# Single-Layer Helical Round Wire Coil Inductor Calculator

Round wire coil with dimensions and
its current-sheet approximation[2]

ENTER:

 D =  mm Mean diameter of the air core coil, measured from wire centre to wire centre N = Number of turns ℓ =  mm Lenght of the coil, measured from the connecting wires centre to centre d =  mm Wire or tubing diameter Cu, annealed Cu, hard-drawn Ag Al Plating material ρ =  nΩ·m Plating conductivity μr = Plating permeability f =  MHz Design frequency

INTERMEDIATE RESULTS:

 p =  mm Winding pitch Φ = Proximity factor according to empirical Medhurst data[2,3] Deff =  mm Effective coil diameter according to Stroobandt (see below) Correction factors: kL = Field non-uniformity correction factor according to Lundin[2,4] ks = Round wire self-inductance correction factor according to Rosa[2,5,6] km = Round wire mutual-inductance correction factor according to Grover and Knight[2,7] Wire: ℓwire, phys =  mm Physical wire length ℓwire, eff =  mm Effective wire length δi =  µm Skin depth at design frequency Sheath helix waveguide mode: ψ =  ° Effective pitch angle β =  rad/m Axial propagation factor of n=0 sheath helix waveguide mode at design frequency[1,8] Zc =  Ω Characteristic impedance of n=0 sheath helix waveguide mode at design frequency[1]

RESULTS:

 Leff,s =  µH Effective series inductance at design frequency from Corum & Corum's sheath helix waveguide formula, corrected for field non-uniformity and round wire[1,2,4-7] Xeff,s =  Ω Effective series reactance of round wire coil at design frequency Reff,s =  Ω Effective series AC resistance of round wire coil at design frequency Qeff,ul = Effective unloaded quality factor of round wire coil at design frequency
 Lumped circuit equivalent: Ls =  µH Frequency-independent series inductance from the current-sheet coil geometrical formula, corrected for field non-uniformity and round wire[2,4-7] XL,s =  Ω Series reactance of round wire coil RL,s =  Ω Series AC resistance of round wire coil at design frequency QL,ul = Unloaded quality factor of round wire coil at design frequency CL,p =  pF Parallel stray capacitance at design frequency[1] Self-resonant frequency: fres,L =  MHz λ/4 (parallel) self-resonant frequency of n=0 sheath helix mode[1,8]

## In what does this inductance calculator differ from the rest?

Dr. James F. Corum,
K1AON

The inductor calculator presented on this page is unique in that it employs the n=0 sheath helix waveguide mode to determine the inductance of a coil, irrespective of its electrical length. This allows for more accurate inductance predictions at high frequencies. Furthermore, the calculator closely follows the National Institute of Standards and Technology (NIST) methodology for applying round wire and non-uniformity correction factors and takes into account both the proximity effect and the skin effect.

The development of this calculator has been primarily based on the 2001 IEEE Microwave Review article by the Corum brothers[1] and the correction formulas presented in David Knight's, G3YNH, theoretical overview[2], extended with a couple of personal additions.

## Which equivalent circuit should be used?

Both equivalent circuits yield exactly the same coil impedance at the design frequency.

For narrowband applications around a single design frequency the effective equivalent circuit may be used. When needed, additional equivalent circuits may be calculated for additional design frequencies.

The lumped equivalent circuit is given here mainly for the purpose of comparing with other calculators. By adding a lumped stray capacitance in parallel, this equivalent circuit tries to mimic the frequency response of the coil impedance. This will be accurate only for a limited band of frequencies centred around the design frequency.

## What can this calculator be used for?

helical antenna

The calculator returns values for the axial propagation factor β and characteristic impedance Zc of the n=0 (T0) sheath helix waveguide mode for any helix dimensions at any frequency.

This information is useful for designing:

travelling wave tube

## What is the problem with other inductor calculators?

Tom Rauch, W8JI

Tom Rauch, W8JI, in his well-known high-Q inductor study once complained that many inductor modeling programs fail to consider two important effects:

• The stray capacitance across the inductor,
• The proximity effect, causing Q to decrease as turns are brought closer together.
For example, the RF Coil Design Javascript calculator by VE3KL ignores these effects.

Likewise, a once very popular computer program, named «COIL» by Brian Beezley ex-K6STI happened to be based on a set of faulty formulas[2,9,10], as has been carefully pointed out by David Knight, G3YNH. The program is luckily no longer commercially available.

## How is the helical waveguide mode being calculated?

Friedrich Wilhelm Bessel

A coil can be best seen as a helical waveguide with a kind of helical surface wave propagating along it. The phase propagation velocity of such a helical waveguide is dispersive, meaning it is different for different frequencies. (This is not the case with ordinary transmission lines like coax or open wire.) Lower frequencies propagate slower along a coil. The actual phase velocity at a specific frequency for a specific wave mode is obtained by solving a transcendental eigenvalue equation involving modified Bessel functions of the first (In) and second kind (Kn) for, respectively, the inside and the outside of the helix.[1,8]

Dr. T. J. Dekker

Since such an equation cannot be solved by ordinary analytical means, the present calculator will determine the phase velocity of the lowest order (n=0) sheath helix mode at the design frequency using T. J. Dekker's combined bisection-secant numerical root-finding technique.[11] The original FORTRAN code was translated to JavaScript.

A similar algorithm is also employed to home in on the frequency for which the coil appears as a quarter-wave resonator. This is the first self-resonant frequency of the coil.

the secant root-finding method

More details about the employed formulas and algorithms can be obtained immediatly from the Javascript source code of this calculator.

## Why are correction factors still needed?

Eventhough the sheath helix waveguide model is the most accurate model available to date, it certainly suffers from a number of limitations. These have their origin in the very definition of a sheath helix, being: an idealised anisotropically conducting cylindrical surface that conducts only in the helical direction.[1]

By chosing a sheath helix as the model for a cylindrical round wire coil, the following assumptions are made:

1. The wire is perfectly conducting.
2. The wire is infinitesimal thin.
3. The coil´s turns are infinitely closed-spaced.
4. Finally, since higher (n>0) sheath helix waveguide modes are disregarded, end-effects in the form of field non-uniformities are not dealt with. Hence, the sheath helix needs to be assumed to be very long and relatively thin in order for the end-effects to become negligible.

Luckily, these assumptions happen to be identical to the assumptions made when using a geometrical inductance formula. This implies that the very same correction factors may be applied to the results of the sheath helix waveguide model, being:

• A field non-uniformity correction factor according to Lundin[2,4] for modelling the end-effects of short & thick coils (high D/ℓ-ratio),
• A round wire self-inductance correction factor according to Rosa[2,5,6],
• A round wire mutual-inductance correction factor according to Grover and Knight[2,7],
• A reduced effective coil diameter for modelling the current quenching in the wire under the proximity effect of nearby windings,
• A series AC resistance for modelling the skin effect including an additional end-correction for the two end-turns that are subject to only half the proximity effect.

## Is there a small discontinuity in calculated inductances around ℓ = D?

Yes, there is. Thomas Bruhns, K7ITM discovered this little flaw. It is entirely due to the intrinsically discontinuous formulation of Lundin's Handbook Formula, used to obtain the field non-uniformity correction factor kL. Nonetheless, this is a state-of-the-art formulation, since alternative formulations produce even larger errors[2]. The discontinuity in calculated inductances appears only around ℓ = D. It is only pronounced when the sum (ks + km) is relatively small in comparison to kL; i.e. for really thick, closed-spaced turns. However, even then, the error in calculated inductances amounts to only approximately 0.5%.

Dr. David W. Knight,
G3YNH

## Does this calculator rely on any empirical data?

David Knight's empirical formula[2] is employed to determine the round wire mutual-inductance correction factor km, whereas the proximity factor Φ, used for calculating the AC resistance of the coil, is interpolated from Medhurst's table of experimental data[2,3].

## How is the effecitve diameter Deff related to the proximity factor Φ?

Other inductor calculators typically employ the mean and inner physical diameters (respectively: D and D-d) to bracket the inductance of the coil between two widely-spaced theoretical limits[2]. However, it has often been alluded that the actual effective diameter Deff of a coil aught to be linked to the proximity factor Φ. In order to do away with the ambiguity of an inductance range result, a new formula [PDF] (Stroobandt's formula?) is deduced in which the effective diameter Deff is a mere function of the proximity factor Φ. As a result of this effort, this inductance calculator will return only a single value of inductance.

the effective diameter and its relation to the proximity factor Φ

Although being more determined than effective diameter bracketing, inductance calculations based on this formula remain approximative for two reasons:

1. As is the case with diameter bracketing, this formula equally assumes that current under the proximity effect will quench while retaining a circular transversal distribution. This is most probably not the case.
2. As with diameter bracketing, the formula does not take into account frequency-dependent disturbances of the transversal current distribution, such as the skin effect.

These transversal current distribution disturbances which are not accounted for, may move the current centre further inward thereby reducing the effective diameter and the resulting inductance even more. The inductance obtained by this calculator will therefore be slightly overstated. Nonetheless, the result will be in most cases more accurate and certainly less ambiguous than the theoretical extremes (or average of these) given by other inductance calculators.

## The inductance, and hence the Q-factor, near resonance are enormous; Can this be right?

As explained before, the correct way to see an inductor is as a helical waveguide, short-circuited at one end (because it is assumed to be fed with a voltage-source). At any frequency, the axial propagation factor β and the characteristic impedance Zc of an equivalent transmission line can be determined.

The input impedance seen at the other end will be a tangential function of the coil's electrical length. Therefore, when the electrical length approaches a quarter wave length or Π/2 rad, the resulting input impedance, and hence inductance, will be extremely high.

A Tesla coil makes intelligent use of this phenomenon to produce extremely high voltages in the range of several hundreds of kV.[1] Inductance calculators that do not show this real-world behaviour are based on geometrical formulas.

voltage-breakdown of the air at one end of a Tesla coil

## The calculated inductance is negative; Can this be right?

You are operating the coil above its first self-resonant frequency. A coil with an electrical length in-between 90° and 180° (and odd multiples of this) will behave like a capacitor instead of an inductor. This is once more an indication that the first self-resonance of a coil is in fact a parallel resonance. Because of its capacitive behaviour, the coil's inductance will be stated as negative in this range.

## Stray capacitance is much higher than that of other calculators; Can this be right?

The concept of stray capacitance is what it is; a correction element in a coil's lumped circuit equivalent in order to deal with the frequency-independency of the inferior geometrical inductance formula.[1]

By employing helical waveguide-based formulas, the present calculator performs much better in estimating inductances at high frequencies. Therefore, it knows it needs to compensate the lumped circuit equivalent even more than others calculators do and does so by stating higher values for CL,s.

At the end of the day, one can really better do away with the old-fashioned concept of lumped circuit equivalent and instead focus on the effective values of inductance, reactance, series resistance and unloaded Q at the design frequency, values which this calculator evidently delivers.

## What is the problem with designing coils using EZNEC?

The pragmatists among us would probably like to employ EZNEC, an antenna modelling program, which allows you to define in a very user-friendly way helical coil structures. However, this valuable method is not without its pitfalls:

EZNEC coil model

• Coils in EZNEC are in fact defined as polyline structures. The projected polygon of the polyline should have the same area as the projected circle of the real helical coil. I will leave it up to the reader to do this trigonometric calculation.
• The more segments in the polyline, the more accurate the model would respresent the real helical coil. However, EZNEC has a lower limit for the segment length which is proportional to the wavelength.
• With respect to the two previous points, it is quite cumbersome to optimise a coil design using EZNEC.
In practice, it is often much easier, quicker and more accurate to resort to the above calculator instead of creating a geometrical EZNEC coil model. Once the results are obtained from the calculator, the coil can be easily modeled as a Laplace load within any EZNEC antenna model.

# References

1. Corum K. L. and Corum J. F., "RF coils, helical resonators and voltage magnification by coherent spatial modes," Microwave Review, IEEE, Vol. 7, No. 2, Sep. 2001, pp. 36-45
2. Knight David W., G3YNH, "Inductors and transformers," From Transmitter to Antenna, http://www.g3ynh.info/zdocs/magnetics/part_1.html
3. Medhurst R. G., "H.F. resistance and self-capacitance of single-layer solenoids," Wireless Engineer, Feb. 1947, pp. 35-43 & Mar. 1947 pp. 80-92
4. Lundin R., "A handbook formula for the inductance of a single-layer circular coil," Proc. IEEE, Vol. 73, No. 9, Sep. 1985, pp.1428-1429.
5. Rosa E. B., Bulletin of the Bureau of Standards, Vol. 2, 1906, pp. 161-187
6. Rosa E. B. and Grover F. W., Formulas and Tables for the Calculation of Mutual and Self Induction," [Revised], Bulletin of the Bureau of Standards, Vol. 8, No. 1, 1911, p. 122
7. Grover F. W., Inductance Calculations: Working Formulas and Tables, 1946 & 1973, Dover Phoenix Edition, 2004, p. 150
8. Collin Robert E., Foundations for Microwave Engineering, 2nd Edition, pp. 580-583
9. Meyer Hank, W6GGV, "Accurate single-layer-solenoid inductance calculations," QST, April 1992, pp. 76-77
10. Meyer Hank, W6GGV, "Corrections to accurate single-layer solenoid inductance calculations," QST, July 1992, p. 73
11. Dekker T. J., "FZERO.F", SLATEC Common Mathematical Library, Version 4.1, July 1993, http://www.netlib.org/slatec/src/fzero.f