Mathematical models and physical parameters for Earth and three transmitter types described in Comparative Study

Models

Earth (parameters)

Hertz system (parameters)

Marconi system (parameters)

Tesla system (parameters)

System Parameters

General parameters

Electric permittivity of free space (_o): 8.85 x 10^-12 F/m

Magnetic permeability of free space (µ_o): 1.257 x 10^-6 H/m

Impedance of free space: E/H = 377 Ω

Inner conducting sphere (earth) [http://en.wikipedia.org/wiki/Earth]

Magnetic field: about 0.3 gauss

Magnetic permeability (μ): 1.3 x 10^-6 H/m

Electrical conductivity (σ, sigma): ____ (mhos)

$\begin{displaymath} \sigma = \sigma_0 \cdot exp(-\displaystyle\frac {\Delta H}{k T}) \end{displaymath}$

$\sigma_{0}$ is a pre-exponential factor

is temperature

is the Boltzmann constant (in electron volt per kelvin)

$\Delta H = \Delta U + P\Delta V$ (activation enthalpy)

(where $\Delta U$ is activation energy, $\Delta V$ is activation volume, and P is pressure).

[Source: Modeling of Mantle Electrical Conductivity Anomalies Associated with an Upwelling Hot Plume; Modeling Electrical Conductivity in the Mantle]

Resistance, earth (R_e; as if measured between antipodes; assumed): 1 Ω

Resistance, ionsphere (R_i; resistive hydrodynamics model): ____ Ω

Length (L; volumetric mean radius x 2): 12,742 km

Width (W; volumetric mean radius x 2): 12,742 km

Capacitance (C; as if measured between the ground and the ionosphere): ____ μμfd

Leakage conductance (G): ____ Ω

Skin depth (δ; 25 x 10³ Hz): ____ m

(√(ρ/πμf), gives the penetration of fields in a plane conductor, amplitudes proportional to e-z/δ.)

Surface resistance (Rs): ____ Ω

ρ/δ in ohms, gives the resistance of a rectangular surface region as R = RsL/W. L is distance in the direction of the current, and W is the width. This is equivalent to assuming uniform current within the skin depth. For copper of resistivity 1.7 x 10-8 Ω-m, δ = 0.0660/√f and Rs = 2.6 x 10-7√f Ω. For iron with an initial permeability of 250 and resistivity 9 x 10-8 Ω-m, δ = 0.00955/√f and Rs = 9.424 x 10-6 Ω. The AC resistance of a wire of diameter d is Rs/πd, while the DC resistance is ρ(4L/πd2). The boundary can be taken at the frequency for which δ = d/2.

Internal inductance (Li): ____ H

μ/8π. μ is the permeability times 4π x 10-7 H/m. This does not depend on the size of the conductor. For a nonpermeable conductor, such as copper or aluminum, Li = 5 x 10-8 H/m, which is normally insignificant. For a conductor such as iron, which has an initial permeability of 250, Li = 1.25 x 10-4 H/m, which will be dominant.

Current density profile:

Skin Depth is defined as the distance below the surface where the current density has fallen to 1/e or 37% of its value at the surface. Because of Skin Effect, the AC to DC resistance of round wire is dependent on the ratio of the wire diameter to skin depth as can be seen in the equation below:

Rratio = Rac/Rdc = (pi*r^2) / (pi*r^2 - pi(r-S)^2)
where S < r ;
S = 2837 / sqrt(f) (Skin Depth)

Skin depth is inversely proportional to the square root of frequency.

Different size wires will have different AC to DC ratios, and these ratios will increase with frequency.

Length: 10 ft.

Diameter: 80.8 mils (12 AWG)

F: 20kHz

Area of a circle = pi*r^2.

The cross sectional area of solid core 12 AWG wire showing the radius of the wire (r) and skin depth (S).

This is a conservative calculation for increase in AC resistance of a solid cable due to skin effect because it assumes the current density profile is uniform within the first skin depth. Actual measured increase in AC Resistance of a stranded cable due to Skin Effect at 20 kHz is less than 3%.

Source: http://www.audioholics.com/techtips/audioprinciples/interconnects/SkinEffect_Cables.htm

Impedance parameters R, L, C and G, are specified per metre of line. This must be done with care if the calculations are to be meaningful. The resistance R and inductance L comprise the series impedance Z, while the capacitance C and the leakage conductance G form the shunt admittance Y. Resistance and inductance depend on the frequency because of the skin effect, as does the part of G due to losses in the dielectric, while C is largely independent of frequency. The formulas for L and C, which may be found in Ramo, Whinnery and van Duzer, give the inductance and capacitance for fields outside the conductors. The impedance due to fields within the conductors must be added to L.

Iron wire is an interesting example because of the internal inductance. The frequency for δ = 2.6 x 10-3 m is f = 13.5 Hz. Below this frequency, we must use the internal impedance and find L = 1.25 x 10-4 H/m. The resistance R will be the DC resistance. Since we have a single wire, we need L and R only for it, assuming the earth perfectly conducting. If we assume that the wire is suspended at a height of 15 ft above the conducting earth, it is equivalent to a parallel-wire line with d = 0.5156 cm and s = 920 cm. For these measurements, cosh-1(s/d) = 8.18. Then, C = πε/8.18 = 3.40 x 10-12 F/m and the external L = 8.18(μ/π) = 3.27 x 10-6 H/m for two wires. For one wire, we find C = 6.8 x 10-12 F/m, L (external) = 1.64 x -6 H/m. The wire is supported in air, so the dielectric constant is unity.

At high frequencies, the characteristic impedance will be Zo = √(L/C) = 491Ω, or about 250Ω wire to ground. The phase velocity will be v = 1/√(LC) = 3 x 108 m/s, a velocity factor of unity. At low frequencies, Zo = 4300Ω, while v = 6.85 x 107 m/s, a velocity factor of only 0.23. This variation of line parameters through an important frequency interval surely had an effect on communication over iron wires.

Let's consider a metallic telephone circuit composed of two #8 BWG copper conductors spaced at 12", as was used on the first transcontinental long-distance line. For this line, d = 4.19 mm, s = 304.8 mm, so s/d = 727.5 and cosh-1(s/d) = 7.283. At f = 988 Hz, the skin depth is 2.1 mm. The DC resistance would be (2)(1.7 x 10-8)/(1.38 x 10-5) = 2.46 x 10-3 Ω/m. The AC resistance at about 10 kHz would be about 3.89 x 10-3 Ω/m. From the usual formulas, C = πε/7.283 = 3.82 x 10-12 F/m and L = (μ/π)(7.283) = 2.91 x 10-6 H/m. Note that the capacitance is actually less than that of the telegraph pole line, because in that case the earth formed a large plane electrode. We assume G = 10-10 S/m.

Source: http://www.du.edu/~jcalvert/tech/cable.htm

In LF radio propagation the earth current is confined a surface layer with a depth equal to the “skin depth.” Using values of 2 mS/m for typical U.S. soil conductivity (mS = millisiemen or millimho), and a transmitter frequency of 30 kHz, the skin depth is 65 metres.

In the ocean it is only 1.3 metres due to the greater conductivity of seawater.

The skin depth is inversely proportional to the square root of the product of the frequency and the conductivity of the medium.

30,000 x .002 = 60

sqrt 60 = 7.7459

7.7459 / 1 = 0.1291

I assume this relationship is based upon the interaction of 30 kHz electromagnetic radiation the earth’s surface. Lambda at 30 kHz = 10,000 m

Another approach would be to assume a point-to-point (transmitter-to-antipode) resistance of low value, say 1 ohm, and then calculate the increase in AC resistance due to Skin Effect.

> Therefore, at a frequency of 300 Hz, the skin depths would increase to 650 and 13 metres respectively, and at around the lowest Earth resonance frequencies, about 4000 [4828 meters = 3 miles] and 80 metres, respectively. So you do not have to worry about the conductivity of the core of the Earth. Tesla was aware of the skin depth phenomenon, and said that the current “dives down a few [say three] miles”.

Crust (lithosphere)

Depth: 0 – 5, 35, 70 km, 1.6 (mid-ocean ridges), 130 (older oceanic crust), 150 (continental plates)

Temperature: 14 – 15^oC (average)

Composition (upper, middle, and lower): Silicates; O, 46.6; Si, 27.7; Al, 8.1, Fe, 5.0; Mg, 2.1 Ca, 3.6; Na, 2.8; K, 2.6; Ti, 0.4; Ni, 0.08; S 0.03, Cr, 0.01 [http://earth.leeds.ac.uk/dynamicearth/composition/comptable/]

Lower: mafic, approaching that of aprimitive mantle-derived basalt, intermediate bulk compositions in some regions.

Middle: intermediate in bulk composition, containing significant K, Th, and U contents.

Average: intermediate in composition, containing a significant proportion of the bulk silicate Earth’s incompatible trace element budget (35–55% of Rb, Ba, K, Pb, Th, and U). [Nature and composition of the continental crust: A lower crustal perspective] [See also Geochemical Earth Reference Model; http://earthref.org/index.html]

Mantle, upper (Asthenosphere) [http://en.wikipedia.org/wiki/Asthenosphere]

Depth: 30, 70 – 250, 100 – 200, 400 km

Temperature:

Composition: Silicates

Mantle, lower [http://en.wikipedia.org/wiki/Mantle_%28geology%29]

Depth: 250 – 2886, 2900 km

Temperature:

Composition: Dense silicates; O, 44.7; Si, 19.1; Al, 1.3; Fe, 4.6; Mg, 20.3; Ca, 1.6; Na, 0.8; K, 0.0; Ti, 0.06; Ni, 0.2; S, 0.04; Cr, 0.56

Core, outer

Depth: 2886 – 5771 km; 1250 – 3500 km

Temperature:

Composition: Ni, Fe; Si, 8.7; Fe, 79.3; Ni, 5.0; S, 6.3

Core, inner

Depth: 5771 – 6371 km; 0 – 1250 km

Temperature: 4982^o C

Composition: Ni, Fe; Si, 8.7; Fe, 79.3; Ni, 5.0; S, 6.3

Volumetric mean radius: 6371.0 km

Core radius: 3485 km

Mean density: 5515 kg/m³

Mass: 5.9736 10²⁴ kg

Volume: 108.321 10¹⁰ km³

Equatorial radius: 6378.1 km

Polar radius: 6356.8 km

Ellipticity (Flattening): 0.00335

Surface gravity: 9.78 m/s²

Second concentric sphere (troposphere) [http://en.wikipedia.org/wiki/Earth%27s_atmosphere]

Atmospheric pressure vs.

Elevation:

Dielectric constant: ____ K (permittivity; http://my.execpc.com/~endlr/dielectric_const_.html)

Breakdown potential: ____ volts AC at 25 x 10³ Hz

Surface pressure: ____ Pa; 14.7 pounds per square inch

Dielectric constant at Earth’s surface: 1.0059 K (10³ Hz, 20^o c)

Depth: 0 to 7 to 9 km at polar regions, 17 km in tropics [divide into n regions]

Pressure gradient: ___ to ___ mm of mercury

Composition: 78% nitrogen, 21% oxygen, 0.9% argon, 0.03% carbon dioxide, trace elements

Third concentric sphere (stratosphere)

Depth: 17 to 50 km above the troposphere and below the mesosphere

Pressure gradient: ___ to ___ mm of mercury

Composition: basically the same as that of the troposphere, with the addition of ozone.

Fourth concentric sphere (mesosphere)

Depth: 50 to 80 km

Pressure gradient: ___ to ___ mm of mercury

Composition: [same as stratosphere?]

Outer concentric sphere (ionosphere; D, E, F1, F2, and topside)

Ionospheric Regions (Structures)

D-Region: 75-95 km, relatively weak ionization is mainly responsible for absorption of high-frequency radio waves.

E-Region: 95-150 km, regular daytime E-layer; E2 thick layer; Sporadic E variable thin layer, ions mainly O2+.

F-Region: 150 km; F2 reflecting layer; F1 temperate-latitude regular stratification; F1.5 low-latitude, semi-regular stratification. Ions in lower part of F-layer mainly are NO+, predominantly O+ in the upper part, of primary interest to radio communications.

Topside: Starts at the height of the maximum density of the F2 layer of the Ionosphere and extends upward with decreasing density to a transition height where O+ ions become less numerous than H+ and He+. The transition height varies but seldom drops below 500 km at night or 800 km in daytime, although it may lie as high as 1100 km. Above the transition height, the weak ionization has little influence on radio signals.

Depth: 70-80 to 640 km

Pressure gradient: ___ to ___ mm of mercury

Composition: contains many ions and free electrons, i.e. plasma

Plasma density vs. elevation

Resistance: ___Ω

Ion density vs. elevation:

Exosphere: 640 to 1,280 km

Magnetosphere: ____

Electromagnetic propagation

Distance vs. frequency

Sky-wave transmission mode: [output parameter]

Ground-wave transmission mode: [output parameter]

Norton surface-wave transmission mode: [output parameter]

Zenneck surface-wave transmission mode: [output parameter]

Distance vs. field strength

Sky-wave transmission mode: [output parameter]

Ground-wave transmission mode: [output parameter]

Norton surface-wave transmission mode: [output parameter]

Zenneck surface-wave transmission mode: [output parameter]

Ionospheric skip

Critical angle of incidence vs. frequency: [output parameter]

Lowest angle of incidence vs. frequency: [output parameter]

Maximum Usable Frequency (MUF): [output parameter]

General transmitter parameters

Geographic coordinates

Longitude: ____deg., ____deg.

Latitude: ____deg., ____deg.

Time: ____ UTC

Frequency: 25 x 10³ Hz; also 7.8 Hz; 1, 75, 150, & 1,800 x 10³ Hz

Lambda (25 x 10³ Hz): 12,000 m

Peak envelope power (PEP): ____ watts

General receiver parameters

E-field probe

Impedance (Z): ____ Ω

Geographic coordinates

Longitude: ____deg., ____deg.

Latitude: ____deg., ____deg.

Frequency: 25 x 10³ Hz; also 7.8 Hz; 1, 75, 150, & 1,800 x 10³ Hz

Hertz transmitter parameters

Launching structure (vertical ½-wave dipole in free space)

Effective radiated power (ERP): ____ db(i)

Height above surface (center of radiation): ____ m

Antenna current: ___ amperes

Antenna

Length: ____ m (resonant at 25 x 10³ Hz)

Diameter: ____ mm

Antenna current: ____ amperes

Maximum antenna current: ____ amperes

Impedance of launching structure relative to free space ____ Ω (free space E/H = 377 Ω)

Maximum potential: at ends: ____ volts

Maximum instantaneous electric charge: ___ coulombs

Master oscillator

Input to final stage: ____ watts

Waveform: Sinusoidal

Hertz receiver parameters

Receiving structure (vertical ½-wave dipole in free space)

Antenna current: [output parameter]

Antenna

Length: ____ m (resonant at 25 x 10³ Hz)

Diameter: ____ mm

Height above surface (center of radiation): ____ m

Antenna current: ___ amperes

Maximum antenna current: ___ amperes

Maximum potential: at ends: ____ volts

Maximum instantaneous electric charge: ___ coulombs

Impedance relative to free space ____ Ω (free space E/H = 377 Ω)

Local oscillator

Detector (resistor across antenna feedpoint)

Load resistance: ____ Ω

Current through load: [output parameter]

Idealized Marconi transmitter parameters

Launching structure (vertical ¼-wave monopole on or near surface)

Effective radiated power (ERP): ____ db(i)

Height above surface (center of radiation): 0 and 2 m

Antenna current: ___ amperes

Antenna

Height: ____ m (resonant at 25 x 10³ Hz)

Diameter: ____ mm

Antenna current (RMS): ___Amperes

Maximum antenna current: ___Amperes

Maximum potential: at free end: ____ volts

Maximum instantaneous electric charge: ___ coulombs

Electrodynamic cathode reaction force: ____

Elevated counterpoise

Height above surface: 2 m

Diameter: ____ m

Capacitance: ____ ufd.

Buried counterpoise

Depth below surface: .5 m

Diameter: ____ m

Ground terminal resistance/impedance: ___ Ω

Impedance relative to free space ____ Ω (free space E/H = 377 Ω)

Master oscillator

Input to final stage: ____ watts

Peak envelope power: ____ watts

Waveform: Sinusoidal AC

Marconi receiver parameters

Receiving structure (vertical ¼-wave monopole on surface)

Antenna

Height: ____ m (resonant at 25 x 10³ Hz)

Diameter: ____ mm

Antenna current (RMS): ___Amperes

Maximum antenna current: ___Amperes

Maximum potential: at free end: ____ volts

Maximum instantaneous electric charge: ___ coulombs

Elevated counterpoise

Height above surface: 2 m

Diameter: ____ m

Capacitance: ____ ufd.

Buried counterpoise

Depth below surface: .5 m

Diameter: ____ m

Ground terminal resistance/impedance: ___ Ω

Impedance relative to free space ____ Ω (free space: E/H = 377 Ω)

Local oscillator

Detector (resistor across antenna feedpoint)

Load resistance: ____ Ω

Current through load: [output parameter]

Tesla transmitter parameters

Launching structure (top loaded ¼-wave helical resonator)

Effective radiated power (ERP): ____ db(i)

Height above surface (center of radiation): 0 m

Antenna current: ___ amperes

Resonator

Form height: 18 m

Diameter: 1.8 m

Wire diameter: ____ mm

Number of turns: ____

Turn Spacing: ____ cm

Mass: ____ kg

Self inductance: ____ H

Maximum magnetic field strength: ___ T

Elevated terminal (spherical frame)

Diameter: 21 m

Mean height above terrain: 50 m

Vacuum tube attachments

Terminal

Diameter: .46 m

Length: .46 m

Overall length: 1.25 m

Capacity: ____ ufd

Potential: 10⁷; also 10³ and 10⁸ volts

Maximum instantaneous electric charge: ___ coulombs

Electrodynamic cathode reaction force: ____

Ground terminal connection resistance/impedance: ___ Ω

Impedance relative to free space ____ Ω (free space E/H = 377 Ω)

Master oscillator

Input to final stage: ____ watts

Form height: 5 m

Diameter: 14 m

Primary #1

Wire diameter: ____ mm

Number of turns: ____

Turn Spacing: ____ cm

Mass: ____ kg

Inductance: ____ H

Capacitance: ____ ufd.

Self inductance: ____ H

Excitation pulse

Voltage: ____ kv dc

Current: ____ amp

Waveform: squarewave, negative-going pulsed dc

Repetition rate: 25 x 10³ Hz

Duration or dwell: ____ nanoseconds; ____%

Maximum magnetic field strength: ___ T

Primary #2

Wire diameter: ____ mm

Number of turns: ____

Turn Spacing: ____ cm

Mass: ____ kg

Inductance: ____ H

Capacitance: ____ ufd.

Excitation pulse

Voltage: 60 kv dc

Current: ____ amp

Waveform: transient, negative-going pulsed dc

Repetition rate: 7.8 Hz

Duration or dwell: ____ nanoseconds; ____%

Maximum magnetic field strength: ___ T

Secondary

Inductance: ____ H

Capacitance: ____ ufd.

Mutual inductance: ____ K

Tesla receiver parameters (passive mode)

Receiving structure (top loaded ¼-wave helical resonator)

Antenna current: [output parameter]

Elevated terminal

Mean height above terrain: 50 m

Maximum potential: [output parameter]

Impedance of receiving structure relative to free space: ____ Ω (Impedance of free space: E/H = 377 Ω)

Resonator

Form height: 18 m

Diameter: 1.8 m

Inductance: ____ H

Maximum magnetic field strength: [output parameter]

Ground terminal connection resistance/impedance: ___ Ω

Impedance relative to free space ____ Ω (free space E/H = 377 Ω)

Detector

Secondary with shunt resistor [associate with helical resonator]

Coefficient of coupling: ____ K

Load resistance: ____ Ω

Current through load: [output parameter]

E-field probe [associate with elevated capacitance]

Tesla receiver parameters (active mode)

Receiving structure (top loaded ¼-wave helical resonator)

Effective radiated power (ERP): ____ db(i)

Height above surface (center of radiation): 0 m

Antenna current: ___ amperes

Ground terminal connection resistance/impedance: ___ Ω

Impedance relative to free space ____ Ω (free space E/H = 377 Ω)

Resonator

Form height: 18 m

Diameter: 1.8 m

Wire diameter: ____ mm

Number of turns: ____

Turn Spacing: ____ cm

Mass: ____ kg

Self inductance: ____ H

Maximum magnetic field strength: ___ T

Elevated terminal (spherical frame)

Diameter: 21 m

Mean height above terrain: 50 m

Vacuum tube attachments

Overall length: 1.25 m

Terminal

Diameter: .46 m

Length: .46 m

Capacity: ____ ufd

Potential: 10⁷; also 10³ and 10⁸ volts

Maximum instantaneous electric charge: ___ coulombs

Electrodynamic cathode reaction force: ____

Master oscillator (active mode only)

Input to final stage: ____ watts

Form height: 5 m

Diameter: 14 m

Primary #1

Wire diameter: ____ mm

Number of turns: ____

Turn Spacing: ____ cm

Mass: ____ kg

Inductance: ____ H

Capacitance: ____ ufd.

Self inductance: ____ H

Excitation pulse

Voltage (magnitude): ____ kv dc

Current: ____ amp

Waveform: squarewave, positive-going pulsed dc

Repetition rate: 25 x 10³ Hz

Duration or dwell: ____ nanoseconds; ____%

Maximum magnetic field strength: ___ T

Primary #2

Wire diameter: ____ mm

Number of turns: ____

Turn Spacing: ____ cm

Mass: ____ kg

Inductance: ____ H

Capacitance: ____ ufd.

Excitation pulse

Voltage: 60 kv dc

Current: ____ amp

Waveform: transient, negative-going pulsed dc

Repetition rate: 7.8 Hz

Duration or dwell: ____ nanoseconds; ____%

Maximum magnetic field strength: ___ T

Secondary

Inductance: ____ H

Capacitance: ____ ufd.

Mutual inductance: ____ K

Detector

Secondary with shunt resistor [associate with helical resonator]

Coefficient of coupling: ____ K

Load resistance: ____ Ω

Current through load: [output parameter]

E-field probe:

Air-glow

Elevation: [output parameter]

Dimensions: [output parameter]

Spectrum: [output parameter]

Intensity: [output parameter]

The approximate value of the magnetic field in a greater than 5:1 solenoid is given by B = u₀nI where,

B is the magnetic field (T)

u₀ is the permeability of free space

n is the number of turns of wire per unit length

I is the current through the wire

Source: http://www.oz.net/~coilgun/theory/solenoidphysics.htm

The magnitude of the magnetic field around a current element is given by

B = m I l / 4 p r²,

m is the magnetic permeability

m₀ = 4 p* 10^-7 N / A ² (an exact value) is the magnetic permeability of the vacuum.

The SI unit of B is the Tesla (denoted by T), or N / A m. It is a very large unit; the Earth's magnetic field is typically a fraction times 10^-4 T (10^-4 T is also called 1 Gauss).

Source: http://www.rwc.uc.edu/koehler/biophys.2ed/magnet.html

Table of Physical Units

Cartesian coordinates – Use Cartesian or Spherical coordinates?

Software Defined Radio

“The universe is not algebraic.”

Two-half-space solution for a grounded helical resonator

cold cathode discharge ion source

skin depth 30 kHz equation