Some Notes on Transmission Line Representations of Tesla’s Transmitters

Some Notes on Transmission Line Representations of Tesla’s Transmitters 1,2

Zoran Blažević1, Dragan Poljak2 University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia.

1 zblaz@fesb.hr, 2dpoljak@fesb.hr

Abstract: In this paper, we present a transmission line representation of various Tesla’s oscillating transmitters and of propagation based on “true conduction”, a Tesla’s concept of non-Hertzian wireless transmission. We follow bits of Tesla’s papers, mainly “The True Wireless”, from which is obvious that his theory is in essence compatible to the transmission line theory, and compare results obtained by the model and those given by Tesla, as well as predict some theoretical characteristics of his resonating coils. Although his conception of a more efficient radio transmission by "non-Hertzian waves" has never been recognized exactly what he claimed it to be, his concept of radio apparatus prevailed as they were the first patented true radio systems.

1. INTRODUCTION In the world of today, there is a rapid development in both fundamental sciences and technology. This, as Tesla would say, rapid increase in human velocity would not be possible without exceptional individuals in past as Faraday, Maxwell, Ampere, Hertz, Tesla and many others, whose work in electromagnetism set a foundation for modern physic and technology. However, the one among them that continuously inspires scientific minds with his inventions and ideas even after more than 100 years from his time is Nikola Tesla, a man whose achievements are so great and inventions and concepts so puzzling that they persistently refuse to retreat to the well deserved place in history but rather seek a sense in future. Besides the invention of the rotational magnetic field and his great work on X-rays1 one of his greatest achievements is by all means the solution to the efficient utilization of radio, what he presented in many papers [1-6], more than one patent [7,8] and lectures [9,10]. In order to repeat the first successful radio transmission experiment of Dr. Hertz and at the same time to build more efficient apparatus than Hertz’s, progress in the investigation led him to the design of what is known as Tesla Magnifying Transmitter (TMT), the first device at the time able to provide a 1

In the matter of fact, it is well known that the fire in his laboratory in 1895 delayed Tesla to patent his work on X-rays prior to Rontgen himself. Moreover, Tesla's footages were of a much higher quality than Rontgen's. And it is not the end of the story. Well before the appearance of quantum mechanics, Tesla found that the Rontgen rays are actually streams of particles [29].

long-distance wireless transmission. However, as Tesla was never flowing with the mainstream, he had his own views about the matter contradicted with a majority of scientific community.

Namely, he considered radio and, consequently, light waves as longitudinal waves in luminiferous ether rather than transversal waves in Maxwell's uncompressible ether, not to say that he opposed both the Einstein's relativity and quantum mechanics.

Latter he claimed [13] that he fully encircled his own Dynamic Theory of Gravity by which he united gravity and electromagnetism. Unfortunately, this theory never saw the light of day because of his death in 1943, but it is known it is the theory that predicts the existence of gravitational waves and comprehends gravity as a field [14].

But whatever the Tesla’s theory might be, it is a fact that he left as his legacy many successful experiments and inventions related to many branches of science and designed an enormous number useful technical systems and devices. On the other hand this look upon radio science provided him a lot of headache because the invention of radio was for several decades being contributed to Marconi. It was until the US Supreme Court finally decided that the invention of radio belongs to Nikola Tesla but the ruling became official after they both passed.

With his transmitters, Tesla aimed to transmit without wires not only voice, data and pictures, but also energy even for industrial scale. However, he claimed that his radio system operate via so called true conduction, a mode of the operation of the transmitter in which electromagnetic radiation is suppressed as much as possible, and the transfer of electromagnetic momentum occurs. This concept, recognised by official science to a significant degree only for (very) low frequency transmissions [15,16], we today relate to the surface waves (ground waves). As to use his magnifying transmitter for generating the "Hertzian radiations" (space waves), Tesla depicted this idea as “cutting the glass with Kohinoor”. That is what really produced the big confusion about the invention of radio. In the Tesla’s concept of radio, the Hertzian radiations or electromagnetic waves are not the most suitable means for the construction of a world-wide radio system, as opposite to his true conduction. He apparently preferred the distributed helical oscillators that he believed are able to spread ground currents, over capacitive antennas as transmitters. This debate would be ended quite fast if there were no some experimental evidences,

many of them from Colorado Springs [6,11] and Long Island, in favour to Tesla vision that in his time were quite unexplainable. However, after (re)discovering the TeslaSchumann cavity it is quite obvious that the effects of Earth resonance noted by Tesla originate from the low frequency cavity effects. Nevertheless, although many cavity parameters measured by Tesla more or less correspond to the measured results of others [17], there are some questions that intrigue.

Namely, Tesla claimed that he had sent a non-Hertzian wave with the mean speed of π/2 times the velocity of light by the Colorado Springs transmitter to the opposite side of Earth (antipode) from which it returned with almost undiminished strength, (return losses less than a half percent [2]). First, the Qfactor of the cavity is nowhere enough to support a low-loss transmission by standing electromagnetic waves [18]. Second, even mentioning a slight possibility for superluminal wave velocities is drawing attention instantly, regardless of the fact that this concept is successfully explained by the propagation of electromagnetic waves in the Schumann cavity with the speed of light [17]. Many, strengthened by the Tesla’s firm standpoint about the feasibility of his scheme, think that other more affirmative explanations are needed, see e.g. [19], [20], but such works seem to be highly speculative. Anyway, a century ago Tesla sparked a continuous scientific discussion and, on the other hand, commenced successful experiments on transmission of energy without wires2 that was followed by microwave power transmission experiments in the second half of 20th century [21]. In addition, he pointed the way for the project HAARP (High Frequency Active Auroral Research Program), based on the patents of B. Eastlund issued as improvements on the patents first issued to Tesla [14]. Not to say that Tesla made the first step in development of radio-astronomy by detecting with his Colorado Springs receivers according to [22] the Jovian signals, the extraterrestrial radio signals emanating from Jupiter-Io system for which he, although due to reasonable grounds, wrongly assumed to originate from Mars. This observation well precedes the work of Jansky.

There are already a lot of papers and books that explains Tesla’s apparatus and experimental achievements, from which the work of Corum & Corum should be highlighted (some are listed as references, [22-25]). A jet another comprehensive approach to the electric circuits of Tesla by nonlinear oscillatorshuttle circuit (OSC) theory is given in [26], in which Tesla OSCs are considered to be analogues of quaternionic systems.

2

There is a claim in [30], but never confirmed independently, that Tesla successfully lighted 200 incandescent lamps of the Edison type at a distance of 26 miles from his Colorado Springs transmitter.

However, Tesla claims in [3] that effects of this transmitter could be noted at a distance of 600 miles, more than enough to confirm an efficient long-distance radio-transmission.

This work was originally disclosed in document No. 225395 U.S.P.O. 19883. In this paper it is explained how Tesla's approach to electromagnetism can be considered to be vastly different from the classical one. Different applications of the presented Tesla circuitry are claimed to be accomplishable: power control, energy control, phase control, noise control and reduction of power loss in power transmission.

It is our intention to present in this paper a working model based on the transmission line (TL) theory for representation of Tesla’s transmitters and his concept of true conduction. This model, explained in Sec. 2, provides relations for predicting the basic effects and parameters of Tesla’s resonating transmitters, and is able to describe his concept of propagation of ground currents and elucidate receiver arrangements depicted in [1]. In Sec. 3 the transmitters used by Tesla in Colorado Springs [11] are modelled as transmission lines in different modes of excitation. It is followed by a TL representation of the propagation by true conduction in Sec. 4.

2. TRANSMISSION LINE MODEL Magnifying transmitter (Fig. 1) is a passive unipolar oscillator of great power in which the magnification of voltage and current is achieved by standing waves. It is a special version of the well-known Tesla coil, which lies in the hearth of the World System for wireless transmission of energy and electrical signals conceived by Tesla [2]. In Fig. 2, we also show some dipole-like transmitting devices that he experimented with in Colorado Springs laboratory in year 1899 [11]. Basically, the concept of Tesla’s apparatus is based on the transmission thru a single wire without return [1]. A simplified application of this model for a transmission of power has been depicted in [27], and a more detailed version in [28].

The analysis of a Tesla coil by a TL model relays on the slow-wave helical resonator analysis [23]. It is assumed that only TL T0 mode propagates through the secondary helix, which of course presumes the analysis simplified to a degree that depends on the type of the coil and the coil height. The line is of characteristic impedance Z0 and β the phase constant determined by the design (geometry and material), and is considered to be concluded with (capacitive) impedances on both ends, as in Fig. 3.a). A small portion of the line is drawn in Fig. 3b with a voltage source Δug that exists only in the case when we assume a linearly distributed excitation over the helix.

Taking that: 3

It is interesting to note that this patent is listed as an important reference in some UWB patents (e.g. U.S.P.O. 6,937,674 B2, and 6,947, 492 B2; both accepted in 2005.). This shows how easy it is to trace Tesla's impact even to state of the art radiocommunications.

⎧ 0, case a) ⎪ u g ( t ) = U g e jωt , Δu g = ⎨ Δx jωt , ⎪⎩U g l e , case b) ∂u ∂i Δu = Δx , Δi = Δx , ∂x ∂x

⎛

γ = α + j β = jω LC ⎜1 + ⎝

R ⎞⎛ G ⎞ ⎟⎜ 1 + ⎟, jω L ⎠⎝ jωC ⎠ case a)

0, ⎧ ⎪ 2 , I0 = ⎨ γ U g Ug ⎪ l R + jω L = ( G + jωC ) l , case b) ) ⎩ ( a) b) Figure 1 - a) Apparatus for transmitting electrical energy [8]; b) System of transmission of electrical energy [7]

(1) (2)

(3)

(4)

where ug is the voltage source on end 1 in serial connection to Z1 if the source is lumped (case a), or the voltage between the top and bottom of the coil due to the alternate current I0 induced in the secondary wire of length l if the source is distributed (case b). γ is the complex propagation constant, ω = 2πf angular frequency and the remaining parameters are as indicated in Fig. 3b. Following the well-known analysis, wave equations for the both TL scenarios can be written as:

∂ 2u − γ 2u = 0 , ∂x 2 ⎧ 0, case a) ∂ 2i − γ 2i = ⎨ , 2 ∂x ⎩ I 0 , case b)

(5) (6)

By implementing the gauge conditions: Figure 2 - Various arrangements for producing disturbances in the natural media [11]

⎧U − I Z , case a) V1 = u x = l = ⎨ g 1 1 , I1 = i x = l , 2 2 ⎩ − I1 Z1 , case b) V2 = u x =− l = − I 2 Z 2 , I 2 = − i x =− l , 2

(7a) (7b)

2

taking that the TL is of characteristic impedance Z0, and defining the reflection coefficients of load impedances as: a)

Γ1 =

Z1 − Z 0 Z − Z0 = Γ1 e jϕ1 , Γ 2 = 2 = Γ 2 e jϕ2 , Z1 + Z 0 Z2 + Z0

(8)

the solutions for current and potential on the line with a lumped voltage source excitation on end 1 are:

b) Figure 3 - a) TL model with lumped or distributed source; b) A small portion of TL of length Δx

u ( x, t ) =

Ug

(1 − Γ1 )

2 1 − Γ1Γ 2 e −2γ l

⎛ l⎞ ⎛ l⎞ ⎡ −2 γ ⎜ x + ⎟ ⎤ γ ⎜ x − ⎟ jω t ⎝ 2⎠ (9) ⎢1 + Γ 2 e ⎥ e ⎝ 2 ⎠e ⎢⎣ ⎥⎦

Ug

(1 − Γ1 )

2Z 0 1 − Γ1Γ 2 e

−2 γ l

l⎞ ⎛ ⎛ l⎞ ⎡ −2 γ ⎜ x + ⎟ ⎤ γ ⎜ x − ⎟ jω t ⎝ 2⎠ e e 1 − Γ ⎢ ⎥ ⎝ 2 ⎠ e (10) 2 ⎣⎢ ⎦⎥

and in the case of distributed voltage excitation: u ( x, t ) =

− I 0 Z 0 Aeγ x − Be −γ x − γ 2l jωt e e , 2 1 − Γ1Γ 2 e −2γ l

⎛ 1 Aeγ x + Be − γ x − γ 2l ⎞ jωt i ( x, t ) = I 0 ⎜ 1 − e ⎟e , −2 γ l ⎝ 2 1 − Γ1Γ 2 e ⎠ A = (1 + Γ1 ) − Γ1 (1 + Γ 2 ) e −γ l ,

B = (1 + Γ 2 ) − Γ 2 (1 + Γ1 ) e−γ l .

(11) (12)

(13) (14)

Clearly, in both cases, the necessary condition for resonance is accomplished when the parameter: ϒ = 1 − Γ1Γ 2 e −2γ l

(15)

is minimum, or zero if the losses are excluded and the load impedances on ends are purely reactive (⎪Γ1⎪ = ⎪Γ2⎪ = 1).

In order to derive the Tesla Equation [24], let us propose somewhat different form of the magnifying ratio M definition for this TL model. Here the top voltage V2 for an arbitrary load is going to be related with the top voltage V20 in the case of a perfectly matched top load. By using (9)-(14) and certain mathematical manipulations one may arrive to the following relation (regardless of the model of voltage source): u ( x = − 2l ; Γ1 , Γ 2 ) V2 1 + Γ2 = = .

V20 u ( x = − 2l ; Γ1 , Γ 2 = 0 ) 1 − Γ1Γ 2 e −2γ l

capacity) on end 1 of the TL. Obviously, for the TL electrically shorter than λ/4, in order to maintain the magnifying effect the top capacity has to be increased accordingly. The diagrams of voltage and current on a lossless line normalised to the maximum values versus different TL lengths are presented in Fig. 4 and are valid for both scenarios in Fig. 2. Note that the top voltage, discrepancy between maximum and minimum current, and current slope decrease as the TL becomes shorter.

The connection between the magnifying factor M for the TL with end 1 short and end 2 open and the normalised top capacity XC = 1/j ωC2Z0 to be added in order to achieve resonance can be expressed as:

(16)

M = 1 − X C2 .

It has been shown so far that the both types of excitations show some identical properties, e.g. regarding the resonance condition or magnification ratio. However, the standing wave patterns are different for the two scenarios for systems out of resonance. Contrary, provided that the resonance condition is maintained on a lossless TL, the node of standing wave for the signal of wavelength λ is determined for both scenarios as: x0 =

3

M =

γl

αl

e e 1 . = ≈ cosh γ l sinh (α l ) α l

(17)

Let us first consider the effect of electrical shortening of the line. For that matter we assume the short circuit (an infinite

ϕ 2 − ϕ1 λ.

8π

(19)

This means that while in resonance the distributed oscillator can be comprehended as two lumped oscillators on the opposite ends vibrating on the same frequency, as explained in Fig. 5. It is easy to derive the following ratio for the scenario (i = 1,2; li ≤ λ/4): 2 2

1: 2: 3: 4: 5: 6: 7: 8:

1

Voltage

1.5 4 1 5 6

λ /4 7λ/32 3λ/16 5λ/32 λ /8 3λ/32 λ /16 λ /32

0.5

1 2 3 4

2

1.5

5 6

1

7

0.5

8

7 8 0 -0.5

0 Relative position

0.5

0 -0.5

0 Relative position

0.5

2 Top voltage (Umax) Bottom current (Imax) Top current (Imin) Difference Imax - Imin

1.5 Extermes

Clearly, for the short circuit on end 1 (Γ1 = –1), equation (16) takes the equivalent form of the Tesla equation given in [11]. If furthermore a pure capacitance is assumed on end 2 (|Γ2| = 1), the square of this ratio could be interpreted as ratio of the power that a resonator is capable to preserve within the system and the power that can theoretically be transferred by the same resonator to the matched load on one end and spent as heat, radiation or like. The magnifying factor is then obtained for a quarter-wave resonator (βl = π/2) as:

(18)

Current

i ( x, t ) =

1

0.5

0

0

0.05

0.1 0.15 Eletrical length (λ )

0.2

0.25

Figure 4 - Voltage, current and the extreme values for resonant lines of lengths equal or less than quarter of wavelength

1

Figure 5 - Lossless distributed resonator seen from opposite ends as two lumped resonators with same resonant frequency Li = Z 0 tg ( β li ) .

Ci

(20)

3&2

Note that the greater the characteristic impedance or the closer is the electrical length of the line to a quarter of wavelength, the greater is its inductivity or smaller the top capacity required for resonance.

1

Voltage

0.5 0

Fig. 1

φ 1 = -55 deg φ 2 = -5 deg

-0.5 -1 -0.5

-0.4

-0.3

-0.2

-0.1 0 0.1 Relative position along the line

0.2

0.3

0.4

0.5

-0.3

-0.2

-0.1 0 0.1 Relative position along the line

0.2

0.3

0.4

0.5

0.4

0.5

1 0.5 Voltage

Now, we turn to modelling some Tesla’s transmitters from the Colorado Springs investigations in Fig. 2 as TLs. Corresponding TL models for these oscillators are presented in Fig. 6. Unfortunately, except a comment: “This seems to be very effective.” there are no performance measurements of these transmitters anywhere in the Notes. However, there is a brief but good explanation of the operating mode in [11]. The important is that all three systems: primary, secondary and extra coil have the same period of vibration. Exploiting this fact, the diagrams of estimated standing wave patterns obtained for each selected transmitter by (9)-(14) are given in Fig. 7.

Note that the standing wave patterns that correspond to the same top capacities are same for distributed and lumped excitation. Furthermore, the primary coil in each case, as shown in Fig. 2 [11], seems to be settled near the standing wave node.

Let us now turn our attention to Tesla’s solution for eliminating the drawbacks of distributed capacity. Tesla used as an approximate representation of the distributed capacity of a coil a scheme as in Fig. 8 [11].

The issue of distributed capacity has already been addressed with a great clarity in [23]. If the distributed capacity were possible to be zero, then the TL model would have a little physical sense. Through such an ideal coil flows a constant current, whereas the voltage rises linearly (see Appendix). Note that in that case wave through the lossless coil of a finite selfinductance must have an infinite speed. There can be no standing waves in that case. This means that we now have an idealized lumped coil with finite inductance (without internal capacity) that connected with top capacity forms a lumped oscillating circle.

Figure 6 - TL models of Tesla’s transmitters in Fig. 2

0

Fig. 2

φ1 = -55 deg φ2 = -5 deg

-0.5 -1 -0.5

-0.4

1 0.5 Voltage

3. MODELS OF TESLA’S RESONANT TRANSMITTERS

Fig. 3

φ 1 = -55 deg φ 2 = -145 deg

0 -0.5 -1 -0.5

-0.4

-0.3

-0.2

-0.1 0 0.1 Relative position along the line

0.2

0.3

Figure 7 - Estimated standing wave patterns for the three case of oscillating transmitters in Fig. 2

Figure 8 - Approximate model for distributed capacity of a coil proposed by Tesla [11]

Being aware that the distributed capacity cannot be avoided because of the laws of physics, Tesla introduced a scheme as in Fig. 9.a). The capacity C, properly sized, is inserted between each turn of the grounded coil. These condensers serve to hold the charge on their ideal paths to the top capacity that would else be collected by the distributed capacity of the coil and lost. This means that the TL representation will incorporate N condensers, N being the number of turns, as in Fig. 9b.

We now turn to the Smith chart in Fig. 10. Let the length of the secondary S wire be less than or else equal to a quarter of the wavelength. At the beginning, we have a short circuit (Tp) and in the point 2 of the line we have a normalized impedance Z2 = jB > j0 that corresponds to the electrical length of single turn (T2) and is purely inductive. The point 2 obviously marks a position of the maximum potential relative to the ground plane depicted as the position 1 on the line in Fig. 9.b). Now if C is selected such that just counteract this phase shift we are back again to the point Tp that corresponds to the short circuit (Z2' = 0 or Γ2' = -1) that is to a position where the current is in the maximum and the potential drops to zero. Following the same logic, by travelling along the line we arrive to the point (N + 1) of top capacity. The last condenser sees only a low inductance and hence, in order to maintain the resonance, the coil requires it to be of a high top capacitance. Besides other considerations, Tesla has left the following note on that: “This capacity ought to be so large that it can take up all the current of the secondary at the frequency used.”

Extending this discrete Tesla's scheme, purely theoretically, to a continuous process, the reflection coefficient remains Γ(x) = -1 regardless of the position along the line, which is to a degree equivalent to the case of a coil of infinite characteristic impedance i.e. of diminished distributive capacity (see (8)). However, the velocity of the wave may considered to be unaltered as, regardless the lumped condensers have ability to annihilate the phase shift of the travelling wave due to the shift through a distance, they are not able to compensate for the corresponding delay physically. If a condenser, (or, for that matter, an equivalent open transmission line) added in serial could introduce a negative time shift of the wave, only then the distributed capacity could really be counteracted, which is, to the present knowledge, impossible. Thus, this method of overcoming the drawbacks of distributed capacity should not be taken literary as a method for achieving some kind of an idealized lumped coil, but just as a method to decrease losses when high voltages are used.

Note that in the structure on Fig. 9.a) condensers cannot physically be added to the turns as symmetrically as to accept the total energy travelling across a turn. Hence, this TL representation of the method should be taken just as a crude approximation, used to elucidate the principle.

a)

b) Figure 9 - a) Tesla’s method of overcoming the drawbacks of distributed capacity [11]; b) Corresponding TL model

Figure 10 - Smith chart: shift “toward generator” from the short circuit thru a distance Nonetheless, by inspecting the diagrams of the current on Fig. 4 and provided that the inserted condensers are uniformly distributed as in Fig. 9, the electrical length between the turns should still remain significant, in order for the formation of standing waves to be allowed. By inspecting the notes on p. 120 of [11], it is clear that Tesla applied this inventive method to the high-voltage high-frequency coils of large diameter, for which the velocity of the travelling waves are low enough so that this request could acceptably be traded-off. Moreover, taking into the consideration that his prime goal was to prevent the loss of energy in the secondary due to the sparks, this approach, properly applied, can in practice lead to the significant increase in the performances of a high-voltage coil.

Tesla reported in [11] a practical reduction in distributed capacity by such a scheme of 95%. This part of Tesla’s investigations is commented by Marinčić in Colorado Springs Notes as follows.

“His analysis of the distributed capacity of the secondary is a good illustration of his inventiveness in a little known field and how he sought to reduce problems to a simple but mathematically and physically sufficiently accurate model.”

4. PROPAGATION THEORY OF TESLA

The electromagnetic wave propagation concept of Tesla is presented in Fig. 11 [4]. It relays on the assumption of the charge redistribution across the globe [16,18], in which the earth current driven by the TMT voltage passes through the Earth along the diameter with a velocity equal or close to the speed of light c.4 The instant velocity v of Tesla surface wave is then determined from the simple geometry, assuming Earth as an ideal smooth conducting sphere of diameter D as: v=

c = c csc φ . sin φ

(21)

The spread of the currents across the surface of Earth (Tesla surface wave) was often compared by Tesla with the passage of the moon’s shadow over the globe. As stated in [7]: “…the law of motion can be expressed by stating that the waves on the terrestrial surface sweep in equal intervals of time over equal areas, but it must be understood the current penetrates deep into the earth and the effects produced on the receivers are the same as if the whole flow was confined to the earth’s axis joining the transmitter with the antipode”.

In other words, the propagation law states that at any point of the earth surface the projections of wavelengths on the earth diameter passing through the pole are all the same [2].

This means that in this conception the surface of the earth could be viewed as a TL (the characteristic impedance ZE = 58,3 Ω according to [16]) but with variable wave velocities ranging from approximately infinity at the electric poles down to the speed of light at the electric equator (the mean velocity vmean ≈ c⋅π/2). Taking that the whole current can be taken as to pass along the diameter of Earth with speed c as depicted in Fig. 12, the impedance Z, potential V, and axial current I distribution would follow from (9), (10) as: Z ( x ) = Z E cth γ e ( d − x ) ,

(22)

Figure 11 - Theory [4] v=∞ TMT

φ =45°

v = √2c

φ v=c

EARTH D

It is interesting that Tesla arrived to the speed of light by modelling the Earth globe as a LC oscillating circle [16].

v=c

v = √2c

v = √2c AP v=∞

4

v = √2c

v≅c

Figure 12 - Propagation of Tesla wave (left) and a simplified TL representation (right)

ch γ e d

sh γ e ( d − x ) ch γ e d

,

,

(23) (24)

where d is the diameter D expressed in wavelength λ, γe = αe + jβe would be the complex propagation constant of Earth and ZE the characteristic impedance. Current ITMT is the current that driven by the great TMT voltage ought to penetrate the earth.

The standing wave patterns of current and potential that should be formed on the surface of the lossless Earth if the Tesla condition f = c/(4D) ≈ 6 Hz [7] is fulfilled are similar to the case 1 (l = λ/4) in Fig. 4. See from (22) that, for frequencies below approximately 6 Hz, the Earth behaves as a capacity that increases with lowering the frequency, as similarly stated in [7].

In such a system, if one could exist, the energy would not be radiated, it would be conserved in a TMT-Earth system from where it may be drawn by properly adjusted antennas. The receiver arrangements explained in Fig. 13 [1], become clear from this standpoint. It must be noted that in the case of a loaded receiver the Poynting flux would be formed chiefly along the orthodromic line (the shortest line between the transmitter and receiver) as stated in [2]. By accepting the fact that the path-loss increases with frequency due to a finite conductivity of the earth, it would be preferable to use lower frequencies (less than 20 kHz [7, 25]) for power transmission.

From this point on, we cite Tesla (Fig. 5 in [1] is Fig. 13 here).

“Granted, then, that an economic system of power transmission thru a single wire is practicable, the question arises how to collect the energy in the receivers. With this object attention is called to Fig. 5, in which a conductor is shown excited by an oscillator joined to it at one end.

Evidently, as the periodic impulses pass thru the wire, differences of potential will be created along the same as well as at right angles to it in the surrounding medium and either of these may be usefully applied. Thus at a, a circuit comprising an inductance and capacity is resonantly excited in the transverse, and at b, in the longitudinal sense. At c, energy is collected in a circuit parallel to the conductor but not in contact with it, and again at d, in a circuit which is partly sunk into the conductor and may be, or not, electrically connected to the same. It is important to keep these typical dispositions in mind, for however the distant actions of the oscillator might be modified thru the immense extent of the globe the principles involved are the same.”

He continues as follows. (Fig. 6 in [1] is Fig. 14.a here.)

“Consider now the effect of such a conductor of vast dimensions on a circuit exciting it. The upper diagram of Fig. 6 illustrates a familiar oscillating system comprising a straight

rod of self-inductance 2L with small terminal capacities cc and a node in the center. In the lower diagram of the figure a large capacity C is attached to the rod at one end with the result of shifting the node to the right, thru a distance corresponding to self-inductance X. As both parts of the system on either side of the node vibrate at the same rate, we have evidently, C −c . When the ( L + X ) c = ( L − X ) C from which X = L C +c capacity C becomes commensurate to that of the earth, X approximates L, in other words, the node is close to the ground connection.”

This can easily be related to the previous analysis, eq.

(19), Figs. 5-7. A comparison of similar standing wave patterns obtained by the TL modelling presented here and those given

by Tesla in [1] is given in Fig. 14.

Figure 13 - Illustrating typical arrangements for collecting energy in a system of transmission thru a single wire [1]

a) Effect of equal (small) condensers on both ends 1

0.5 Voltage

I ( x ) = ITMT

ch γ e ( d − x )

0

abs(Γ1) = abs(Γ2) = 1 φ1 = φ2 = -5 deg

-0.5

-1 -0.5

-0.25

0

0.25

0.5

0.25

0.5

Relative position along the line Effect of big condenser on one end 1

0.5 Voltage

V ( x ) = ITMT Z E

0

abs(Γ1) = abs(Γ2) = 1 φ1 = -5 deg φ2 = -135 deg

-0.5

-1 -0.5

-0.25

0

b) Figure 14 - a) Diagram elucidating effect of large capacity on one end [1]; b) Same diagram obtained by resonant TL model Relative position along the line

remarkable and inexplicable aberrations of the scientific mind which has ever been recorded in history” has been missed as depicted in many historian books. How did he feel about all this at the time is best depicted by his own words: “… My project was retarded by laws of nature. The world was not prepared for it. It was too far ahead of time. But the same laws will prevail in the end and make it a triumphal success.”

Well, one way or the other, success it was.

REFERENCES

Figure 15 - Tesla’s world-wide wireless transmission of electrical signals, as well as light and power, is here illustrated in theory, analogy and realization [4] In elucidations of the true conduction concept for a worldwide multipurpose radio as opposite to Hertzian transmission, he often used “a crude but correct approximation of his wireless system” as depicted in Fig. 15 taken from [4]. Tesla perceived this system during the Colorado Springs radio investigations [6], [11] and any of his numerous inventions as AC generators, bladeless turbine, magnifying transmitters and receivers, light bulbs, teleautomatons, vacuum tubes etc. can be considered to be just bricks in the wall of his integrated radio network, compatible to the existing electric systems. And he conceived most of this in 19th century! On the other hand, his antenna systems were the first, the most (and only) efficient radioapparatus at the time and, as such, the firm base for the fast development of radiocommunications in the past century.

5. CONCLUSION

In this paper, we compare several Tesla's high-frequency resonators to the corresponding transmission line models. If in a transmission line regime, the behaviour of a helix with a significant electrical length such as a Tesla coil can efficiently be predicted by a Smith chart knowing only two parameters, the characteristic impedance and the propagation constant.

Most of these resonant structures that Tesla experimented with are actually poor radiators. On the other hand he made a variety of successful and valuable experiments with electromagnetic waves and designed many efficient radioantennas. Thus it still remains a mystery not completely solved what exactly Tesla meant by his radio-transmission concept, at least when wireless transmission of energy is in question. A fair chance for Tesla to experimentally prove by his Long Island TMT that his view on electromagnetism was correct while Hertz’s was, as he did not hesitate to say, “one of the most

[1] N. Tesla: The true wireless. Electrical Experimenter, May 1919, pp. 28-30, 61-63, 87.

[2] N. Tesla: World system of wireless transmission of energy, Telegraph and Telephone Age, October 16, 1927, 457-460.

[3] N. Tesla: The problem of increasing human energy, The Century Illustrated Monthly Magazine, June 1900.

[4] N. Tesla: Famous scientific illusions. Electrical Experimenter, February, 1919.

[5] N. Tesla: My Inventions. V. The Magnifying Transmitter.

Electrical Experimenter, June 1919, 112-113, 148, 173, 176-178.

[6] N. Tesla: The transmission of electrical energy without wires.

Electrical World and Engineer, March 5, 1904.

[7] N. Tesla: Art of transmitting electrical energy through the natural mediums. Specification forming Patent No. 787,412, USPO, April 18, 1905.

[8] N. Tesla: Apparatus for transmitting electrical energy, Patent No. 1,119,732, USPO, December 1, 1914.

[9] N. Tesla: Experiments with alternate currents of high potential and high frequency. Lecture delivered before the I.E.E., London, February 1892.

[10] Nikola Tesla on his work with alternating currents and their application to wireless telegraphy, telephony and transmission of power: An Extended Interview, ISBN: 1-893817-01-6, 1916.

[11] N. Tesla, ed. & commentaries by A. Marinčić: Colorado Springs Notes 1899-1900, Nolit, Beograd, Yugoslavia, 1978.

[12] Nikola Tesla tells of new radio theories. An interview with Nikola Tesla. NY Herald Tribune, September 22, 1929.

[13] Prepared statement by Nikola Tesla. (Prior to interviews with the press on his 81st birthday observance), July 10, 1937.

[14] R. Lomas: Spark of genius. Independent Magazine, 21. August, 1999.

[15] A. Marinčić: Some recent recognition of pioneering role of Nikola Tesla in the development of radio, In: The Life and Work of Nikola Tesla Proceedings, Zagreb, June, 2006, pp. 32-48.

[16] A. Marinčić: Research of Nikola Tesla in Long Island laboratory.

In: Energy and Development at the International Scientific Conference in Honor of the 130th Anniversary of the Birth of Nikola Tesla, 1986.

[17] T. Grotz: The true meaning of wireless transmission of power, A Journal of Modern Science [18] P. Nicholson: The real science of non-Hertzian waves, August 2002, 〈http://205.243.100.155/frames/Non-Herzian_Waves.html 〉

[19] O. Nichelson: Tesla’s self-sustaining electrical generator. In: Proceedings Tesla Centennial Symposium 1984.

[20] K. J. Vlaenderen: A generalization of classical electrodynamics for the prediction of scalar field effects, 〈arXiv:physics/0305098 v1〉 [21] W. C. Brown: The history of power transmission by radio waves.

IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-32, No. 9, September 1984, pp. 1230-1242.

[22] K. L. Corum, J. F. Corum: Nikola Tesla and the planetary radio signals. In: Proceedings Tesla III Millennium, October 15-19, 1996, Belgrade, Yugoslavia [23] K. L. Corum, J. F. Corum: RF coils, helical resonators and voltage magnification by coherent spatial modes. Microwave Review, Vol. 7, No. 2.

[24] K. L. Corum, J. F. Corum: Tesla Coils: 1890-1990. 100 Years of Cavity Resonator Development. In: Proceedings Tesla Symposium, Colorado 1990, Sec. 2, pp. 1-18.

[25] K. L. Corum, J. F. Corum: Tesla & the magnifying transmitter. A popular study for engineers.

[26] T. W. Barret: Tesla’s nonlinear oscillator-shuttle-circuit (OSC) theory compared with linear, nonlinear-feedback and nonlinearelement electrical engineering circuit theory. Annales de la Fondation Luis de Broglie, Vol. 16, No. 1, 1991.

[27] S. Avramenko: Method of industrially significant transmission of electrical power along a single-wire line. In: Proceedings Tesla III Millennium, Belgrade 1996, Sec. VI, pp. 57-64.

[28] D. S. Strebkov: Nikola Tesla and future of electric power engineering, Proceedings of the Sixth International Symposium Nikola Tesla, Belgrade, Serbia, October 18-20, 2006, pp. 121125.

[29] D. Bajić: Teslin doprinos medicini, Zbornik radova Tesla III Millennium, Beograd 1996, Sec. IV, pp. 1-64.

[30] J. J. O'Neill: Prodigal genius. Biography of Nikola Tesla, New York, 1944.

A Transmission Line Model of an Ideal Lumped Coil

Consider an ideal lossless lumped coil of a significant electrical length and without distributed capacity, which is grounded on one end. Referring to Fig. 3, taking that C = 0 one may write the following TL relations:

∂x ∂i ( x ) ∂x

=

∂u g ( x )

= 0.

∂x

+ jω Li ( x ) ,

(a1) (a2)

From (a2) the current is constant through the line, so we may put i(x) = I. Integrating (a1) from l/2 to x, taking that u(l/2) = ug(l/2) = 0, we obtain a linear rise of potential along the coil:

(a3)

From (7) we also obtain: V2 1 u l. = Z 2 Z 2 x =− 2

I = −I2 =

(a4)

Hence, taking that: V2 = U g − jω LlI ,

(a5)

where Ug = ug(x = –l/2), and supposing a purely capacitive load CL on end 2 we easily obtain the current as: I=

Ug 1 jω LC + jωCL

,

(a6)

where LC = Ll is the total self-inductivity of the coil. The top voltage follows from (a5) and (a6) as: V2 =

Ug 1−

ω2 ω02

,

(a7)

where:

ω0 =

APPENDIX

∂u ( x )

l⎞ ⎛ u ( x ) = u g ( x ) + jω L ⎜ x − ⎟ I .

2⎠ ⎝

1 LC CL

,

(a8)

is the resonant angular frequency of the lossless lumped LCCL circuit.