## Litzendraht Wire

Litzendraht Wire (Litz wire) is a bundle of multiple insulated strands which, in theory, has lower ac resistance than a single strand of the same cross sectional area. This is due to the reduction of the skin effect (i.e. the current of an ac signal doesn't penetrate all the way into a conductor), but is limited by the mutual coupling between the strands. Much of the text below is taken from the NBS Circular 74, and there are some more links at the bottom.

Resistance ratio is the ratio between the AC resistance and the DC resistance. It is always >=1. My comments are in italics.

From p 306, NBS Circular 74

### Stranded Wire

The use of conductors consisting of a number of fine wires to reduce the skin effect is common. The resistance ratio for a stranded conductor is, however, always considerably larger than the value calculated by Table 19, p 311, and Table 17, p309 for a single one of the strands. Only when the strands are at impracticably large distances from one another is this condition even approximately realized.

Formulas have been propesd for calculating the reistance ratio of starnded conductors, but although they enable qualitatively correct conclusions to be drawn as to the effect of changing the frequency and some of the other variables, they do not give numerical values which agree at all closely with experiment. The cause for this lies, probably, to a large extent in the importance of small changes in the arrangement of the strands. The following general statements will serve as a rough guide as to what may be expected for the order of magnitude of the resitance ratio as an aid in design, but when a precise knowledge of the resistance ratio is required in any given case it should be measured. (See methods given in sections 46 to 50.)

#### Bare Strands in Contact

The resistance ratio of n strands of bare wire placed parallel and making contact with one another is found by experiment to be the same as for a round solid wire which has the same area of cross section as the sum of the cross sectional areas of the strands; that is, n times the cross section of a single strand. [note this doesn't say that the resistance is n times, it is the resistance ratio we are tallking about here. The DC resistance will be R/n, where R is the resistance of a single strand. What this says is that the AC resistance will be, generally, R, so that Rac/Rdc (n parallel bare strands) = n * Rac/Rdc (single strand). ] This will be essentially the case in conductors that are in contact and are poorly insulated, except that at high frequencies the additional loss of energy due to heating of the imperfect contacts by the passage of the current from one strand to another may raise the [ac] resistance still higher.

Note also that if the lengths of the strands, or their spacings, start to be a significant fraction of a wavelength, simple analysis is impossible.

Insulated Strands

As the distance between the strands is increased, the resistance ratio falls, rapidly at first, and then more slowly toward the limit which holds for a single isolated strand. A very moderate thickness of insulation between the strands will quite materially reduce the resistance ratio, provided conduction in the dielectric is negligible.

Spiraling or twisting the strands has the effect of increasing the resistance ratio slightly, the distance between the strands being unchanged.

Transposition of the strands so that each takes up successively all possible positions in the cross section - as for example, by thorough braiding - reduces the resitance ratio but not so low as the value for a single strand.

Twisting together conductors, each of which is made up of a number of strands twisted together, the resulting composite conductor being twisted together with other similar composite conductors, etc.[much like large steel cables -jl], is a common method for transposing the strands in the cross section. Such conductors do not have a resistance ratio very much different from a simple bundle of well-insulated strands.

The most efficient method of transposition is to combine the strands in a holow tube of basket weave. Such a conductor is naturally more costly than other forms of stranded conductor.

#### Effect of Number of Strands

With respect to the choice of the number of strands, experiment shows that the absolute rise of the resistance in ohms depends on the diameter of a single strand, but is independent of the number of strands. Since, however, the direct-current resistance of the conductor is smaller the greater the number of strands, the resistance ratio is greater the greater the number of strands. Reducing the diameter of the strands reduces the resistance ratio, the number of strands remaining unchanged, but to obtain a given current-carrying capacity, or a small enough total resistance, the total cross section must not be lowered below a certain limit, so that, in general, reducing the diameter of the strands means an increase in the number of strands.

[- here i have paraphrased the original text in the circular - ]

p308, Circular 74

To calculate the order of magnitude of the resistance ratio considering a composite of enameled strands of 0.07 mm diameter (ca 41-42 ga) twisted together.

Calculate the resistance ratio for a single strand at the desired frequency (it will be close to 1, with such fine wire).

Calculate the resistance ratio for a single strand with the same area (that is, the diameter is D * SQRT(N), where D is the diameter of a single strand, and N is the number of wires)

[end paraphrase]

The resistance ratio for the composite stranded wire (for moderate frequencies) will be about 1/4 to 1/3 of the way between the two values. This holds for straight wires up to higher frequencies than for solenoids. (See critical frequncy mentioned in second paragraph below) Not all so-called Litzendraht is as good as this by any means.For a woven tube, the ratio may be as close as 1/10 of the way from the lower to the higher.

#### Coils of stranded wire

For coils wound with stranded conductors, the resistance ratio is always larger than for the straight conductor, and at high frequencies may be two or three times as great. It is appreciably greater for a very short coil than for a long solenoid.

For moderate frequencies the resistance ratio is less than for a similar coil of solid wire of the same cross section as just stated, but for every strnaded-conductor coil there is a critical frequency above which the stranded conductor has the larger resistance ratio. This critical frequecny lies higher, the finer the strands and the smaller their number. For 100 strands of say 0.07 mm diameter, this limit lies above the more usual radio frequencies [in 1937 - jl].

This supposes that losses in the dielectric are not important, which is the case for single layer coils with strands well insulated. In multiple layer coils of stranded wire, dielectric losses are not negligble at high frequencies.

### For further information:

A paper by Charles Sullivan at Dartmouth, with outstanding presentation of theory, equations, design charts, etc: "Optimal Choice for Number of Strands in a Litz-Wire Transformer Winding"
http://www.dartmouth.edu/~sullivan/litzwire/litz.html

#### Retail (ish) sources

http://www.surplussales.com/Wire-Cable/LitzWire.html - surplus sales

hv/litz.htm - 3 Feb 2000 Jim  Lux