4116:) 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,
4139:
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.
1179:
2950:
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.
4081:, 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.
2918:
4156:
50:
2902:
2982:
170:
4398:
1163:= 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.
38:
213:, 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
1056:(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
4109:
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.
4201:
2063:
3648:). 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
2894:(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
4143:
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
545:
2949:
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
3116:, 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,
1182:
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
45:, 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.
2945:
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
2936:
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.
4142:
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
4138:
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
4088:
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
4108:
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
4021:
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
1158:
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
1879:
1364:
5181:
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
4393:{\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}}}
259:
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
4012:
4057:
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
4093:
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.
2921:
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
3132:
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.
2574:
441:
4994:
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.
4890:
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
2927:
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
2932:
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.
2817:
4743:
107:
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
2690:
1154:
1189:
3396:
1670:
3511:
2422:
3580:
3883:
4041:
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
2861:
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
1028:
208:
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
1721:
2058:{\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.}
3711:
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.
363:
2495:
2127:
2849:; 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
2319:
5258:
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.
4206:
4026:(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.
859:
3877:
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.
217:(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.
435:
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.
412:
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
3624:
5097:, 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.
4826:
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
2712:
4668:
5091:
2892:
1834:
1098:
972:
60:) 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.
5186:
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.
1506:
1460:
723:
926:
890:
3425:
3292:
3226:
3166:
2234:
4588:
4484:
5061:
5034:
3706:
3136:
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.
2941:
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
4783:
3679:
3325:
3259:
1533:
818:
2625:
2602:
2255:
2200:
1103:
1050:
1785:
3052:). 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
4663:
4515:
3092:, 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 (
790:
754:
4813:
4565:
4452:
4427:
3193:
1748:
1601:
1567:
637:
602:
4123:
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.
383:
3334:
574:
236:
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
4540:
3438:
2490:
2324:
2154:
682:
540:{\displaystyle \delta ={\sqrt {{\frac {\,2\rho \,}{\omega \mu }}\left({\sqrt {1+\left({\rho \omega \varepsilon }\right)^{2}\,}}+\rho \omega \varepsilon \right)\,}}}
1422:
1395:
431:
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
4620:
3629:
Most discussions of coaxial transmission lines assume they will be used for radio frequencies, so equations are supplied corresponding only to the latter case.
2178:
1858:
657:
3124:, 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
2970:
be the inner conductor radius, the shield (outer conductor) inside radius and the shield outer radius respectively, as seen in the crossection of figure
255:
impinging on a conductor will therefore generally produce such a current; this explains the attenuation of electromagnetic waves in metals. Although the term
977:
3084:
high frequency. There are three regions that may contain induced magnetic fields: the center conductor, the dielectric and the outer conductor. In stage
5638:, 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)
224:
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
1608:
1183:
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.
4105:; 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.
3518:
2905:
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
1063:
However, in very poor conductors, at sufficiently high frequencies, the factor under the large radical increases. At frequencies much higher than
294:
5109:
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.
2068:
1359:{\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)}}}
3108:) the magnetic field in the dielectric region is unchanged as it is proportional to the total current flowing in the center conductor. In
5036:. The intensity of the wave is proportional to the square of the amplitude, and thus the depth at which the intensity has diminished by
84:
is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. It is caused by opposing
2985:
Four stages of skin effect in a coax showing the effect on inductance. Diagrams show a cross-section of the coaxial cable. Color code:
1175:. The current density inside round wire away from the influences of other fields, as function of distance from the axis is given by:
4007:{\displaystyle L(f)={\frac {\ell _{0}+\ell _{\infty }\left({\frac {f}{f_{m}}}\right)^{b}}{1+\left({\frac {f}{f_{m}}}\right)^{b}}}\,}
2436:
The most important effect of skin effect on the impedance of a single wire is the increase of the wire's resistance, and consequent
1681:
123:. Because the interior of a large conductor carries little of the current, tubular conductors can be used to save weight and cost.
5009:
In electromagnetic waves, the skin depth is the depth at which the amplitude of the electric and magnetic fields have reduced by
4037:
its permeability is about 10,000 times greater. This reduces the skin depth for iron to about 1/38 that of copper, about 220
5837:
5811:
5772:
5739:
5703:
5660:
5440:
5279:
5251:
4815:
as was pointed out above for the case of iron, despite its poorer conductivity. A practical consequence is seen by users of
4102:
2260:
5105:
For high frequencies and low temperatures, the usual formulas for skin depth break down. This effect was first noticed by
5161:
5126:
2699:
826:
5721:
5360:
5309:
4022:
between its AC and DC resistance, when compared with a conductor of higher resistivity. For example, at 60 Hz, a
1860:
is complex, the Bessel functions are also complex. The amplitude and phase of the current density varies with depth.
5756:
5680:
2157:
3588:
2694:
This formula for the increase in AC resistance is accurate only for an isolated wire. For nearby wires, e.g. in a
1171:
When skin depth is not small with respect to the radius of the wire, current density may be described in terms of
4182:
4030:
2569:{\displaystyle R\approx {{\ell \rho } \over {\pi (D-\delta )\delta }}\approx {{\ell \rho } \over {\pi D\delta }}}
2202:
is a constant phasor. To satisfy the boundary condition for the current density at the surface of the conductor,
684:
4159:
Skin depth vs. frequency for some materials at room temperature, red vertical line denotes 50 Hz frequency:
5883:
2440:. The effective resistance due to a current confined near the surface of a large conductor (much thicker than
389:
which is defined as the depth below the surface of the conductor at which the current density has fallen to 1/
119:
Increased AC resistance caused by skin effect can be mitigated by using a specialized multistrand wire called
4755:, about 9 times that of gold. Its skin depth at 50 Hz is likewise found to be about 33 mm, or
1869:
5066:
2865:
2448:
based on the DC resistivity of that material. The effective cross-sectional area is approximately equal to
1796:
1066:
940:
2452:
times the conductor's circumference. Thus a long cylindrical conductor such as a wire, having a diameter
214:
147:
143:
1475:
1429:
1178:
690:
4069:
At 1 megahertz skin effect depth in wet soil is about 5.0 m; in seawater it is about 0.25 m.
929:
901:
865:
793:
17:
3402:
3269:
3203:
3143:
2205:
5878:
5110:
4570:
4457:
4033:
of the conductor. In the case of iron, its conductivity is about 1/7 that of copper. However being
2850:
1100:
it can be shown that skin depth, rather than continuing to decrease, approaches an asymptotic value:
390:
248:
233:
5039:
5012:
3684:
5820:
Xi Nan; Sullivan, C. R. (2005). "An equivalent complex permeability model for litz-wire windings".
5156:
5146:
4758:
3651:
3297:
3231:
1513:
800:
5822:
Fourtieth IAS Annual
Meeting. Conference Record of the 2005 Industry Applications Conference, 2005
2581:
2238:
2183:
1035:
974:
the quantity inside the large radical is close to unity and the formula is more usually given as:
116:
in copper, skin depth is about 8.5 mm. At high frequencies, skin depth becomes much smaller.
5522:
2812:{\displaystyle \mathbf {Z} _{\text{int}}={\frac {k\rho }{2\pi R}}{\frac {J_{0}(kR)}{J_{1}(kR)}}.}
1758:
157:
in 1883 for the case of spherical conductors, and was generalized to conductors of any shape by
4738:{\displaystyle \delta =503\,{\sqrt {\frac {2.44\cdot 10^{-8}}{1\cdot 50}}}=11.1\,\mathrm {mm} }
4638:
4490:
765:
729:
4791:
4547:
4434:
4409:
3632:
As skin effect increases, the currents are concentrated near the outside the inner conductor (
3427:
is not changed by the skin effect and is given by the frequently cited formula for inductance
3171:
1728:
1574:
1540:
1052:
would have a large imaginary part) and at frequencies that are much below both the material's
610:
584:
1788:
757:
368:
109:
5235:
1032:
This formula is valid at frequencies away from strong atomic or molecular resonances (where
556:
4522:
3720:
Representative parameter data for 24 gauge PIC telephone cable at 21 °C (70 °F).
2826:
2706:
2475:
2132:
664:
252:
77:
4130:
to take advantage of silver's higher conductivity. This technique is particularly used at
1402:
1375:
8:
4895:
thick evaporated on a piece of glass is thus an excellent conductor at such frequencies.
4098:
126:
Skin effect has practical consequences in the analysis and design of radio-frequency and
73:
42:
4602:
400:. The imaginary part of the exponent indicates that the phase of the current density is
5851:
5692:
5575:
5286:
To understand skin effect, you must first understand how eddy currents operate ...
5224:
4747:
Lead, in contrast, is a relatively poor conductor (among metals) with a resistivity of
2618:
of a wire of circular cross-section whose resistance will increase by 10% at frequency
2163:
1843:
642:
92:
field resulting from the alternating current. The electric current flows mainly at the
5299:
4788:
Highly magnetic materials have a reduced skin depth owing to their large permeability
41:
Distribution of current flow in a cylindrical conductor, shown in cross section. For
5855:
5843:
5833:
5807:
5768:
5752:
5735:
5717:
5699:
5676:
5656:
5567:
5547:
5356:
5305:
5275:
5247:
5228:
5151:
5114:
5004:
4898:
In copper, skin depth can be seen to fall according to the square root of frequency:
2685:{\displaystyle D_{\mathrm {W} }={\frac {200~\mathrm {mm} }{\sqrt {f/\mathrm {Hz} }}}}
2065:
The solution to this equation is, for finite current in the center of the conductor,
1149:{\displaystyle \delta \approx {2\rho }{\sqrt {{\frac {\,\varepsilon \,}{\mu }}\,}}~.}
604:
277:
261:
204:
which partially cancel the current flow in the center and reinforce it near the skin.
139:
131:
5579:
438:
The general formula for skin depth when there is no dielectric or magnetic loss is:
30:"Skin depth" redirects here. For the depth (layers) of biological/organic skin, see
5825:
5790:
5559:
5216:
5207:
Lamb, Horace (1883-01-01). "XIII. On electrical motions in a spherical conductor".
4816:
4164:
2917:
2695:
2608:
1053:
158:
65:
53:
5670:
5334:
The formula as shown is algebraically equivalent to the formula found on page 130
2895:
4820:
4176:
4090:
1172:
221:
135:
81:
3261:, the region between the two conductors (containing a dielectric, possibly air).
2444:) can be solved as if the current flowed uniformly through a layer of thickness
4127:
4050:
3391:{\displaystyle L_{\text{total}}=L_{\text{cen}}+L_{\text{shd}}+L_{\text{ext}}\,}
2822:
1665:{\displaystyle k={\sqrt {\frac {-j\omega \mu }{\rho }}}={\frac {1-j}{\delta }}}
432:
421:
417:
237:
5563:
3506:{\displaystyle L/D={\frac {\mu _{0}}{2\pi }}\ln \left({\frac {b}{a}}\right)\,}
2417:{\displaystyle \mathbf {J} (r)=\mathbf {J} (R){\frac {J_{0}(kr)}{J_{0}(kR)}}.}
5872:
5847:
5829:
5571:
4034:
424:
in a vacuum. For example, a 1 MHz radio wave has a wavelength in vacuum
240:
which usually will be distributed evenly over the cross-section of the wire.
3585:
At high frequencies, only the dielectric region has magnetic flux, so that
3575:{\displaystyle L_{\text{DC}}=L_{\text{cen}}+L_{\text{shd}}+L_{\text{ext}}\,}
247:
in a conductor due to an alternating magnetic field according to the law of
5220:
5131:
5106:
4063:
4023:
2901:
1873:
893:
820:
85:
49:
4155:
4120:
conductors) in which the effects from corners are effectively eliminated.
5141:
4085:
4059:
2841:
A portion of a wire's inductance can be attributed to the magnetic field
2437:
1750:
1673:
576:
401:
185:. If the current increases, as in this figure, the resulting increase in
154:
5094:
4827:
4043:
4038:
2830:
1023:{\displaystyle \delta ={\sqrt {{\frac {\,2\rho \,}{\omega \mu }}\,}}~.}
405:
3708:
contributes to the electrical inductance at these higher frequencies.
2981:
2937:
However, when only a single wire is involved, then in addition to the
5639:
5136:
4188:
4135:
127:
120:
104:
57:
5506:
3515:
At low frequencies, all three inductances are fully present so that
169:
2702:, which can cause an additional increase in the AC resistance. The
210:
146:
and distribution systems. It is one of the reasons for preferring
96:
of the conductor, between the outer surface and a level called the
89:
5548:"Study on Cu/Ni nano superlattice conductors for reduced RF loss"
4054:
3331:
The net electrical inductance is due to all three contributions:
2825:
quantity corresponding to a resistance (real) in series with the
3880:
Chen gives an equation of this form for telephone twisted pair:
3168:
is associated with the magnetic field in the region with radius
232:
is a measure of the depth at which the current density falls to
5669:
Feynman, Richard P; Leighton, Robert B; Sands, Matthew (1964).
4991:
4113:
2468:
carrying direct current. The AC resistance of a wire of length
1463:
413:
4665:
1, so its skin depth at a frequency of 50 Hz is given by
4198:
We can derive a practical formula for skin depth as follows:
2857:
of the wire produced by the current in the wire. Unlike that
1716:{\displaystyle \delta ={\sqrt {\frac {2\rho }{\omega \mu }}}}
113:
37:
5523:"Spinning Carbon Nanotubes Spawns New Wireless Applications"
4892:
4847:
4112:
In high current situations where conductors (round or flat
404:
1 radian for each skin depth of penetration. One full
31:
4823:
cookware are unusable because they are not ferromagnetic.
4084:
Litz wire is often used in the windings of high-frequency
5802:
Vander Vorst, Andre; Rosen, Arye; Kotsuka, Youji (2006),
5765:
5209:
4131:
4029:
Skin depth also varies as the inverse square root of the
358:{\displaystyle J=J_{\mathrm {S} }\,e^{-{(1+j)d/\delta }}}
5801:
5323:
2709:
per unit length of a segment of round wire is given by:
27:
Tendency of AC current flow in a conductor's outer layer
4144:
2122:{\displaystyle \mathbf {J} (r)=\mathbf {C} J_{0}(kr),}
5069:
5042:
5015:
4794:
4761:
4671:
4641:
4605:
4573:
4550:
4525:
4493:
4460:
4437:
4412:
4204:
3886:
3687:
3654:
3591:
3521:
3441:
3405:
3337:
3300:
3272:
3234:
3206:
3174:
3146:
2913:
as skin depth becomes small (as frequency increases).
2868:
2715:
2628:
2584:
2498:
2478:
2327:
2314:{\displaystyle {\frac {\mathbf {J} (R)}{J_{0}(kR)}}.}
2263:
2241:
2208:
2186:
2166:
2135:
2071:
1882:
1846:
1799:
1761:
1731:
1684:
1611:
1577:
1543:
1516:
1478:
1432:
1405:
1378:
1192:
1106:
1069:
1038:
980:
943:
904:
868:
829:
803:
768:
732:
693:
667:
645:
613:
587:
559:
444:
371:
297:
179:
flowing through a conductor induces a magnetic field
5113:
was developed for this specific case for metals and
3294:
is associated with the magnetic field in the region
3228:
is associated with the magnetic field in the region
5541:
5539:
3042:coming out of the diagram, dashed black lines with
5691:
5353:Transmission and Distribution of Electrical Energy
5272:High-Speed Signal propagation Advanced Black Magic
5085:
5055:
5028:
4807:
4777:
4737:
4657:
4614:
4582:
4559:
4534:
4509:
4478:
4446:
4421:
4392:
4077:A type of cable called litz wire (from the German
4006:
3700:
3673:
3618:
3574:
3505:
3419:
3390:
3319:
3286:
3253:
3220:
3187:
3160:
2886:
2811:
2684:
2596:
2568:
2484:
2416:
2313:
2249:
2228:
2194:
2172:
2148:
2121:
2057:
1852:
1828:
1779:
1742:
1715:
1664:
1595:
1561:
1527:
1500:
1454:
1416:
1389:
1358:
1148:
1092:
1044:
1022:
966:
920:
884:
854:{\displaystyle \varepsilon _{r}\,\varepsilon _{0}}
853:
812:
784:
748:
717:
676:
651:
631:
596:
568:
539:
377:
357:
5473:(11th ed.), McGraw Hill, p. Table 18–21
2698:or a coil, the AC resistance is also affected by
1508:current density phasor at the surface of the wire
5870:
5804:RF/Microwave Interaction with Biological Tissues
5536:
5469:Fink, Donald G.; Beatty, H. Wayne, eds. (1978),
5164:, a method for estimating skin effect resistance
4046:in power transformers, increasing their losses.
4016:
5392:
5274:(3rd ed.). Prentice Hall. pp. 58–78.
4627:Gold is a good conductor with a resistivity of
5819:
5762:
5552:IEEE Microwave and Wireless Components Letters
5495:
5483:
5346:
5344:
5507:Central Electricity Generating Board (1982).
5269:
5246:(14th ed.). McGraw-Hill. pp. 2–50.
153:The effect was first described in a paper by
5439:harvtxt error: no target: CITEREFReeve1995 (
5317:
4990:, Hayt points out that in a power station a
4567:the conductivity of the medium (for copper,
4103:aluminum cable with a steel reinforcing core
3619:{\displaystyle L_{\infty }=L_{\text{ext}}\,}
2829:(imaginary) due to the wire's internal self-
420:in a conductor is very much slower than the
191:induces separate, circulating eddy currents
5732:Electromagnetic Waves and Radiating Systems
5404:
5341:
5545:
5471:Standard Handbook for Electrical Engineers
5468:
5263:
5244:Standard Handbook for Electrical Engineers
4126:Solid or tubular conductors may be silver-
4066:rods are used for high-frequency welding.
2464:that of a hollow tube with wall thickness
1603:Bessel function of the first kind, order 1
1569:Bessel function of the first kind, order 0
5500:
5452:
5450:
5242:Fink, Donald G.; Beaty, H. Wayne (2000).
5241:
4726:
4681:
4357:
4326:
4003:
3670:
3615:
3571:
3502:
3416:
3387:
3327:, the region inside the shield conductor.
3316:
3283:
3250:
3217:
3195:, the region inside the center conductor.
3184:
3157:
1137:
1130:
1126:
1011:
999:
992:
840:
704:
534:
514:
463:
456:
316:
5780:
5672:The Feynman Lectures on Physics Volume 2
5428:
5398:
5324:Vander Vorst, Rosen & Kotsuka (2006)
5270:Johnson, Howard; Graham, Martin (2003).
5100:
4998:
4154:
2980:
2916:
2900:
1177:
168:
48:
36:
5763:Popovic, Zoya; Popovic, Branko (1999),
5711:
5668:
5622:
5610:
5416:
5380:
14:
5871:
5789:
5729:
5546:Rahimi, A.; Yoon, Y.-K. (2016-03-16).
5447:
5374:
5335:
2578:The final approximation above assumes
150:for long-distance power transmission.
5434:
5350:
5328:
5291:
5086:{\displaystyle {\frac {\delta }{2}}.}
4836:Skin depths at microwave frequencies
2887:{\displaystyle {\frac {\mu }{8\pi }}}
1829:{\displaystyle \mu =\mu _{r}\mu _{0}}
1093:{\displaystyle 1/(\rho \varepsilon )}
967:{\displaystyle 1/(\rho \varepsilon )}
220:Regardless of the driving force, the
173:Cause of skin effect. A main current
5689:
5650:
5598:
5456:
5422:
5410:
5386:
5297:
5206:
4930:8420
4922:9220
2845:the wire itself which is termed the
2607:A convenient formula (attributed to
76:(AC) to become distributed within a
5824:. Vol. 3. pp. 2229–2235.
5162:Wheeler incremental inductance rule
5127:Proximity effect (electromagnetism)
4946:206
4938:652
243:An alternating current may also be
24:
5749:Oliver Heaviside: Sage in Solitude
5640:https://www.jstor.org/stable/99095
4731:
4728:
4622:the frequency of the current in Hz
3923:
3715:
3597:
2675:
2672:
2657:
2654:
2635:
1999:
1995:
1950:
1940:
1884:
1501:{\displaystyle \mathbf {J} (R)={}}
1455:{\displaystyle \mathbf {J} (r)={}}
1397:distance from the axis of the wire
718:{\displaystyle \mu _{r}\,\mu _{0}}
310:
25:
5895:
5864:
2158:Bessel function of the first kind
921:{\displaystyle \varepsilon _{0}=}
885:{\displaystyle \varepsilon _{r}=}
4635:and is essentially nonmagnetic:
3640:) and the inside of the shield (
3420:{\displaystyle L_{\text{ext}}\,}
3287:{\displaystyle L_{\text{shd}}\,}
3221:{\displaystyle L_{\text{ext}}\,}
3161:{\displaystyle L_{\text{cen}}\,}
2953:
2718:
2346:
2329:
2268:
2243:
2229:{\displaystyle \mathbf {J} (R),}
2210:
2188:
2090:
2073:
2036:
2009:
1967:
1921:
1894:
1518:
1480:
1434:
1291:
1217:
1194:
5628:
5616:
5604:
5592:
5515:
5489:
5477:
5462:
4583:{\displaystyle \sigma \approx }
4479:{\displaystyle {\frac {Np}{m}}}
4183:grain-oriented electrical steel
5751:. New York: IEEE Press, 1988.
5730:Jordan, Edward Conrad (1968),
5525:. Sciencedaily.com. 2009-03-09
5368:
5200:
5175:
5056:{\displaystyle {\frac {1}{e}}}
5029:{\displaystyle {\frac {1}{e}}}
4542:the permeability of the medium
4517:the permeability of free space
4282:
4259:
4256:
4244:
3896:
3890:
3701:{\displaystyle L_{\text{ext}}}
2800:
2791:
2776:
2767:
2531:
2519:
2405:
2396:
2381:
2372:
2356:
2350:
2339:
2333:
2302:
2293:
2278:
2272:
2220:
2214:
2113:
2104:
2083:
2077:
2046:
2040:
2019:
2013:
1977:
1971:
1931:
1925:
1904:
1898:
1490:
1484:
1444:
1438:
1350:
1341:
1326:
1317:
1301:
1295:
1281:
1272:
1257:
1248:
1204:
1198:
1087:
1078:
961:
952:
338:
326:
291:from the surface, as follows:
280:from its value at the surface
13:
1:
5806:, John Wiley and Sons, Inc.,
5698:(4th ed.), McGraw-Hill,
5509:Modern Power Station Practice
5304:(5th ed.), McGraw-Hill,
5193:
4778:{\displaystyle {\sqrt {9}}=3}
4072:
4017:Material effect on skin depth
3674:{\displaystyle b<r<c\,}
3320:{\displaystyle b<r<c\,}
3254:{\displaystyle a<r<b\,}
2836:
2431:
1870:electromagnetic wave equation
1863:
1528:{\displaystyle \mathbf {I} =}
1166:
813:{\displaystyle \varepsilon =}
5712:Jackson, John David (1999),
5694:Engineering Electromagnetics
5301:Engineering Electromagnetics
4988:Engineering Electromagnetics
2946:case at higher frequencies.
2597:{\displaystyle D\gg \delta }
2426:
2250:{\displaystyle \mathbf {C} }
2195:{\displaystyle \mathbf {C} }
1045:{\displaystyle \varepsilon }
74:alternating electric current
7:
5783:Electric Transmission Lines
5120:
4150:
4097:High-voltage, high-current
1780:{\displaystyle \mu _{r}={}}
1470:, from the axis of the wire
408:in the conductor requires 2
215:counter-electromotive force
148:high-voltage direct current
144:electric power transmission
10:
5900:
5781:Skilling, Hugh H. (1951),
5496:Xi Nan & Sullivan 2005
5484:Popovic & Popovic 1999
5002:
4187:Fe-Ni – high-permeability
4163:Mn-Zn – magnetically soft
937:At frequencies much below
930:permittivity of free space
794:permeability of free space
267:
138:. It is also important at
103:Skin depth depends on the
29:
5795:Radio Engineers' Handbook
5714:Classical Electrodynamics
5636:The Anomalous Skin Effect
5588:– via ResearchGate.
5564:10.1109/LMWC.2016.2537780
5351:Weeks, Walter L. (1981),
5298:Hayt, William H. (1989),
4658:{\displaystyle \mu _{r}=}
4510:{\displaystyle \mu _{0}=}
785:{\displaystyle \mu _{0}=}
749:{\displaystyle \mu _{r}=}
5830:10.1109/IAS.2005.1518758
5651:Chen, Walter Y. (2004),
5168:
5157:Magnetic Reynolds number
4927:60 Hz
4919:50 Hz
4808:{\displaystyle \mu _{r}}
4560:{\displaystyle \sigma =}
4447:{\displaystyle \alpha =}
4429:the skin depth in meters
4422:{\displaystyle \delta =}
4049:Iron rods work well for
3188:{\displaystyle r<a\,}
1743:{\displaystyle \rho ={}}
1596:{\displaystyle J_{1}={}}
1562:{\displaystyle J_{0}={}}
632:{\displaystyle =2\pi f,}
597:{\displaystyle \omega =}
164:
88:induced by the changing
5797:, New York: McGraw-Hill
5716:(3rd ed.), Wiley,
3755:1 Hz
1723:also called skin depth.
416:, or equivalently, the
378:{\displaystyle \delta }
287:according to the depth
278:decreases exponentially
272:The AC current density
5690:Hayt, William (1981),
5653:Home Networking Basics
5221:10.1098/rstl.1883.0013
5087:
5057:
5030:
4819:, where some types of
4809:
4779:
4739:
4659:
4616:
4584:
4561:
4536:
4511:
4480:
4448:
4423:
4394:
4195:
4169:Al – metallic aluminum
4008:
3702:
3675:
3620:
3576:
3507:
3421:
3392:
3321:
3288:
3255:
3222:
3189:
3162:
3129:
3076:middle frequency, and
2923:
2914:
2888:
2813:
2686:
2598:
2570:
2486:
2418:
2315:
2251:
2230:
2196:
2174:
2150:
2123:
2059:
1854:
1830:
1781:
1744:
1717:
1666:
1597:
1563:
1529:
1502:
1456:
1418:
1391:
1360:
1184:
1150:
1094:
1046:
1024:
968:
922:
886:
855:
814:
786:
750:
719:
678:
653:
633:
598:
570:
569:{\displaystyle \rho =}
541:
379:
359:
205:
142:(50–60 Hz) in AC
72:is the tendency of an
61:
46:
5884:Electrical parameters
5111:Mattis–Bardeen theory
5101:Anomalous skin effect
5088:
5058:
5031:
4999:Electromagnetic waves
4904:Skin depth in copper
4810:
4780:
4740:
4660:
4617:
4585:
4562:
4537:
4535:{\displaystyle \mu =}
4512:
4481:
4449:
4424:
4395:
4175:steel 410 – magnetic
4158:
4009:
3703:
3676:
3621:
3577:
3508:
3422:
3393:
3322:
3289:
3256:
3223:
3190:
3163:
3044:arrowheads = magnetic
2984:
2920:
2904:
2889:
2853:from magnetic fields
2814:
2687:
2599:
2571:
2487:
2485:{\displaystyle \rho }
2419:
2316:
2252:
2231:
2197:
2175:
2151:
2149:{\displaystyle J_{0}}
2124:
2060:
1855:
1831:
1789:magnetic permeability
1782:
1745:
1718:
1667:
1598:
1564:
1530:
1503:
1457:
1419:
1392:
1361:
1181:
1151:
1095:
1047:
1025:
969:
923:
887:
856:
815:
787:
758:magnetic permeability
751:
720:
679:
677:{\displaystyle \mu =}
654:
634:
599:
571:
542:
380:
360:
172:
134:(or waveguides), and
52:
40:
5355:, Harper & Row,
5067:
5040:
5013:
4967:100 MHz
4943:100 kHz
4792:
4785:times that of gold.
4759:
4669:
4639:
4603:
4571:
4548:
4523:
4491:
4458:
4435:
4410:
4202:
4172:Cu – metallic copper
4099:overhead power lines
3884:
3806:100 kHz
3685:
3652:
3589:
3519:
3439:
3435:of a coaxial cable:
3403:
3335:
3298:
3270:
3232:
3204:
3172:
3144:
2866:
2821:This impedance is a
2713:
2626:
2582:
2496:
2476:
2325:
2261:
2239:
2206:
2184:
2164:
2133:
2069:
1880:
1844:
1797:
1759:
1729:
1682:
1609:
1575:
1541:
1535:total current phasor
1514:
1476:
1430:
1417:{\displaystyle R={}}
1403:
1390:{\displaystyle r={}}
1376:
1190:
1104:
1067:
1036:
978:
941:
902:
866:
827:
801:
766:
730:
691:
665:
643:
611:
585:
557:
442:
369:
295:
253:electromagnetic wave
56:use stranded coils (
5413:, pp. 434–439)
5401:, pp. 157–159)
4959:10 MHz
4935:10 kHz
4905:
4837:
4454:the attenuation in
3789:10 kHz
3126:external inductance
2994:insulating sheath,
2958:Let the dimensions
2943:internal inductance
2939:external inductance
2847:internal inductance
2833:, per unit length.
2611:) for the diameter
2460:, has a resistance
43:alternating current
5675:. Addison-Wesley.
5083:
5053:
5026:
4975:1 GHz
4962:20.6
4954:65.2
4951:1 MHz
4903:
4835:
4805:
4775:
4735:
4655:
4615:{\displaystyle f=}
4612:
4580:
4557:
4532:
4507:
4476:
4444:
4419:
4390:
4388:
4196:
4004:
3857:5 MHz
3840:2 MHz
3823:1 MHz
3772:1 kHz
3698:
3671:
3616:
3572:
3503:
3417:
3388:
3317:
3284:
3251:
3218:
3185:
3158:
3130:
3031:into the diagram,
2924:
2915:
2884:
2809:
2682:
2594:
2566:
2482:
2456:large compared to
2414:
2311:
2247:
2226:
2192:
2170:
2146:
2119:
2055:
1850:
1826:
1777:
1740:
1713:
1662:
1593:
1559:
1525:
1498:
1452:
1424:radius of the wire
1414:
1387:
1356:
1185:
1146:
1090:
1042:
1020:
964:
918:
882:
851:
823:of the conductor,
810:
782:
746:
715:
687:of the conductor,
674:
649:
629:
594:
566:
537:
375:
355:
206:
132:transmission lines
62:
47:
5839:978-0-7803-9208-3
5813:978-0-471-73277-8
5774:978-0-201-32678-9
5767:, Prentice-Hall,
5741:978-0-13-249995-8
5734:, Prentice Hall,
5705:978-0-07-027395-5
5662:978-0-13-016511-4
5655:, Prentice Hall,
5511:. Pergamon Press.
5281:978-0-13-084408-8
5253:978-0-07-022005-8
5152:Induction heating
5147:Induction cooking
5078:
5051:
5024:
5005:Penetration depth
4982:
4981:
4886:
4885:
4817:induction cookers
4767:
4718:
4717:
4474:
4381:
4380:
4349:
4348:
4318:
4317:
4287:
4286:
4227:
4001:
3988:
3948:
3873:
3872:
3695:
3612:
3568:
3555:
3542:
3529:
3496:
3476:
3413:
3384:
3371:
3358:
3345:
3280:
3214:
3154:
2882:
2804:
2752:
2725:
2680:
2679:
2652:
2564:
2538:
2409:
2306:
2173:{\displaystyle 0}
2006:
1991:
1964:
1853:{\displaystyle k}
1711:
1710:
1660:
1639:
1638:
1354:
1285:
1233:
1142:
1138:
1135:
1016:
1012:
1009:
659:is the frequency.
652:{\displaystyle f}
605:angular frequency
535:
515:
473:
140:mains frequencies
54:Induction cookers
16:(Redirected from
5891:
5879:Electromagnetism
5859:
5816:
5798:
5786:
5777:
5744:
5726:
5708:
5697:
5686:
5665:
5642:
5634:R. G. Chambers,
5632:
5626:
5625:, p. 32-11)
5620:
5614:
5608:
5602:
5596:
5590:
5589:
5587:
5586:
5543:
5534:
5533:
5531:
5530:
5519:
5513:
5512:
5504:
5498:
5493:
5487:
5481:
5475:
5474:
5466:
5460:
5454:
5445:
5444:
5432:
5426:
5420:
5414:
5408:
5402:
5396:
5390:
5384:
5378:
5372:
5366:
5365:
5348:
5339:
5332:
5326:
5321:
5315:
5314:
5295:
5289:
5288:
5267:
5261:
5260:
5239:
5233:
5232:
5204:
5187:
5179:
5092:
5090:
5089:
5084:
5079:
5071:
5062:
5060:
5059:
5054:
5052:
5044:
5035:
5033:
5032:
5027:
5025:
5017:
4906:
4902:
4838:
4834:
4814:
4812:
4811:
4806:
4804:
4803:
4784:
4782:
4781:
4776:
4768:
4763:
4754:
4752:
4744:
4742:
4741:
4736:
4734:
4719:
4716:
4705:
4704:
4703:
4684:
4683:
4664:
4662:
4661:
4656:
4651:
4650:
4634:
4632:
4621:
4619:
4618:
4613:
4596:
4594:
4589:
4587:
4586:
4581:
4566:
4564:
4563:
4558:
4541:
4539:
4538:
4533:
4516:
4514:
4513:
4508:
4503:
4502:
4485:
4483:
4482:
4477:
4475:
4470:
4462:
4453:
4451:
4450:
4445:
4428:
4426:
4425:
4420:
4399:
4397:
4396:
4391:
4389:
4382:
4373:
4372:
4363:
4359:
4350:
4347:
4343:
4342:
4329:
4328:
4319:
4304:
4300:
4292:
4288:
4285:
4281:
4280:
4271:
4270:
4242:
4234:
4233:
4228:
4220:
4147:communications.
4091:carbon nanotubes
4013:
4011:
4010:
4005:
4002:
4000:
3999:
3998:
3993:
3989:
3987:
3986:
3974:
3960:
3959:
3958:
3953:
3949:
3947:
3946:
3934:
3927:
3926:
3914:
3913:
3903:
3725:
3724:
3707:
3705:
3704:
3699:
3697:
3696:
3693:
3680:
3678:
3677:
3672:
3625:
3623:
3622:
3617:
3614:
3613:
3610:
3601:
3600:
3581:
3579:
3578:
3573:
3570:
3569:
3566:
3557:
3556:
3553:
3544:
3543:
3540:
3531:
3530:
3527:
3512:
3510:
3509:
3504:
3501:
3497:
3489:
3477:
3475:
3467:
3466:
3457:
3449:
3426:
3424:
3423:
3418:
3415:
3414:
3411:
3397:
3395:
3394:
3389:
3386:
3385:
3382:
3373:
3372:
3369:
3360:
3359:
3356:
3347:
3346:
3343:
3326:
3324:
3323:
3318:
3293:
3291:
3290:
3285:
3282:
3281:
3278:
3266:The inductance
3260:
3258:
3257:
3252:
3227:
3225:
3224:
3219:
3216:
3215:
3212:
3200:The inductance
3194:
3192:
3191:
3186:
3167:
3165:
3164:
3159:
3156:
3155:
3152:
3140:The inductance
3122:
3121:
3114:
3113:
3106:
3105:
3098:
3097:
3090:
3089:
3082:
3081:
3074:
3073:
3066:
3065:
3058:
3057:
3050:
3045:
3041:
3039:
3038:
3030:
3028:
3027:
3019:
3017:
3015:
3014:
3004:
3002:
3001:
2993:
2991:
2976:
2975:
2969:
2965:
2961:
2912:
2893:
2891:
2890:
2885:
2883:
2881:
2870:
2818:
2816:
2815:
2810:
2805:
2803:
2790:
2789:
2779:
2766:
2765:
2755:
2753:
2751:
2740:
2732:
2727:
2726:
2723:
2721:
2700:proximity effect
2691:
2689:
2688:
2683:
2681:
2678:
2670:
2662:
2661:
2660:
2650:
2645:
2640:
2639:
2638:
2621:
2614:
2603:
2601:
2600:
2595:
2575:
2573:
2572:
2567:
2565:
2563:
2552:
2544:
2539:
2537:
2514:
2506:
2491:
2489:
2488:
2483:
2472:and resistivity
2471:
2467:
2459:
2455:
2451:
2447:
2443:
2423:
2421:
2420:
2415:
2410:
2408:
2395:
2394:
2384:
2371:
2370:
2360:
2349:
2332:
2320:
2318:
2317:
2312:
2307:
2305:
2292:
2291:
2281:
2271:
2265:
2256:
2254:
2253:
2248:
2246:
2235:
2233:
2232:
2227:
2213:
2201:
2199:
2198:
2193:
2191:
2179:
2177:
2176:
2171:
2155:
2153:
2152:
2147:
2145:
2144:
2128:
2126:
2125:
2120:
2103:
2102:
2093:
2076:
2064:
2062:
2061:
2056:
2039:
2034:
2033:
2012:
2007:
2005:
1994:
1992:
1984:
1970:
1965:
1963:
1962:
1961:
1948:
1947:
1938:
1924:
1919:
1918:
1897:
1892:
1891:
1859:
1857:
1856:
1851:
1835:
1833:
1832:
1827:
1825:
1824:
1815:
1814:
1791:of the conductor
1786:
1784:
1783:
1778:
1776:
1771:
1770:
1753:of the conductor
1749:
1747:
1746:
1741:
1739:
1722:
1720:
1719:
1714:
1712:
1709:
1701:
1693:
1692:
1676:in the conductor
1671:
1669:
1668:
1663:
1661:
1656:
1645:
1640:
1634:
1620:
1619:
1602:
1600:
1599:
1594:
1592:
1587:
1586:
1568:
1566:
1565:
1560:
1558:
1553:
1552:
1534:
1532:
1531:
1526:
1521:
1507:
1505:
1504:
1499:
1497:
1483:
1462:current density
1461:
1459:
1458:
1453:
1451:
1437:
1423:
1421:
1420:
1415:
1413:
1396:
1394:
1393:
1388:
1386:
1365:
1363:
1362:
1357:
1355:
1353:
1340:
1339:
1329:
1316:
1315:
1305:
1294:
1286:
1284:
1271:
1270:
1260:
1247:
1246:
1236:
1234:
1232:
1221:
1220:
1211:
1197:
1173:Bessel functions
1162:
1155:
1153:
1152:
1147:
1140:
1139:
1136:
1131:
1124:
1122:
1120:
1099:
1097:
1096:
1091:
1077:
1059:
1054:plasma frequency
1051:
1049:
1048:
1043:
1029:
1027:
1026:
1021:
1014:
1013:
1010:
1008:
1000:
990:
988:
973:
971:
970:
965:
951:
927:
925:
924:
919:
914:
913:
896:of the conductor
891:
889:
888:
883:
878:
877:
860:
858:
857:
852:
850:
849:
839:
838:
819:
817:
816:
811:
791:
789:
788:
783:
778:
777:
760:of the conductor
755:
753:
752:
747:
742:
741:
724:
722:
721:
716:
714:
713:
703:
702:
683:
681:
680:
675:
658:
656:
655:
650:
638:
636:
635:
630:
603:
601:
600:
595:
579:of the conductor
575:
573:
572:
567:
546:
544:
543:
538:
536:
533:
529:
516:
513:
512:
507:
503:
481:
474:
472:
464:
454:
452:
427:
411:
396:
393:(about 0.37) of
384:
382:
381:
376:
364:
362:
361:
356:
354:
353:
352:
348:
315:
314:
313:
290:
283:
275:
262:normal incidence
202:
201:
195:
189:
183:
177:
159:Oliver Heaviside
66:electromagnetism
21:
5899:
5898:
5894:
5893:
5892:
5890:
5889:
5888:
5869:
5868:
5867:
5862:
5840:
5814:
5775:
5747:Nahin, Paul J.
5742:
5724:
5706:
5683:
5663:
5646:
5645:
5633:
5629:
5621:
5617:
5609:
5605:
5601:, pp. 401)
5597:
5593:
5584:
5582:
5544:
5537:
5528:
5526:
5521:
5520:
5516:
5505:
5501:
5494:
5490:
5482:
5478:
5467:
5463:
5455:
5448:
5438:
5433:
5429:
5421:
5417:
5409:
5405:
5397:
5393:
5389:, pp. 303)
5385:
5381:
5373:
5369:
5363:
5349:
5342:
5333:
5329:
5322:
5318:
5312:
5296:
5292:
5282:
5268:
5264:
5254:
5240:
5236:
5205:
5201:
5196:
5191:
5190:
5180:
5176:
5171:
5123:
5115:superconductors
5103:
5070:
5068:
5065:
5064:
5043:
5041:
5038:
5037:
5016:
5014:
5011:
5010:
5007:
5001:
4913:
4845:
4821:stainless steel
4799:
4795:
4793:
4790:
4789:
4762:
4760:
4757:
4756:
4750:
4748:
4727:
4706:
4696:
4692:
4685:
4682:
4670:
4667:
4666:
4646:
4642:
4640:
4637:
4636:
4630:
4628:
4625:
4604:
4601:
4600:
4592:
4590:
4572:
4569:
4568:
4549:
4546:
4545:
4524:
4521:
4520:
4498:
4494:
4492:
4489:
4488:
4463:
4461:
4459:
4456:
4455:
4436:
4433:
4432:
4411:
4408:
4407:
4387:
4386:
4368:
4364:
4358:
4338:
4334:
4333:
4327:
4299:
4290:
4289:
4276:
4272:
4266:
4262:
4243:
4235:
4232:
4219:
4212:
4205:
4203:
4200:
4199:
4194:
4177:stainless steel
4153:
4075:
4019:
3994:
3982:
3978:
3973:
3969:
3968:
3961:
3954:
3942:
3938:
3933:
3929:
3928:
3922:
3918:
3909:
3905:
3904:
3902:
3885:
3882:
3881:
3749:
3744:
3739:
3734:
3729:
3718:
3716:Telephone cable
3692:
3688:
3686:
3683:
3682:
3653:
3650:
3649:
3609:
3605:
3596:
3592:
3590:
3587:
3586:
3565:
3561:
3552:
3548:
3539:
3535:
3526:
3522:
3520:
3517:
3516:
3488:
3484:
3468:
3462:
3458:
3456:
3445:
3440:
3437:
3436:
3410:
3406:
3404:
3401:
3400:
3381:
3377:
3368:
3364:
3355:
3351:
3342:
3338:
3336:
3333:
3332:
3299:
3296:
3295:
3277:
3273:
3271:
3268:
3267:
3233:
3230:
3229:
3211:
3207:
3205:
3202:
3201:
3173:
3170:
3169:
3151:
3147:
3145:
3142:
3141:
3119:
3118:
3111:
3110:
3103:
3102:
3095:
3094:
3087:
3086:
3079:
3078:
3071:
3070:
3068:low frequency,
3063:
3062:
3060:non-energized,
3055:
3054:
3048:
3043:
3034:
3033:
3032:
3023:
3022:
3021:
3010:
3009:
3007:
3006:
2997:
2996:
2995:
2987:
2986:
2973:
2972:
2967:
2963:
2959:
2956:
2926:
2910:
2874:
2869:
2867:
2864:
2863:
2839:
2785:
2781:
2780:
2761:
2757:
2756:
2754:
2741:
2733:
2731:
2722:
2717:
2716:
2714:
2711:
2710:
2671:
2666:
2653:
2646:
2644:
2634:
2633:
2629:
2627:
2624:
2623:
2619:
2617:
2612:
2583:
2580:
2579:
2553:
2545:
2543:
2515:
2507:
2505:
2497:
2494:
2493:
2477:
2474:
2473:
2469:
2465:
2457:
2453:
2449:
2445:
2441:
2434:
2429:
2390:
2386:
2385:
2366:
2362:
2361:
2359:
2345:
2328:
2326:
2323:
2322:
2287:
2283:
2282:
2267:
2266:
2264:
2262:
2259:
2258:
2242:
2240:
2237:
2236:
2209:
2207:
2204:
2203:
2187:
2185:
2182:
2181:
2165:
2162:
2161:
2140:
2136:
2134:
2131:
2130:
2098:
2094:
2089:
2072:
2070:
2067:
2066:
2035:
2029:
2025:
2008:
1998:
1993:
1983:
1966:
1957:
1953:
1949:
1943:
1939:
1937:
1920:
1914:
1910:
1893:
1887:
1883:
1881:
1878:
1877:
1866:
1845:
1842:
1841:
1838:
1820:
1816:
1810:
1806:
1798:
1795:
1794:
1775:
1766:
1762:
1760:
1757:
1756:
1738:
1730:
1727:
1726:
1702:
1694:
1691:
1683:
1680:
1679:
1646:
1644:
1621:
1618:
1610:
1607:
1606:
1591:
1582:
1578:
1576:
1573:
1572:
1557:
1548:
1544:
1542:
1539:
1538:
1517:
1515:
1512:
1511:
1496:
1479:
1477:
1474:
1473:
1450:
1433:
1431:
1428:
1427:
1412:
1404:
1401:
1400:
1385:
1377:
1374:
1373:
1335:
1331:
1330:
1311:
1307:
1306:
1304:
1290:
1266:
1262:
1261:
1242:
1238:
1237:
1235:
1222:
1216:
1212:
1210:
1193:
1191:
1188:
1187:
1169:
1160:
1125:
1123:
1121:
1113:
1105:
1102:
1101:
1073:
1068:
1065:
1064:
1057:
1037:
1034:
1033:
1001:
991:
989:
987:
979:
976:
975:
947:
942:
939:
938:
935:
909:
905:
903:
900:
899:
873:
869:
867:
864:
863:
845:
841:
834:
830:
828:
825:
824:
802:
799:
798:
773:
769:
767:
764:
763:
737:
733:
731:
728:
727:
709:
705:
698:
694:
692:
689:
688:
666:
663:
662:
644:
641:
640:
612:
609:
608:
586:
583:
582:
558:
555:
554:
508:
493:
489:
488:
480:
479:
475:
465:
455:
453:
451:
443:
440:
439:
430:
425:
409:
399:
394:
370:
367:
366:
344:
325:
321:
317:
309:
308:
304:
296:
293:
292:
288:
286:
281:
276:in a conductor
273:
270:
222:current density
203:
199:
198:
193:
187:
181:
175:
167:
82:current density
35:
28:
23:
22:
15:
12:
11:
5:
5897:
5887:
5886:
5881:
5866:
5865:External links
5863:
5861:
5860:
5838:
5817:
5812:
5799:
5787:
5778:
5773:
5760:
5745:
5740:
5727:
5723:978-0471309321
5722:
5709:
5704:
5687:
5681:
5666:
5661:
5647:
5644:
5643:
5627:
5615:
5613:, p. 353)
5603:
5591:
5558:(4): 258–260.
5535:
5514:
5499:
5488:
5476:
5461:
5446:
5437:, p. 558)
5427:
5425:, p. 434)
5415:
5403:
5399:Skilling (1951
5391:
5379:
5367:
5362:978-0060469825
5361:
5340:
5338:, p. 130)
5327:
5316:
5311:978-0070274068
5310:
5290:
5280:
5262:
5252:
5234:
5198:
5197:
5195:
5192:
5189:
5188:
5173:
5172:
5170:
5167:
5166:
5165:
5159:
5154:
5149:
5144:
5139:
5134:
5129:
5122:
5119:
5102:
5099:
5082:
5077:
5074:
5050:
5047:
5023:
5020:
5000:
4997:
4984:
4983:
4980:
4979:
4976:
4972:
4971:
4968:
4964:
4963:
4960:
4956:
4955:
4952:
4948:
4947:
4944:
4940:
4939:
4936:
4932:
4931:
4928:
4924:
4923:
4920:
4916:
4915:
4910:
4888:
4887:
4884:
4883:
4880:
4876:
4875:
4872:
4868:
4867:
4864:
4860:
4859:
4856:
4852:
4851:
4842:
4802:
4798:
4774:
4771:
4766:
4733:
4730:
4725:
4722:
4715:
4712:
4709:
4702:
4699:
4695:
4691:
4688:
4680:
4677:
4674:
4654:
4649:
4645:
4624:
4623:
4611:
4608:
4598:
4579:
4576:
4556:
4553:
4543:
4531:
4528:
4518:
4506:
4501:
4497:
4486:
4473:
4469:
4466:
4443:
4440:
4430:
4418:
4415:
4404:
4385:
4379:
4376:
4371:
4367:
4362:
4356:
4353:
4346:
4341:
4337:
4332:
4325:
4322:
4316:
4313:
4310:
4307:
4303:
4298:
4295:
4293:
4291:
4284:
4279:
4275:
4269:
4265:
4261:
4258:
4255:
4252:
4249:
4246:
4241:
4238:
4231:
4226:
4223:
4218:
4215:
4213:
4211:
4208:
4207:
4193:
4192:
4185:
4179:
4173:
4170:
4167:
4160:
4152:
4149:
4074:
4071:
4062:itself. Only
4051:direct-current
4018:
4015:
3997:
3992:
3985:
3981:
3977:
3972:
3967:
3964:
3957:
3952:
3945:
3941:
3937:
3932:
3925:
3921:
3917:
3912:
3908:
3901:
3898:
3895:
3892:
3889:
3875:
3874:
3871:
3870:
3867:
3864:
3861:
3858:
3854:
3853:
3850:
3847:
3844:
3841:
3837:
3836:
3833:
3830:
3827:
3824:
3820:
3819:
3816:
3813:
3810:
3807:
3803:
3802:
3799:
3796:
3793:
3790:
3786:
3785:
3782:
3779:
3776:
3773:
3769:
3768:
3765:
3762:
3759:
3756:
3752:
3751:
3746:
3741:
3736:
3731:
3717:
3714:
3691:
3669:
3666:
3663:
3660:
3657:
3608:
3604:
3599:
3595:
3564:
3560:
3551:
3547:
3538:
3534:
3525:
3500:
3495:
3492:
3487:
3483:
3480:
3474:
3471:
3465:
3461:
3455:
3452:
3448:
3444:
3409:
3380:
3376:
3367:
3363:
3354:
3350:
3341:
3329:
3328:
3315:
3312:
3309:
3306:
3303:
3276:
3263:
3262:
3249:
3246:
3243:
3240:
3237:
3210:
3197:
3196:
3183:
3180:
3177:
3150:
2955:
2952:
2880:
2877:
2873:
2838:
2835:
2808:
2802:
2799:
2796:
2793:
2788:
2784:
2778:
2775:
2772:
2769:
2764:
2760:
2750:
2747:
2744:
2739:
2736:
2730:
2720:
2677:
2674:
2669:
2665:
2659:
2656:
2649:
2643:
2637:
2632:
2615:
2593:
2590:
2587:
2562:
2559:
2556:
2551:
2548:
2542:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2513:
2510:
2504:
2501:
2481:
2433:
2430:
2428:
2425:
2413:
2407:
2404:
2401:
2398:
2393:
2389:
2383:
2380:
2377:
2374:
2369:
2365:
2358:
2355:
2352:
2348:
2344:
2341:
2338:
2335:
2331:
2310:
2304:
2301:
2298:
2295:
2290:
2286:
2280:
2277:
2274:
2270:
2245:
2225:
2222:
2219:
2216:
2212:
2190:
2169:
2143:
2139:
2118:
2115:
2112:
2109:
2106:
2101:
2097:
2092:
2088:
2085:
2082:
2079:
2075:
2054:
2051:
2048:
2045:
2042:
2038:
2032:
2028:
2024:
2021:
2018:
2015:
2011:
2004:
2001:
1997:
1990:
1987:
1982:
1979:
1976:
1973:
1969:
1960:
1956:
1952:
1946:
1942:
1936:
1933:
1930:
1927:
1923:
1917:
1913:
1909:
1906:
1903:
1900:
1896:
1890:
1886:
1868:Combining the
1865:
1862:
1849:
1837:
1836:
1823:
1819:
1813:
1809:
1805:
1802:
1792:
1774:
1769:
1765:
1754:
1737:
1734:
1724:
1708:
1705:
1700:
1697:
1690:
1687:
1677:
1659:
1655:
1652:
1649:
1643:
1637:
1633:
1630:
1627:
1624:
1617:
1614:
1604:
1590:
1585:
1581:
1570:
1556:
1551:
1547:
1536:
1524:
1520:
1509:
1495:
1492:
1489:
1486:
1482:
1471:
1449:
1446:
1443:
1440:
1436:
1425:
1411:
1408:
1398:
1384:
1381:
1370:
1352:
1349:
1346:
1343:
1338:
1334:
1328:
1325:
1322:
1319:
1314:
1310:
1303:
1300:
1297:
1293:
1289:
1283:
1280:
1277:
1274:
1269:
1265:
1259:
1256:
1253:
1250:
1245:
1241:
1231:
1228:
1225:
1219:
1215:
1209:
1206:
1203:
1200:
1196:
1168:
1165:
1159:100 kHz (
1145:
1134:
1129:
1119:
1116:
1112:
1109:
1089:
1086:
1083:
1080:
1076:
1072:
1041:
1019:
1007:
1004:
998:
995:
986:
983:
963:
960:
957:
954:
950:
946:
934:
933:
917:
912:
908:
897:
881:
876:
872:
861:
848:
844:
837:
833:
809:
806:
796:
781:
776:
772:
761:
745:
740:
736:
725:
712:
708:
701:
697:
673:
670:
660:
648:
628:
625:
622:
619:
616:
593:
590:
580:
565:
562:
551:
532:
528:
525:
522:
519:
511:
506:
502:
499:
496:
492:
487:
484:
478:
471:
468:
462:
459:
450:
447:
428:
422:speed of light
418:phase velocity
397:
385:is called the
374:
351:
347:
343:
340:
337:
334:
331:
328:
324:
320:
312:
307:
303:
300:
284:
269:
266:
238:direct current
197:
166:
163:
80:such that the
26:
9:
6:
4:
3:
2:
5896:
5885:
5882:
5880:
5877:
5876:
5874:
5857:
5853:
5849:
5845:
5841:
5835:
5831:
5827:
5823:
5818:
5815:
5809:
5805:
5800:
5796:
5792:
5791:Terman, F. E.
5788:
5785:, McGraw-Hill
5784:
5779:
5776:
5770:
5766:
5761:
5758:
5757:0-87942-238-6
5754:
5750:
5746:
5743:
5737:
5733:
5728:
5725:
5719:
5715:
5710:
5707:
5701:
5696:
5695:
5688:
5684:
5682:0-201-02117-X
5678:
5674:
5673:
5667:
5664:
5658:
5654:
5649:
5648:
5641:
5637:
5631:
5624:
5623:Feynman (1964
5619:
5612:
5611:Jackson (1999
5607:
5600:
5595:
5581:
5577:
5573:
5569:
5565:
5561:
5557:
5553:
5549:
5542:
5540:
5524:
5518:
5510:
5503:
5497:
5492:
5486:, p. 385
5485:
5480:
5472:
5465:
5459:, p. 26)
5458:
5453:
5451:
5442:
5436:
5431:
5424:
5419:
5412:
5407:
5400:
5395:
5388:
5383:
5376:
5371:
5364:
5358:
5354:
5347:
5345:
5337:
5331:
5325:
5320:
5313:
5307:
5303:
5302:
5294:
5287:
5283:
5277:
5273:
5266:
5259:
5255:
5249:
5245:
5238:
5230:
5226:
5222:
5218:
5214:
5210:
5203:
5199:
5185:
5178:
5174:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5124:
5118:
5116:
5112:
5108:
5098:
5096:
5080:
5075:
5072:
5048:
5045:
5021:
5018:
5006:
4996:
4993:
4989:
4977:
4974:
4973:
4969:
4966:
4965:
4961:
4958:
4957:
4953:
4950:
4949:
4945:
4942:
4941:
4937:
4934:
4933:
4929:
4926:
4925:
4921:
4918:
4917:
4911:
4908:
4907:
4901:
4900:
4899:
4896:
4894:
4881:
4878:
4877:
4873:
4870:
4869:
4865:
4862:
4861:
4857:
4854:
4853:
4849:
4843:
4840:
4839:
4833:
4832:
4831:
4829:
4824:
4822:
4818:
4800:
4796:
4786:
4772:
4769:
4764:
4745:
4723:
4720:
4713:
4710:
4707:
4700:
4697:
4693:
4689:
4686:
4678:
4675:
4672:
4652:
4647:
4643:
4609:
4606:
4599:
4577:
4574:
4554:
4551:
4544:
4529:
4526:
4519:
4504:
4499:
4495:
4487:
4471:
4467:
4464:
4441:
4438:
4431:
4416:
4413:
4406:
4405:
4403:
4400:
4383:
4377:
4374:
4369:
4365:
4360:
4354:
4351:
4344:
4339:
4335:
4330:
4323:
4320:
4314:
4311:
4308:
4305:
4301:
4296:
4294:
4277:
4273:
4267:
4263:
4253:
4250:
4247:
4239:
4236:
4229:
4224:
4221:
4216:
4214:
4209:
4191:(80%Ni-20%Fe)
4190:
4186:
4184:
4180:
4178:
4174:
4171:
4168:
4166:
4162:
4161:
4157:
4148:
4146:
4140:
4137:
4133:
4129:
4124:
4121:
4119:
4115:
4110:
4106:
4104:
4100:
4095:
4092:
4087:
4082:
4080:
4070:
4067:
4065:
4061:
4056:
4052:
4047:
4045:
4040:
4036:
4035:ferromagnetic
4032:
4027:
4025:
4014:
3995:
3990:
3983:
3979:
3975:
3970:
3965:
3962:
3955:
3950:
3943:
3939:
3935:
3930:
3919:
3915:
3910:
3906:
3899:
3893:
3887:
3878:
3868:
3865:
3862:
3859:
3856:
3855:
3851:
3848:
3845:
3842:
3839:
3838:
3834:
3831:
3828:
3825:
3822:
3821:
3817:
3814:
3811:
3808:
3805:
3804:
3800:
3797:
3794:
3791:
3788:
3787:
3783:
3780:
3777:
3774:
3771:
3770:
3766:
3763:
3760:
3757:
3754:
3753:
3747:
3742:
3737:
3732:
3727:
3726:
3723:
3722:
3721:
3713:
3709:
3689:
3667:
3664:
3661:
3658:
3655:
3647:
3644: =
3643:
3639:
3636: =
3635:
3630:
3627:
3606:
3602:
3593:
3583:
3562:
3558:
3549:
3545:
3536:
3532:
3523:
3513:
3498:
3493:
3490:
3485:
3481:
3478:
3472:
3469:
3463:
3459:
3453:
3450:
3446:
3442:
3434:
3430:
3407:
3398:
3378:
3374:
3365:
3361:
3352:
3348:
3339:
3313:
3310:
3307:
3304:
3301:
3274:
3265:
3264:
3247:
3244:
3241:
3238:
3235:
3208:
3199:
3198:
3181:
3178:
3175:
3148:
3139:
3138:
3137:
3134:
3127:
3123:
3115:
3107:
3099:
3091:
3083:
3075:
3067:
3059:
3051:
3037:
3026:
3013:
3000:
2990:
2983:
2979:
2977:
2954:Coaxial cable
2951:
2947:
2944:
2940:
2934:
2931:
2919:
2908:
2903:
2899:
2897:
2878:
2875:
2871:
2860:
2856:
2852:
2848:
2844:
2834:
2832:
2828:
2824:
2819:
2806:
2797:
2794:
2786:
2782:
2773:
2770:
2762:
2758:
2748:
2745:
2742:
2737:
2734:
2728:
2708:
2705:
2701:
2697:
2692:
2667:
2663:
2647:
2641:
2630:
2610:
2605:
2591:
2588:
2585:
2576:
2560:
2557:
2554:
2549:
2546:
2540:
2534:
2528:
2525:
2522:
2516:
2511:
2508:
2502:
2499:
2479:
2463:
2462:approximately
2439:
2424:
2411:
2402:
2399:
2391:
2387:
2378:
2375:
2367:
2363:
2353:
2342:
2336:
2308:
2299:
2296:
2288:
2284:
2275:
2223:
2217:
2167:
2159:
2141:
2137:
2116:
2110:
2107:
2099:
2095:
2086:
2080:
2052:
2049:
2043:
2030:
2026:
2022:
2016:
2002:
1988:
1985:
1980:
1974:
1958:
1954:
1944:
1934:
1928:
1915:
1911:
1907:
1901:
1888:
1875:
1871:
1861:
1847:
1821:
1817:
1811:
1807:
1803:
1800:
1793:
1790:
1772:
1767:
1763:
1755:
1752:
1735:
1732:
1725:
1706:
1703:
1698:
1695:
1688:
1685:
1678:
1675:
1657:
1653:
1650:
1647:
1641:
1635:
1631:
1628:
1625:
1622:
1615:
1612:
1605:
1588:
1583:
1579:
1571:
1554:
1549:
1545:
1537:
1522:
1510:
1493:
1487:
1472:
1469:
1466:at distance,
1465:
1447:
1441:
1426:
1409:
1406:
1399:
1382:
1379:
1372:
1371:
1369:
1366:
1347:
1344:
1336:
1332:
1323:
1320:
1312:
1308:
1298:
1287:
1278:
1275:
1267:
1263:
1254:
1251:
1243:
1239:
1229:
1226:
1223:
1213:
1207:
1201:
1180:
1176:
1174:
1164:
1156:
1143:
1132:
1127:
1117:
1114:
1110:
1107:
1084:
1081:
1074:
1070:
1061:
1055:
1039:
1030:
1017:
1005:
1002:
996:
993:
984:
981:
958:
955:
948:
944:
931:
915:
910:
906:
898:
895:
879:
874:
870:
862:
846:
842:
835:
831:
822:
807:
804:
797:
795:
779:
774:
770:
762:
759:
743:
738:
734:
726:
710:
706:
699:
695:
686:
671:
668:
661:
646:
626:
623:
620:
617:
614:
606:
591:
588:
581:
578:
563:
560:
553:
552:
550:
547:
530:
526:
523:
520:
517:
509:
504:
500:
497:
494:
490:
485:
482:
476:
469:
466:
460:
457:
448:
445:
436:
434:
423:
419:
415:
407:
403:
392:
388:
372:
349:
345:
341:
335:
332:
329:
322:
318:
305:
301:
298:
279:
265:
263:
258:
254:
250:
246:
241:
239:
235:
231:
227:
223:
218:
216:
212:
196:
190:
184:
178:
171:
162:
160:
156:
151:
149:
145:
141:
137:
133:
129:
124:
122:
117:
115:
111:
106:
101:
99:
95:
91:
87:
86:eddy currents
83:
79:
75:
71:
67:
59:
55:
51:
44:
39:
33:
19:
5821:
5803:
5794:
5782:
5764:
5748:
5731:
5713:
5693:
5671:
5652:
5635:
5630:
5618:
5606:
5594:
5583:. Retrieved
5555:
5551:
5527:. Retrieved
5517:
5508:
5502:
5491:
5479:
5470:
5464:
5430:
5418:
5406:
5394:
5382:
5377:, p. ??
5370:
5352:
5336:Jordan (1968
5330:
5319:
5300:
5293:
5285:
5271:
5265:
5257:
5243:
5237:
5212:
5208:
5202:
5183:
5177:
5132:Eddy current
5107:Heinz London
5104:
5008:
4987:
4985:
4978:2.06
4970:6.52
4897:
4889:
4825:
4787:
4746:
4626:
4401:
4197:
4141:
4125:
4122:
4117:
4111:
4107:
4096:
4089:composed of
4086:transformers
4083:
4078:
4076:
4068:
4064:non-magnetic
4048:
4031:permeability
4028:
4020:
3879:
3876:
3719:
3710:
3645:
3641:
3637:
3633:
3631:
3628:
3584:
3514:
3432:
3428:
3399:
3330:
3135:
3131:
3125:
3117:
3109:
3101:
3093:
3085:
3077:
3069:
3061:
3053:
3047:
3035:
3024:
3018:= dielectric
3011:
2998:
2988:
2971:
2957:
2948:
2942:
2938:
2935:
2929:
2925:
2906:
2858:
2854:
2846:
2842:
2840:
2820:
2703:
2693:
2606:
2577:
2461:
2435:
1867:
1839:
1467:
1367:
1186:
1170:
1157:
1062:
1031:
936:
894:permittivity
821:permittivity
685:permeability
548:
437:
386:
271:
256:
244:
242:
229:
225:
219:
207:
192:
186:
180:
174:
152:
125:
118:
102:
97:
93:
69:
63:
5435:Reeve (1995
5375:Terman 1943
5215:: 519–549.
5142:Transformer
4753:10 Ω·m
4633:10 Ω·m
4595:10 S/m
4079:Litzendraht
4044:laminations
4039:micrometers
3431:per length
3003:= conductor
2896:table below
2609:F.E. Terman
1751:resistivity
1674:wave number
607:of current
577:resistivity
433:Snell's law
257:skin effect
226:skin effect
155:Horace Lamb
70:skin effect
5873:Categories
5599:Hayt (1981
5585:2020-12-22
5529:2011-11-08
5457:Chen (2004
5423:Hayt (1981
5411:Hayt (1981
5387:Hayt (1981
5194:References
5095:waveguides
5003:See also:
4912:Skin depth
4844:Skin depth
4828:micrometer
4101:often use
4073:Mitigation
2837:Inductance
2831:inductance
2432:Resistance
1864:Derivation
1167:Round wire
1058:10 Hz
406:wavelength
387:skin depth
230:skin depth
130:circuits,
110:resistance
98:skin depth
18:Skin depth
5856:114947614
5848:0197-2618
5572:1531-1309
5229:111283238
5184:magnitude
5137:Litz wire
5073:δ
4909:Frequency
4841:Conductor
4797:μ
4711:⋅
4698:−
4690:⋅
4673:δ
4644:μ
4578:≈
4575:σ
4552:σ
4527:μ
4496:μ
4439:α
4414:δ
4378:σ
4366:μ
4352:≈
4336:μ
4331:ρ
4321:≈
4315:σ
4312:μ
4306:π
4274:μ
4264:μ
4251:π
4240:ρ
4225:α
4210:δ
4189:permalloy
4136:microwave
3924:∞
3920:ℓ
3907:ℓ
3728:Frequency
3598:∞
3482:
3473:π
3460:μ
3040:= current
3029:= current
2992:= overall
2879:π
2872:μ
2851:induction
2827:reactance
2746:π
2738:ρ
2707:impedance
2592:δ
2589:≫
2561:δ
2555:π
2550:ρ
2547:ℓ
2541:≈
2535:δ
2529:δ
2526:−
2517:π
2512:ρ
2509:ℓ
2503:≈
2480:ρ
2427:Impedance
2160:of order
2000:∂
1996:∂
1951:∂
1941:∂
1885:∇
1876:produces
1874:Ohm's law
1818:μ
1808:μ
1801:μ
1787:relative
1764:μ
1733:ρ
1707:μ
1704:ω
1699:ρ
1686:δ
1658:δ
1651:−
1636:ρ
1632:μ
1629:ω
1623:−
1227:π
1133:μ
1128:ε
1118:ρ
1111:≈
1108:δ
1085:ε
1082:ρ
1040:ε
1006:μ
1003:ω
997:ρ
982:δ
959:ε
956:ρ
907:ε
892:relative
871:ε
843:ε
832:ε
805:ε
771:μ
756:relative
735:μ
707:μ
696:μ
669:μ
621:π
589:ω
561:ρ
527:ε
524:ω
521:ρ
501:ε
498:ω
495:ρ
470:μ
467:ω
461:ρ
446:δ
373:δ
350:δ
323:−
249:induction
211:electrons
161:in 1885.
128:microwave
121:litz wire
105:frequency
78:conductor
58:Litz wire
5793:(1943),
5580:30187468
5121:See also
4855:Aluminum
4181:Fe-Si –
4151:Examples
4024:2000 MCM
3750:(nF/km)
3128:remains.
2859:external
2704:internal
2257:must be
228:and the
136:antennas
112:. At 60
90:magnetic
4891:3
4165:ferrite
4055:welding
3866:118.074
3745:(μS/km)
3740:(mH/km)
3681:. Only
3016:
3008:
2978:below.
2855:outside
2823:complex
402:delayed
268:Formula
245:induced
5854:
5846:
5836:
5810:
5771:
5755:
5738:
5720:
5702:
5679:
5659:
5578:
5570:
5359:
5308:
5278:
5250:
5227:
4992:busbar
4882:0.634
4879:Silver
4874:0.753
4866:0.652
4863:Copper
4858:0.820
4402:where
4128:plated
4118:ribbon
4114:busbar
3869:51.57
3863:0.4675
3860:999.41
3852:51.57
3849:53.205
3846:0.4862
3843:643.14
3835:51.57
3832:29.111
3829:0.5062
3826:463.59
3818:51.57
3812:0.5807
3809:191.63
3801:51.57
3795:0.6099
3792:172.70
3784:51.57
3778:0.6125
3775:172.28
3767:51.57
3761:0.6129
3758:172.24
3735:(Ω/km)
3046:flux (
2966:, and
2930:inside
2922:depth.
2843:inside
2651:
2438:losses
2321:Thus,
2129:where
1840:Since
1464:phasor
1368:where
1141:
1015:
639:where
549:where
414:vacuum
365:where
5852:S2CID
5576:S2CID
5225:S2CID
5169:Notes
4914:(μm)
4053:(DC)
3815:3.327
3798:0.531
3781:0.072
3764:0.000
3344:total
3100:and
3025:green
3012:white
2989:black
2696:cable
2156:is a
251:. An
165:Cause
5844:ISSN
5834:ISBN
5808:ISBN
5769:ISBN
5753:ISBN
5736:ISBN
5718:ISBN
5700:ISBN
5677:ISBN
5657:ISBN
5568:ISSN
5441:help
5357:ISBN
5306:ISBN
5276:ISBN
5248:ISBN
4871:Gold
4724:11.1
4687:2.44
4629:2.44
4591:58.5
3730:(Hz)
3665:<
3659:<
3311:<
3305:<
3245:<
3239:<
3179:<
3036:blue
2622:is:
2492:is:
2180:and
1872:and
1672:the
928:the
792:the
94:skin
32:skin
5826:doi
5560:doi
5217:doi
5213:174
5093:In
5063:is
4986:In
4749:2.2
4679:503
4355:503
4324:503
4134:to
4132:VHF
4060:arc
3694:ext
3611:ext
3567:ext
3554:shd
3541:cen
3412:ext
3383:ext
3370:shd
3357:cen
3279:shd
3213:ext
3153:cen
2999:tan
2724:int
2648:200
234:1/e
64:In
5875::
5850:.
5842:.
5832:.
5574:.
5566:.
5556:26
5554:.
5550:.
5538:^
5449:^
5343:^
5284:.
5256:.
5223:.
5211:.
5117:.
4893:μm
4850:)
4848:μm
4830::
4714:50
4694:10
4145:5G
3626:.
3582:.
3528:DC
3479:ln
3020:,
3005:,
2962:,
2909:/8
2898:.
2604:.
2053:0.
1060:.
264:.
114:Hz
100:.
68:,
5858:.
5828::
5759:.
5685:.
5562::
5532:.
5443:)
5231:.
5219::
5081:.
5076:2
5049:e
5046:1
5022:e
5019:1
4846:(
4801:r
4773:3
4770:=
4765:9
4751:×
4732:m
4729:m
4721:=
4708:1
4701:8
4676:=
4653:=
4648:r
4631:×
4610:=
4607:f
4597:)
4593:×
4555:=
4530:=
4505:=
4500:0
4472:m
4468:p
4465:N
4442:=
4417:=
4384:,
4375:f
4370:r
4361:1
4345:f
4340:r
4309:f
4302:1
4297:=
4283:)
4278:r
4268:0
4260:(
4257:)
4254:f
4248:2
4245:(
4237:2
4230:=
4222:1
4217:=
3996:b
3991:)
3984:m
3980:f
3976:f
3971:(
3966:+
3963:1
3956:b
3951:)
3944:m
3940:f
3936:f
3931:(
3916:+
3911:0
3900:=
3897:)
3894:f
3891:(
3888:L
3748:C
3743:G
3738:L
3733:R
3690:L
3668:c
3662:r
3656:b
3646:b
3642:r
3638:a
3634:r
3607:L
3603:=
3594:L
3563:L
3559:+
3550:L
3546:+
3537:L
3533:=
3524:L
3499:)
3494:a
3491:b
3486:(
3470:2
3464:0
3454:=
3451:D
3447:/
3443:L
3433:D
3429:L
3408:L
3379:L
3375:+
3366:L
3362:+
3353:L
3349:=
3340:L
3314:c
3308:r
3302:b
3275:L
3248:b
3242:r
3236:a
3209:L
3182:a
3176:r
3149:L
3120:D
3112:C
3104:D
3096:C
3088:B
3080:D
3072:C
3064:B
3056:A
3049:B
2974:A
2968:c
2964:b
2960:a
2911:π
2907:μ
2876:8
2807:.
2801:)
2798:R
2795:k
2792:(
2787:1
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2777:)
2774:R
2771:k
2768:(
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2729:=
2719:Z
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2673:H
2668:/
2664:f
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2655:m
2642:=
2636:W
2631:D
2620:f
2616:W
2613:D
2586:D
2558:D
2532:)
2523:D
2520:(
2500:R
2470:ℓ
2466:δ
2458:δ
2454:D
2450:δ
2446:δ
2442:δ
2412:.
2406:)
2403:R
2400:k
2397:(
2392:0
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2334:(
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2309:.
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2221:)
2218:R
2215:(
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2142:0
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2108:k
2105:(
2100:0
2096:J
2091:C
2087:=
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2081:r
2078:(
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2050:=
2047:)
2044:r
2041:(
2037:J
2031:2
2027:k
2023:+
2020:)
2017:r
2014:(
2010:J
2003:r
1989:r
1986:1
1981:+
1978:)
1975:r
1972:(
1968:J
1959:2
1955:r
1945:2
1935:=
1932:)
1929:r
1926:(
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1899:(
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1889:2
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1822:0
1812:r
1804:=
1773:=
1768:r
1736:=
1696:2
1689:=
1654:j
1648:1
1642:=
1626:j
1616:=
1613:k
1589:=
1584:1
1580:J
1555:=
1550:0
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1523:=
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1494:=
1491:)
1488:R
1485:(
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1442:r
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1410:=
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1383:=
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1318:(
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1288:=
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1202:r
1199:(
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1161:λ
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1115:2
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1079:(
1075:/
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1018:.
994:2
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953:(
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932:.
916:=
911:0
880:=
875:r
847:0
836:r
808:=
780:=
775:0
744:=
739:r
711:0
700:r
672:=
647:f
627:,
624:f
618:2
615:=
592:=
564:=
531:)
518:+
510:2
505:)
491:(
486:+
483:1
477:(
458:2
449:=
429:o
426:λ
410:π
398:S
395:J
391:e
346:/
342:d
339:)
336:j
333:+
330:1
327:(
319:e
311:S
306:J
302:=
299:J
289:d
285:S
282:J
274:J
200:W
194:I
188:H
182:H
176:I
34:.
20:)
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