Inductance and Magnetic Field of Multi-Layer Air Coils

1Model of the coil

A real winding is a helix. For inductance purposes it is replaced by N closed, coaxial circular rings. Ring i has the radius R_i (to the wire centre) and sits at the axial position z_i. All rings carry the same current I because they are connected in series.

The error of this approximation comes from the pitch angle of the helix, which is \tan\alpha = p/(2\pi R). For typical coils with p \ll R this is well below one percent of the inductance.

2Inductance from magnetic energy

For a system of N current loops with currents I_k the stored magnetic energy is

The diagonal entries L_{ii} are the self inductances of the individual rings, the off-diagonal entries L_{ij}=M_{ij} are the mutual inductances. Reciprocity gives a symmetric matrix, L_{ij}=L_{ji}.

In a series coil every ring carries the same current, I_i=I. The energy of the coil as a single inductor is W=\tfrac12 L I^2. Comparing both expressions:

So the coil inductance is the sum of all matrix entries. What remains is to compute L_{ii} and M_{ij}.

3Neumann formula

In the Coulomb gauge (\nabla\cdot\vec A = 0) the vector potential obeys the Poisson equation

Its free-space solution, using the Green's function 1/(4\pi|\vec r-\vec r^{\prime}|), is for a thin loop C_1 carrying I_1:

The flux through a second loop C_2 follows from \vec B=\nabla\times\vec A and Stokes' theorem:

With M_{21}=\Phi_{21}/I_1 this gives the Neumann formula, which is symmetric in both loops:

4Two coaxial rings

Parametrise both rings by their azimuth angles \varphi_1,\varphi_2:

The line elements and the distance depend only on the difference \varphi=\varphi_2-\varphi_1:

One integration is trivial and yields 2\pi. Using the symmetry of the integrand about \varphi=\pi:

Substitute \varphi=\pi-2\theta, so d\varphi=-2\,d\theta and \cos\varphi = 2\sin^2\theta-1. The root becomes

with the modulus

Since \sqrt D = 2\sqrt{R_1R_2}/k, the prefactor is 2\mu_0R_1R_2/\sqrt D = \mu_0 k\sqrt{R_1R_2}, and

Write the numerator as 2\sin^2\theta - 1 = \frac{2}{k^2}\bigl[1-(1-k^2\sin^2\theta)\bigr]-1. With the complete elliptic integrals of the first and second kind

the integral evaluates to \left(\frac{2}{k^2}-1\right)K-\frac{2}{k^2}E, which gives Maxwell's formula:

This is exact for two filamentary rings. For round wires of finite radius it is still very accurate as long as the wires do not overlap, because the field of a round conductor outside it equals that of a filament at its centre.

5Self inductance of a ring

The Neumann integral diverges for a filament with itself (r_{12}\to 0), so the finite wire radius a=d/2 must enter. Split the self inductance into the part from the field inside the wire and the part from the field outside:

Internal inductance

For a straight round wire with uniform current density, Ampère's law gives inside the conductor (\rho\le a)

The energy per unit length stored inside the wire is

It is independent of the wire radius. For a bend radius much larger than a the ring behaves locally like a straight wire, so along the circumference 2\pi R:

External inductance

The external part is the flux outside the conductor per unit current. Replace the current by a filament on the wire axis (radius R). The flux that this filament produces through the hole of the ring, up to the inner surface of the wire, is the flux through a second filament of radius R-a in the same plane. That is exactly a mutual inductance with b=0:

Together:

Thin-wire limit

For a\ll R the modulus approaches 1. With k'^2 = 1-k^2 \approx a^2/(4R^2) and the asymptotics K\approx\ln(4/k'), E\approx1:

This is the classic Kirchhoff/Rosa result for a round-wire loop. The tool uses the exact elliptic form, so no approximation enters here.

6Winding geometry

The coil is built from layers stacked along the axis. Each layer is a plane perpendicular to the axis and holds the same number of turns, the turns per layer m, side by side from the inner radius outward. Two independent distances describe the grid of wire centres:

• Layer distance p_z: the axial centre-to-centre distance between neighbouring layers. It is entered directly; the coil length follows from it.
• Radial turn distance p_r: the centre-to-centre distance between neighbouring turns within one layer. It is an input of its own. With the option Couple p_r = p_z it is set equal to the layer pitch, so the wire centres form a square grid.

Layer 1 lies at z=0, every further layer is placed above it. Turn j of layer k (both counted from 0) lies at

R_0 is the inner ring radius from the axis to the centre of the innermost wire rings. The winding extends from z=-d/2 to z=(n_L-1)\,p_z+d/2 and radially from R_0-d/2 to R_0+(m-1)\,p_r+d/2. The surface of the coil former lies at R_0-d/2, which must be positive. The wires fit without overlap only if p_z\ge d and p_r\ge d; the calculator warns otherwise. Insulation thickness belongs into d if it should be accounted for in the spacing.

Only distances between rings enter the inductance, so the choice z_0=0 for the first layer is a convention for the drawing and does not change any result.

7The inductance matrix

With N rings the inductance matrix is the N\times N matrix of all self and mutual inductances,

It links the vector of ring currents \vec I to the vector of flux linkages \vec\Psi: entry L_{ij} is the flux through ring i caused by a unit current in ring j. The magnetic energy is the quadratic form W=\tfrac12\,\vec I^{\mathsf T}\mathbf L\,\vec I.

Properties

• Symmetric: L_{ij}=L_{ji}, because the Neumann integral is symmetric in both loops. Only N(N+1)/2 entries have to be calculated.
• Positive definite: the energy \tfrac12\vec I^{\mathsf T}\mathbf L\vec I is positive for every non-zero current distribution. Consequently every off-diagonal entry is smaller than the geometric mean of the two self inductances, M_{ij}<\sqrt{L_{ii}L_{jj}}; the ratio is the coupling factor k_{ij}.
• All entries positive: coaxial rings with currents in the same direction always link positive flux. The coupling falls off with distance, roughly as 1/b^3 once the axial distance b is large compared to the radii.
• Diagonal: the self inductances L_{ii} grow with the ring radius, approximately as R\ln(8R/a). They are highlighted in the table.

Block structure

With the numbering By layer the rings of layer 1 come first, then those of layer 2 and so on. The matrix splits into n_L\times n_L blocks of size m\times m:

A diagonal block \mathbf L_{kk} describes one layer alone: self inductances on its diagonal, mutual inductances between rings in the same plane (b=0) elsewhere. An off-diagonal block \mathbf L_{kl} holds the couplings between layer k and layer l at the axial distance |k-l|\,p_z. Because all layers are identical and equally spaced, a block depends only on |k-l|:

The matrix is therefore block-Toeplitz, and the blocks themselves are symmetric. This is the pattern visible in Fig. 5 of the Thomson ring page and in the table of the calculator.

With the numbering By radius rows and columns are reordered so that all rings of the same radius follow each other. This is a simultaneous permutation \mathbf L' = \mathbf P\,\mathbf L\,\mathbf P^{\mathsf T} of rows and columns: every entry keeps its value, and the sum of all entries is unchanged.

From the matrix to the coil inductance

In the series connection all rings carry the same current, \vec I = I\,(1,1,\ldots,1)^{\mathsf T}. The flux linkage of ring j is then the column sum times I, and the total flux linkage of the coil is the sum of all column sums:

The bottom row of the table shows these column sums. Rings in the middle of the winding have the largest sums because they are surrounded by the most neighbours. The results panel splits the total into the diagonal part (self) and the off-diagonal part (mutual); for compact multi-layer coils the mutual part usually dominates.

8Numerical evaluation

K and E are computed with the arithmetic-geometric mean (AGM). Starting from a_0=1,\;b_0=\sqrt{1-k^2},\;c_0=k:

The iteration converges quadratically: five to six steps reach machine precision even for k close to 1, which occurs for the self inductance and for neighbouring turns.

The matrix has N^2 entries, but because it is symmetric only N(N+1)/2 of them are evaluated. A coil with 1000 turns needs about half a million elliptic-integral evaluations, which a browser handles in well under a second.

9Validation and limits

Reference example

The example is taken from Fig. 5 in the article Thomson Ring Experiment (chapter 3.2, inductance matrix of a multi-layer coil). Three planar layers of four rings each (R = 5, 15, 25, 35 mm, layer spacing 10 mm, wire radius 1 mm) give a total of 2770 nH with the internal term and 2695 nH without it. The difference is exactly 3\cdot\frac{\mu_0}{4}(5+15+25+35)\,\text{mm} = 75.4 nH. Single entries: L_{11}=10.61 nH (9.03 nH without internal term), M(5,15,0)=3.44 nH, M(5,5,10)=0.71 nH. To reproduce it, enter the layer distance p_z=10 mm, 3 layers, 4 turns per layer, d=2 mm, p_r=10 mm (or coupled), R_0=5 mm with By layer; the matrix then has the same block layout as Fig. 5. These are the start values of the program.

Wheeler cross-check

The results panel shows Wheeler's empirical multilayer formula in SI form,

with mean radius r_m and winding depth c. It is accurate to about 1 % for compact coils whose three terms in the denominator are of similar size, and less accurate for very short or very long coils. Deviations of a few percent are therefore expected and are not an error of the ring model.

Limits of the model

• The internal inductance assumes a uniform (DC) current density. At high frequency or in fast pulse discharges the skin effect pushes current to the surface and L_\text{in} tends towards zero; untick the option for that case.
• Proximity effects between neighbouring turns redistribute current inside the wires and are not modelled.
• The helix pitch is neglected (see section 1).
• Air core only, no magnetic materials, no conductive parts nearby.

10Magnetic field of the coil

The Magnetic Field tab computes the field strength vector \vec H of the same ring model that is used for the inductance. Each ring carries the coil current I, positive in the right-hand sense about +z. Because the arrangement is rotationally symmetric, it is enough to evaluate the field in the half plane y=0; the field there has only a radial and an axial component.

Biot–Savart law for one ring

A stationary current I in a thin conductor C produces at the point \vec r the magnetic field strength

Each element d\vec{l}^{\,\prime} at the source point \vec{r}^{\,\prime} contributes a vector perpendicular to the line element and to the distance vector \vec R=\vec r-\vec{r}^{\,\prime}, decaying as 1/R^2. The flux density follows from \vec B=\mu_0\vec H (see below). The geometry of all vectors is shown in the sketch Field point geometry on the Magnetic Field tab.

For a ring of radius a at height z^{\prime} the source point is \vec{r}^{\,\prime}=(a\cos\varphi,\,a\sin\varphi,\,z^{\prime}) and the line element d\vec{l}^{\,\prime}=(-a\sin\varphi,\,a\cos\varphi,\,0)\,d\varphi. For the field point \vec r=(\rho,0,z) and \zeta=z-z^{\prime} the distance is R^2=a^2+\rho^2+\zeta^2-2a\rho\cos\varphi and the cross product becomes

The y-component integrates to zero because \sin\varphi is odd about \varphi=\pi. Two integrals remain:

Reduction to elliptic integrals

The same substitution as for the mutual inductance, \varphi=\pi-2\theta, gives R^2=Q\,(1-k^2\sin^2\theta) with

Because R^3 now appears in the denominator, two auxiliary integrals with \Delta=\sqrt{1-k^2\sin^2\theta} are needed. Differentiating \sin\theta\cos\theta/\Delta and splitting \sin^2\theta=(1-\Delta^2)/k^2 reduces them to K and E:

Inserting I_1 and I_2 and expressing k by Q and D gives the closed form of the ring field:

For a point at negative x the radial component changes its sign, H_x(-\rho,z)=-H_\rho(\rho,z), while H_z is even. The colour maps show both half planes.

From field strength H to flux density B

The Biot–Savart integral above is written for \vec H, the magnetic field strength in A/m. It depends only on the currents and the geometry. The quantity that acts on charges and conductors, enters Faraday's law and is measured by a Hall probe is the magnetic flux density \vec B in tesla (1 T = 1 V·s/m²). Both are linked by the constitutive relation of the medium:

\vec M is the magnetisation of the material. In air, and to an excellent approximation in copper (\mu_r-1\approx-10^{-5}) and aluminium (\mu_r-1\approx+2\cdot10^{-5}), the magnetisation is negligible and \mu_r=1. The calculator therefore uses

As a rule of thumb, 1 kA/m corresponds to 1.257 mT and 1 T to 795.8 kA/m. With a ferromagnetic core (iron, ferrite) \mu_r becomes large and field dependent, and the simple factor no longer holds; such cores are outside the scope of this model. The display can be switched between B (default) and H; colour maps, probe values and plots change only by the constant factor \mu_0.

Superposition over all rings

The Biot–Savart integral is linear in the current, and all rings share the same axis. The field of the coil is therefore the plain sum of the ring fields, with a\to R_i and z^{\prime}\to z_i from the winding geometry (section 6):

On the axis this reduces to the textbook result H_z(0,z)=\frac{I}{2}\sum_i R_i^2/\bigl(R_i^2+(z-z_i)^2\bigr)^{3/2}, which is what the axial plot shows. The flux density follows as \vec B=\mu_0\vec H in air.

Plausibility checks

• Axis: for \rho\to0, k\to0 and K=E=\pi/2; the formula gives H_z=Ia^2/\bigl(2(a^2+\zeta^2)^{3/2}\bigr) and H=I/(2a) at the ring centre.
• Near the axis: the bracket of H_\rho and the prefactor 1/\rho cancel. The program uses the series H_\rho\approx\frac{3Ia^2\zeta\rho}{4(a^2+\zeta^2)^{5/2}} for \rho<10^{-6}a, which also follows from \nabla\cdot\vec B=0.
• Near the wire: the field approaches that of a straight conductor, |H|\approx I/(2\pi d) at the distance d from the wire axis.
• Far field: for distances much larger than the coil the field becomes a dipole field with the moment m=I\sum_i\pi R_i^2.
• Long solenoid: inside a long, closely wound coil H\approx NI/l; the results panel shows this value for comparison.

Inside the conductor

For a filament the field diverges at the wire (D\to0, K\to\infty). A real wire of radius w=d/2 carries the current spread over its cross-section. For points with a distance d_i<w from the centre of ring i, that ring's contribution is taken as the field of a uniformly filled round conductor: the value at the wire surface in the same direction, scaled linearly with d_i/w. The field is then continuous at the surface and zero on the wire axis, as Ampère's law requires.

Numerical verification

For a single ring with a=1 m, z^{\prime}=0, I=1 A the program reproduces the values obtained by direct numerical integration of the Biot–Savart integral, for example B_x=0.597367\cdot10^{-7} T and B_z=6.250318\cdot10^{-7} T at (x,z)=(0.3,\,0.2) m, and B_x=38.494812\cdot10^{-7} T, B_z=11.809170\cdot10^{-7} T directly beside the wire at (0.99,\,0.05) m. K and E come from the same AGM routine as in section 8.