UnitMath Examples ( Skin Depth )

Last Modified: 2/16/2000

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- What is Skin Depth?
- Where is the Skin Depth Formula Valid?
- Useful Metal Properties
- Simple Skin Depth Examples
- Skin Depth and Temperature Examples

Neither the electric field (E) nor the magnetic field (H) penetrate far into a "good" conductor. The point where these fields are reduced by a factor of 1/e ~ 1/2.71 is called the skin depth. The figure at the right shows a good conductor and how a pulse traveling along this conductor is attenuated going into the conductor.

Skin depth is dependent on the type of metal in the conductor and the frequency fields applied to the conductor. At high frequencies the skin depth is very shallow, and the field are often considered to be 0 in a few millimeters. A general rule is that at 5 * skin depth the fields can be considered to be 0 ( The actual value is ( 1/e )^5 = 0.674 %; which, is indeed quite small ).

Skin depth is important in many pulsed power applications because it changes the effective resistance in a conductor, and that only the surface of the conductor matters.

Generally the skin depth formula is valid for frequencies below several hundred GHz. A more precise value can be found by insuring the following conditions are met:

frequency << conductivity / eo

frequency << 1 / tau

where:

tau: conductivity electron_mass / N / sq electron_charge

N: density * Avogadro's# * mole / (atomicWt gram)

density: the metals density

atomicWt: the metals atomic weight

conductivity: 1 / ( the metals resistivity )

eo: the permitivity of free space

Find the valid frequency range for annealed copper.

`
density: 8.89 gram/cc;
atomicWt: 63.54;
resistivity: 1.7241 microOhm cm;
conductivity: 1/resistivity; ~ 5.800e7 / ( m ohm )
`

`
N: density * Avogadro's# * mole / (atomicWt gram) as 1/cu m ~ ( 8.4203 to 8.4311 )e28 1/cu m
tau: conductivity electron_mass / N / sq electron_charge ~ 2.44e-14 s
`

`
conductivity / eo as hz ~ 6.551e18 hz
1 / tau as hz ~ ( 4.091 to 4.096 )e13 hz
`

`
f: 1/ tau / 100 as Ghz ~ ( 409.1 to 409.6 ) Ghz
`

So the skin depth formula is valid for all frequencies up to about 400 GHz.

- The following table lists useful information for common metals used as conductors. In the table:
- Ro is the metal's resistance, and a is the coefficient of resistivity both are measured at 20 degrees C. Ro and a are used to find the metal resistance at a given temperature using the formula: R = Ro ( 1 + a deltaT ). Where R is the resistance and deltaT is the temperature difference from 20 degrees C.
- f is the frequency beyond which the skin depth formula becomes inaccurate, and sd is skin depth at f.

Metal | Density | Ro | a | f in GHz | sd |
---|---|---|---|---|---|

Aluminum | 2.70 g/cc | 2.824 microOhm cm; | 0.0039 / K | 478.59 to 480.54 | 0.12 microns |

Annealed Copper | 8.89 g/cc | 1.7241 microOhm cm; | 0.00393 / K | 409.1 to 409.6 | 0.1033 microns |

Gold | 19.3 g/cc | 2.44 microOhm cm; | 0.0034 / K | 403.8 to 407.6 | 0.12 microns |

Mercury | 13.546 g/cc | 95.783 microOhm cm; | 0.00089 / K | 10,975. to 10,978. | 0.15 microns |

Silver | 10.5 g/cc | 1.59 microOhm cm; | 0.0038 / K | 2.6e2 | 0.12 microns |

Find the skin depth and resistivity per square of copper when subjected to a 100 MHz field.

`
resistivity: 1.7241 microOhm cm;
conductivity: 1/resistivity; ~ 5.80e7 / ( m ohm )
µ: 1 µo;
f: 100 Mhz;
`

`
skin_depth: 1/sqrt( conductivity pi µ f ) as cm ~ 6.61 e-4 cm
`

`
Rsq:1/( conductivity * skin_depth ) as mOhm ~ 2.61 mOhm
`

Find the skin depth and resistivity per square of aluminum when subjected to a 1.6 MHz field.

`
resistivity: 2.824 microOhm cm;
conductivity: 1/resistivity; ~ ( 3.5 to 3.5 to 3.5 )e7 / ( m ohm )
µ: 1 µo;
f: 1.6 Mhz;
w: 2 pi f; ~ ( 1.0 to 1.0 to 1.0 )e7 / s
`

`
skin_depth: 1/sqrt( conductivity pi µ f ) as microns ~ ( 65.8 to 66.9 to 67.9 ) microns
`

`
Rsq:1/( conductivity *skin_depth ) as mOhm ~ ( 0.4 to 0.4 to 0.4 ) mOhm
`

Find the frequency needed to have a skin depth of 1 mm in Silver.

`
resistivity: 1.59 microOhm cm;
conductivity: 1/resistivity; ~ 6.3e7 / ( m ohm )
µ: 1 µo;
`

`
skin_depth: 1/sqrt( conductivity pi µ f ) as microns;
f: 1 khz to 100 Mhz as kHz;
solve( skin_depth = 1 mm; f: 10 Hz ) ~ f: ( 4.01 to 4.05 ) kHz;
`

So a 4 kHz field will have a 1 mm skin depth in silver.

Find the skin depth and resistance of a copper coated ( 4 ft by 8 ft ) sheet of plywood at 212 degrees F, when subjected to a 10 MHz field.

`
R0: 1.7241 microOhm cm;
tempCoef:0.00393/Cdeg;
temperature: 212 degF;
tempChangeFromStd : temperature - 20 degC;
`

`
Rf: R0( 1 + tempCoef * tempChangeFromStd ) as microOhm cm ~ 2.3 microOhm cm
conductivity: 1/Rf; ~ 4.4e7 / ( m ohm )
µ: 1 µo;
f: 10 Mhz;
`

`
skin_depth: 1/sqrt( conductivity pi µ f ) as cm ~ 2.4 e-3 cm
`

`
Rsq: 1 / ( conductivity * skin_depth ) as mOhm ~ 0.9 mOhm
`

`
length: 8 ft; width: 4 ft;
Resistance: Rsq length / width as mOhm ~ 1.9 mOhm
`

So the sheet will have a resistance of 1.9 mOhms at 212 degF with a 10 MHz field.

Find the skin depth and resistance of a copper coated ( 4 ft by 8 ft ) sheet of plywood at -40 degrees F, when subjected to a 10 MHz field.

`
R0: 1.7241 microOhm cm;
tempCoef:0.00393/Cdeg;
temperature: -40 degF;
tempChangeFromStd : temperature - 20 degC;
Rf: R0( 1 + tempCoef * tempChangeFromStd ) as microOhm cm ~ 1.3 microOhm cm
conductivity: 1/Rf; ~ 7.6e7 / ( m ohm )
µ: 1 µo;
f: 10 Mhz;
`

`
skin_depth: 1/sqrt( conductivity pi µ f ) as cm ~ 1.8 e-3 cm
`

`
Rsq: 1 / ( conductivity * skin_depth ) as mOhm ~ 0.7 mOhm
`

`
length: 8 ft; width: 4 ft;
Resistance: Rsq length / width as mOhm ~ 1.4 mOhm
`

So the sheet will have a resistance of 1.4 mOhms at -40 degF in a 10 MHz field.