4105:) may be between 5 and 50 mm thick skin effect also occurs at sharp bends where the metal is compressed inside the bend and stretched outside the bend. The shorter path at the inner surface results in a lower resistance, which causes most of the current to be concentrated close to the inner bend surface. This causes an increase in temperature at that region compared with the straight (unbent) area of the same conductor. A similar skin effect occurs at the corners of rectangular conductors (viewed in cross-section), where the magnetic field is more concentrated at the corners than in the sides. This results in superior performance (i.e. higher current with lower temperature rise) from wide thin conductors (for example,
4128:
resistive losses affecting the accompanying eddy currents; skin effect confines such eddy currents to a very thin surface layer of the waveguide structure. Skin effect itself is not actually combatted in these cases, but the distribution of currents near the conductor's surface makes the use of precious metals (having a lower resistivity) practical. Although it has a lower conductivity than copper and silver, gold plating is also used, because unlike copper and silver, it does not corrode. A thin oxidized layer of copper or silver would have a low conductivity, and so would cause large power losses as the majority of the current would still flow through this layer.
1168:
2939:
self-inductance) regardless of the wire's length, so that the inductance decrease due to skin effect can still be important. For instance, in the case of a telephone twisted pair, below, the inductance of the conductors substantially decreases at higher frequencies where skin effect becomes important. On the other hand, when the external component of the inductance is magnified due to the geometry of a coil (due to the mutual inductance between the turns), the significance of the internal inductance component is even further dwarfed and is ignored.
4070:, braided wire) is used to mitigate skin effect for frequencies of a few kilohertz to about one megahertz. It consists of a number of insulated wire strands woven together in a carefully designed pattern, so that the overall magnetic field acts equally on all the wires and causes the total current to be distributed equally among them. With skin effect having little effect on each of the thin strands, the bundle does not suffer the same increase in AC resistance that a solid conductor of the same cross-sectional area would due to skin effect.
2907:
4145:
39:
2891:
2971:
159:
4387:
1152:= 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: Any current is equally distributed throughout the material's cross-section, regardless of its frequency.
27:
202:, 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
1045:(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. For example, in the case of copper, this would be true for frequencies much below
4098:
strength but low weight of tubes substantially increases span capability. Tubular conductors are typical in electric power switchyards where the distance between supporting insulators may be several meters. Long spans generally exhibit physical sag but this does not affect electrical performance. To avoid losses, the conductivity of the tube material must be high.
4190:
2052:
3637:). Since there is essentially no current deeper in the inner conductor, there is no magnetic field beneath the surface of the inner conductor. Since the current in the inner conductor is balanced by the opposite current flowing on the inside of the outer conductor, there is no remaining magnetic field in the outer conductor itself where
2883:(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
4132:
results in fields and/or currents that oppose those generated by relatively nonmagnetic materials, but more work is needed to verify the exact mechanisms. As experiments have shown, this has potential to greatly improve the efficiency of conductors operating in tens of GHz or higher. This has strong ramifications for
534:
2938:
For a single wire, this reduction becomes of diminishing significance as the wire becomes longer in comparison to its diameter, and is usually neglected. However, the presence of a second conductor in the case of a transmission line reduces the extent of the external magnetic field (and of the total
3105:, however, there is a reduced magnetic field in the deeper sections of the inner conductor and the outer sections of the shield (outer conductor). Thus there is less energy stored in the magnetic field given the same total current, corresponding to a reduced inductance. At an even higher frequency,
1171:
Current density in round wire for various skin depths. Numbers shown on each curve are the ratio of skin depth to wire radius. The curve shown with the infinity sign is the zero frequency (DC) case. All curves are normalized so that the current density at the surface is the same. The horizontal
34:, current density decreases exponentially from the surface towards the inside. Skin depth, δ, is defined as the depth where the current density is just 1/e (about 37%) of the value at the surface; it depends on the frequency of the current and the electrical and magnetic properties of the conductor.
2934:
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
2925:
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.
4131:
Recently, a method of layering non-magnetic and ferromagnetic materials with nanometer scale thicknesses has been shown to mitigate the increased resistance from skin effect for very high-frequency applications. A working theory is that the behavior of ferromagnetic materials in high frequencies
4127:
frequencies where the small skin depth requires only a very thin layer of silver, making the improvement in conductivity very cost effective. Silver plating is similarly used on the surface of waveguides used for transmission of microwaves. This reduces attenuation of the propagating wave due to
4077:
to increase their efficiency by mitigating both skin effect and proximity effect. Large power transformers are wound with stranded conductors of similar construction to litz wire, but employing a larger cross-section corresponding to the larger skin depth at mains frequencies. Conductive threads
4097:
In applications where high currents (up to thousands of amperes) flow, solid conductors are usually replaced by tubes, eliminating the inner portion of the conductor where little current flows. This hardly affects the AC resistance, but considerably reduces the weight of the conductor. The high
4010:
In a good conductor, skin depth is proportional to square root of the resistivity. This means that better conductors have a reduced skin depth. The overall resistance of the better conductor remains lower even with the reduced skin depth. However the better conductor will show a higher ratio
1147:
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
1868:
1353:
5170:
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
4382:{\displaystyle {\begin{aligned}\delta &={\frac {1}{\alpha }}={\sqrt {{2\rho } \over {(2\pi f)(\mu _{0}\mu _{r})}}}\\&={\frac {1}{\sqrt {\pi f\mu \sigma }}}\approx 503\,{\sqrt {\frac {\rho }{\mu _{r}f}}}\approx 503\,{\frac {1}{\sqrt {\mu _{r}f\sigma }}},\end{aligned}}}
248:
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
4001:
4046:
but it is difficult to use them at frequencies much higher than 60 Hz. At a few kilohertz, an iron welding rod would glow red hot as current flows through the greatly increased AC resistance resulting from skin effect, with relatively little power remaining for the
4082:
have been demonstrated as conductors for antennas from medium wave to microwave frequencies. Unlike standard antenna conductors, the nanotubes are much smaller than the skin depth, allowing full use of the thread's cross-section resulting in an extremely light antenna.
2910:
The ratio AC resistance to DC resistance of a round wire versus the ratio of the wire's radius to the skin depth. As skin depth becomes small relative to the radius, the ratio of AC to DC resistance approaches one half of the ratio of the radius to the skin
3121:
For a given current, the total energy stored in the magnetic fields must be the same as the calculated electrical energy attributed to that current flowing through the inductance of the coax; that energy is proportional to the cable's measured inductance.
2563:
430:
4983:
for alternating current at 60 Hz with a radius larger than one-third of an inch (8 mm) is a waste of copper, and in practice bus bars for heavy AC current are rarely more than half an inch (12 mm) thick except for mechanical reasons.
4879:
Thus at microwave frequencies, most of the current flows in an extremely thin region near the surface. Ohmic losses of waveguides at microwave frequencies are therefore only dependent on the surface coating of the material. A layer of silver
2916:
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
2921:
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.
2806:
4732:
96:
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
2679:
1143:
1178:
3385:
1659:
3500:
2411:
3569:
3872:
4030:
at 60 Hz. Iron wire is impractical for AC power lines (except to add mechanical strength by serving as a core to a non-ferromagnetic conductor like aluminum). Skin effect also reduces the effective thickness of
2850:
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
1017:
197:
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
1710:
2047:{\displaystyle \nabla ^{2}\mathbf {J} (r)+k^{2}\mathbf {J} (r)={\frac {\partial ^{2}}{\partial r^{2}}}\mathbf {J} (r)+{\frac {1}{r}}{\frac {\partial }{\partial r}}\mathbf {J} (r)+k^{2}\mathbf {J} (r)=0.}
3700:
Although the geometry is different, a twisted pair used in telephone lines is similarly affected: at higher frequencies, the inductance decreases by more than 20% as can be seen in the following table.
352:
2484:
2116:
2838:; this 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
2308:
5247:
These emf's are greater at the center than at the circumference, so the potential difference tends to establish currents that oppose the current at the center and assist it at the circumference.
4195:
4015:(1000 square millimeter) copper conductor has 23% more resistance than it does at DC. The same size conductor in aluminum has only 10% more resistance with 60 Hz AC than it does with DC.
848:
3866:
More extensive tables and tables for other gauges, temperatures and types are available in Reeve. Chen gives the same data in a parameterized form that he states is usable up to 50 MHz.
206:(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.
424:
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.
401:
skin depths, at which point the current density is attenuated to e (1.87×10, or −54.6 dB) of its surface value. The wavelength in the conductor is much shorter than the wavelength in
3613:
5086:, losses due to induced currents occur mostly within one skin depth of the surface. Thus, plating the surface of a waveguide with a material which has a low skin depth reduces losses.
4815:
At very high frequencies skin depth for good conductors becomes tiny. For instance, skin depths of some common metals at a frequency of 10 GHz (microwave region) are less than a
2701:
4657:
5080:
2881:
1823:
1087:
961:
49:) to reduce heating of the coil itself due to skin effect. The AC frequencies used in induction cookers are much higher than standard mains frequency ‒ typically around 25–50 kHz.
5175:
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.
1495:
1449:
712:
915:
879:
3414:
3281:
3215:
3155:
2223:
4577:
4473:
5050:
5023:
3695:
3125:
The magnetic field inside a coaxial cable can be divided into three regions, each of which will therefore contribute to the electrical inductance seen by a length of cable.
2930:
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
4772:
3668:
3314:
3248:
1522:
807:
2614:
2591:
2244:
2189:
1092:
1039:
1774:
3041:). The width of the dashed black lines is intended to show relative strength of the magnetic field integrated over the circumference at that radius. The four stages are
4652:
4504:
3081:, current covers the conductors uniformly and there is a significant magnetic field in all three regions. As the frequency is increased and the skin effect takes hold (
779:
743:
4802:
4554:
4441:
4416:
3182:
1737:
1590:
1556:
626:
591:
4112:
It follows that a transformer with a round core will be more efficient than an equivalent-rated transformer having a square or rectangular core of the same material.
372:
3323:
563:
225:
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
4529:
3427:
2479:
2313:
2143:
671:
529:{\displaystyle \delta ={\sqrt {{\frac {\,2\rho \,}{\omega \mu }}\left({\sqrt {1+\left({\rho \omega \varepsilon }\right)^{2}\,}}+\rho \omega \varepsilon \right)\,}}}
1411:
1384:
420:
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
4609:
3618:
Most discussions of coaxial transmission lines assume they will be used for radio frequencies, so equations are supplied corresponding only to the latter case.
2167:
1847:
646:
3113:, the skin depth is tiny: All current is confined to the surface of the conductors. The only magnetic field is in the regions between the conductors; only the
2959:
be the inner conductor radius, the shield (outer conductor) inside radius and the shield outer radius respectively, as seen in the crossection of figure
244:
impinging on a conductor will therefore generally produce such a current; this explains the attenuation of electromagnetic waves in metals. Although the term
966:
3073:
high frequency. There are three regions that may contain induced magnetic fields: the center conductor, the dielectric and the outer conductor. In stage
5627:, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 215, No. 1123 (Dec. 22, 1952), pp. 481–497 (17 pages)
213:
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
1597:
1172:
axis is the position within the wire with the left and right extremes being the surface of the wire. The vertical axis is relative current density.
4094:; the higher resistance of the steel core is of no consequence since it is located far below the skin depth where essentially no AC current flows.
3507:
2894:
The internal component of a round wire's inductance vs. the ratio of skin depth to radius. That component of the self inductance is reduced below
1052:
However, in very poor conductors, at sufficiently high frequencies, the factor under the large radical increases. At frequencies much higher than
283:
5098:
in 1940, who correctly suggested that it is due to the mean free path length of the electrons reaching the range of the classical skin depth.
2057:
1348:{\displaystyle \mathbf {J} (r)={\frac {k\mathbf {I} }{2\pi R}}{\frac {J_{0}(kr)}{J_{1}(kR)}}=\mathbf {J} (R){\frac {J_{0}(kr)}{J_{0}(kR)}}}
3097:) the magnetic field in the dielectric region is unchanged as it is proportional to the total current flowing in the center conductor. In
5025:. The intensity of the wave is proportional to the square of the amplitude, and thus the depth at which the intensity has diminished by
73:
is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. It is caused by opposing
2974:
Four stages of skin effect in a coax showing the effect on inductance. Diagrams show a cross-section of the coaxial cable. Color code:
1164:. The current density inside round wire away from the influences of other fields, as function of distance from the axis is given by:
3996:{\displaystyle L(f)={\frac {\ell _{0}+\ell _{\infty }\left({\frac {f}{f_{m}}}\right)^{b}}{1+\left({\frac {f}{f_{m}}}\right)^{b}}}\,}
2425:
The most important effect of skin effect on the impedance of a single wire is the increase of the wire's resistance, and consequent
1670:
112:. Because the interior of a large conductor carries little of the current, tubular conductors can be used to save weight and cost.
4998:
In electromagnetic waves, the skin depth is the depth at which the amplitude of the electric and magnetic fields have reduced by
4026:
its permeability is about 10,000 times greater. This reduces the skin depth for iron to about 1/38 that of copper, about 220
5826:
5800:
5761:
5728:
5692:
5649:
5429:
5268:
5240:
4804:
as was pointed out above for the case of iron, despite its poorer conductivity. A practical consequence is seen by users of
4091:
2249:
5094:
For high frequencies and low temperatures, the usual formulas for skin depth break down. This effect was first noticed by
5150:
5115:
2688:
815:
5710:
5349:
5298:
4011:
between its AC and DC resistance, when compared with a conductor of higher resistivity. For example, at 60 Hz, a
1849:
is complex, the Bessel functions are also complex. The amplitude and phase of the current density varies with depth.
5745:
5669:
2146:
3577:
2683:
This formula for the increase in AC resistance is accurate only for an isolated wire. For nearby wires, e.g. in a
1160:
When skin depth is not small with respect to the radius of the wire, current density may be described in terms of
4171:
4019:
2558:{\displaystyle R\approx {{\ell \rho } \over {\pi (D-\delta )\delta }}\approx {{\ell \rho } \over {\pi D\delta }}}
2191:
is a constant phasor. To satisfy the boundary condition for the current density at the surface of the conductor,
673:
4148:
Skin depth vs. frequency for some materials at room temperature, red vertical line denotes 50 Hz frequency:
5872:
2429:. The effective resistance due to a current confined near the surface of a large conductor (much thicker than
378:
which is defined as the depth below the surface of the conductor at which the current density has fallen to 1/
108:
Increased AC resistance caused by skin effect can be mitigated by using a specialized multistrand wire called
4744:, about 9 times that of gold. Its skin depth at 50 Hz is likewise found to be about 33 mm, or
1858:
5055:
2854:
2437:
based on the DC resistivity of that material. The effective cross-sectional area is approximately equal to
1785:
1055:
929:
2441:
times the conductor's circumference. Thus a long cylindrical conductor such as a wire, having a diameter
203:
136:
132:
1464:
1418:
1167:
679:
4058:
At 1 megahertz skin effect depth in wet soil is about 5.0 m; in seawater it is about 0.25 m.
918:
890:
854:
782:
3391:
3258:
3192:
3132:
2194:
5867:
5099:
4559:
4446:
4022:
of the conductor. In the case of iron, its conductivity is about 1/7 that of copper. However being
2839:
1089:
it can be shown that skin depth, rather than continuing to decrease, approaches an asymptotic value:
379:
237:
222:
5028:
5001:
3673:
5809:
Xi Nan; Sullivan, C. R. (2005). "An equivalent complex permeability model for litz-wire windings".
5145:
5135:
4747:
3640:
3286:
3220:
1502:
789:
5811:
Fourtieth IAS Annual
Meeting. Conference Record of the 2005 Industry Applications Conference, 2005
2570:
2227:
2172:
1024:
963:
the quantity inside the large radical is close to unity and the formula is more usually given as:
105:
in copper, skin depth is about 8.5 mm. At high frequencies, skin depth becomes much smaller.
5511:
2801:{\displaystyle \mathbf {Z} _{\text{int}}={\frac {k\rho }{2\pi R}}{\frac {J_{0}(kR)}{J_{1}(kR)}}.}
1747:
146:
in 1883 for the case of spherical conductors, and was generalized to conductors of any shape by
4727:{\displaystyle \delta =503\,{\sqrt {\frac {2.44\cdot 10^{-8}}{1\cdot 50}}}=11.1\,\mathrm {mm} }
4627:
4479:
754:
718:
4780:
4536:
4423:
4398:
3621:
As skin effect increases, the currents are concentrated near the outside the inner conductor (
3416:
is not changed by the skin effect and is given by the frequently cited formula for inductance
3160:
1717:
1563:
1529:
1041:
would have a large imaginary part) and at frequencies that are much below both the material's
599:
573:
1777:
746:
357:
98:
5224:
1021:
This formula is valid at frequencies away from strong atomic or molecular resonances (where
545:
4511:
3709:
Representative parameter data for 24 gauge PIC telephone cable at 21 °C (70 °F).
2815:
2695:
2464:
2121:
653:
241:
66:
4119:
to take advantage of silver's higher conductivity. This technique is particularly used at
1391:
1364:
8:
4884:
thick evaporated on a piece of glass is thus an excellent conductor at such frequencies.
4087:
115:
Skin effect has practical consequences in the analysis and design of radio-frequency and
62:
31:
4591:
389:. The imaginary part of the exponent indicates that the phase of the current density is
5840:
5681:
5564:
5275:
To understand skin effect, you must first understand how eddy currents operate ...
5213:
4736:
Lead, in contrast, is a relatively poor conductor (among metals) with a resistivity of
2607:
of a wire of circular cross-section whose resistance will increase by 10% at frequency
2152:
1832:
631:
81:
field resulting from the alternating current. The electric current flows mainly at the
5288:
4777:
Highly magnetic materials have a reduced skin depth owing to their large permeability
30:
Distribution of current flow in a cylindrical conductor, shown in cross section. For
5844:
5832:
5822:
5796:
5757:
5741:
5724:
5706:
5688:
5665:
5645:
5556:
5536:
5345:
5294:
5264:
5236:
5217:
5140:
5103:
4993:
4887:
In copper, skin depth can be seen to fall according to the square root of frequency:
2674:{\displaystyle D_{\mathrm {W} }={\frac {200~\mathrm {mm} }{\sqrt {f/\mathrm {Hz} }}}}
2054:
The solution to this equation is, for finite current in the center of the conductor,
1138:{\displaystyle \delta \approx {2\rho }{\sqrt {{\frac {\,\varepsilon \,}{\mu }}\,}}~.}
593:
266:
250:
193:
which partially cancel the current flow in the center and reinforce it near the skin.
128:
120:
5568:
427:
The general formula for skin depth when there is no dielectric or magnetic loss is:
19:"Skin depth" redirects here. For the depth (layers) of biological/organic skin, see
5814:
5779:
5548:
5205:
5196:
Lamb, Horace (1883-01-01). "XIII. On electrical motions in a spherical conductor".
4805:
4153:
2906:
2684:
2597:
1042:
147:
54:
42:
5659:
5323:
The formula as shown is algebraically equivalent to the formula found on page 130
2884:
4809:
4165:
4079:
1161:
210:
124:
70:
3250:, the region between the two conductors (containing a dielectric, possibly air).
2433:) can be solved as if the current flowed uniformly through a layer of thickness
4116:
4039:
3380:{\displaystyle L_{\text{total}}=L_{\text{cen}}+L_{\text{shd}}+L_{\text{ext}}\,}
2811:
1654:{\displaystyle k={\sqrt {\frac {-j\omega \mu }{\rho }}}={\frac {1-j}{\delta }}}
421:
410:
406:
226:
5552:
3495:{\displaystyle L/D={\frac {\mu _{0}}{2\pi }}\ln \left({\frac {b}{a}}\right)\,}
2406:{\displaystyle \mathbf {J} (r)=\mathbf {J} (R){\frac {J_{0}(kr)}{J_{0}(kR)}}.}
5861:
5836:
5818:
5560:
4023:
413:
in a vacuum. For example, a 1 MHz radio wave has a wavelength in vacuum
229:
which usually will be distributed evenly over the cross-section of the wire.
3574:
At high frequencies, only the dielectric region has magnetic flux, so that
3564:{\displaystyle L_{\text{DC}}=L_{\text{cen}}+L_{\text{shd}}+L_{\text{ext}}\,}
236:
in a conductor due to an alternating magnetic field according to the law of
5209:
5120:
5095:
4052:
4012:
2890:
1862:
882:
809:
74:
38:
4144:
4109:
conductors) in which the effects from corners are effectively eliminated.
5130:
4074:
4048:
2830:
A portion of a wire's inductance can be attributed to the magnetic field
2426:
1739:
1662:
565:
390:
174:. If the current increases, as in this figure, the resulting increase in
143:
5083:
4816:
4032:
4027:
2819:
1012:{\displaystyle \delta ={\sqrt {{\frac {\,2\rho \,}{\omega \mu }}\,}}~.}
394:
3697:
contributes to the electrical inductance at these higher frequencies.
2970:
2926:
However, when only a single wire is involved, then in addition to the
5628:
5125:
4177:
4124:
116:
109:
93:
46:
5495:
3504:
At low frequencies, all three inductances are fully present so that
158:
2691:, which can cause an additional increase in the AC resistance. The
199:
135:
and distribution systems. It is one of the reasons for preferring
85:
of the conductor, between the outer surface and a level called the
78:
5537:"Study on Cu/Ni nano superlattice conductors for reduced RF loss"
4043:
3320:
The net electrical inductance is due to all three contributions:
2814:
quantity corresponding to a resistance (real) in series with the
3869:
Chen gives an equation of this form for telephone twisted pair:
3157:
is associated with the magnetic field in the region with radius
221:
is a measure of the depth at which the current density falls to
5658:
Feynman, Richard P; Leighton, Robert B; Sands, Matthew (1964).
4980:
4102:
2457:
carrying direct current. The AC resistance of a wire of length
1452:
402:
4654:
1, so its skin depth at a frequency of 50 Hz is given by
4187:
We can derive a practical formula for skin depth as follows:
2846:
of the wire produced by the current in the wire. Unlike that
1705:{\displaystyle \delta ={\sqrt {\frac {2\rho }{\omega \mu }}}}
102:
26:
5512:"Spinning Carbon Nanotubes Spawns New Wireless Applications"
4881:
4836:
4101:
In high current situations where conductors (round or flat
393:
1 radian for each skin depth of penetration. One full
20:
4812:
cookware are unusable because they are not ferromagnetic.
4073:
Litz wire is often used in the windings of high-frequency
5791:
Vander Vorst, Andre; Rosen, Arye; Kotsuka, Youji (2006),
5754:
5198:
4120:
4018:
Skin depth also varies as the inverse square root of the
347:{\displaystyle J=J_{\mathrm {S} }\,e^{-{(1+j)d/\delta }}}
5790:
5312:
2698:
per unit length of a segment of round wire is given by:
16:
Tendency of AC current flow in a conductor's outer layer
4133:
2111:{\displaystyle \mathbf {J} (r)=\mathbf {C} J_{0}(kr),}
5058:
5031:
5004:
4783:
4750:
4660:
4630:
4594:
4562:
4539:
4514:
4482:
4449:
4426:
4401:
4193:
3875:
3676:
3643:
3580:
3510:
3430:
3394:
3326:
3289:
3261:
3223:
3195:
3163:
3135:
2902:
as skin depth becomes small (as frequency increases).
2857:
2704:
2617:
2573:
2487:
2467:
2316:
2303:{\displaystyle {\frac {\mathbf {J} (R)}{J_{0}(kR)}}.}
2252:
2230:
2197:
2175:
2155:
2124:
2060:
1871:
1835:
1788:
1750:
1720:
1673:
1600:
1566:
1532:
1505:
1467:
1421:
1394:
1367:
1181:
1095:
1058:
1027:
969:
932:
893:
857:
818:
792:
757:
721:
682:
656:
634:
602:
576:
548:
433:
360:
286:
168:
flowing through a conductor induces a magnetic field
5102:
was developed for this specific case for metals and
3283:
is associated with the magnetic field in the region
3217:
is associated with the magnetic field in the region
5530:
5528:
3031:coming out of the diagram, dashed black lines with
5680:
5342:Transmission and Distribution of Electrical Energy
5261:High-Speed Signal propagation Advanced Black Magic
5074:
5044:
5017:
4796:
4766:
4726:
4646:
4603:
4571:
4548:
4523:
4498:
4467:
4435:
4410:
4381:
4066:A type of cable called litz wire (from the German
3995:
3689:
3662:
3607:
3563:
3494:
3408:
3379:
3308:
3275:
3242:
3209:
3176:
3149:
2875:
2800:
2673:
2585:
2557:
2473:
2405:
2302:
2238:
2217:
2183:
2161:
2137:
2110:
2046:
1841:
1817:
1768:
1731:
1704:
1653:
1584:
1550:
1516:
1489:
1443:
1405:
1378:
1347:
1137:
1081:
1033:
1011:
955:
909:
873:
843:{\displaystyle \varepsilon _{r}\,\varepsilon _{0}}
842:
801:
773:
737:
706:
665:
640:
620:
585:
557:
528:
366:
346:
5462:(11th ed.), McGraw Hill, p. Table 18–21
2687:or a coil, the AC resistance is also affected by
1497:current density phasor at the surface of the wire
5859:
5793:RF/Microwave Interaction with Biological Tissues
5525:
5458:Fink, Donald G.; Beatty, H. Wayne, eds. (1978),
5153:, a method for estimating skin effect resistance
4035:in power transformers, increasing their losses.
4005:
5381:
5263:(3rd ed.). Prentice Hall. pp. 58–78.
4616:Gold is a good conductor with a resistivity of
5808:
5751:
5541:IEEE Microwave and Wireless Components Letters
5484:
5472:
5335:
5333:
5496:Central Electricity Generating Board (1982).
5258:
5235:(14th ed.). McGraw-Hill. pp. 2–50.
142:The effect was first described in a paper by
5428:harvtxt error: no target: CITEREFReeve1995 (
5306:
4979:, Hayt points out that in a power station a
4556:the conductivity of the medium (for copper,
4092:aluminum cable with a steel reinforcing core
3608:{\displaystyle L_{\infty }=L_{\text{ext}}\,}
2818:(imaginary) due to the wire's internal self-
409:in a conductor is very much slower than the
180:induces separate, circulating eddy currents
5721:Electromagnetic Waves and Radiating Systems
5393:
5330:
5534:
5460:Standard Handbook for Electrical Engineers
5457:
5252:
5233:Standard Handbook for Electrical Engineers
4115:Solid or tubular conductors may be silver-
4055:rods are used for high-frequency welding.
2453:that of a hollow tube with wall thickness
1592:Bessel function of the first kind, order 1
1558:Bessel function of the first kind, order 0
5489:
5441:
5439:
5231:Fink, Donald G.; Beaty, H. Wayne (2000).
5230:
4715:
4670:
4346:
4315:
3992:
3659:
3604:
3560:
3491:
3405:
3376:
3316:, the region inside the shield conductor.
3305:
3272:
3239:
3206:
3184:, the region inside the center conductor.
3173:
3146:
1126:
1119:
1115:
1000:
988:
981:
829:
693:
523:
503:
452:
445:
305:
5769:
5661:The Feynman Lectures on Physics Volume 2
5417:
5387:
5313:Vander Vorst, Rosen & Kotsuka (2006)
5259:Johnson, Howard; Graham, Martin (2003).
5089:
4987:
4143:
2969:
2905:
2889:
1166:
157:
37:
25:
5752:Popovic, Zoya; Popovic, Branko (1999),
5700:
5657:
5611:
5599:
5405:
5369:
5860:
5778:
5718:
5535:Rahimi, A.; Yoon, Y.-K. (2016-03-16).
5436:
5363:
5324:
2567:The final approximation above assumes
139:for long-distance power transmission.
5423:
5339:
5317:
5280:
5075:{\displaystyle {\frac {\delta }{2}}.}
4825:Skin depths at microwave frequencies
2876:{\displaystyle {\frac {\mu }{8\pi }}}
1818:{\displaystyle \mu =\mu _{r}\mu _{0}}
1082:{\displaystyle 1/(\rho \varepsilon )}
956:{\displaystyle 1/(\rho \varepsilon )}
209:Regardless of the driving force, the
162:Cause of skin effect. A main current
5678:
5639:
5587:
5445:
5411:
5399:
5375:
5286:
5195:
4919:8420
4911:9220
2834:the wire itself which is termed the
2596:A convenient formula (attributed to
65:(AC) to become distributed within a
5813:. Vol. 3. pp. 2229–2235.
5151:Wheeler incremental inductance rule
5116:Proximity effect (electromagnetism)
4935:206
4927:652
232:An alternating current may also be
13:
5738:Oliver Heaviside: Sage in Solitude
5629:https://www.jstor.org/stable/99095
4720:
4717:
4611:the frequency of the current in Hz
3912:
3704:
3586:
2664:
2661:
2646:
2643:
2624:
1988:
1984:
1939:
1929:
1873:
1490:{\displaystyle \mathbf {J} (R)={}}
1444:{\displaystyle \mathbf {J} (r)={}}
1386:distance from the axis of the wire
707:{\displaystyle \mu _{r}\,\mu _{0}}
299:
14:
5884:
5853:
2147:Bessel function of the first kind
910:{\displaystyle \varepsilon _{0}=}
874:{\displaystyle \varepsilon _{r}=}
4624:and is essentially nonmagnetic:
3629:) and the inside of the shield (
3409:{\displaystyle L_{\text{ext}}\,}
3276:{\displaystyle L_{\text{shd}}\,}
3210:{\displaystyle L_{\text{ext}}\,}
3150:{\displaystyle L_{\text{cen}}\,}
2942:
2707:
2335:
2318:
2257:
2232:
2218:{\displaystyle \mathbf {J} (R),}
2199:
2177:
2079:
2062:
2025:
1998:
1956:
1910:
1883:
1507:
1469:
1423:
1280:
1206:
1183:
5617:
5605:
5593:
5581:
5504:
5478:
5466:
5451:
4572:{\displaystyle \sigma \approx }
4468:{\displaystyle {\frac {Np}{m}}}
4172:grain-oriented electrical steel
5740:. New York: IEEE Press, 1988.
5719:Jordan, Edward Conrad (1968),
5514:. Sciencedaily.com. 2009-03-09
5357:
5189:
5164:
5045:{\displaystyle {\frac {1}{e}}}
5018:{\displaystyle {\frac {1}{e}}}
4531:the permeability of the medium
4506:the permeability of free space
4271:
4248:
4245:
4233:
3885:
3879:
3690:{\displaystyle L_{\text{ext}}}
2789:
2780:
2765:
2756:
2520:
2508:
2394:
2385:
2370:
2361:
2345:
2339:
2328:
2322:
2291:
2282:
2267:
2261:
2209:
2203:
2102:
2093:
2072:
2066:
2035:
2029:
2008:
2002:
1966:
1960:
1920:
1914:
1893:
1887:
1479:
1473:
1433:
1427:
1339:
1330:
1315:
1306:
1290:
1284:
1270:
1261:
1246:
1237:
1193:
1187:
1076:
1067:
950:
941:
327:
315:
280:from the surface, as follows:
269:from its value at the surface
1:
5795:, John Wiley and Sons, Inc.,
5687:(4th ed.), McGraw-Hill,
5498:Modern Power Station Practice
5293:(5th ed.), McGraw-Hill,
5182:
4767:{\displaystyle {\sqrt {9}}=3}
4061:
4006:Material effect on skin depth
3663:{\displaystyle b<r<c\,}
3309:{\displaystyle b<r<c\,}
3243:{\displaystyle a<r<b\,}
2825:
2420:
1859:electromagnetic wave equation
1852:
1517:{\displaystyle \mathbf {I} =}
1155:
802:{\displaystyle \varepsilon =}
5701:Jackson, John David (1999),
5683:Engineering Electromagnetics
5290:Engineering Electromagnetics
4977:Engineering Electromagnetics
2935:case at higher frequencies.
2586:{\displaystyle D\gg \delta }
2415:
2239:{\displaystyle \mathbf {C} }
2184:{\displaystyle \mathbf {C} }
1034:{\displaystyle \varepsilon }
63:alternating electric current
7:
5772:Electric Transmission Lines
5109:
4139:
4086:High-voltage, high-current
1769:{\displaystyle \mu _{r}={}}
1459:, from the axis of the wire
397:in the conductor requires 2
204:counter-electromotive force
137:high-voltage direct current
133:electric power transmission
10:
5889:
5770:Skilling, Hugh H. (1951),
5485:Xi Nan & Sullivan 2005
5473:Popovic & Popovic 1999
4991:
4176:Fe-Ni – high-permeability
4152:Mn-Zn – magnetically soft
926:At frequencies much below
919:permittivity of free space
783:permeability of free space
256:
127:. It is also important at
92:Skin depth depends on the
18:
5784:Radio Engineers' Handbook
5703:Classical Electrodynamics
5625:The Anomalous Skin Effect
5577:– via ResearchGate.
5553:10.1109/LMWC.2016.2537780
5340:Weeks, Walter L. (1981),
5287:Hayt, William H. (1989),
4647:{\displaystyle \mu _{r}=}
4499:{\displaystyle \mu _{0}=}
774:{\displaystyle \mu _{0}=}
738:{\displaystyle \mu _{r}=}
5819:10.1109/IAS.2005.1518758
5640:Chen, Walter Y. (2004),
5157:
5146:Magnetic Reynolds number
4916:60 Hz
4908:50 Hz
4797:{\displaystyle \mu _{r}}
4549:{\displaystyle \sigma =}
4436:{\displaystyle \alpha =}
4418:the skin depth in meters
4411:{\displaystyle \delta =}
4038:Iron rods work well for
3177:{\displaystyle r<a\,}
1732:{\displaystyle \rho ={}}
1585:{\displaystyle J_{1}={}}
1551:{\displaystyle J_{0}={}}
621:{\displaystyle =2\pi f,}
586:{\displaystyle \omega =}
153:
77:induced by the changing
5786:, New York: McGraw-Hill
5705:(3rd ed.), Wiley,
3744:1 Hz
1712:also called skin depth.
405:, or equivalently, the
367:{\displaystyle \delta }
276:according to the depth
267:decreases exponentially
261:The AC current density
5679:Hayt, William (1981),
5642:Home Networking Basics
5210:10.1098/rstl.1883.0013
5076:
5046:
5019:
4808:, where some types of
4798:
4768:
4728:
4648:
4605:
4573:
4550:
4525:
4500:
4469:
4437:
4412:
4383:
4184:
4158:Al – metallic aluminum
3997:
3691:
3664:
3609:
3565:
3496:
3410:
3381:
3310:
3277:
3244:
3211:
3178:
3151:
3118:
3065:middle frequency, and
2912:
2903:
2877:
2802:
2675:
2587:
2559:
2475:
2407:
2304:
2240:
2219:
2185:
2163:
2139:
2112:
2048:
1843:
1819:
1770:
1733:
1706:
1655:
1586:
1552:
1518:
1491:
1445:
1407:
1380:
1349:
1173:
1139:
1083:
1035:
1013:
957:
911:
875:
844:
803:
775:
739:
708:
667:
642:
622:
587:
559:
558:{\displaystyle \rho =}
530:
368:
348:
194:
131:(50–60 Hz) in AC
61:is the tendency of an
50:
35:
5873:Electrical parameters
5100:Mattis–Bardeen theory
5090:Anomalous skin effect
5077:
5047:
5020:
4988:Electromagnetic waves
4893:Skin depth in copper
4799:
4769:
4729:
4649:
4606:
4574:
4551:
4526:
4524:{\displaystyle \mu =}
4501:
4470:
4438:
4413:
4384:
4164:steel 410 – magnetic
4147:
3998:
3692:
3665:
3610:
3566:
3497:
3411:
3382:
3311:
3278:
3245:
3212:
3179:
3152:
3033:arrowheads = magnetic
2973:
2909:
2893:
2878:
2842:from magnetic fields
2803:
2676:
2588:
2560:
2476:
2474:{\displaystyle \rho }
2408:
2305:
2241:
2220:
2186:
2164:
2140:
2138:{\displaystyle J_{0}}
2113:
2049:
1844:
1820:
1778:magnetic permeability
1771:
1734:
1707:
1656:
1587:
1553:
1519:
1492:
1446:
1408:
1381:
1350:
1170:
1140:
1084:
1036:
1014:
958:
912:
876:
845:
804:
776:
747:magnetic permeability
740:
709:
668:
666:{\displaystyle \mu =}
643:
623:
588:
560:
531:
369:
349:
161:
123:(or waveguides), and
41:
29:
5344:, Harper & Row,
5056:
5029:
5002:
4956:100 MHz
4932:100 kHz
4781:
4774:times that of gold.
4748:
4658:
4628:
4592:
4560:
4537:
4512:
4480:
4447:
4424:
4399:
4191:
4161:Cu – metallic copper
4088:overhead power lines
3873:
3795:100 kHz
3674:
3641:
3578:
3508:
3428:
3424:of a coaxial cable:
3392:
3324:
3287:
3259:
3221:
3193:
3161:
3133:
2855:
2810:This impedance is a
2702:
2615:
2571:
2485:
2465:
2314:
2250:
2228:
2195:
2173:
2153:
2122:
2058:
1869:
1833:
1786:
1748:
1718:
1671:
1598:
1564:
1530:
1524:total current phasor
1503:
1465:
1419:
1406:{\displaystyle R={}}
1392:
1379:{\displaystyle r={}}
1365:
1179:
1093:
1056:
1025:
967:
930:
891:
855:
816:
790:
755:
719:
680:
654:
632:
600:
574:
546:
431:
358:
284:
242:electromagnetic wave
45:use stranded coils (
5402:, pp. 434–439)
5390:, pp. 157–159)
4948:10 MHz
4924:10 kHz
4894:
4826:
4443:the attenuation in
3778:10 kHz
3115:external inductance
2983:insulating sheath,
2947:Let the dimensions
2932:internal inductance
2928:external inductance
2836:internal inductance
2822:, per unit length.
2600:) for the diameter
2449:, has a resistance
32:alternating current
5664:. Addison-Wesley.
5072:
5042:
5015:
4964:1 GHz
4951:20.6
4943:65.2
4940:1 MHz
4892:
4824:
4794:
4764:
4724:
4644:
4604:{\displaystyle f=}
4601:
4569:
4546:
4521:
4496:
4465:
4433:
4408:
4379:
4377:
4185:
3993:
3846:5 MHz
3829:2 MHz
3812:1 MHz
3761:1 kHz
3687:
3660:
3605:
3561:
3492:
3406:
3377:
3306:
3273:
3240:
3207:
3174:
3147:
3119:
3020:into the diagram,
2913:
2904:
2873:
2798:
2671:
2583:
2555:
2471:
2445:large compared to
2403:
2300:
2236:
2215:
2181:
2159:
2135:
2108:
2044:
1839:
1815:
1766:
1729:
1702:
1651:
1582:
1548:
1514:
1487:
1441:
1413:radius of the wire
1403:
1376:
1345:
1174:
1135:
1079:
1031:
1009:
953:
907:
871:
840:
812:of the conductor,
799:
771:
735:
704:
676:of the conductor,
663:
638:
618:
583:
555:
526:
364:
344:
195:
121:transmission lines
51:
36:
5828:978-0-7803-9208-3
5802:978-0-471-73277-8
5763:978-0-201-32678-9
5756:, Prentice-Hall,
5730:978-0-13-249995-8
5723:, Prentice Hall,
5694:978-0-07-027395-5
5651:978-0-13-016511-4
5644:, Prentice Hall,
5500:. Pergamon Press.
5270:978-0-13-084408-8
5242:978-0-07-022005-8
5141:Induction heating
5136:Induction cooking
5067:
5040:
5013:
4994:Penetration depth
4971:
4970:
4875:
4874:
4806:induction cookers
4756:
4707:
4706:
4463:
4370:
4369:
4338:
4337:
4307:
4306:
4276:
4275:
4216:
3990:
3977:
3937:
3862:
3861:
3684:
3601:
3557:
3544:
3531:
3518:
3485:
3465:
3402:
3373:
3360:
3347:
3334:
3269:
3203:
3143:
2871:
2793:
2741:
2714:
2669:
2668:
2641:
2553:
2527:
2398:
2295:
2162:{\displaystyle 0}
1995:
1980:
1953:
1842:{\displaystyle k}
1700:
1699:
1649:
1628:
1627:
1343:
1274:
1222:
1131:
1127:
1124:
1005:
1001:
998:
648:is the frequency.
641:{\displaystyle f}
594:angular frequency
524:
504:
462:
129:mains frequencies
43:Induction cookers
5880:
5868:Electromagnetism
5848:
5805:
5787:
5775:
5766:
5733:
5715:
5697:
5686:
5675:
5654:
5631:
5623:R. G. Chambers,
5621:
5615:
5614:, p. 32-11)
5609:
5603:
5597:
5591:
5585:
5579:
5578:
5576:
5575:
5532:
5523:
5522:
5520:
5519:
5508:
5502:
5501:
5493:
5487:
5482:
5476:
5470:
5464:
5463:
5455:
5449:
5443:
5434:
5433:
5421:
5415:
5409:
5403:
5397:
5391:
5385:
5379:
5373:
5367:
5361:
5355:
5354:
5337:
5328:
5321:
5315:
5310:
5304:
5303:
5284:
5278:
5277:
5256:
5250:
5249:
5228:
5222:
5221:
5193:
5176:
5168:
5081:
5079:
5078:
5073:
5068:
5060:
5051:
5049:
5048:
5043:
5041:
5033:
5024:
5022:
5021:
5016:
5014:
5006:
4895:
4891:
4827:
4823:
4803:
4801:
4800:
4795:
4793:
4792:
4773:
4771:
4770:
4765:
4757:
4752:
4743:
4741:
4733:
4731:
4730:
4725:
4723:
4708:
4705:
4694:
4693:
4692:
4673:
4672:
4653:
4651:
4650:
4645:
4640:
4639:
4623:
4621:
4610:
4608:
4607:
4602:
4585:
4583:
4578:
4576:
4575:
4570:
4555:
4553:
4552:
4547:
4530:
4528:
4527:
4522:
4505:
4503:
4502:
4497:
4492:
4491:
4474:
4472:
4471:
4466:
4464:
4459:
4451:
4442:
4440:
4439:
4434:
4417:
4415:
4414:
4409:
4388:
4386:
4385:
4380:
4378:
4371:
4362:
4361:
4352:
4348:
4339:
4336:
4332:
4331:
4318:
4317:
4308:
4293:
4289:
4281:
4277:
4274:
4270:
4269:
4260:
4259:
4231:
4223:
4222:
4217:
4209:
4136:communications.
4080:carbon nanotubes
4002:
4000:
3999:
3994:
3991:
3989:
3988:
3987:
3982:
3978:
3976:
3975:
3963:
3949:
3948:
3947:
3942:
3938:
3936:
3935:
3923:
3916:
3915:
3903:
3902:
3892:
3714:
3713:
3696:
3694:
3693:
3688:
3686:
3685:
3682:
3669:
3667:
3666:
3661:
3614:
3612:
3611:
3606:
3603:
3602:
3599:
3590:
3589:
3570:
3568:
3567:
3562:
3559:
3558:
3555:
3546:
3545:
3542:
3533:
3532:
3529:
3520:
3519:
3516:
3501:
3499:
3498:
3493:
3490:
3486:
3478:
3466:
3464:
3456:
3455:
3446:
3438:
3415:
3413:
3412:
3407:
3404:
3403:
3400:
3386:
3384:
3383:
3378:
3375:
3374:
3371:
3362:
3361:
3358:
3349:
3348:
3345:
3336:
3335:
3332:
3315:
3313:
3312:
3307:
3282:
3280:
3279:
3274:
3271:
3270:
3267:
3255:The inductance
3249:
3247:
3246:
3241:
3216:
3214:
3213:
3208:
3205:
3204:
3201:
3189:The inductance
3183:
3181:
3180:
3175:
3156:
3154:
3153:
3148:
3145:
3144:
3141:
3129:The inductance
3111:
3110:
3103:
3102:
3095:
3094:
3087:
3086:
3079:
3078:
3071:
3070:
3063:
3062:
3055:
3054:
3047:
3046:
3039:
3034:
3030:
3028:
3027:
3019:
3017:
3016:
3008:
3006:
3004:
3003:
2993:
2991:
2990:
2982:
2980:
2965:
2964:
2958:
2954:
2950:
2901:
2882:
2880:
2879:
2874:
2872:
2870:
2859:
2807:
2805:
2804:
2799:
2794:
2792:
2779:
2778:
2768:
2755:
2754:
2744:
2742:
2740:
2729:
2721:
2716:
2715:
2712:
2710:
2689:proximity effect
2680:
2678:
2677:
2672:
2670:
2667:
2659:
2651:
2650:
2649:
2639:
2634:
2629:
2628:
2627:
2610:
2603:
2592:
2590:
2589:
2584:
2564:
2562:
2561:
2556:
2554:
2552:
2541:
2533:
2528:
2526:
2503:
2495:
2480:
2478:
2477:
2472:
2461:and resistivity
2460:
2456:
2448:
2444:
2440:
2436:
2432:
2412:
2410:
2409:
2404:
2399:
2397:
2384:
2383:
2373:
2360:
2359:
2349:
2338:
2321:
2309:
2307:
2306:
2301:
2296:
2294:
2281:
2280:
2270:
2260:
2254:
2245:
2243:
2242:
2237:
2235:
2224:
2222:
2221:
2216:
2202:
2190:
2188:
2187:
2182:
2180:
2168:
2166:
2165:
2160:
2144:
2142:
2141:
2136:
2134:
2133:
2117:
2115:
2114:
2109:
2092:
2091:
2082:
2065:
2053:
2051:
2050:
2045:
2028:
2023:
2022:
2001:
1996:
1994:
1983:
1981:
1973:
1959:
1954:
1952:
1951:
1950:
1937:
1936:
1927:
1913:
1908:
1907:
1886:
1881:
1880:
1848:
1846:
1845:
1840:
1824:
1822:
1821:
1816:
1814:
1813:
1804:
1803:
1780:of the conductor
1775:
1773:
1772:
1767:
1765:
1760:
1759:
1742:of the conductor
1738:
1736:
1735:
1730:
1728:
1711:
1709:
1708:
1703:
1701:
1698:
1690:
1682:
1681:
1665:in the conductor
1660:
1658:
1657:
1652:
1650:
1645:
1634:
1629:
1623:
1609:
1608:
1591:
1589:
1588:
1583:
1581:
1576:
1575:
1557:
1555:
1554:
1549:
1547:
1542:
1541:
1523:
1521:
1520:
1515:
1510:
1496:
1494:
1493:
1488:
1486:
1472:
1451:current density
1450:
1448:
1447:
1442:
1440:
1426:
1412:
1410:
1409:
1404:
1402:
1385:
1383:
1382:
1377:
1375:
1354:
1352:
1351:
1346:
1344:
1342:
1329:
1328:
1318:
1305:
1304:
1294:
1283:
1275:
1273:
1260:
1259:
1249:
1236:
1235:
1225:
1223:
1221:
1210:
1209:
1200:
1186:
1162:Bessel functions
1151:
1144:
1142:
1141:
1136:
1129:
1128:
1125:
1120:
1113:
1111:
1109:
1088:
1086:
1085:
1080:
1066:
1048:
1043:plasma frequency
1040:
1038:
1037:
1032:
1018:
1016:
1015:
1010:
1003:
1002:
999:
997:
989:
979:
977:
962:
960:
959:
954:
940:
916:
914:
913:
908:
903:
902:
885:of the conductor
880:
878:
877:
872:
867:
866:
849:
847:
846:
841:
839:
838:
828:
827:
808:
806:
805:
800:
780:
778:
777:
772:
767:
766:
749:of the conductor
744:
742:
741:
736:
731:
730:
713:
711:
710:
705:
703:
702:
692:
691:
672:
670:
669:
664:
647:
645:
644:
639:
627:
625:
624:
619:
592:
590:
589:
584:
568:of the conductor
564:
562:
561:
556:
535:
533:
532:
527:
525:
522:
518:
505:
502:
501:
496:
492:
470:
463:
461:
453:
443:
441:
416:
400:
385:
382:(about 0.37) of
373:
371:
370:
365:
353:
351:
350:
345:
343:
342:
341:
337:
304:
303:
302:
279:
272:
264:
251:normal incidence
191:
190:
184:
178:
172:
166:
148:Oliver Heaviside
55:electromagnetism
5888:
5887:
5883:
5882:
5881:
5879:
5878:
5877:
5858:
5857:
5856:
5851:
5829:
5803:
5764:
5736:Nahin, Paul J.
5731:
5713:
5695:
5672:
5652:
5635:
5634:
5622:
5618:
5610:
5606:
5598:
5594:
5590:, pp. 401)
5586:
5582:
5573:
5571:
5533:
5526:
5517:
5515:
5510:
5509:
5505:
5494:
5490:
5483:
5479:
5471:
5467:
5456:
5452:
5444:
5437:
5427:
5422:
5418:
5410:
5406:
5398:
5394:
5386:
5382:
5378:, pp. 303)
5374:
5370:
5362:
5358:
5352:
5338:
5331:
5322:
5318:
5311:
5307:
5301:
5285:
5281:
5271:
5257:
5253:
5243:
5229:
5225:
5194:
5190:
5185:
5180:
5179:
5169:
5165:
5160:
5112:
5104:superconductors
5092:
5059:
5057:
5054:
5053:
5032:
5030:
5027:
5026:
5005:
5003:
5000:
4999:
4996:
4990:
4902:
4834:
4810:stainless steel
4788:
4784:
4782:
4779:
4778:
4751:
4749:
4746:
4745:
4739:
4737:
4716:
4695:
4685:
4681:
4674:
4671:
4659:
4656:
4655:
4635:
4631:
4629:
4626:
4625:
4619:
4617:
4614:
4593:
4590:
4589:
4581:
4579:
4561:
4558:
4557:
4538:
4535:
4534:
4513:
4510:
4509:
4487:
4483:
4481:
4478:
4477:
4452:
4450:
4448:
4445:
4444:
4425:
4422:
4421:
4400:
4397:
4396:
4376:
4375:
4357:
4353:
4347:
4327:
4323:
4322:
4316:
4288:
4279:
4278:
4265:
4261:
4255:
4251:
4232:
4224:
4221:
4208:
4201:
4194:
4192:
4189:
4188:
4183:
4166:stainless steel
4142:
4064:
4008:
3983:
3971:
3967:
3962:
3958:
3957:
3950:
3943:
3931:
3927:
3922:
3918:
3917:
3911:
3907:
3898:
3894:
3893:
3891:
3874:
3871:
3870:
3738:
3733:
3728:
3723:
3718:
3707:
3705:Telephone cable
3681:
3677:
3675:
3672:
3671:
3642:
3639:
3638:
3598:
3594:
3585:
3581:
3579:
3576:
3575:
3554:
3550:
3541:
3537:
3528:
3524:
3515:
3511:
3509:
3506:
3505:
3477:
3473:
3457:
3451:
3447:
3445:
3434:
3429:
3426:
3425:
3399:
3395:
3393:
3390:
3389:
3370:
3366:
3357:
3353:
3344:
3340:
3331:
3327:
3325:
3322:
3321:
3288:
3285:
3284:
3266:
3262:
3260:
3257:
3256:
3222:
3219:
3218:
3200:
3196:
3194:
3191:
3190:
3162:
3159:
3158:
3140:
3136:
3134:
3131:
3130:
3108:
3107:
3100:
3099:
3092:
3091:
3084:
3083:
3076:
3075:
3068:
3067:
3060:
3059:
3057:low frequency,
3052:
3051:
3049:non-energized,
3044:
3043:
3037:
3032:
3023:
3022:
3021:
3012:
3011:
3010:
2999:
2998:
2996:
2995:
2986:
2985:
2984:
2976:
2975:
2962:
2961:
2956:
2952:
2948:
2945:
2915:
2899:
2863:
2858:
2856:
2853:
2852:
2828:
2774:
2770:
2769:
2750:
2746:
2745:
2743:
2730:
2722:
2720:
2711:
2706:
2705:
2703:
2700:
2699:
2660:
2655:
2642:
2635:
2633:
2623:
2622:
2618:
2616:
2613:
2612:
2608:
2606:
2601:
2572:
2569:
2568:
2542:
2534:
2532:
2504:
2496:
2494:
2486:
2483:
2482:
2466:
2463:
2462:
2458:
2454:
2446:
2442:
2438:
2434:
2430:
2423:
2418:
2379:
2375:
2374:
2355:
2351:
2350:
2348:
2334:
2317:
2315:
2312:
2311:
2276:
2272:
2271:
2256:
2255:
2253:
2251:
2248:
2247:
2231:
2229:
2226:
2225:
2198:
2196:
2193:
2192:
2176:
2174:
2171:
2170:
2154:
2151:
2150:
2129:
2125:
2123:
2120:
2119:
2087:
2083:
2078:
2061:
2059:
2056:
2055:
2024:
2018:
2014:
1997:
1987:
1982:
1972:
1955:
1946:
1942:
1938:
1932:
1928:
1926:
1909:
1903:
1899:
1882:
1876:
1872:
1870:
1867:
1866:
1855:
1834:
1831:
1830:
1827:
1809:
1805:
1799:
1795:
1787:
1784:
1783:
1764:
1755:
1751:
1749:
1746:
1745:
1727:
1719:
1716:
1715:
1691:
1683:
1680:
1672:
1669:
1668:
1635:
1633:
1610:
1607:
1599:
1596:
1595:
1580:
1571:
1567:
1565:
1562:
1561:
1546:
1537:
1533:
1531:
1528:
1527:
1506:
1504:
1501:
1500:
1485:
1468:
1466:
1463:
1462:
1439:
1422:
1420:
1417:
1416:
1401:
1393:
1390:
1389:
1374:
1366:
1363:
1362:
1324:
1320:
1319:
1300:
1296:
1295:
1293:
1279:
1255:
1251:
1250:
1231:
1227:
1226:
1224:
1211:
1205:
1201:
1199:
1182:
1180:
1177:
1176:
1158:
1149:
1114:
1112:
1110:
1102:
1094:
1091:
1090:
1062:
1057:
1054:
1053:
1046:
1026:
1023:
1022:
990:
980:
978:
976:
968:
965:
964:
936:
931:
928:
927:
924:
898:
894:
892:
889:
888:
862:
858:
856:
853:
852:
834:
830:
823:
819:
817:
814:
813:
791:
788:
787:
762:
758:
756:
753:
752:
726:
722:
720:
717:
716:
698:
694:
687:
683:
681:
678:
677:
655:
652:
651:
633:
630:
629:
601:
598:
597:
575:
572:
571:
547:
544:
543:
497:
482:
478:
477:
469:
468:
464:
454:
444:
442:
440:
432:
429:
428:
419:
414:
398:
388:
383:
359:
356:
355:
333:
314:
310:
306:
298:
297:
293:
285:
282:
281:
277:
275:
270:
265:in a conductor
262:
259:
211:current density
192:
188:
187:
182:
176:
170:
164:
156:
71:current density
24:
17:
12:
11:
5:
5886:
5876:
5875:
5870:
5855:
5854:External links
5852:
5850:
5849:
5827:
5806:
5801:
5788:
5776:
5767:
5762:
5749:
5734:
5729:
5716:
5712:978-0471309321
5711:
5698:
5693:
5676:
5670:
5655:
5650:
5636:
5633:
5632:
5616:
5604:
5602:, p. 353)
5592:
5580:
5547:(4): 258–260.
5524:
5503:
5488:
5477:
5465:
5450:
5435:
5426:, p. 558)
5416:
5414:, p. 434)
5404:
5392:
5388:Skilling (1951
5380:
5368:
5356:
5351:978-0060469825
5350:
5329:
5327:, p. 130)
5316:
5305:
5300:978-0070274068
5299:
5279:
5269:
5251:
5241:
5223:
5187:
5186:
5184:
5181:
5178:
5177:
5162:
5161:
5159:
5156:
5155:
5154:
5148:
5143:
5138:
5133:
5128:
5123:
5118:
5111:
5108:
5091:
5088:
5071:
5066:
5063:
5039:
5036:
5012:
5009:
4989:
4986:
4973:
4972:
4969:
4968:
4965:
4961:
4960:
4957:
4953:
4952:
4949:
4945:
4944:
4941:
4937:
4936:
4933:
4929:
4928:
4925:
4921:
4920:
4917:
4913:
4912:
4909:
4905:
4904:
4899:
4877:
4876:
4873:
4872:
4869:
4865:
4864:
4861:
4857:
4856:
4853:
4849:
4848:
4845:
4841:
4840:
4831:
4791:
4787:
4763:
4760:
4755:
4722:
4719:
4714:
4711:
4704:
4701:
4698:
4691:
4688:
4684:
4680:
4677:
4669:
4666:
4663:
4643:
4638:
4634:
4613:
4612:
4600:
4597:
4587:
4568:
4565:
4545:
4542:
4532:
4520:
4517:
4507:
4495:
4490:
4486:
4475:
4462:
4458:
4455:
4432:
4429:
4419:
4407:
4404:
4393:
4374:
4368:
4365:
4360:
4356:
4351:
4345:
4342:
4335:
4330:
4326:
4321:
4314:
4311:
4305:
4302:
4299:
4296:
4292:
4287:
4284:
4282:
4280:
4273:
4268:
4264:
4258:
4254:
4250:
4247:
4244:
4241:
4238:
4235:
4230:
4227:
4220:
4215:
4212:
4207:
4204:
4202:
4200:
4197:
4196:
4182:
4181:
4174:
4168:
4162:
4159:
4156:
4149:
4141:
4138:
4063:
4060:
4051:itself. Only
4040:direct-current
4007:
4004:
3986:
3981:
3974:
3970:
3966:
3961:
3956:
3953:
3946:
3941:
3934:
3930:
3926:
3921:
3914:
3910:
3906:
3901:
3897:
3890:
3887:
3884:
3881:
3878:
3864:
3863:
3860:
3859:
3856:
3853:
3850:
3847:
3843:
3842:
3839:
3836:
3833:
3830:
3826:
3825:
3822:
3819:
3816:
3813:
3809:
3808:
3805:
3802:
3799:
3796:
3792:
3791:
3788:
3785:
3782:
3779:
3775:
3774:
3771:
3768:
3765:
3762:
3758:
3757:
3754:
3751:
3748:
3745:
3741:
3740:
3735:
3730:
3725:
3720:
3706:
3703:
3680:
3658:
3655:
3652:
3649:
3646:
3597:
3593:
3588:
3584:
3553:
3549:
3540:
3536:
3527:
3523:
3514:
3489:
3484:
3481:
3476:
3472:
3469:
3463:
3460:
3454:
3450:
3444:
3441:
3437:
3433:
3398:
3369:
3365:
3356:
3352:
3343:
3339:
3330:
3318:
3317:
3304:
3301:
3298:
3295:
3292:
3265:
3252:
3251:
3238:
3235:
3232:
3229:
3226:
3199:
3186:
3185:
3172:
3169:
3166:
3139:
2944:
2941:
2869:
2866:
2862:
2827:
2824:
2797:
2791:
2788:
2785:
2782:
2777:
2773:
2767:
2764:
2761:
2758:
2753:
2749:
2739:
2736:
2733:
2728:
2725:
2719:
2709:
2666:
2663:
2658:
2654:
2648:
2645:
2638:
2632:
2626:
2621:
2604:
2582:
2579:
2576:
2551:
2548:
2545:
2540:
2537:
2531:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2502:
2499:
2493:
2490:
2470:
2422:
2419:
2417:
2414:
2402:
2396:
2393:
2390:
2387:
2382:
2378:
2372:
2369:
2366:
2363:
2358:
2354:
2347:
2344:
2341:
2337:
2333:
2330:
2327:
2324:
2320:
2299:
2293:
2290:
2287:
2284:
2279:
2275:
2269:
2266:
2263:
2259:
2234:
2214:
2211:
2208:
2205:
2201:
2179:
2158:
2132:
2128:
2107:
2104:
2101:
2098:
2095:
2090:
2086:
2081:
2077:
2074:
2071:
2068:
2064:
2043:
2040:
2037:
2034:
2031:
2027:
2021:
2017:
2013:
2010:
2007:
2004:
2000:
1993:
1990:
1986:
1979:
1976:
1971:
1968:
1965:
1962:
1958:
1949:
1945:
1941:
1935:
1931:
1925:
1922:
1919:
1916:
1912:
1906:
1902:
1898:
1895:
1892:
1889:
1885:
1879:
1875:
1857:Combining the
1854:
1851:
1838:
1826:
1825:
1812:
1808:
1802:
1798:
1794:
1791:
1781:
1763:
1758:
1754:
1743:
1726:
1723:
1713:
1697:
1694:
1689:
1686:
1679:
1676:
1666:
1648:
1644:
1641:
1638:
1632:
1626:
1622:
1619:
1616:
1613:
1606:
1603:
1593:
1579:
1574:
1570:
1559:
1545:
1540:
1536:
1525:
1513:
1509:
1498:
1484:
1481:
1478:
1475:
1471:
1460:
1438:
1435:
1432:
1429:
1425:
1414:
1400:
1397:
1387:
1373:
1370:
1359:
1341:
1338:
1335:
1332:
1327:
1323:
1317:
1314:
1311:
1308:
1303:
1299:
1292:
1289:
1286:
1282:
1278:
1272:
1269:
1266:
1263:
1258:
1254:
1248:
1245:
1242:
1239:
1234:
1230:
1220:
1217:
1214:
1208:
1204:
1198:
1195:
1192:
1189:
1185:
1157:
1154:
1148:100 kHz (
1134:
1123:
1118:
1108:
1105:
1101:
1098:
1078:
1075:
1072:
1069:
1065:
1061:
1030:
1008:
996:
993:
987:
984:
975:
972:
952:
949:
946:
943:
939:
935:
923:
922:
906:
901:
897:
886:
870:
865:
861:
850:
837:
833:
826:
822:
798:
795:
785:
770:
765:
761:
750:
734:
729:
725:
714:
701:
697:
690:
686:
662:
659:
649:
637:
617:
614:
611:
608:
605:
582:
579:
569:
554:
551:
540:
521:
517:
514:
511:
508:
500:
495:
491:
488:
485:
481:
476:
473:
467:
460:
457:
451:
448:
439:
436:
417:
411:speed of light
407:phase velocity
386:
374:is called the
363:
340:
336:
332:
329:
326:
323:
320:
317:
313:
309:
301:
296:
292:
289:
273:
258:
255:
227:direct current
186:
155:
152:
69:such that the
15:
9:
6:
4:
3:
2:
5885:
5874:
5871:
5869:
5866:
5865:
5863:
5846:
5842:
5838:
5834:
5830:
5824:
5820:
5816:
5812:
5807:
5804:
5798:
5794:
5789:
5785:
5781:
5780:Terman, F. E.
5777:
5774:, McGraw-Hill
5773:
5768:
5765:
5759:
5755:
5750:
5747:
5746:0-87942-238-6
5743:
5739:
5735:
5732:
5726:
5722:
5717:
5714:
5708:
5704:
5699:
5696:
5690:
5685:
5684:
5677:
5673:
5671:0-201-02117-X
5667:
5663:
5662:
5656:
5653:
5647:
5643:
5638:
5637:
5630:
5626:
5620:
5613:
5612:Feynman (1964
5608:
5601:
5600:Jackson (1999
5596:
5589:
5584:
5570:
5566:
5562:
5558:
5554:
5550:
5546:
5542:
5538:
5531:
5529:
5513:
5507:
5499:
5492:
5486:
5481:
5475:, p. 385
5474:
5469:
5461:
5454:
5448:, p. 26)
5447:
5442:
5440:
5431:
5425:
5420:
5413:
5408:
5401:
5396:
5389:
5384:
5377:
5372:
5365:
5360:
5353:
5347:
5343:
5336:
5334:
5326:
5320:
5314:
5309:
5302:
5296:
5292:
5291:
5283:
5276:
5272:
5266:
5262:
5255:
5248:
5244:
5238:
5234:
5227:
5219:
5215:
5211:
5207:
5203:
5199:
5192:
5188:
5174:
5167:
5163:
5152:
5149:
5147:
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5113:
5107:
5105:
5101:
5097:
5087:
5085:
5069:
5064:
5061:
5037:
5034:
5010:
5007:
4995:
4985:
4982:
4978:
4966:
4963:
4962:
4958:
4955:
4954:
4950:
4947:
4946:
4942:
4939:
4938:
4934:
4931:
4930:
4926:
4923:
4922:
4918:
4915:
4914:
4910:
4907:
4906:
4900:
4897:
4896:
4890:
4889:
4888:
4885:
4883:
4870:
4867:
4866:
4862:
4859:
4858:
4854:
4851:
4850:
4846:
4843:
4842:
4838:
4832:
4829:
4828:
4822:
4821:
4820:
4818:
4813:
4811:
4807:
4789:
4785:
4775:
4761:
4758:
4753:
4734:
4712:
4709:
4702:
4699:
4696:
4689:
4686:
4682:
4678:
4675:
4667:
4664:
4661:
4641:
4636:
4632:
4598:
4595:
4588:
4566:
4563:
4543:
4540:
4533:
4518:
4515:
4508:
4493:
4488:
4484:
4476:
4460:
4456:
4453:
4430:
4427:
4420:
4405:
4402:
4395:
4394:
4392:
4389:
4372:
4366:
4363:
4358:
4354:
4349:
4343:
4340:
4333:
4328:
4324:
4319:
4312:
4309:
4303:
4300:
4297:
4294:
4290:
4285:
4283:
4266:
4262:
4256:
4252:
4242:
4239:
4236:
4228:
4225:
4218:
4213:
4210:
4205:
4203:
4198:
4180:(80%Ni-20%Fe)
4179:
4175:
4173:
4169:
4167:
4163:
4160:
4157:
4155:
4151:
4150:
4146:
4137:
4135:
4129:
4126:
4122:
4118:
4113:
4110:
4108:
4104:
4099:
4095:
4093:
4089:
4084:
4081:
4076:
4071:
4069:
4059:
4056:
4054:
4050:
4045:
4041:
4036:
4034:
4029:
4025:
4024:ferromagnetic
4021:
4016:
4014:
4003:
3984:
3979:
3972:
3968:
3964:
3959:
3954:
3951:
3944:
3939:
3932:
3928:
3924:
3919:
3908:
3904:
3899:
3895:
3888:
3882:
3876:
3867:
3857:
3854:
3851:
3848:
3845:
3844:
3840:
3837:
3834:
3831:
3828:
3827:
3823:
3820:
3817:
3814:
3811:
3810:
3806:
3803:
3800:
3797:
3794:
3793:
3789:
3786:
3783:
3780:
3777:
3776:
3772:
3769:
3766:
3763:
3760:
3759:
3755:
3752:
3749:
3746:
3743:
3742:
3736:
3731:
3726:
3721:
3716:
3715:
3712:
3711:
3710:
3702:
3698:
3678:
3656:
3653:
3650:
3647:
3644:
3636:
3633: =
3632:
3628:
3625: =
3624:
3619:
3616:
3595:
3591:
3582:
3572:
3551:
3547:
3538:
3534:
3525:
3521:
3512:
3502:
3487:
3482:
3479:
3474:
3470:
3467:
3461:
3458:
3452:
3448:
3442:
3439:
3435:
3431:
3423:
3419:
3396:
3387:
3367:
3363:
3354:
3350:
3341:
3337:
3328:
3302:
3299:
3296:
3293:
3290:
3263:
3254:
3253:
3236:
3233:
3230:
3227:
3224:
3197:
3188:
3187:
3170:
3167:
3164:
3137:
3128:
3127:
3126:
3123:
3116:
3112:
3104:
3096:
3088:
3080:
3072:
3064:
3056:
3048:
3040:
3026:
3015:
3002:
2989:
2979:
2972:
2968:
2966:
2943:Coaxial cable
2940:
2936:
2933:
2929:
2923:
2920:
2908:
2897:
2892:
2888:
2886:
2867:
2864:
2860:
2849:
2845:
2841:
2837:
2833:
2823:
2821:
2817:
2813:
2808:
2795:
2786:
2783:
2775:
2771:
2762:
2759:
2751:
2747:
2737:
2734:
2731:
2726:
2723:
2717:
2697:
2694:
2690:
2686:
2681:
2656:
2652:
2636:
2630:
2619:
2599:
2594:
2580:
2577:
2574:
2565:
2549:
2546:
2543:
2538:
2535:
2529:
2523:
2517:
2514:
2511:
2505:
2500:
2497:
2491:
2488:
2468:
2452:
2451:approximately
2428:
2413:
2400:
2391:
2388:
2380:
2376:
2367:
2364:
2356:
2352:
2342:
2331:
2325:
2297:
2288:
2285:
2277:
2273:
2264:
2212:
2206:
2156:
2148:
2130:
2126:
2105:
2099:
2096:
2088:
2084:
2075:
2069:
2041:
2038:
2032:
2019:
2015:
2011:
2005:
1991:
1977:
1974:
1969:
1963:
1947:
1943:
1933:
1923:
1917:
1904:
1900:
1896:
1890:
1877:
1864:
1860:
1850:
1836:
1810:
1806:
1800:
1796:
1792:
1789:
1782:
1779:
1761:
1756:
1752:
1744:
1741:
1724:
1721:
1714:
1695:
1692:
1687:
1684:
1677:
1674:
1667:
1664:
1646:
1642:
1639:
1636:
1630:
1624:
1620:
1617:
1614:
1611:
1604:
1601:
1594:
1577:
1572:
1568:
1560:
1543:
1538:
1534:
1526:
1511:
1499:
1482:
1476:
1461:
1458:
1455:at distance,
1454:
1436:
1430:
1415:
1398:
1395:
1388:
1371:
1368:
1361:
1360:
1358:
1355:
1336:
1333:
1325:
1321:
1312:
1309:
1301:
1297:
1287:
1276:
1267:
1264:
1256:
1252:
1243:
1240:
1232:
1228:
1218:
1215:
1212:
1202:
1196:
1190:
1169:
1165:
1163:
1153:
1145:
1132:
1121:
1116:
1106:
1103:
1099:
1096:
1073:
1070:
1063:
1059:
1050:
1044:
1028:
1019:
1006:
994:
991:
985:
982:
973:
970:
947:
944:
937:
933:
920:
904:
899:
895:
887:
884:
868:
863:
859:
851:
835:
831:
824:
820:
811:
796:
793:
786:
784:
768:
763:
759:
751:
748:
732:
727:
723:
715:
699:
695:
688:
684:
675:
660:
657:
650:
635:
615:
612:
609:
606:
603:
595:
580:
577:
570:
567:
552:
549:
542:
541:
539:
536:
519:
515:
512:
509:
506:
498:
493:
489:
486:
483:
479:
474:
471:
465:
458:
455:
449:
446:
437:
434:
425:
423:
412:
408:
404:
396:
392:
381:
377:
361:
338:
334:
330:
324:
321:
318:
311:
307:
294:
290:
287:
268:
254:
252:
247:
243:
239:
235:
230:
228:
224:
220:
216:
212:
207:
205:
201:
185:
179:
173:
167:
160:
151:
149:
145:
140:
138:
134:
130:
126:
122:
118:
113:
111:
106:
104:
100:
95:
90:
88:
84:
80:
76:
75:eddy currents
72:
68:
64:
60:
56:
48:
44:
40:
33:
28:
22:
5810:
5792:
5783:
5771:
5753:
5737:
5720:
5702:
5682:
5660:
5641:
5624:
5619:
5607:
5595:
5583:
5572:. Retrieved
5544:
5540:
5516:. Retrieved
5506:
5497:
5491:
5480:
5468:
5459:
5453:
5419:
5407:
5395:
5383:
5371:
5366:, p. ??
5359:
5341:
5325:Jordan (1968
5319:
5308:
5289:
5282:
5274:
5260:
5254:
5246:
5232:
5226:
5201:
5197:
5191:
5172:
5166:
5121:Eddy current
5096:Heinz London
5093:
4997:
4976:
4974:
4967:2.06
4959:6.52
4886:
4878:
4814:
4776:
4735:
4615:
4390:
4186:
4130:
4114:
4111:
4106:
4100:
4096:
4085:
4078:composed of
4075:transformers
4072:
4067:
4065:
4057:
4053:non-magnetic
4037:
4020:permeability
4017:
4009:
3868:
3865:
3708:
3699:
3634:
3630:
3626:
3622:
3620:
3617:
3573:
3503:
3421:
3417:
3388:
3319:
3124:
3120:
3114:
3106:
3098:
3090:
3082:
3074:
3066:
3058:
3050:
3042:
3036:
3024:
3013:
3007:= dielectric
3000:
2987:
2977:
2960:
2946:
2937:
2931:
2927:
2924:
2918:
2914:
2895:
2847:
2843:
2835:
2831:
2829:
2809:
2692:
2682:
2595:
2566:
2450:
2424:
1856:
1828:
1456:
1356:
1175:
1159:
1146:
1051:
1020:
925:
883:permittivity
810:permittivity
674:permeability
537:
426:
375:
260:
245:
233:
231:
218:
214:
208:
196:
181:
175:
169:
163:
141:
114:
107:
91:
86:
82:
58:
52:
5424:Reeve (1995
5364:Terman 1943
5204:: 519–549.
5131:Transformer
4742:10 Ω·m
4622:10 Ω·m
4584:10 S/m
4068:Litzendraht
4033:laminations
4028:micrometers
3420:per length
2992:= conductor
2885:table below
2598:F.E. Terman
1740:resistivity
1663:wave number
596:of current
566:resistivity
422:Snell's law
246:skin effect
215:skin effect
144:Horace Lamb
59:skin effect
5862:Categories
5588:Hayt (1981
5574:2020-12-22
5518:2011-11-08
5446:Chen (2004
5412:Hayt (1981
5400:Hayt (1981
5376:Hayt (1981
5183:References
5084:waveguides
4992:See also:
4901:Skin depth
4833:Skin depth
4817:micrometer
4090:often use
4062:Mitigation
2826:Inductance
2820:inductance
2421:Resistance
1853:Derivation
1156:Round wire
1047:10 Hz
395:wavelength
376:skin depth
219:skin depth
119:circuits,
99:resistance
87:skin depth
5845:114947614
5837:0197-2618
5561:1531-1309
5218:111283238
5173:magnitude
5126:Litz wire
5062:δ
4898:Frequency
4830:Conductor
4786:μ
4700:⋅
4687:−
4679:⋅
4662:δ
4633:μ
4567:≈
4564:σ
4541:σ
4516:μ
4485:μ
4428:α
4403:δ
4367:σ
4355:μ
4341:≈
4325:μ
4320:ρ
4310:≈
4304:σ
4301:μ
4295:π
4263:μ
4253:μ
4240:π
4229:ρ
4214:α
4199:δ
4178:permalloy
4125:microwave
3913:∞
3909:ℓ
3896:ℓ
3717:Frequency
3587:∞
3471:
3462:π
3449:μ
3029:= current
3018:= current
2981:= overall
2868:π
2861:μ
2840:induction
2816:reactance
2735:π
2727:ρ
2696:impedance
2581:δ
2578:≫
2550:δ
2544:π
2539:ρ
2536:ℓ
2530:≈
2524:δ
2518:δ
2515:−
2506:π
2501:ρ
2498:ℓ
2492:≈
2469:ρ
2416:Impedance
2149:of order
1989:∂
1985:∂
1940:∂
1930:∂
1874:∇
1865:produces
1863:Ohm's law
1807:μ
1797:μ
1790:μ
1776:relative
1753:μ
1722:ρ
1696:μ
1693:ω
1688:ρ
1675:δ
1647:δ
1640:−
1625:ρ
1621:μ
1618:ω
1612:−
1216:π
1122:μ
1117:ε
1107:ρ
1100:≈
1097:δ
1074:ε
1071:ρ
1029:ε
995:μ
992:ω
986:ρ
971:δ
948:ε
945:ρ
896:ε
881:relative
860:ε
832:ε
821:ε
794:ε
760:μ
745:relative
724:μ
696:μ
685:μ
658:μ
610:π
578:ω
550:ρ
516:ε
513:ω
510:ρ
490:ε
487:ω
484:ρ
459:μ
456:ω
450:ρ
435:δ
362:δ
339:δ
312:−
238:induction
200:electrons
150:in 1885.
117:microwave
110:litz wire
94:frequency
67:conductor
47:Litz wire
5782:(1943),
5569:30187468
5110:See also
4844:Aluminum
4170:Fe-Si –
4140:Examples
4013:2000 MCM
3739:(nF/km)
3117:remains.
2848:external
2693:internal
2246:must be
217:and the
125:antennas
101:. At 60
79:magnetic
4880:3
4154:ferrite
4044:welding
3855:118.074
3734:(μS/km)
3729:(mH/km)
3670:. Only
3005:
2997:
2967:below.
2844:outside
2812:complex
391:delayed
257:Formula
234:induced
5843:
5835:
5825:
5799:
5760:
5744:
5727:
5709:
5691:
5668:
5648:
5567:
5559:
5348:
5297:
5267:
5239:
5216:
4981:busbar
4871:0.634
4868:Silver
4863:0.753
4855:0.652
4852:Copper
4847:0.820
4391:where
4117:plated
4107:ribbon
4103:busbar
3858:51.57
3852:0.4675
3849:999.41
3841:51.57
3838:53.205
3835:0.4862
3832:643.14
3824:51.57
3821:29.111
3818:0.5062
3815:463.59
3807:51.57
3801:0.5807
3798:191.63
3790:51.57
3784:0.6099
3781:172.70
3773:51.57
3767:0.6125
3764:172.28
3756:51.57
3750:0.6129
3747:172.24
3724:(Ω/km)
3035:flux (
2955:, and
2919:inside
2911:depth.
2832:inside
2640:
2427:losses
2310:Thus,
2118:where
1829:Since
1453:phasor
1357:where
1130:
1004:
628:where
538:where
403:vacuum
354:where
5841:S2CID
5565:S2CID
5214:S2CID
5158:Notes
4903:(μm)
4042:(DC)
3804:3.327
3787:0.531
3770:0.072
3753:0.000
3333:total
3089:and
3014:green
3001:white
2978:black
2685:cable
2145:is a
240:. An
154:Cause
5833:ISSN
5823:ISBN
5797:ISBN
5758:ISBN
5742:ISBN
5725:ISBN
5707:ISBN
5689:ISBN
5666:ISBN
5646:ISBN
5557:ISSN
5430:help
5346:ISBN
5295:ISBN
5265:ISBN
5237:ISBN
4860:Gold
4713:11.1
4676:2.44
4618:2.44
4580:58.5
3719:(Hz)
3654:<
3648:<
3300:<
3294:<
3234:<
3228:<
3168:<
3025:blue
2611:is:
2481:is:
2169:and
1861:and
1661:the
917:the
781:the
83:skin
21:skin
5815:doi
5549:doi
5206:doi
5202:174
5082:In
5052:is
4975:In
4738:2.2
4668:503
4344:503
4313:503
4123:to
4121:VHF
4049:arc
3683:ext
3600:ext
3556:ext
3543:shd
3530:cen
3401:ext
3372:ext
3359:shd
3346:cen
3268:shd
3202:ext
3142:cen
2988:tan
2713:int
2637:200
223:1/e
53:In
5864::
5839:.
5831:.
5821:.
5563:.
5555:.
5545:26
5543:.
5539:.
5527:^
5438:^
5332:^
5273:.
5245:.
5212:.
5200:.
5106:.
4882:μm
4839:)
4837:μm
4819::
4703:50
4683:10
4134:5G
3615:.
3571:.
3517:DC
3468:ln
3009:,
2994:,
2951:,
2898:/8
2887:.
2593:.
2042:0.
1049:.
253:.
103:Hz
89:.
57:,
5847:.
5817::
5748:.
5674:.
5551::
5521:.
5432:)
5220:.
5208::
5070:.
5065:2
5038:e
5035:1
5011:e
5008:1
4835:(
4790:r
4762:3
4759:=
4754:9
4740:×
4721:m
4718:m
4710:=
4697:1
4690:8
4665:=
4642:=
4637:r
4620:×
4599:=
4596:f
4586:)
4582:×
4544:=
4519:=
4494:=
4489:0
4461:m
4457:p
4454:N
4431:=
4406:=
4373:,
4364:f
4359:r
4350:1
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4286:=
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4257:0
4249:(
4246:)
4243:f
4237:2
4234:(
4226:2
4219:=
4211:1
4206:=
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3980:)
3973:m
3969:f
3965:f
3960:(
3955:+
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3945:b
3940:)
3933:m
3929:f
3925:f
3920:(
3905:+
3900:0
3889:=
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3883:f
3880:(
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3732:G
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3627:a
3623:r
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3592:=
3583:L
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3535:+
3526:L
3522:=
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3483:a
3480:b
3475:(
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3453:0
3443:=
3440:D
3436:/
3432:L
3422:D
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3397:L
3368:L
3364:+
3355:L
3351:+
3342:L
3338:=
3329:L
3303:c
3297:r
3291:b
3264:L
3237:b
3231:r
3225:a
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3171:a
3165:r
3138:L
3109:D
3101:C
3093:D
3085:C
3077:B
3069:D
3061:C
3053:B
3045:A
3038:B
2963:A
2957:c
2953:b
2949:a
2900:π
2896:μ
2865:8
2796:.
2790:)
2787:R
2784:k
2781:(
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2757:(
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2647:m
2644:m
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2625:W
2620:D
2609:f
2605:W
2602:D
2575:D
2547:D
2521:)
2512:D
2509:(
2489:R
2459:ℓ
2455:δ
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2443:D
2439:δ
2435:δ
2431:δ
2401:.
2395:)
2392:R
2389:k
2386:(
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2100:r
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2070:r
2067:(
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2039:=
2036:)
2033:r
2030:(
2026:J
2020:2
2016:k
2012:+
2009:)
2006:r
2003:(
1999:J
1992:r
1978:r
1975:1
1970:+
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1961:(
1957:J
1948:2
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1924:=
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1918:r
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1911:J
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1762:=
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1725:=
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1678:=
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1578:=
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1068:(
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797:=
769:=
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733:=
728:r
700:0
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661:=
636:f
616:,
613:f
607:2
604:=
581:=
553:=
520:)
507:+
499:2
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480:(
475:+
472:1
466:(
447:2
438:=
418:o
415:λ
399:π
387:S
384:J
380:e
335:/
331:d
328:)
325:j
322:+
319:1
316:(
308:e
300:S
295:J
291:=
288:J
278:d
274:S
271:J
263:J
189:W
183:I
177:H
171:H
165:I
23:.
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