LF: Re ELF…12.67Hz, ELF Resistance Model

[James Hollander <mrsocion@aol.com>]

        Hi ELF,    Do KV’s of ELF on a TX antenna offset or divert a tiny amount of atmospheric current entering the RX antenna?  Suppose the atmosphere has equipotential surface(s) high in a an electrically resistive air mass in which the ELF TX and RX antennas are embedded at ground level.   If so, what happens quantitatively? 

        I’m trying out different math models to describe the atmosphere around the DK7FC 12.67Hz site to see if any model might be supported by actual numbers and performance Stefan may develop.

        Call one an “ELF Resistance Model” as if the atmosphere has some fixed resistance R1 coming down from an equipotential plane at altitude.  In the model, R1 connects to a disk of air represented by a large number N of parallel resistances R_sky going down to earth. Each such parallel resistance represents  the resistance of a vertical column of air having cross-sectional area proportional to antenna height-squared  A = Co h_Ant².  Currents in the parallel resistances R_sky compete with each other to divide up the current coming down from the equipotential plane high above.

        The TX antenna and RX antenna have equal height h_Ant.  If the antennas instead have a top hat area A that is bigger than h_Ant² , then area A is whichever top hat mesh area or square of top hat wire length applies.  (The horizontal length of an inverted-L is considered its top hat wire length.)

        Picture the disk of air several kilometers radius with DK7FC 12.67Hz TX antenna at its center and RX antenna at a distance r meters away, say at 3500m or you name it.   The TX antenna delivers many KV of 12.67Hz ELF voltage to just one of the resistances R_sky.

        After the math is done, the results say that the voltage on the RX antenna declines slowly at shorter distances and then declines as an inverse-square (1/r^2) farther away.  That’s a lot less favorable than a radio antenna.  In a radio antenna the voltage merely declines inversely (1/r).  Signal strength dB in this ELF model would fall as  -40 log₁₀ r at the farther distances, which is -12dB for every doubling of distance.  Ouch!

       If Stefan were able to set up a second identical RX antenna and RX-integrator setup at a second distance r₂ and compare its performance with his setup at current distance r₁, the model allows one to calculate the value of a constant K that the model needs to predict signal strength at any distance r.   Basically, this constant K lumps together sky resistances with the antenna-based area A.   

      Specifically, the model formula for antenna voltage at any distance r says:

 

       V_RX(r) = V_RX(r₁)  (K+r₁²) / (K+r²).

 

In dB, a model formula recognizes power ratio V-squared and the inverse-squared distance:

 

        V_RX(r) / V_RX(r₁) in dB = -40 log₁₀[ (r / r₁) sqrt[(1+K/r²)/(1+K/r₁²)] ].

 

Constant K is defined to be (1/π) (R_sky/R1) A.

Constant K is calculated from reception voltages (volts, not dB) at two RX stations using equal integration times, according to this calculation:

 

         K = [V_RX2 r₂² -  V_RX1 r₁²] / [V_RX1 -  V_RX2].      

 

        I hope the equation formats haven’t been too badly mangled by the email and the LF reflector!    I’m also considering a disk-shaped lossy RLC transmission-line model of the ELF atmosphere, which probably would predict different behavior.   Maybe someday, amateur work can tell us which model(s) are garbage or useful material for advancing ham ELF aspirations. 

GL & 73, Jim H   W5EST