**A Measurement Method For**

**Ground****
Conductivity At Radio Frequencies**

**James E. Lanier K4BPM**

** The
method described in this paper uses a parallel conductor transmission line
buried in the earth**** to determine ground parameters.
The line parameters in free space are compared to the parameters of the same line in the complex
medium of earth.**

**The equivalent circuit of a transmission line
is shown here in figure 1.**

**figure 1**

**Where:**

**L=inductance per unit length (henries)**

**R=resistance per unit length (ohms)**

**C=capacitance per unit length (farads)**

**G= conductance per unit length (siemens)**

**The characteristic impedance of the line is
found from**

**Equation 1**

**where ZO is a
complex impedance in the form R (+ -) jX.**

**It is convenient to refer to the parameters
per unit length normalized to meters. **

**Where:**

**L'=inductance per meter (henries/m)**

**R'=resistance per meter (ohms/m)**

**C'=capacitance per meter (farads/m)**

**G'= conductance per meter (siemens/m)**

**Equation 2**

**In the case of the line in free space it will
be assumed that G and R are very small quantities and will be neglected.**

**The characteristic impedance of the line in
free space is:**

**Equation ****3**

**ZO
is real with no imaginary component.**

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**Transmission Line ****Probe****
Parameters in Free Space**

**Equation 4**

**Where:**

**a = distance between conductors in
meters**

**b = radius of conductors in meters**

*e**0 *=* *
permittivity of free space =
8.854187817 × 10^{−12} F/m

*c**0 *=* ***speed of light ****=****
299,792,458 m/s**

**
C' and L'**** of the TL probe in free space are easily
determined using the following equations:**

**Equation ****5**

**Equation ****6**

**Where:**

** u0 =
permeability of free space =**:

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**Transmission Line ****Probe****
Parameters in Earth Medium**

** The transmission line probe
is driven into the earth vertically using an alignment jig in order to keep the
two lines parallel. The line will now be in a complex medium and will take on
new characteristics as a lossy transmission line. **

**It is necessary to find the new values of G',
C' and L' in order to calculate conductivity and dielectric constant of the
earth medium. **

*C' *=* C'**earth
* =* *** ***C'**air
* **e**r *

*L' *=* L'**earth
* =* *** ***L**'**air
* **u**r *

*e**r ***is the
relative permittivity of the medium (dielectric constant)**

*u**r ***is the
relative permeability of the medium.**

*G' ***will be
used to calculate the conductivity of the medium.**

**By determining the characteristic impedance of
the lossy line, the elementary components of the equivalent circuit can be
determined. Given the physical length of the line and the input impedance
Zin, Z0
can be calculated.**

**For a lossy line terminated into a known
impedance, ****Zin is found using
equation 7.**

**Equation 7**

**Where:**

*l
= *

**g****=
****propagation constant of line **

**It should be noted that Zo, Zin and **
**g are complex
numbers.**

**ZL in this case is
infinite as the probe is not terminated. Dividing numerator and
denominator by ZL (infinite),
equation 7 is simplified to**

**that of equation 8.**

**Equation 8**

**Again Zo, Zin and ****
g are complex numbers.
Equation 1 can be simplified by assuming R' to be zero.**

**
**

**Equation 9**

**The propagation constant of the line can be
found from equation 10. **

**
**

**Equation ****10**

**Again, if R' is assumed to be zero, equation
10 can be simplified to that of equation 11.**

**Equation ****11**

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**Earth Parameter Determination**

** By substituting equations 9
and 11 into equation 8, the input impedance of the line can be determined. In
practice, it is the input impedance that is known, and the elementary elements
of the equivalent transmission line circuit will need to be determined. I have
developed software to accomplish this task. It is worth noting that L' can
not be determined from a single measurement of a specified length of probe.
It would be possible to find solution by making more than one measurement using
different length probes. However, most soils have a relative permeability that
is close to that of free space. It may be easier to make a permeability
measurement of the soil by using an inductor with a soil core using solenoid
equations.**

** After the elementary values
have been found, conductivity and dielectric constant of the earth can be
calculated using the following simple relationships.**

*e**r = ***
Dielectric Constant =*** C'
/ ***C'air**

**s =
**