In electromagnetism, skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. It is caused by opposing eddy currents induced by the changing magnetic field resulting from the alternating current. The electric current flows mainly at the skin of the conductor, between the outer surface and a level called the skin depth.
Skin depth depends on the frequency of the alternating current; as frequency increases, current flow becomes more concentrated near the surface, resulting in less skin depth. Skin effect reduces the effective cross-section of the conductor and thus increases its effective resistance. At 60 Hz in copper, skin depth is about 8.5 mm. At high frequencies, skin depth becomes much smaller.
Increased AC resistance caused by skin effect can be mitigated by using a specialized multistrand wire called litz wire. Because the interior of a large conductor carries little of the current, tubular conductors can be used to save weight and cost.
Skin effect has practical consequences in the analysis and design of radio-frequency and microwave circuits, transmission lines (or waveguides), and antennas. It is also important at mains frequencies (50–60 Hz) in AC electric power transmission and distribution systems. It is one of the reasons for preferring high-voltage direct current for long-distance power transmission.
The effect was first described in a paper by Horace Lamb in 1883 for the case of spherical conductors, and was generalized to conductors of any shape by Oliver Heaviside in 1885.
Cause
Conductors, typically in the form of wires, may be used to transfer electrical energy or signals using an alternating current flowing through that conductor. The charge carriers constituting that current, usually electrons, are driven by an electric field due to the source of electrical energy. A current in a conductor produces a magnetic field in and around the conductor. When the intensity of current in a conductor changes, the magnetic field also changes. The change in the magnetic field, in turn, creates an electric field that opposes the change in current intensity. This opposing electric field is called counter-electromotive force (back EMF). The back EMF is strongest / most concentrated at the center of the conductor, allowing current only near the outside skin of the conductor, as shown in the diagram on the right.
Regardless of the driving force, the current density is found to be greatest at the conductor's surface, with a reduced magnitude deeper in the conductor. That decline in current density is known as the skin effect and the skin depth is a measure of the depth at which the current density falls to 1/e of its value near the surface. Over 98% of the current will flow within a layer 4 times the skin depth from the surface. This behavior is distinct from that of direct current which usually will be distributed evenly over the cross-section of the wire.
An alternating current may also be induced in a conductor due to an alternating magnetic field according to the law of induction. An electromagnetic wave impinging on a conductor will therefore generally produce such a current; this explains the attenuation of electromagnetic waves in metals. Although the term skin effect is most often associated with applications involving transmission of electric currents, skin depth also describes the exponential decay of the electric and magnetic fields, as well as the density of induced currents, inside a bulk material when a plane wave impinges on it at normal incidence.
Formula The AC current density in a conductor decreases exponentially from its value at the surface according to the depth from the surface, as follows:
where is called the skin depth which is defined as the depth below the surface of the conductor at which the current density has fallen to 1/e (about 0.37) of S. The imaginary part of the exponent indicates that the phase of the current density is delayed 1 radian for each skin depth of penetration. One full wavelength in the conductor requires 2 skin depths, at which point the current density is attenuated to e−2 (1.87×, or −54.6 dB) of its surface value. The wavelength in the conductor is much shorter than the wavelength in vacuum, or equivalently, the phase velocity in a conductor is very much slower than the speed of light in vacuum. For example, a 1 MHz radio wave has a wavelength in vacuum of about 300 m, whereas in copper, the wavelength is reduced to only about 0.5 mm with a phase velocity of only about 500 m/s. As a consequence of Snell's law and this very tiny phase velocity in a conductor, any wave entering a conductor, even at grazing incidence, refracts essentially in the direction perpendicular to the conductor's surface.
The general formula for skin depth when there is no dielectric or magnetic loss is:The formula as shown is algebraically equivalent to the formula found on page 130
where
At frequencies much below the quantity inside the large parentheses is close to unity and the formula is more usually given as:
This formula is valid at frequencies away from strong atomic or molecular resonances (where would have a large imaginary part) and at frequencies that are much below both the material's plasma frequency (dependent on the density of free electrons in the material) and the reciprocal of the mean time between collisions involving the conduction electrons. In good conductors such as metals all of those conditions are ensured at least up to microwave frequencies, justifying this formula's validity.Note that the above equation for the current density inside the conductor as a function of depth applies to cases where the usual approximation for skin depth holds. In the extreme cases where it doesn't, the exponential decrease with respect to skin depth still applies to the magnitude of the induced currents, however the imaginary part of the exponent in that equation, and thus the phase velocity inside the material, are altered with respect to that equation. For example, in the case of copper, this would be true for frequencies much below .
However, in very poor conductors, at sufficiently high frequencies, the quantity inside the large parentheses increases. At frequencies much higher than , skin depth approaches the asymptotic value:
This departure from the usual formula only applies for materials of rather low conductivity and at frequencies where the vacuum wavelength is not much larger than the skin depth itself. For instance, bulk silicon (undoped) is a poor conductor and has a skin depth of about 40 meters at 100 kHz ( = 3 km). However, as the frequency is increased well into the megahertz range, its skin depth never falls below the asymptotic value of 11 meters. The conclusion is that in poor solid conductors, such as undoped silicon, skin effect does not need to be taken into account in most practical situations.
Round wire
When skin depth is not small with respect to the radius of the wire, current density may be described in terms of Bessel functions. The current density inside round wire away from the influences of other fields, as function of distance from the axis is given by:
where
Since is complex, the Bessel functions are also complex. The amplitude and phase of the current density varies with depth.
Derivation Combining the electromagnetic wave equation and Ohm's law produces
The solution to this equation is, for finite current in the center of the conductor,
where is a Bessel function of the first kind of order and is a constant phasor. To satisfy the boundary condition for the current density at the surface of the conductor, must be Thus,
Impedance
Resistance The most important effect of skin effect on the impedance of a single wire is the increase of the wire's resistance, and consequent losses. The effective resistance due to a current confined near the surface of a large conductor (much thicker than ) can be solved as if the current flowed uniformly through a layer of thickness based on the DC resistivity of that material. The effective cross-sectional area is approximately equal to times the conductor's circumference. Thus a long cylindrical conductor such as a wire, having a diameter large compared to , has a resistance approximately that of a hollow tube with wall thickness carrying direct current. The AC resistance of a wire of length and resistivity is:
The final approximation above assumes .
A convenient formula (attributed to F.E. Terman) for the diameter of a wire of circular cross-section whose resistance will increase by 10% at frequency is:
This formula for the increase in AC resistance is accurate only for an isolated wire. For nearby wires, e.g. in a cable or a coil, the AC resistance is also affected by proximity effect, which can cause an additional increase in the AC resistance. The internal impedance per unit length of a segment of round wire is given by:
This impedance is a complex quantity corresponding to a resistance (real) in series with the reactance (imaginary) due to the wire's internal self-inductance, per unit length.
Inductance
The portion of a wire's inductance that can be attributed to the magnetic field inside the wire itself is called the internal inductance, which accounts for the inductive reactance (imaginary part of the impedance) given by the above formula. In most cases this is a small portion of a wire's inductance which includes the effect of induction from magnetic fields outside of the wire produced by the current in the wire. Unlike that external inductance, the internal inductance is reduced by skin effect, that is, at frequencies where skin depth is no longer large compared to the conductor's size. This small component of inductance approaches a value of (50 nH/m for non-magnetic wire) at low frequencies, regardless of the wire's radius. Its reduction with increasing frequency, as the ratio of skin depth to the wire's radius falls below about 1, is plotted in the accompanying graph, and accounts for the reduction in the telephone cable inductance with increasing frequency in the table below.
Refer to the diagram below showing the inner and outer conductors of a coaxial cable. Since skin effect causes a current at high frequencies to flow mainly at the surface of a conductor, it can be seen that this will reduce the magnetic field inside the wire, that is, beneath the depth at which the bulk of the current flows. It can be shown that this will have a minor effect on the self-inductance of the wire itself; see Skilling or Hayt for a mathematical treatment of this phenomenon.
The inductance considered in this context refers to a bare conductor, not the inductance of a coil used as a circuit element. The inductance of a coil is dominated by the mutual inductance between the turns of the coil which increases its inductance according to the square of the number of turns. However, when only a single wire is involved, then in addition to the external inductance involving magnetic fields outside the wire (due to the total current in the wire) as seen in the white region of the figure below, there is also a much smaller component of internal inductance due to the portion of the magnetic field inside the wire itself, the green region in figure B. That small component of the inductance is reduced when the current is concentrated toward the skin of the conductor, that is, when skin depth is not much larger than the wire's radius, as will become the case at higher frequencies.