# Resistivity and permeability

Resistivity of a material is the resistance from face to opposite face of a 1m cube of a material. Resistivity varies with different metals, a few common conductors are listed below.

Permeability is the ratio of flux density to magnetising force, self inductance is proportional to permeability.

 Conductor Resistivity Relative (µΩ-m) Permeability (µr) Conductivity (MS/m) Copper 0.0168 59.52 1 Silver 0.0159 62.89 1 Aluminium 0.0282 35.46 1 Aluminium 7xxx annealed 0.045 22.22 1 ZA ZAL 0.059 17 1 Zinc 0.059 16.95 1 Brass 0.064 15.63 1 Nickel 0.064 15.63 1240 Iron 0.100 10.00 200 Tin 0.109 9.17 1 Solder (63/37 Eutectic) 0.145 6.90 1 Mild steel 0.20 5.00 800 Lead 0.22 4.55 1 Stainless steel 316 0.75 1.33 1 Stainless steel 17-7 PH 0.82 1.22 120

A note about stainless steels. Pure austenitic steels do not exhibit ferro magnetism, relative permeability is unity. Many stainless steels are austenitic, but the process of hardening them, or the hardening as a result of working them (eg drawing wire) can create martensite, and they may exhbit µr significant greater than unity. Be suspicious of hardened stainless steel, check it with a magnet. If it is attracted to the magnet, it may have high µr and as a result may have quite high RF resistance. Stainless 316 tends to not develop ferromagnetism with work, 304 a little, and in the table above, 17-7PH has quite high µr in the hardened state (you wouldn't use it in the annealed state).

# Skin effect

AC current tends to distribute itself in conductors so that current density tends to be higher near the surface of conductors than deep inside the conductors. Skin effect is very pronounced at radio frequencies and must be considered when determining the effective resistance of conductors at radio frequencies. The effective resistance for a round conductor with much larger than skin depth δ is approximately that of a cylinder of thickness δ carrying DC current.

Skin depth is calculated as follows.

δ=((2ρ)/(ωμ))0.5

Conductors with high resistivity and high permeability will have high effective resistance at radio frequencies. However, because of skin effect, a hollow tube of low resistivity material of sufficient thickness, or a poor conductor clad in material of low resistivity of sufficient thickness may have low effective RF resistance, similar to a solid conductor of the good conductor material. For this to be true, the cladding thickness needs to be more than about three skin depths in thickness.

When the outer 'layer' of a circular conductor is more than about 3δ, its resistance can be approximated as π/4(d2-(d-2δ)2)ρ.

Note that some materials used to coat wire might be higher resistivity than the base material, an example is solder tinned copper wire.

Resistance is proportional to diameter, resistivity and skin depth, which is proportional to the square root of resistivity, and inversely to square root of frequency and permeability.

# Resistive losses in antenna conductors

Current flowing in antenna conductors generates heat because of I2R losses. Since current varies along the conductors, the total amount of heat generated can found by segmenting the conductors, calculating the segment currents and totalling I2R for each segment.

In a half wave dipole, the current distribution is approximately the cosine of the phase displacement from centre , and the total heat generated is I02R/2 where I0 is the current at the centre, and R is the end to end resistance of the dipole conductor. R/2 can be considered the equivalent conductor resistance referred to the feedpoint. The same factor is applicable to a quarter wave monopole over perfect ground.

In the case of an unloaded very short end fed monopole (over perfect ground) or centre fed dipole, the current tapers approximately linearly from feed point to the end, and total heat generated is I02R/3 where I0 is the current at the feedpoint, and R is the end to end resistance of the radiator conductor. R/3 can be considered the equivalent conductor resistance referred to the feedpoint. Under some circumstances, the current may taper even more quickly and heat generated less than I02R/3.

For the purpose of this discussion, lets call the equivalent conductor resistance referred to the feedpoint Rc'.

The portion of RF power delivered to the feedpoint can be calculated as Rc'/Rtot, and so acceptable values for Rc' depend on the magnitude of Rtot. As a dipole is shortened from a half wave in length, Rtot falls more quickly that Rc', so the portion of input power lost as heat in the conductors increases. Likewise as a monopole is shortened from a quarter wave in length.

Because of skin effect, quadrupling frequency increases R per unit length by a factor of two, but the radiator is one quarter of the length, so Rc' is doubled. Likewise, reducing frequency by a factor of two increases Rc' by a factor of square root of two (1.414). Conductor losses tend to be more significant as frequency is reduced.

Conductor losses tend to be more significant:

• with electrically short antennas; and
• at lower frequencies.

# Bringing it all together

The loss performance of a conductor depends on all these effects. The suitability of a material depends on its characteristics and its application

Radiator Skin Depth (µm) Rc' (Ω) R'(Ω) Conductor loss (%)
20mm Aluminium quarter wave monopole at 20m
23
0.05
35.1
0.1
12mm Stainless Steel 316 folded dipole at 2m
36
0.55
300
0.2
2mm HDC half wave dipole at 20m 18
0.78
70.8
1.1
3mm Stainless Steel 316 quarter wave monopole at 2m
36
0.56
35.5
1.6
2mm HDC at 160m 49
2.3
77.3
3.0
2.7mm Aluminium (Gallagher XL fence wire) dipole at 160m 63
2.2
72.2
3.0
1.6mm Zinc half wave dipole at 40m (heavy gal fence wire)
46
2.6
72.6
3.6
40m dipole 0.2mm Copper (#32)
25
6.3
76.3
8.2
3mm Stainless Steel 17-7PH quarter wave monopole at 2m 3.8
5.8
41.8
13.8
0.9mm Stainless Steel 316 half wave dipole at 40m 165
19.8
94.7
20.8
1.6mm Stainless Steel 316 half wave dipole at 160m 325
23.1
98.1
23.5
4.8mm Stainless Steel 17-7PH unloaded 2.6m vertical at 80m over perfect ground
25
1.94
2.3
84.7
0.9mm steel MIG wire half wave dipole at 80m
4.9
397
63
84.1

Table notes: Rc' is the conductor resistance referred to the feed point, R' is the assumed resistance component of feedpoint impedance. Monopoles are assumed to be over perfect ground, results will be different for on-vehicle mounts. The model uses the simple method given above for estimating effective resistance. NEC4 uses a more accurate method for calculating resistance and will give better results.

The table above gives some applications and expected conductor loss. Note the very wide range of loss figures, and whilst some materials are very lossy in some applications, they may be quite good in others.

Some of the examples exploit skin effect in a low resistivity cladding over a high resistivity, and sometimes ferromagnetic, core. It is critical in these applications that the cladding thickness is at least several skin depths in thickness. For example, whilst heavy galvanised steel fence wire may perform well, modern Zal clad fence wire with a polymer coating has insufficient Zal thickness to be effective at low HF and will perform quite poorly due to the high resistivity ferromagnetic core.

You might be surprised how well the 40m dipole using 0.2mm copper performs (though not to be recommended for structural reasons), yet the 0.9mm Stainless Steel (MIG wire) dipole is quite thicker but poorer. Stealth antennas often sacrifice performance for low visibility.

The 0.9mm steel MIG wire is often confused with Copper Clad Steel because of the flash of copper on MIG wire to stop it rusting. The copper flash is insignificant, and it is an extremely poor antenna wire. This is certainly the makings of a stealth antenna, not only can no one see the antenna, no one can hear it!

You might be surprised how poorly the popular 2.6m (102") 17-7PH Stainless Steel whip performs as a short unloaded antenna. Still, it is very robust, and RF loss is one of the trade offs. Note that in a typical mobile installation, R' may be more like 7Ω than 2.4Ω, and loss more like 30%.