[TowerTalk] Ground wire impedance

[K4SAV]

All this grounding talk has got me thinking again.

Most of us know that the impedance of a wire is increased by its skin 
resistance, and that a wire with more skin area (such as a strap) will 
provide a lower impedance. But how much lower? I decided to break out my 
spread sheet I made for calculating these things and take a look. The 
data is tabulated below.

Some interesting conclusions can be drawn from this data. For example, 
at 330 HZ the impedance of a #4 wire due to its inductance is the same 
as its skin resistance, but the DC resistance still dominates.  At 1660 
Hz the impedance due to inductance is equal to the total DC resistance 
(skin plus DC resistance). That means for any frequency above 1660 Hz, 
the impedance of the wire is due primarily to its inductance, not its DC 
resistance or skin resistance. As a matter of fact, above about 20 KHz, 
the DC resistance and skin resistances are insignificant for this wire, 
even though they are continually increasing. Since all these parameters 
are linear with length, this conclusion is the same regardless of the 
length of the wire. For a #2 wire, this same point happens at 1100 HZ 
instead of 1660 Hz.

Note that when applying these conclusions to other things besides ground 
wires, there is a difference between impedance due to inductance and 
impedance due to resistance.

---------------------
Calculated data:
#4 wire, 10 ft length, L = 0.43 uH (straight wire in free space)
Note: A wire in the ground will appear as a higher inductance than shown 
here, because of the decreased velocity factor of the medium.
Z(L) represents impedance calculated from inductance only.

Freq       Z(L)       DC res    Skin res
330 Hz    8.86e-4    2.49e-3    8.86e-4
920 Hz    2.47e-3    2.49e-3    1.48e-3
1660 Hz   4.46e-3    2.49e-3    1.99e-3
10 kHz    2.69e-2    2.49e-3    4.88e-3
100 kHz   0.269      2.49e-3    1.54e-2
1 MHz     2.69       2.49e-3    4.88e-2


I didn't have a spreadsheet already made up to calculate the skin 
resistance of a strap, but I do have one to calculate its inductance. 
Since the inductance is the predominate parameter, it's probable all you 
will need anyway. The calculations are for a strap thickness of 0.05 
inches, and a length of 10 ft.  Since the thickness doesn't effect the 
inductance very much, it wasn't included as a variable parameter. 
Compare these numbers to a #4 wire, same length, which was 0.43 uH.

Strap width   Inductance uH
 0.5 in          0.40
 1 in            0.36
 2 in            0.32
 3 in            0.30
 4 in            0.28
 5 in            0.27
 6 in            0.26

The formula for the wire inductance and strap inductance came from the 
Polyphaser book, Grounds for Lightning & EMP Protection.
 
One other note of significance: None of these calculations include 
resonant effects. For  wires that are long compared to the frequencies 
being considered, resonance effects can increase the impedance by a huge 
amount compared to an impedance value calculated from wire inductance.

Jerry, K4SAV