## Overall View

## Total resistance of litz wire

The total resistance of a given litz wire construction is determined by the specific resistance of the conductor material, the nominal diameter and number of single wires, the number of bunching steps, the chosen length of lay, and additional process-specific influences.

The resistance value of the single wire can be obtained from technical data provided by Elektrisola. Using the procedure outlined in DIN EN 60317-11 the total resistance of a litz wire can be calculated:

**Nominal value of resistance = (Nominal value of resistance of single wire)/(Number of single wires) * k _{1.}**

The factor k_{1} is 1,02 and represents the shortening of length due to the bunching process used to produce the litz wire construction.

**Minimum value of resistance = (Minimum value of resistance of single wire)/(Number of single wires).**

**Maximum value****of resistance:**

*a) For number of single wires up to and including 25*

**(Maximum value of resistance of single wire)/(Number of single wires) * k _{1}.**

The factor **k**_{1} is 1,02 and represents the shortening of length due to bunching process.

*b) For number of single wires above 25*

(**Maximum value of resistance of single wire)/(Number of single wires) * k _{1} * k_{2}.**

The factor **k**_{1} represents the shortening of length due to bunching process.

**k**_{1} for 1 x bunching is 1,02

2x bunching is 1,04

3x bunching and more is 1,06

**k**_{2} is 1,03 and represents a factor for broken wires, which may occur.

## Outer diameter (OD) of litz wire

The nominal outer diameter depends upon the method of twisting (directly, freely bunched / concentrically bunched), number of bunching steps, pitch direction, length of lay and selected nominal diameter of single wires. It is also influenced by process-specific factors.

Due to the litz wire natural flexibility, bending radius and winding tension-dependent dimensional stability, the nominal outer diameter is approximated by an average value in combination with a defined measuring method.

The nominal outer diameter of a bunched litz wire can be calculated acc. DIN EN 60317-11 with the following formula:

** OD = p * √n * d** + increase in diameter by yarn serving

^{ 1)}

With:

** OD** nominal outer diameter of the bunched litz wire;

** p** packing factor;

** n** number of single wires;

** d** nominal value for the outer diameter of the single wire.

_{(Comment: Approximately 0,040 mm can be taken as increase of diameter per each layer of silk serving. This calculation can also be taken for nylon serving.) }

_{}

_{ 1) The nominal value for the outer diameter of the single wire is the nominal diameter of the single wire plus 2/3 of the maximum value for the increase in grade 1 acc. IEC 60317-0-1. The nominal value for the outer diameter of yarn served litz wires is the nominal value for outer diameter of the litz wire final construction (without serving) plus increase of the diameter due to the yarn serving.}

The nominal dimensions for the outer diameters of the individual wires can be obtained from the technical data provided by Elektrisola.

Packing factor | |
---|---|

no. of wires | packing factor |

3 through 12 | 1,25 |

16 | 1,26 |

20 | 1,27 |

25 through 400 | 1,28 |

__ __

## Total copper cross section of litz wire

This cross sections results from the sum of the single wire cross sections.

**A _{Cu} = n * π/4 * d^{2}**

^{ }with

**n** number of single wires

**d** nominal Diameter of single bare wire

## Total cross section of litz wire

The total cross section results from the calculated outer diameter OD of the litz wire

**A _{ges} = π/4 * **D

_{a}

^{2}with D

_{a}= OD = Total outer diameter of litz

## Litz filling factor

The litz wire filling factor describes the relationship between the electrical conducting cross section of a litz wire construction to its total cross sectional area expressed in %.

This factor depends on the choice of nominal single wire diameter, the number of bunching steps, the length of lay, the pitch direction, the thickness of insulation material, as well as the influence of other process parameters.

The litz wire filling factor is approximately described by the Ratio

**A _{Cu }/A_{ges} * _{p}**

whereas p is a product- and process specific factor.

Basically the filling factor decreases at constant total copper cross section with single wires getting finer. Since the area associated with intermediate air spaces and enamel increases disproportionately, also the litz wire outer diameter and total cross section rises. The same is true for a constant given outer diameter, since here conversely the copper cross section has to be reduced successively.

The charts below show this relationship by means of a litz wire with constant copper cross section and several single wires of different diameter.

By compacting round shaped litz wires to square profiles the filling factor can be further increased, see succeeding chart, green line.

In this case the closer proximity of neighboring windings make it possible to increase the coil-filling factor once again. Use of single wire diameters thicker than 0,100 mm (38 AWG) are preferred, since litz wires constructed of finer individual wires become more sensitive in their ability to absorb mechanical stress.

## Coil filling factor

This factor depends upon the litz wire filling factor and the packing factor of the coil windings.

**Filling factor bobbin [%] = (N x A _{Cu,Li})/A_{Sp }* 100_{ } **with

**N** number of windings

**A _{Cu,Li}**

_{ }litz wire copper cross section

**A _{Sp}** winding window cross section of coil

Profiled litz wires as well as litz wire constructions using Smartbond have more efficient performance due to their high filling factor.

## Right-hand-rule

A current **I** flowing through a straight conductor creates a magnetic field **B**,

whose field lines are placed concentrically around the conductor. If a straight conductor is gripped with the right hand and the thumb points in direction of the flowing current **I**, then the fingers point in the direction of the circulating magnet field **B**. The item **B** is also called magnetic flux density, which is proportional to the magnetic field intensity **H** and the material dependent magnetic permeability **µ**:

B = µ_{0} * µ_{r} * H = µ * H | with |

µ_{0 }= 4 π * 10^{-7} N/A^{2} | magnetic field constant, permeablility of free space |

µ_{r} | relative permeability |

## Skin-effect and skin depth

The current causes concentric magnetic fields, both internal and external to the conductor. This is shown in the scheme below by the magnetic field strength **H**. The portion of the magnetic field within the conductor itself create concentric and interfering eddy currents, which influences the current flow towards the outer surface area of the cross section with rising frequency **f**. Due to this effect the so called skin depth **δ** of the current decreases, where **δ** is the distance from the conductors surface, at which the current density has dropped to 1/e (e = Euler’s constant) of the amplitude value (see below). Thus the measurable ohmic resistance becomes frequency dependent and rises in value with increasing frequency. Consequently heating of the conductor and additional electrical losses result as frequency increases. More HF-losses are also caused by the so-called external and internal proximity effect.

Skin depth **δ = 1/√(π * µ0 * б * f)** with

Frequency | δ (copper) |

10 kHz | 0,66 mm |

50 kHz | 0,30 mm |

100 kHz | 0,21 mm |

500 kHz | 0,094 m = 94 μm |

1 MHz | 0,066 mm = 66 μm |

10 MHz | 0,021 mm = 21 μm |

100 MHz | 0,0066 mm = 6,6 μm |

µ_{0} magnetic field constant, permeability of free space

б conductivity of conductor material

f frequency

This simplified formula describes the skin-effect only in those cases where **δ** is lss than or equal to a third of the minimal conductor diameter and smaller than a quarter for square constructions.

## Outer Proximity-effect

The effect of current displacement can also be caused by the influence of outer alternating magnetic fields of neighboring conductors or other electrical components.

In contrast to eddy currents, which are induced by the skin-effect, the outer proximity effect is not rotationally symmetric to the center of the conductor. The reason is the alternating magnetic field is created by an external current. Thus it has nearly the same direction at any place on the affected conductor. Also here eddy currents cause ohmic losses, which lead to an apparent increase of the ohmic resistance. The necessary energy for moving these eddy currents is delivered by the causing magnetic field of the external current. Due to this general interference between eddy currents and its causing magnetic field, additional high frequency losses can also occur in any other neighboring conductive material.

## Internal Proximity-effect

The alternating magnetic fields of the single wires (strands) of a litz wire also create losses in neighboring strands by eddy currents. Since these fields are created internally by the strands itself, this effect is called internal proximity-effect but formally seen as belonging to the skin-effect, see scheme of current displacement below. As a consequence the electrical losses of a litz wire increase with rising frequencies and can in certain cases even exceed the losses of a solid conductor with the same DC-resistance.

As an example, the following picture shows the non-homogeneous distribution of current between neighboring single wires (current density increasing from blue to red color).

This effect demonstrates there is an optimal range of frequency for litz wires, in which the losses are lower than for a solid conductor. Beyond this range the use of multiple single wires such as a litz wire can have negative effects.

Thus both the skin- and proximity-effects are the most important aspects for considering HF-losses in electrical conductors, whereby combined influence of inner and outer proximity-effects are dominating. For a specified working frequency in most cases only a litz wire construction can help reduce these losses. In this case the construction parameters such as number of single wires, single wire diameter, number of bunching steps, length of lay (pitch) and lay direction have to be specified for each application. At the same time care has to be taken that each single wire occupies each place of the litz wire cross section consistently within a defined length. In combination with enameled single wires litz wires are called high frequency (HF) litz wires in this context.

## Relationship of single wire diameter to range of frequency

The design and construction of a high frequency litz wire and its resulting electrical performance depends upon many factors. Different design approaches can generate similar performance values but experience is required to correctly specify litz construction that can be manufactured economically and consistently. The correct choice of the single wire diameter is therefore an important consideration for each specific application.

The table below shows the relationship between recommended single wire diameter and frequency range.

range of frequency [kHz] | nominal Diameter of single wire [mm] | ||
---|---|---|---|

from | to | from | to |

0.06 | 1 | 0.400 | 0.254 |

1 | 10 | 0.254 | 0.200 |

10 | 20 | 0.200 | 0.127 |

20 | 50 | 0.127 | 0.102 |

50 | 100 | 0.102 | 0.079 |

100 | 200 | 0.079 | 0.063 |

200 | 350 | 0.063 | 0.050 |

350 | 850 | 0.050 | 0.040 |

850 | 1400 | 0.040 | 0.030 |

1400 | 3000 | 0.030 | 0.020 |

## Calculation of high frequency litz wire losses

High frequency losses depend upon the cumulative influences of the different loss mechanisms, as well as the expected working conditions of each individual litz application. Therefore, a simple differentiated formula-like calculation is not possible without a deeper understanding and additional tools.

## Rac/Rdc-Ratio

With increasing frequency the current flows more and more along the outer surface of the conductor with increasing frequency. The measured alternating current resistance R_{AC} rises compared against the direct current resistance R_{DC}. With increasing resistance values the ohmic losses rise and can even exceed the DC-losses at high frequencies.

The R_{AC}/R_{DC}-factor now describes the alternating current resistance normalized to the direct current resistance (R_{AC}/R_{DC} ≥ 1) and is an indicator for the high-frequency performance of a litz wire. The R_{AC}/R_{DC}-factor can be measured or calculated with sufficient accuracy in most cases for a given litz wire construction and is desired to be typically between 1-12 for the respective range of frequency. Along with the correct choice of the single wire dimension, the design of the litz wire construction plays an equally important role.

The following graph shows the calculated frequency dependent R_{AC}/R_{DC}-trend of five different litz wire constructions with the same copper cross sectional area. It shows the AC-resistance and the HF-losses increase with the frequency, the thicker the single wire diameter selected. At a target frequency of 1 MHz the construction with 50 µm single wires has the best results. Nevertheless the related R_{AC}/R_{DC}-factor of 1,29 is still significantly higher from the optimum value of 1,0.

In this case for example, a first step of improvement could be the selection of a smaller single wire diameter and/or optimization of the bunching construction.

## Coil Q-factor

The quality (Q-)factor measures the freedom of losses of a swinging electrical or mechanical system. (1) As an example, a higher Q factor indicates a lower rate of energy loss relative to the stored energy of the resonator, the oscillations die out more slowly. A pendulum suspended from a high quality bearing, oscillating in air, has a high Q. While a pendulum immersed in oil has a low Q.

(1) Wikipedia definition

In an electrical oscillating circuit consisting of an air coil (inductance **L**), capacitor (capacity **C**) and ohmic resistance (**R**), the Q-factor measures the relationship between the total energy of an oscillation and its loss of energy per oscillation.

An important characteristic of a high quality system is the use of a coil with high Q-factor (coil Q-factor Q_{Coil}).

The basic loss factor of the coil is its resistance R_{L,Coil}. This resistance increases with growing frequency, influenced by the frequency dependent skin- and proximity-effect.

One has:

Q _{Coil} = ~ f * L / R_{L,Coil} (f) | with | f = frequency [Hz] | ||

L = coil inductance [nH] | ||||

R_{L,Coil} = resistance of coil [Ohm] |

and also approximately for the example of a single layer planar coil:

L = L _{planar} = (21,5 * N^{2} * 2a) / (1 + 2,72 * w/2a) | with | w = width of winding area [cm] | ||

a = average radius [cm] | ||||

N = number of windings |

Different influencing factors interfere with each other and lead to a frequency dependent trend of coil Q-factor.

In particular these are:

**frequency f:**

The coil Q-factor increases with growing frequency and decreases again at a certain point due to disproportional rising HF-losses; positive influence by litz wire construction (number of single wires, nominal diameter, length of lay) is possible.

**inductance L:**

The coil Q-factor increases with growing inductance, (i.e. with increased number of windings **N**); the negative influence of the resulting increased coil resistance loss R_{L,Coil} compensates for this effect only at higher frequencies. The self-capacitance of the coil rises with increasing number of windings.

**Resistance R _{L,Coil}:**

The coil resistance loss is influenced by the total conductor cross section A_{Cu}. The reduction of R_{L,Coil} leads initially to an increased Q-factor, but at higher frequencies comes a stronger decrease in the Q factor due to higher HF-losses; positive Q factor influence by the litz wire construction my be possible(number of single wires, nominal diameter, length of lay).

The graph below shows the influence of the litz wire- and coil construction on the trend of Q-factor by means of three measured planar coils with 12 windings and differently constructed Smartbond-litz wires.

By reducing the length of lay (SL=10mm red line in graph) the coil Q-factor can be increased over the complete frequency range (in comparison to the blue solid line with lay length SL=26mm). If the increase of coil Q-factor is only necessary for a selective range of frequency (in this example until 150 kHz), it can be sufficient for longer lay length to increase the coil inductance L by choosing a higher number of windings (in this example from 12 to 17). Here the Q-factor inreases for the indicated range of frequency, but drops faster for higher frequencies (compare blue dotted line with red solid line).