The Technician Explains

Radio Record, Volume III, Issue 11, 27 September 1929, Page 28

The Technician Explains

Inductance Coil Design

(By

Cathode

Second Instalment. N case any readers should not have noticed the correction inserted in the issue of September 6, it is desired to point out that in the design charts printed in the issue of August 28, and giving the number of turns necessary to attain certain specified inductances with different dimensions of coil, curve A refers to a coil diameter of 3 inches and not 2 inches, as stated, while curve C relates to a coil diameter of 2 inches. The subject to be dealt with in this paper is that of the most efficient, or, as it is termed, the "optimum" diameter of wire for any .particular coil. The fact that for any particular coil there exists one diameter of wire which will be productive of higher efficieucy than can be obtained from any other diameter of wire is fairly well known, thanks to the vublicity with which the introduction of the original Browning: Drake coils was attended. The statement then made, that the most efficient design entailed having the wire spaced by half its diameter is, of course, nonsense, as Glenn Browning would doubtless be first to admit in the light of fuller knowledge; the error was one that is very common-that of too readily propounding a general rule from 2 particular instance; however, Browning did succeed in focussing attentiojn on the problem of increasin;, coil efficiency, and for this he is deserving of every credit. To appreciate just why the diameter of the wire employed should affect the efficiency of the coil in this manner it is necessary that the readér shotid have some understanding of what is meant by high-frequency resistance. It will be remembered that in the writér’s last paper in this series, it was mentioned that the amplification obtiin-

able from a high-frequency stage depends, other things being equal, on the magnitude of a factor L/CR, where L is tlae inductance of the tuning coil, C the. associated capacity. and R the eoil’s high-frequency resistance in ohyas. It was pointed out that some adwantage could be gained by increasing; L and decrvasing C, but at the cost of: some slight loss of selectivity; what it is desire. to stress now is that a very considerable advantage can be gained by reducing R, not only an advantage in the direction of increased amplification, but also in the direction of enhanced selectivity. If it were desired to reduce the direct-current resistance of a tuning coil, the obvious thing would be to use as heavy a wire as could be accommo dated. Studying Fig..1, it is appareni that increasing the diameter of the wire steadily reduces the d.c. resis tance of the coil. A glance at the curve labelled xf. resistance (radio-frequen ey or high-frequency resistance) how ever, shows that there is a point be yond which any increase in wire diameter is productive of an actual deerease in efficiency (ie, an increase in high-frequency resistance). Obvious- ly, then, the pvint at which the highfrequency resistance is at a minimum represents the best or "optimum" wire diameter. The reason for the increase in resistance which accompanies an increase in wire diameter beyond the optimum will perhaps not be clear without explanation. One factor contributing to this result is what is known as "skin-effect.’ When a wire (over a

certain very small diameter) is carrying a current alternating at a very high frequency, the current is not evenly distributed throughout the wire, but is carried on the outside or "skin" of the wire; moreover, if this wire is wound into a coil, the current will not

even be evenly distributed over the surface, but will crowd to the inner surface (ie., the surface nearest the centre of the coil) of the wire; thus an increase in the diameter or area of the wire is not productive of anything like the expected effect in reducing the high-frequency resistance. Added to this "skin-effect,’’ however is another factor which we may term the "proximity effect." Bach turn of wire, when passing high-frequency current, has its own magnetic field, and the effect of these fields in introducing eddy current losses and current flow distortion in neighbouring turns is sue! as to still further increase the high frequency resistance over the directcurrent resistance. Furthermore, i! will be clear that the nearer the turns approach each other (or what is the same thing, the greater the diameter of the wire) the more serious this effect will become. It is, in fact, this "proximity factor" which is the root cause of the increase in high-frequency resistance when the wire diameter 1s increased beyond the point we have christened the "optimum diameter." The formula for the calculation of high-frequency resistance of solenoids takes into consideration both the effects we have just discussed, and was originally formulated by 8S. Butterworth, of the Admiralty Research Laboratory. It may be stated as follows :- ( = 2) Rur = R(i+F+G ) 4 ; 2D )

Where Rhf is the high-frequency resistance in ohms at the. particular frequency involved; R is the direct current resistance of ‘the winding in ohnis, n is the number of turns; d is the diameter of the wire in millimetres; D is the coil diameter in millimetres. To ascertain the values of the factors F and G, it is necessary firststo solve the subsidiary equation Z Il ms | where f is the frequency in cycles per second, and d, as before, the diameter of the wire in millimetres. Havin figured out the value of Z, the values of F and G may then simply be read off from the charts reproduced in Figs. 2 and 3. MK, the only remaining tor, is a shape factor depending on the ratio of the length of the winding to its diameter, and may be read directly from Fig. 4. In ease the method of working this formula is not immediately apparent, it is proposed to work out the resistance at 300 metres (or one million cycles per second) of a coil having 74 turns of 3-inch diameter and 4 winding length of 23-inch (actually the coil used for preparing Fig. 1). We will assume the wire diameter to be 0.565 millimetres, which reference to Fig. 1 indicates to be the optimum diameter. The direct current resistance of 0.565 m.m. (24 8.W.G.) copper wire, 1ay be ascertained from a wire table to be approximately .02145 ohms per ft.; the d.c. resistance of 74 turns 3 inches in diameter will be .25 x 3.1416 K 74 = 58.1 feet at .02145 ohms per ft. = 1.24 ohms. So R is 1.24 ohms. Before finding F and G we must know Z, and this found to be Vv 1,000,000 Z = 0.565 ---_-__-_- = 6.09 92.8

Then from Fig. 2 we find the value of F corresponding to this value of Z to be 1.483, from Fig. 8 the value of @ corresponding to a figure of 6.09, for Z is 948 So F is 1.48 and G is .946.

The ratio of winding length to diameter is 2.25 _-_-_- = 0.75 3 and from Fig. 4 the value of K corresponding to this ration is 5.9. We know n, the number of turns, 74, and d, the diameter of the wire in millimetres 0.565; . the diameter of the coil, we have to reduce to millimetres-l-inch = 25.4 millimetres

‘approximately, so D is 76.2; 2D will be 152.4. We are now in a position to simplify the term in the minor bracket. The Knd 5.9 x T4 X B65 --- becomes D 152.5 or 1.62. This figure has now to be squared, giving a result of 2.62. Substitution may now be effected throughout the entire equation ( Knd\, 2 ) Rar = R(1+F+G — ) ( 2D ) = 1.24 (1 + 143 + .948 x 2.62) == 1.24 (1 + 143 + 2.48) = 1.24 x 4.91 = 6,09 Thus we have found the high-frequency resistance of this coil at a frequency of 1000 kilocycles per second (300 metres), to be 6.09 ohms, a very different figure from the direct current resistance of 1.24 ohms. Even this figure of 6.09 ohms will be increased slightly jn practice by certain losses (e¢., dielectric losses and eddy current losses jin screening), which will inevitably be -introduced when the coil is located in a receiver. However, the distribution of these additional losses need not be dealt with here. It will be seen that in order to determine the optimum wire diameter for any particular coil, it is necessary to work out the high-frequency resistances for a number of different wire diameters, plotting a curve like that labelled "r.f.-resistance" in Fig. 1, and . reading off the diameter corresponding to the lowest point in the curve. When it is realised that this has in fact been done, albeit with the aid of certain mathematical short cuts, for three different inductances corresponding to those for which the required numbers

of turns were given in the last paper, for three different coil diameters in respect of each inductance, and for quite a number of winding lengths in respect of each coil diameter, and that the whole of these results are summarised in the little charts of Fig. 5, most experimenters will be heartily grateful that it fell to the lot of "Cathode" and not to their lot to prepare this data. Referring to these charts, it will be seen that they correspond to those previously published giving the number of turns. ‘Three inductances are again provided for to accommodate variable condensers of differing maximum capacity. In this connection the writer wishes to make a word of explanation. It was originally intended to publish four sets of charts for coils of 200 m.h., 280 m:h., 820 m.h., and 340 m.h., the last two being intended respectively for variable condensers of .0003 mfd. and .00025 mfd. maximum capacity. Measurements subsequently showed that the minimum capacity of a .0003 condenser was so little different from that of a .00025 condenser that the in‘ductance of 340 m.h. was suitable for both; naturally the charts covering coils of 320 m.h. were cut out. Unfortunately, the text was not amended everywhere, with the result that one or two phrases appeared which may have rather puzzled readers in the absence of this explanation. It must be clearly understood that the optimum wire diameter referred to in this paper is the diameter of the copper section only and that the increase in diameter occasioned by the

insulating covering of the wire must not be taken into consideration. As a matter of fact, the wire covering may frequently be usefully employed in distributing the wire over the allotted winding length. Some slight loss of efficiency will result from using this method of spacing, a better method being to use a grooved former. It will be noted that the charts of Fig. 5, giving the optimum wire diameter, have the B. and 8. gauges corresponding to different diameters noted alongside, so that the best gauge to employ can be read off directly. In New Zealand, however, most of the wire is rated by the Standard Wire Gauge, or S.W.G., which is slightly different from the B. and 8. gauge. A variation from the optimum diameter of a single gauge, or even two, is not likely to affect the coil efficiency very

Fig. 5. : Optimum diameter of wire for coils the number of turns for which were given on August 23. Top.-Coils of 200 microhenries. Middle.-280 microhenries. Bottom.-840 microhenries. Curve A relates to coils of 3 inch diameter. Curve B to coils of 2.5 inch diameter. Curve C to coils of 2 inch diameter. Winding lengths indicated along bottom of each chart. --

Fig. 6. Charts referred to in diagram 5 showing the number of turns for inductance coils, Top.-Coils of 200 microhenries, Middle.-280 microhenries. Bottom.-340 microhenries. Curve A relates to coils of 3 inch diameter. Curve B to coils of 2.5 inch diameter. Curve C to coils of 2 inch diameter. Winding lengths indicated along bottom of each chart. =

seriously; nevertheless, it has been thought advisable to prepare a little chart giving the diameter in millimetres of all gauges likely to be used, and this is as follows:-

Diameter S.W.G& B.&S. in m:m, 2 19 9140 21 20 8124 — 21 -T213 22 — -7108 -- 22 6126 23 — 6093 24 23 5585. 25 24 5078 26 25 -4570 27 26 4062 28 27 3555 29 — 3300 — yo) -3100 . 30 -_ -8046 31 29 -2800