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Faced with difficulties in the calculation of the multilayer coil on a solenoidal former, we have abandoned the simple empirical Wheeler's formula and went by the way of complication of the inductor model and the calculation algorithm. In the case of a multilayer coil on a rectangular former, which is quite often used, it is obvious that we should go by the same way. This coil can be represented as a set of coaxial rectangular loops and we can calculate the total inductance of the coil as the sum of own and mutual inductances of these loops.

But we can go further and imagine the coil as a set of straight wires. Each turn is a set of four straight segments. Moreover, mutually perpendicular segments do not interact with each other and their mutual inductance is zero. Using the fact that the rectangle is a symmetric figure, we can perform calculations only for two of its sides. Doubling the result of the calculation, we obtain the total inductance of the coil. The following figure shows two parallel wires unequal length for the true understanding of formulas.For such calculations, we need to know the following...

1. The formula of self-inductance of straight wire:
 [1]
where:
• L - inductance [µH]
• l = l1 or l = l2 - length of wire [mm];
• r = d/2 - the radius of the conductor [mm];
2. More commonly known formula which is a simplification of the above, but we do not it use, because less accurate:
 [2]
3. The formula for mutual inductance of two wire segments of unequal length:
 [3]
where:
• M - the value of mutual inductance [µH];
• x1 = l1/2 - half the length of the first conductor [mm];
• x2 = l2/2 - the half-length of the second conductor [mm];
• D - the distance between the centers of the conductors [mm];

The formula is true for conductors, the centers of which lie on the same axis as is the case for our coil.

The coil is calculated by a numerical method "virtual winding". When you add a new turn, is calculated self-inductance of all its segments, as well as possible mutual induction with all the other segments, taking into account the mutual directions of current in them. In result, we obtain the self-inductance of the multilayer coil. At the same time, we calculate  DC resistance and length of wire required for winding. In addition to the plug-in "multi_rectangular" to the Coil32 program for Windows, which is available for download from the program menu and calculates such inductor, you can use the online calculator of multilayer coil on a rectangular former.

The real loop has rounded corners and does not represent a perfect rectangle. This circumstance, as well as the limited accuracy of the original formulas, do not allow to achieve high precision of the calculation. The error of calculation is about ±5% of the value of the inductance. We mean that the winding is dense, without gaps and interlayer shims. Despite the low accuracy, this numerical algorithm allows the calculation of arbitrary multi-layer coil on a rectangular former without resorting to searching complicated empirical formulae, are always constrained by the geometry of the winding. Using this algorithm, for example, we can easily calculate the inductance of a multi-turn rectangular loop of large diameter.

References:

Multi-layer coil on a PCB is needed for those designers who want to create a miniature device with the use of inductance sufficiently high value. Single-layer coil on a PCB with inductance more that 10 µh usually has a relatively large size. To solve this problem will help the fabrication of multilayer printed coils. Such coils are used as various sensors, elements of radio frequency filters, etc. However, one should not forget that they have a relatively low quality factor, that is a natural payment for miniaturization. But this is in comparison with wire coils. If we compare with the SMD coils, multilayer printed coils have the best indicators of the Q-factor and smaller the self-capacitance.

# How to calculate inductance?

We can use different methods to calculate the inductance using numerical methods or handbook formulas. They can be conditionally divided into three levels - high, medium and low level.

High level involves the use of programs having a common name - electromagnetic simulators. For example, Comsol Multisystems with RF module, Ansys HFSS, etc. Their work is based on the differential Maxwell's equations for the electromagnetic field providing the boundary conditions.
Advantages: accurate calculation of inductance and other parameters of the coil with any geometry of winding in any frequency range.
Disadvantages: these programs are quite complex and require prior learning, require large computational resources, the calculation takes a long time. Can be used for professional work or if the inductor is used at or above the self-resonance frequency.

The medium level is based on the simplified model of the inductor introduced by J.C.Maxwell.
Advantages: acceptable calculation accuracy for radio-enthusiasts practice, the opportunity of using in simple programs, low demands on the processing power of the computer.
Disadvantages: less accurate calculations than the high level; the calculation isn't possible for any winding geometry and maybe only in the frequency range that does not exceed 60-70% of the self-resonant frequency (or rather 1-resonance) of the coil.

The low level is based on simple handbook formulas. These formulas are based on the simplification of medium level formulas or on the basis of a set of measurements of the actual coils.
Advantages: simple calculation, undemanding to resources of the computer

Disadvantages: the formulas work only with a restricted geometry of the winding and at frequencies much lower than the self-resonance frequency.

More about medium level calculation that Col32 uses... The great physicist J.C. Maxwell has shown in the late XIX century in his famous work - "A Treatise on Electricity and Magnetism."  that the mutual inductance between two infinitely thin circular coaxial conductors can be calculated as follows:

Where

•     M - mutual inductance;
•     r1, r2 - radii of the two circular filaments;
•     x - the distance between the centers of the circles bounded by these filaments;
•     K, E - elliptic integrals of the first and second kind;

A numerical method for the calculation of Maxwell's formula reduced to numerical methods for solving elliptic integrals.

By using Maxwell's equation can be calculated the inductance of a single-layer, multi-layer or flat coil and the mutual inductance of two separate coils. Errors related to the coaxial circular filaments approximation (in fact we deal the round wire helix) can be reduced through additional corrections.

Ferrite cores for inductors have a wide variety species. W-shaped, U-shaped, pot cores of various modifications.  Such cores are also used with the powder iron materials. Calculation of the coils with any ferromagnetic core uses method calculation with a special parameter AL - inductance factor of the core.

# Agreement on the dimensions and notations

Different programs (or different versions of this software)  are using different methods of measurement and notation of coil sizes. In order to avoid confusion and mistakes on this page details the nuances of this issue.

First of all, it concerns a single-layer coil. Numerical methods of the inductance calculation are based on on the assumption that the coil is wound by infinitely thin wire. The real thickness of the wire is taken into account in the future through adding corrections, however, the initial assumption leads to the fact that all dimensions in the original calculation formulas are measured from center to center of wire:

• D - diameter of the winding
• l - the length of the winding
• p - the pitch of the winding
• d - wire diameter
• di - the diameter of the wire in the insulation

For all empirical formulas, this rule is retained. Often in articles on this site, it is possible to meet the concept of the form-factor of the coil. This is the ratio of the length to the diameter of the winding is l/D. Coil with a large l/D ratio is called the solenoid.

To simplify measurements for designers, in the program Coil32 the coil-former dimensions and nominal diameter of the wire (no insulation!)  are measured and then on the basis of these measurements, the program calculates the required dimensions "from center to center", which are used in numerical algorithms and in empirical formulas. This applies to both single-layer and multilayer coils, as well as other types of inductors. All dimensions and notation are clear from the images in the program. As an example, for single-layer coils see figure below:

• Dc - the diameter of the coil
• lc - the length of the coil
• p - the pitch of the winding
• d - nominal diameter of the wire
• di - the diameter of the wire in the insulation

Thus, if we are talking about the size of a coil-former upon which the coil is wound, we will use the notion of the coil diameter and the coil length, if we are talking about the coil, we will use the concepts of the winding diameter and winding length, measured from center to center of wire.

It should be noted that because the insulation thickness is the parameter is not zero, even when winding a coil to a coil, the winding pitch p is always greater than the nominal wire diameter d. This is taken into account in the program.

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