Inductance Calculation

In this section we return to the summation method of inductance calculation, and consider some corrections to account for the fact that the coil is a helix rather than a series of perfectly circular rings. This has several consequences, but we will just consider the following two, for the time being:

- One turn is longer than an equivalent ring;
- The distance between turns is less than the pitch!

We will address these items in the following subsections. However, it will make things easier to follow if we examine the geometry of a coil in greater detail. Consider the following three turn coil:

For this example, the pitch has deliberately been made large in order to accentuate the effects of helicity. Let's look at a cross section of the coil taken through the coil axis:

It can be seen that the cut face of the wire is elliptical, because the cut has been taken parallel to the axis, while the wire lies at an oblique angle.

If we take the original coil and cut it open, along the bottom as shown in the diagram at the right, then unroll it (but otherwise keep all the pieces in the same relative positions) and lay it flat, we end up with the conductors arranged as shown in the diagram below:

Note
that as the coil form is unwrapped, the conductors will stick beyond
it by an amount equal to half of the conductor diameter. But by
convention we measure diameter to the centre of the conductor. And
therefore the circumference is * π d*.

We
see that the pitch * p*
which is measured parallel to the coil axis is not the same as the
conductor centre to centre distance

It
can also be seen that the length of one turn * ℓ*
is greater than the coil circumference

We
will define * ψ*
(Greek letter psi) as the pitch angle as shown in the diagram. From
basic geometry we have the following relationships:

* ψ* = Arctan(

* ℓ*
= √(

* p'* =

From
the above it can be seen that for a close-wound coil, of wire
diameter **d**_{W},
the pitch **p**_{c}
is slightly greater than **d**_{W},
and is given by the close-wound pitch angle **ψ**_{c}:

**ψ**_{c
} = Arcsin(**d**_{W}/(* π d*))

**p**_{c} = **d**_{W}**/**cos **ψ**_{c}

**Turn
Length Correction**

If the coil consisted of a series of parallel perfectly circular rings, then we would use the standard formula for the circumference of a circle to calculate the length of a turn:

* ℓ* =

However, we have already shown above that the actual turn length for a helix is:

* ℓ*
= √(

It's
therefore logical, that the greater length of turn will have an
effect on the mutual inductance calculated by elliptic integral
formulae (10), (11) or (12). We can compensate for this by making an
adjustment to the value of the loop diameter used in the formulae. We
can calculate an adjusted diameter * d'*,
so that it gives a correct helical turn length:

* d'* =

which simplifies to:

* d'* = √((

We
can therefore use * d'*
in place of

**Turn
Spacing Correction**

Similarly,
we see from the above relationships, that the distance between
adjacent turns is not equal to the pitch * p*,
but rather,

* p'* =

We
can therefore use * p'*
in place of

This is as far as I've progressed on the turns spacing correction, for the following reasons:

- For coils that are closely wound,
and**p'**are virtually equal, and so no correction is required.**p** - For coils with considerable space between turns, then the assumption that the turns are circular becomes invalid and it is difficult to justify a simple spacing correction without having some means to test the validity of the correction.

I plan to address this correction in the future once I've developed a means to test it. This will include coding of an independent inductance calculation, which takes helicity into account in order to have a means to cross check the corrections. This work is now partially completed as is discussed in Part 2c.

Continue to:

Part 2c – Snow's Helical Inductance Calculation Back to:

Numerical Methods 2a

Numerical Methods Introduction

Radio Theory

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This page last updated: November 30, 2022

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