Re: [TCML] inductance Vs diameter Vs height

[Paul Nicholson <tcml88@xxxxxxxxxxx>]

Antonio wrote:
> The impedance seen across the coil is inductive at low
> frequency, tending to the DC value of the inductance, but at
> higher frequencies the distributed capacitances contribute
> more and more to the impedance.
>
> If you consider just the value of the reactance and attributes
> it to a single inductor, the equivalent inductor really
> appears to change with the frequency. In the simplest model,
> a grounded vertical coil seen from the top has a capacitor
> in parallel with it. As the frequency rises, the impedance
> grows to a maximum (infinite in the lossless case) at the
> resonance frequency of the combination.

Agreed.   But this is not quite the same as the definition
of Les.  The reactance described above is the impedance seen
across the parallel combination of the coil's inductance
and capacitance, which rises to a high value as resonance is
approached, infinite in the lossless case.

But Les is defined differently, as the ratio of voltage across
the coil, to the current circulating through the coil's L and C.
This doesn't rise to a high value at resonance - instead it
corresponds to the surge impedance or characteristic impedance
of the resonator.

A diagram clarifies:

 http://abelian.org/tssp/efl.gif

This lossless resonator has circulating current Ic with a
voltage V appearing across it.  Les is defined by

 V = 2 * pi * Fres * Les * Ic ;

The impedance Antonio refers to is seen when measuring across
the combination:  V / Ix, where Ix is the measuring current
injected by some external instrument.

In a lumped model there is no distinction between Les and Ldc,
but when we deal with distributed resonators, they acquire
different values.   And not only that, we are also forced to
specify just where in the resonator we choose to define and
measure the circulating current, since the current entering
the coil at one end is not the same as that leaving the coil
at the other.

We choose the coil base as the current measurement point and
call this Ibase or Ib, and the resonator voltage is the topload
voltage Vtop or Vt.

Then Vt = 2 * pi * Fres * Les * Ib ;

In solenoids longer than about h/d = 1, Les is less than Ldc
because not all of the turns have Ib passing through them.
As the current moves up the coil, more and more of it is
being diverted into the coil's distributed capacitance, so
the higher turns are passing less current than the lower turns
and so contribute less to the total induced voltage.

If you measure (at low frequency using some kind of LCR meter)
the coil's L and C, you obtain values we label as Ldc and
Cdc respectively.   The coiler will then examine the coil at
resonance, measuring a frequency Fres and may also measure
the coil's base current Ib and top volts Vt.

It will be found that the formula

  Fres = 1/(2 * pi * sqrt( Ldc * Cdc) ;

doesn't give the right answer.   Nor does

  Vt = 2 * pi * Fres * Ldc * Ib ;

describe correctly the voltage and current. It will be found
instead that

  Vt = 2 * pi * Fres * Les * Ib ;

and also that

  Fres = 1/(2 * pi * sqrt( Les * Ces)) ;

where we must here introduce an effective capacitance Ces
which is different from Cdc, for similar reasons to why Les
is different from Ldc.

At small h/d values, say less than about 1:1, Les actually is
higher than Ldc.  This occurs because current circulates within
the coil due to the close capacitive coupling between the turns.
This circulating current raises the induced voltage but does
not appear at the base terminal to be measured.

Note that Les and Ces are only defined for a specific frequency,
usually we choose the fundamental resonant frequency of
the coil.  This is because their values depend on the voltage
and current distributions present in the coil.  They can of course
be calculated and measured for any of the other (overtone)
resonances and these values will be very different.

Further notes on this topic, section 7 of

 http://abelian.org/tssp/pn2511.html

The ratio of effective inductance to low frequency inductance depends
mostly on the overall shape of the coil, and slightly on its height
above the ground plane.  Here is a table calculated for a range of
coil shapes,

 http://abelian.org/tssp/pn100502.html

The B=0 columns describe the solenoid.
--
Paul Nicholson
--

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