L'effetto pelle (in inglese:skin effect) è la tendenza di una
corrente elettrica alternata a distribuirsi dentro un conduttore in modo non
uniforme: la sua densità è maggiore sulla superficie ed inferiore all'interno.
Questo comporta un aumento della resistenza elettrica del conduttore
particolarmente alle alte frequenze. In altre parole una parte del conduttore
non viene utilizzata: è come se non esistesse. Il fenomeno venne spiegato, per
la prima volta, da Lord Kelvin nel 1887. Anche Nikola Tesla studiò il problema.
L'effetto pelle ha conseguenze pratiche nel progetto di:
Da un punto di vista teorico la densità di corrente J (la corrente che attraversa l'unità di superficie) in un conduttore decresce esponenzialmente man mano che dalla superficie esterna si penetra nel suo interno. Questo vale per conduttori a sezione circolare o di altra forma. Alla profondità d la densità di corrente J è approssimativamente:
dove:
Alla profondità δ la densità della corrente vale 1/e (circa
0.37) volte quella presente sulla superficie esterna.
Per profondità δ, si usa la relazione:
dove
ρ = resistività (detta anche resistenza specifica) del conduttore
ω = frequenza (o pulsazione) della corrente = 2π × frequenza
μ = permeabilità magnetica del materiale conduttore (che, per i conduttori comuni, è uguale a quella del vuoto: μ_{0}).
La resistenza di una lastra piana (di spessore maggiore di δ) al passaggio di una corrente alternata è uguale alla resistenza di una lastra di spessore δ in cui scorre una corrente continua.
Se il conduttore è un filo a sezione circolare la sua resistenza in alternata è circa la stessa che presenta un filo cavo di spessore δ e con lo stesso diametro del filo pieno. L’espressione approssimata per R in questo caso è:
dove:
L = lunghezza del conduttore cavo
D = diametro esterno del conduttore cavo
δ = spessore della corona circolare
Materiale |
Resistività r (in W × mm^{2} / m ) |
Argento |
0.016 |
Rame |
0.017 |
Oro |
0.024 |
Alluminio |
0.028 |
Tungsteno |
0.055 |
Platino |
0.10 |
Ferro |
0.13 |
Acciaio |
0.18 |
Piombo |
0.22 |
Mercurio |
0.94 |
Costantana (lega 80% Cu, 40% Ni) |
0.49 |
Carbonio |
35 |
Germanio |
60 × 10^{2} |
Silicio |
2.3 × 10^{9} |
Ambra |
5 × 10^{20} |
Vetro |
10^{16 }÷ 10^{20} |
Mica |
10^{17 }÷ 10^{21} |
Zolfo |
10^{21} |
Legno secco |
10^{14 }÷ 10^{17} |
There is a considerable amount
of discussion amongst those interested in audio and Hi-Fi about the possible
effects of cables upon ‘sound quality’. This tends to lead to some people
adopting almost ‘theological’ viewpoints that differ fundamentally from the
views held by others. One of the technical factors which is sometimes claimed
to affect sound quality is what is usually called ‘Skin effect’.
In most electronics textbooks, the properties of cables and wires are
considered as a form of transmission line. The text may mention briefly the
skin effect without exploring this in detail. More often, however, the only
parameters that tend to be considered are the capacitance per length,
inductance per length, and their relationship with the signal’s nominal
propagation velocity and the characteristic impedance of the system. The fact
that normal materials have a finite conductance (or resistance) is not usually
considered beyond its effect upon the d.c. (and low frequency) resistance of
the cables and the resulting implied signal power losses.
In reality, when we transmit alternating signals along conductive lines we may
experience effects due to what is generally called ‘skin effects’. This subject
is widely misunderstood, and hence people occasionally tend to invoke this
frequency dependent behaviour as the implied basis for all kinds of claims
regarding the ‘sounds’ of different types of cables. The purpose of this
analysis is to throw some light into this area and help provide some
understanding of the effects of using conductors of finite conductivity.
In engineering textbooks, the consequences of finite conductivity and wire size
are treated in terms of an ‘Internal Impedance’. This term is probably more
useful that ‘skin effect’ as it acts as a reminder that the effects arise due
to the fields internal to the conductor. The internal impedance per unit length
of a wire is considered in Ramo et al. From this was may draw the following
results as a starting point.
The d.c. (i.e. very low frequency) impedance of a wire which has a circular
cross-section and is uniform may be said to consist of a resistance per unit
length of
where is the bulk
conductivity value appropriate for the material used to manufacture the wire,
and is the radius of
the wire. The resistance is in Ohms/metre if we are using S.I. units (which
will be assumed from now on).
The wire will also exhibit an effective inductance per unit length at very low
frequency due to its internal fields. At very low frequencies this has the
value
where is the permeability
of the material. In general we can assume that this equals the value for free
space
The detailed analysis in Ramo
leads to the following expressions which can be used to determine the relevant
wire resistance and inductance per unit length for a conductive wire of
circular cross section at frequencies above d.c.
where
and is the ‘skin depth’
value which may be calculated via
where is the signal
frequency (as opposed to the signal’s
angular frequency).
In fact, using the above expressions we can show that
and specifying this factor in
terms of and may be more
convenient when performing calculations.
Now, , etc are Bessel
functions. We can find numerical expressions for evaluating these in a text
like Abramowitz and Stegun. Using these we can compute values. For the sake of
clarity is is useful to plot values normalised in terms of . Some results of
doing this are illustrated in figure 1. These are plotted versus so that the
relevant nominal skin thickness is also normalised in terms of the wire radius.
The solid lines plotted show the relevant values calculated from the above
expressions.
Unfortunately, the expressions
provided by A & S only cover the range for which roughly
corresponds to . Above this value
the Bessel functions become hard to evaluate and their combination tends to
lead to a situation where a set of large values cancel to give a moderate
result. Hence for computational simplicity we can use a simpler approximation
for the situations where is ‘large’. Here
this may be defined as where this ratio has a value greater than 5.
The standard ‘h.f.’ approximation is that both and will be essentially
equal to . However by
inspecting the results shown in figure 1 we can see that it is possible to do
better than this and a more reliable approximation would be
This approximations are show
in figure 1 by the broken lines. The approximation for is shown by a
short-dashed red line, and that for by a longer-dashed
green line. It can be seen that these approximations are likely to be
reasonably accurate in the region .
In general, electronic signals are conveyed using a ‘pair of wires’.
These are used to form a closed loop (path) between the signal source and the
load around which charge may flow. Broadly speaking we can then define a
‘cable’ to consist of a pair of wires. The most common forms of cable used in
audio are ‘Twin Feeder’ and ‘Co-ax’. The basic properties of these are
discussed elsewhere. Each of the wires may be a single, solid, length of
conductor. More usually, however, each wire will consist of a bundle of
conducting ‘strands’. Multistrand wires have properties which may differ from
that of a single, solid wire of similar cross-section. We can therefore treat
wires as falling into three general categories as outlined below.
For solid-core wires the above analysis can be used immediately to compute the
internal impedance and deduce the effects it may have in a given situation. For
the stranded and Litz wires we need to take the stranding and the effect of
inter-strand contacts or insulation into account.
Litz wires consist of a bundle of very thin, individually insulated conductors.
The insulation ensures that the current flows in all of the wires in the bundle
as the charge cannot migrate towards the surface of the bundle. The entire
cross section of conductor bundle is therefore used by the charge transport.
Provided that the individual strands are thin enough, the strands all have
individual radii that are small compared to the skin depth at audio
frequencies. Hence the overall properties of the Litz bundle tends to be
similar to that of a single wire of the same diameter of the bundle but where
‘skin effect’ is apparently absent.
In practice, most of the multi-strand wires used for audio purposes have no
insulation on the individual strands. This means they do not behave like a Litz
wire. In stranded wires without insulation between the individual strands
charge may cross from strand to strand. Hence current will tend to
preferentially flow near the skin of the bundle of wires, just as it does with
a single solid conductor of similar overall diameter. Hence when the strands
are thin but in electrical contact with their neighbours we can expect the
effect of internal impedance to be similar to that of a solid wire of a
diameter similar to the bundle of strands. There is, however, one factor we
should take into account. This arises when the strands do not ‘fill’ the bundle
and their are air gaps. This is illustrated in figure 2. This shows a
close-packed array of strands, each of radius .
Since the strands all have a
circular cross-section we find that even if they are tightly packed into a
hexagonal array there will be spaces in between the places where they touch.
Hence a bundle of small packed strands that are in electrical contact can be
regarded as similar to a solid but which has some air inclusions which mean
that overall wire cross section is only partly filled with conductor. Given a
value for the ‘fill factor’, , (the fraction of
the cross section which is filled with conductor) we can treat the wire as
being equivalent to a solid conductor of the same wire diameter, , but having an
effective conductivity of . Thus the main
electronic effect of using a bundle of strands is to dilute the effective
conductivity and lower its apparent value.
To estimate the value of the fill factor we can note that the array has a
symmetry similar to a packed array of equilateral triangles whose sides all
have a length of . Taking one of
these elemental triangles we can see that it contains three 60 degree sectors
of conductor. Hence the cross sectional area of conductor in each triangle is
equal to . However the area
of each triangle will be . Hence the fill
factor will be approximately equal to
Real strands will tend to deform slightly when compressed and may not
have perfectly smooth, circular cross sections. They may also not be perfectly
packed. Hence we can expect the actual value of the fill factor to vary
accordingly in practice. In Litz bundles the insulation layers on the strands
will also move the conductors apart by a small amount, thus reducing the fill
factor value. Usually, we can expect this effect to be small as the layer of
insulation is likely to be very thin. In most cases we can therefore tend to
assume that the fill factor is reasonably close to unity so this is a
reasonable assumption for general analysis. That said, in some specific cases
we can take stranding into account by modifying the effective conductivity by
an appropriate fill factor value.