tg}\delta\) of the frame material. The formula was derived by representing the inductance coil as a two-terminal network (Fig. 1), for which the quality factor was determined.

Fig. 1. Equivalent circuit of a two-terminal network—the coil

The formula obtained has the form:

\[ Q= \frac{Q_1(1-\omega^2LC_0)} {1-\omega^2LC'_0+Q_L\omega^2LC_g\operatorname{tg}\delta -Q_L\omega^4C'_0C_gL^2\operatorname{tg}\delta}, \tag{2} \]

where \(Q_1\) is the quality factor of the frameless coil;
\(C'_0\) is the total capacitance of the coil wound on the frame;
\(C_0\) is the capacitance of the frameless coil;
\(C_g\) is the capacitance through the dielectric of the frame;
\(L\) is the inductance;
\(Q_L\) is the quality factor of the capacitance-free coil;
\(\operatorname{tg}\delta\) is the tangent of the dielectric-loss angle of the frame material.

If the fourth term of the denominator in (2) is neglected, a more convenient expression for calculation can be obtained:

\[ Q = Q_1(1 - Q_1 \omega^2 C_g L \operatorname{tg}\delta). \tag{3} \]

Expression (3) is valid under the condition that \(Q_1 \omega^2 C_g L \operatorname{tg}\delta \ll 1\). It was also shown there that, for ceramic formers made of ultraporcelain (\(\operatorname{tg}\delta\) not greater than 0.002), the influence of dielectric losses in the former material on the coil quality factor is insignificant. However, formula (3) had not been subjected to detailed experimental verification.

Experimental Results

For experimental verification of formula (3), two batches of coils were made for different frequency ranges. The first batch was made for the range \(6 \div 20\) MHz with \(L = 1.8\ \mu\text{H}\); the second batch of coils was made for the range \(60 \div 80\) MHz. The inductance of these coils was \(0.2\ \mu\text{H}\).

In both the first and second cases, the coils were wound with bare copper wire. The wire diameter was 1 mm. The coil formers were made with threading from organic dielectrics (polystyrene, ebonite, and getinax).

The selected organic dielectrics were of interest as a structural material from which formers can be made rather easily. In addition, their dielectric constant \((\varepsilon)\) and dielectric-loss tangent \((\operatorname{tg}\delta)\) differed significantly from one another.

Simultaneously with the coils wound on formers, formerless coils similar to the former ones were made.

The coils without formers and with formers were made from identical conductors, with exact repetition of the cross section and other dimensions. The losses in the conductor and the part of the active resistance introduced by them were the same for each batch of formerless coils and coils on formers. The \(\operatorname{tg}\delta\) of the insulating materials and the quality factor \((Q)\) were measured with a Q-meter. The characteristics of the insulating materials used are given in Table 1.

The self-capacitance of the coils through the dielectric of the former was determined by the formula given in [9].

Table 1

Verification of formula (3) was carried out in the frequency range from 6 to 80 MHz. The results of the calculated and experimental data are presented in Table 2.

It can be seen from the table that the measured and calculated values of the quality factors of the coils agree within up to 10%.

The formula (3) obtained made it possible to estimate the influence of dielectric losses in the former material on the quality factor of single-layer cylindrical inductance coils and to establish permissible limits for the absolute values of \(\operatorname{tg}\delta\) of the former material. Analysis of expression (3) shows that, over considerable ranges of frequency variation, for limited values of the loss angle of the former material \((\operatorname{tg}\delta < 5 \cdot 10^{-3})\), the influence of losses in the dielectric of the coil former remains small.

The frequency dependence of the quality factor of the coils is shown in Figs. 2 and 3. It follows from these figures that the quality-factor curves of frameless coils lie above the quality-factor curves of coils wound on formers.

The decrease in the quality factor of coils wound on formers, in comparison with frameless coils, is explained by losses in the former material.

Fig. 2. Dependence of the quality factor on frequency for coils wound on formers made of various dielectrics.
\(x\) — frameless coil,
\(\bullet\) — coil on a polystyrene former,
\(\circ\) — coil on a getinax former,
\(\triangle\) — coil on an ebonite former

In Fig. 4 the dependence of the quality factor of coils on \(\operatorname{tg}\delta\) of the former material is given for three frequencies (6, 10, and 20 MHz), and in Fig. 5 an analogous dependence is given, but only for frequencies of 60 and 80 MHz. In the operating frequency range, the absolute value of the quality factor of any coil is greater at a higher frequency. Therefore

Fig. 3. Dependence of the quality factor on frequency for coils wound on formers made of various dielectrics.
\(x\) — frameless coil,
— coil on a polystyrene former,
— coil on an ebonite former,
\(\triangle\) — coil on a getinax former

the curves \(Q = f(\operatorname{tg}\delta)\), representing the dependence of the quality factor on \(\operatorname{tg}\delta\) of the former material for fixed frequencies, pass at higher frequencies through the region of larger values of \(Q\).

As for the more noticeable decrease in quality factor as a function of \(\operatorname{tg}\delta\) of the former material at higher frequencies—

Table 2

\((10, 20, 60, 80\ \text{MHz})\), then this can be explained by the frequency dependence of the \(\tg \delta\) of getinax, which is what causes this decrease.

From these same figures (4 and 5) and Table 2 it is evident that the \(Q\) factor of coils wound on polystyrene and ebonite forms differs little from the \(Q\) factor of frameless coils \((4 \div 8\%\), at \(f = 10 \div 80\ \text{MHz})\).

Fig. 4. Dependence of the \(Q\) factor of inductance coils on the dielectric-loss angle tangent of the form material

Fig. 5. Dependence of the \(Q\) factor of inductance coils on the dielectric-loss angle tangent of the form material.

The \(Q\) factor of coils wound on getinax forms differs appreciably from the \(Q\) factor of frameless coils \((16—31\%\) at \(f = 10—80\ \text{MHz})\). For illustration, Fig. 6 gives the dependence of the resistance \((R_{\partial.п.})\) introduced by the form material on frequency.

This resistance \((R_{\partial.п.})\) was determined on the basis of measurements of the \(Q\) factor of the coils (frameless and wound on forms).

From the presented Fig. 6 it is evident that the magnitude of the introduced resistance \((R_{\partial.п.})\) for polystyrene and ebonite forms depends on frequency ...

practically does not depend on frequency. For the getinaks frame, however, a noticeable increase of \(R_{\mathrm{d.l.}}\) with increasing frequency is observed.

The experimental data we have presented (see Figs. 4, 5 and Table 2) show that increasing \(\operatorname{tg}\delta\) of the frame material to values of \(50\cdot 10^{-4}\) does not lead to a decrease in \(Q\) by more than 8%.

Fig. 6. Dependence of the resistance introduced by the frame material on frequency.
1 — coil on a polystyrene frame.
2 — coil on an ebonite frame.
3 — coil on a getinaks frame.

When selecting a frame material for stable inductance coils, it is necessary to take into account that \(\operatorname{tg}\delta\) of the frame material must not increase with frequency in the operating range, and its temperature dependence must be small.

At the same time, it is often not so much the absolute value of the dielectric loss-angle tangent of the frame material that is of importance as its specific influence [10]. This specific influence of the losses is proportional to the ratio of the coil’s self-capacitance through the frame dielectric to the total capacitance of the circuit.

For limited values of the loss angle of the frame material, the influence of dielectric losses in the frame material on the quality factor of single-layer cylindrical coils is insignificant. Modern designs of stable coils have a component of the self-capacitance of the order of \(0.5\ \mathrm{pF}\) [9], while the loss-angle tangent of frame materials (quartz glass, steatite, ultraporcelain, celsian) lies within \((3 \div 10)\cdot 10^{-4}\). In addition, it should be noted that \(\operatorname{tg}\delta\) of the indicated dielectrics practically does not depend on frequency in the operating range of the coils, and its temperature dependence in this same range is either absent or insignificant (for example, \(\operatorname{tg}\delta\) of quartz glass up to \(300^\circ\mathrm{C}\) does not depend on temperature).

Conclusions

1. The formula obtained,
   \[ Q = Q_1\left(1 - Q_1\omega^2 C_g L\,\operatorname{tg}\delta\right), \]
   under the condition that
   \[ Q_1\omega^2 C_g L\,\operatorname{tg}\delta \ll 1, \]
   gives good agreement with experimental data. The measured and calculated values of the quality factors of coils in the range \(6 \div 80\ \mathrm{MHz}\) lie within 10%.

2. Increasing \(\operatorname{tg}\delta\) of the frame material to values of \(50\cdot 10^{-4}\) does not lead to a decrease in the quality factor of the coils by more than 8% \((f = 10 \div 80\ \mathrm{MHz})\).

3. Consideration of our experimental data gives grounds to believe that the influence of dielectric losses in the frame material on the quality factor of single-layer cylindrical coils can be reduced to a negligibly small value if modern inorganic dielectrics are used (quartz glass, steatite, ultraporcelain, celsian).

References

1. Abd-el Samil Mostafa and Gohar M. K. Determination of the distribution of voltage, current, and magnetic field, together with the self-capacitance, inductance, and high-frequency resistance of single-layer coils. Proc. G. R. E., 4, No. 4, 537, 1953.

2. Jackson W. Dielectric losses in single-layer coils at radio frequencies. E. W. W. E., p. 225, 1928.

3. Thompson A. Influence of the former material on the quality factor of inductance coils. Radio E. E., p. 41, 1945.

4. Volgov A. V. Components of circuits of radio equipment, Gosenergoizdat, 1954.

5. Pestryakov V. B. and Sachkov D. D. Design of radio components and assemblies of radio equipment, Gosenergoizdat, 1947.

6. Evteev F. E. and Zhukov V. A. Technology of radio equipment, Gosenergoizdat, 1952.

7. Siforov V. I. Radio receiving devices, Military Publishing House, 1954.

8. Siforov V. I. Radio receivers at SHF, 1955.

9. Kessenikh R. M. Self-capacitance of single-layer cylindrical self-inductance coils, Proceedings of NIIIS, issue 3 (38), 24, 1947.

10. Sosunov V. N., Self-inductance coils with a small temperature coefficient, Izvestiya VETAS, vol. XXI, 23, 1940.

ERRATA