Modelling Tesla’s Radio System as a Transmission Line

Tesla - The Scientist Out of Time 2006 - Split, Croatia.

Modelling Tesla’s Radio System as a Transmission Line 1,2,3

Z. Blažević(1), D. Poljak(2), M. Cvetković(3)

University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture 1 zblaz@fesb.hr, 2dpoljak@fesb.hr, 3mcvetkov@fesb.hr

Abstract-In this paper, we present a transmission line model of the Tesla’s radio system concept. It is shown that the radio equipment as well as “propagation of current through the earth” is essentially a transmission line based concept. The basic parameters of a Tesla coil such as the magnification ratio, node position, etc are derived.

I. INTRODUCTION A human of today is very well got used to many electric and radio gadgets, without which our lives would be subsequently different. But, as it turns out, a relatively small number of us are aware that the key inventions on which they are based are not the results of gradual and systematic work of organised scientist and engineers, but rather of a few individuals whose understanding of the phenomena they were involved into provided fast development of those appliances that we now take for granted. The greatest of them was certainly Dr. Nikola Tesla, whose inventions and bravery to expose his outstanding achievements in physics and electrical engineering in the times when carriages pulled by poor animals were the main means for transport, were of the uttermost importance.

Besides the rotational magnetic field, one of the most valuated Tesla’s inventions is by all means the solution to the efficient utilization of radio, based on the experiment conducted by another great scientist Dr. Hertz, the first man who managed to transmit a significant amount of energy by radio waves to the receiver at some distance from transmitter. However, Tesla often disputed Hertz theory, trying to explain his system by means of “true conduction” or “non-hertzian waves” [1-7]. (Later, he also disputed the very existence of Hertzian waves, viewing them as the sound-like longitudinal waves in luminiferous ether, e.g. see [9]). This provided him a lot of headache as mainly because of such explanations and regardless of the successful radio experiments that he already conducted, the invention of radio has been contributed to Marconi, who, by combination of Tesla’s and Hertz’s equipment [10], managed to achieve transatlantic radio transmission of noisy Morse coded letter “S”. It took all Tesla’s life to eventually prove before USA Supreme Court that invention of radio belongs to him and not Marconi. However, although the radio systems of Tesla and those based on Hertzian dipole are explained by modern science as basically the same phenomenon of Hertzian waves, which means that Tesla’s theories about radio are rejected, it is a fact that the radio systems developed on his studies and experiments are in essence those systems universally adopted. This is because an efficient radio transmission is based on the effect of resonance, which reduces static noise as the main nuisance for every radio engineer. From those facts it could seem that

when we are talking about Tesla we are talking about history of radiocommunications engineering rather than present or future. Nonetheless, it should be born in mind the propagation theories of Tesla are not a result of merely attempts to explain radio phenomena by means of old theories of ether and like, but rather of the extensive and enhanced experiments conducted, which some of them present a great puzzle even today.

He contemplated his radio system as diametrically opposite to the Hertz’s in both the operation mode and efficiency, which was depicted by him as a metaphor of “cutting the butter with blunt and sharp edge of a knife” [7]. The key invention on which he based his wireless transmission is the well-known Tesla magnifying transmitter (TMT), a very powerful grounded distributed helical oscillator concluded with an open capacity and driven by a tuned primary circuit, a device for which he claims, if properly adjusted, to be able to transmit energy, voice, pictures and data without wires to all points on Earth without tangible loss and which activity would be noticeable on nearby planets such as Mars or Venus [6]. Moreover, this device (or like ones) was used and experimented by him for many other purposes such as generating Xrays, splitting atoms, burning nitrogen in the atmosphere, even weather control, defensive weaponry, in medicine etc. Consider for example a Livingston chart given in Fig. 7 in [11] that indicates that by the invention of TMT Tesla was constantly about 40 years ahead the technology of his time.

In this paper we shall commence with Tesla propagation concept in Sec. II, and then try to derive by transmission line (TL) models the main parameters important to the design of a TMT in Sec. III. Additionally, in Appendix at the end of the paper we shall present the scattering matrix [21] explanation of the magnifying effect on two examples given in [12] that relates to the analysis presented here. In Sec. IV, TL elucidation of Tesla propagation theory will be presented. It can be fully represented by TL approach, which makes it easily comprehendible.

II. TESLA PROPAGATION THEORY The propagation model to be provided in this paper can be found to be compatible with the one elucidated by Tesla in [2-4]. Basically, it is based on the Tesla’s concept of the electric transmission thru single wire without return [1,13]. In [1-4] the wireless transmission is explained, but the depicted approach has been disputed by science as actually the existence of so-called non-hertzian waves is not recognised (at least not as Tesla viewed them, for further reference see [14]). Thus another explanation of his results such as the ones from Colorado Springs has been required, the soundest being the one that relays on coupling to

Tesla - The Scientist Out of Time 2006 - Split, Croatia.

the Schumann cavity. This is strengthened by the fact that, as pointed out in [15], the cavity parameters measured by Tesla are very similar to those measured by other scientists and even the conclusion about superluminal velocity of the waves can be elucidated.

There are of course other attempts to explain the physic that could be hidden behind Tesla’s conclusions, some of them based on quantum mechanics, e.g. [16] and some of them resided in grey areas of science that still require a great deal of considerations and experimenting. Some may think that the transmission via Schumann cavity provides a satisfying explanation and a means for wireless power transmission but, as pointed in [14], the Q-factor of the cavity is nowhere sufficient enough to support such a task. It is fair to emphasize some other facts about that.

First that the low Q-factor does not necessarily limit the practicability of power transmission [15], second that there seems to be some measurements that report the Q-factor in excess of 1000 as opposite to those measurements that reports Q = 4-5, and third that perhaps Tesla already made distinction between the Schumann cavity resonance and the globe resonance as former being the resonance achieved by electromagnetic (EM) waves and latter by so-called pressure wave (or Tesla wave) [16] (not to mention the round-trip loss reported by Tesla of approximately 0,5 %, [2,4] which is not the case for the wave propagating through the earth-ionosphere waveguide, at least not for the operational frequencies of the order of tens of kilohertz). These are just a fair number of reasons not to definitively reject the Tesla’s claims and to continue the research of this interesting topic further.

There are many explanations how the Tesla theory conforms to the present understanding of radiocommunications. In essence, many authors equalize the term “Hertzian wave” with the term “space wave”, and “Tesla’s wave” to “surface wave”, e.g. [10,15]. However, it must be born in mind that the Tesla surface waves, if we could be allowed to introduce them at least as an assumption, exhibit some different properties than other known types of surface waves. The main distinction is that the former travels with the speed in excess to those of light waves. Thus, although the both types of surface waves are alike, they are by all means not the same phenomenon.

Consider Fig. 1 from [3], which elucidates the Tesla’s propagation conception. Basically, there are two effect of a TMT according to Tesla. First, there are Hertz waves that are practically a waste of power and second, the current through the earth causes the Tesla surface wave that should be applied for radio transmission of power and intelligence. The former travels with the speed of light, whereas the latter with superluminal velocities, in range from approximately infinity at the electric poles down to a velocity of light at the equator.

In this concept, both the TMT and Earth can be represented as TLs, what will be depicted in the next section.

Fig. 1. Propagation theory of Tesla [3]

III. TRANSMISSION LINE MODEL TMT is a passive device, an oscillator of great power in which the magnification of voltage and current is achieved by standing waves. There are already a lot of papers and books that explains this and other Tesla’s apparatus and effects, from which the work of Corum brothers should be highlighted (some are listed in the references). A jet another comprehensive approach to the electric circuits of Tesla by nonlinear oscillator-shuttle circuit (OSC) theory is given in [18], in which Tesla OSCs are considered to be analogues of quaternionic systems.

This work was originally disclosed in document No. 225395 U.S.P.O. 1988. In this paper it is explained how Tesla EM can be considered to be vastly different from the classical EM approach. Different applications of the circuitry are claimed to be accomplishable: power control, energy control, phase control, noise control and reduction of power loss in power transmission.

The analysis to be presented here by a TL model relays on the slow wave helical resonator analysis [19].

It is assumed that only TL T0 mode propagates through the secondary helix. By this model it is possible to estimate the main parameters of such an oscillator, e.g. potential and current distribution across the structure, magnification factor, standing wave formation etc.

Consider two examples of OSCs presented in Fig. 2 (e.g. see p. 198 in [8]). Both circuits are grounded on one end. The first (a) comprises a circuit

Tesla - The Scientist Out of Time 2006 - Split, Croatia.

ex that consists of an extra coil attached to the free end of grounded secondary oscillator s inductively related with primary p and concluded by an open capacity C2.

All three p, s and ex, have the same period of vibration.

The second (b) is realised as two loosely coupled oscillating circuits with the same period of vibration and there is no shunting of the secondary coil by a condenser as in the case (a). These types of apparatus are to be modelled as TLs, the first one with a lumped voltage source at the beginning of the line and the second with a distributed voltage source along the line, as depicted in Fig 3a. However, some generalizations are made regarding grounding on end 1; namely, in case (b) instead of the short circuit or grounding resistance there is a capacitive impedance Z1. A small piece of the line is drawn in Fig. 3b with voltage source ∆ug that exists only in the case depicted in Fig. 2b. Furthermore, there are only thermal losses included in the scheme, whereas the radiation losses can be incorporated by introducing an imaginary inductivity and an imaginary capacity in serial and parallel connection respectively to their real counterparts, which is omitted for the occasion. It is further assumed that there are no discharges, which means no escaping of EM energy from the oscillator.

Taking that: ⎧ 0, case a) ⎪ jω t u g ( t ) = U g e , ∆u g = ⎨ ∆x jωt ⎪⎩U g l e , case b) ∂u ∂i ∆u = ∆x, ∆i = ∆x ∂x ∂x

⎛

γ = α + j β = jω LC ⎜1 + ⎝

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

0, ⎧ ⎪ I0 = ⎨ Ug γ 2U g , case b) G j ω C + = ( ) ⎪ l l ( R + jω L ) ⎩

a) (1) (2) (3)

(4)

b) Fig. 3. a) TL model with lumped or distributed source; b) Small peace of length ∆x of the TL

⎧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

where ug is the voltage on the condenser in the secondary s that is modelled as a lumped source attached on end 1 of the TL in the case (a), and in the case (b) the voltage between the top and bottom of the coil due to the alternate current I0 induced in the secondary wire of length l, γ is the complex propagation constant, ω = 2πf angular frequency and the rest of parameters are as indicated in Fig. 3b.

Following the well-known analysis, wave equations for both cases of TLs can be written as: ∂ 2u + γ 2u = 0 ∂x 2 ∂ 2i + γ 2i = I 0 , 2 ∂x

b) a) Fig. 2. Two approaches for designing Tesla’s oscillators

(5) (6)

(7a)

(7b)

2

and defining the reflection coefficients of load impedances for a TL of characteristic impedance Z0 as: 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 direct voltage source excitation on end 1 are:

u ( x, t ) =

i ( x, t ) =

U g e jωt (1 − Γ1 ) e 2

l⎞ ⎠

1 − Γ1Γ 2 e −2γ l

U g e jωt (1 − Γ1 ) e 2Z 0

⎛ ⎝

γ ⎜ x− ⎟ 2

⎛ ⎝

l⎞ ⎠

γ ⎜ x− ⎟ 2

1 − Γ1Γ 2 e −2γ l

⎛ l⎞ −2 γ ⎜ x + ⎟ ⎤ ⎡ ⎢1 + Γ 2 e ⎝ 2 ⎠ ⎥ , ⎢⎣ ⎥⎦

(9)

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

By implementing the gauge conditions:

and in the case of distributed voltage excitation:

Tesla - The Scientist Out of Time 2006 - Split, Croatia.

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

(11) (12)

3

2

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

1

1.5 4 1 5 6

1 2 3 4

2

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

1.5 Current

⎛ 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 ,

2

Voltage

− I Z Aeγ x − Be −γ x −γ 2l jωt u ( x, t ) = 0 0 e e , 2 1 − Γ1Γ 2 e −2γ l

0.5

5 6

1

7

0.5 8

7 8 0 -0.5

(13) (14)

0 -0.5

0.5

0 Relative position

0.5

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

Extermes

1.5

Clearly, in both cases, the necessary condition for the resonance is accomplished when the parameter:

0 Relative position

1

0.5

(15)

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

The famous Tesla equation for these types of lines will be defined in somewhat different manner than in [11]. Here, we are considering the ratio of voltages on end 2 of line: in the nominator we shall put the top potential V2 and in denominator the voltage V20 measured on the matched top load (Γ2 = 0). In the both considered cases the following form of the Tesla equation applies: u ( x = − 2l ; Γ1 , Γ 2 ) V2 1 + Γ2 = = . (16) l V20 u ( x = − 2 ; Γ1 , Γ 2 = 0 ) 1 − Γ1Γ 2 e −2γ l

Clearly, for the short circuit on end 1 (Γ1 = -1), eq.

(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 and spent as heat, radiation or like. The magnifying factor is obtained for end 1 of a low-loss TL short and end 2 open (Γ2 = 1): eγ l eα l 1 = ≈ . cosh γ l sinh (α l ) α l

0

0.05

0.1 0.15 Eletrical length (λ)

0.2

0.25

Fig. 4. Voltage, current and the extreme values for lines of different lengths equal or less than quarter of wavelength

M = 1 − X C2 .

(18)

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 generally different for the two scenarios. Nonetheless, provided that the resonance condition is maintained on a lossless TL, by inspection of given relations it can be found that those standing wave patterns are identical.

In that case, the node of standing wave for the signal of wavelength λ is determined as: x0 =

ϕ2 − 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.

A comparison of the standing wave patterns obtained by the TL modelling presented here and those given by Tesla in [1] is given in Fig. 6.

(17)

Let us first consider the effect of electrical shortening of the line. For that matter we assume the short circuit (an infinite 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 2 open and the normalised top capacity XC to be added in order to achieve resonance can be expressed as:

Fig. 5. Lossless TL resonator as two resonators seen from opposite ends that resonate at same frequency Effect of equal (small) condensers on both ends 1

0.5 Voltage

M =

0

0

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

-0.5

-1 -0.5

-0.25

0

0.25

0.5

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

0.5 Voltage

ϒ = 1 − Γ1Γ 2 e −2γ l

0

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

-0.5

a) b) Fig. 6. a) Diagram elucidating effect of large capacity on one end [1]; b) A similar explanation obtained by the model of lossless TL in resonance -1 -0.5

-0.25

0

Relative position along the line

0.25

0.5

Tesla - The Scientist Out of Time 2006 - Split, Croatia.

Note that by increasing capacity on one end the node shifts toward it accordingly, in the fashion explained in [1].

IV. TL REPRESENTATION OF TESLA’S WIRELESS SYSTEM The basic effect on which Tesla was to establish his “World System” is propagation of current through the Earth, along the diameter, with a velocity equal or close to the speed of light. The spread of the currents across the surface of Earth (Tesla surface wave) as the consequence was often compared by Tesla with the passage of the moon’s shadow over the globe. As stated in [4], “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 earth can be viewed as a TL (the characteristic impedance Z0 = 58,3 Ω according to [20]) but with variable wave velocity v = c⋅cscφ ranging from infinity at the electric poles down to the speed of light at the electric equator (mean value vmean ≈ c⋅π/2), as illustrated in Fig. 7.

In such a fashion the energy is not radiated, it is conserved in a TMT-Earth system from where it may be drawn by properly adjusted antennas. The receiver arrangements explained in [1] Fig. 5, 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 [4]. By accepting the fact that the earth loss increases with frequency, it is preferable to use lower frequencies (less than 20 kHz) for power transmission. v=∞

REFERENCES [1] [2] [3] [4]

[5] [6] [7]

[8]

TMT

[9]

φ =45° v = √2c

v = √2c

v≅c

φ v=c

media, i.e. the earth, can be regarded as TLs. This can be viewed as compatible to propagation in the earthionosphere cavity to a high degree. Although Tesla contemplated the use of “the upper conducting air strata” in various ways, one of them certainly being the transmission of energy, on the basis of his considerations he often disputed decisively that his radio system can be deduced to the EM radiation in any way. Unfortunately the chance for Tesla to prove by his Wardenclyffe TMT that “he was right and that profession was completely misled” has been missed.

We can only hope that it is not going to be the only attempt and, finally, who could say for certain that people in some future will not enjoy all benefits of wireless power based on Tesla’s World System.

But, regardless, whether right or wrong about this, Tesla nevertheless gave us a reason to dream; he seeded our imagination with a great and profound goal of clean and efficient global transmission of energy without wires, which would have a potential to make the present problems of ecology, economy and poverty solved once and for all. Indeed, a perfect way to increase human energy.

[10] EARTH

D

v=c

[11] v = √2c

v = √2c AP v=∞

[12] [13]

Fig. 7. TL elucidation of Tesla wave propagation

V. CONCLUSION Here, we presented a TL based elucidation of the Tesla's concept of radio. It is clearly shown that in this theory both the radio transmitters and propagation

[14] [15]

N. Tesla, The True Wireless, Electrical Experimenter, May, 1919, pp. 28-30, 61-63, 87.

N. Tesla, Art of Transmitting Electrical Energy Through the Natural Mediums, Spec. P. No. 787,412, USPO, April 18, 1905.

N. Tesla, Famous Scientific Illusions, Electrical Experimenter, February, 1919.

N. Tesla, World System of Wireless Transmission of Energy, Telegraph and Telephone Age, October 16, 1927, pp. 457-460.

N. Tesla, Apparatus for Transmitting Electrical Energy, Pat. No. 1,119,732, USPO, December 1, 1914.

N. Tesla, The Problem of Increasing Human Energy, The Century Illustrated Monthly Magazine, June 1900.

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.

N. Tesla, Colorado Springs Notes 1899-1900, A. Marinčić, ed., Nolit, Beograd, Yugoslavia, 1978.

Nikola Tesla tells of new radio theories, An interview with Nikola Tesla, NY Herald Tribune, Sept. 22, 1929.

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

K. L. Corum, J. F. Corum, Tesla Coils: 1890-1990.

100 Years of Cavity Resonator Development, Tesla Symposium, Colorado 1990.

K. L. Corum, J. F. Corum, Tesla & The Magnifying Transmitter. A Popular Study for Engineers S. Avramenko, Method of Industrially Significant Transmission of Electrical Power along a Single-wire Line, Proceedings Tesla Millennium, Belgrade 1996, Sec. VI, pp. 57-64.

P. Nicholson, The Real Science of Non-Hertzian Waves, August 2002.

T. Grotz, The True Meaning of Wireless Transmission of Power, A Journal of Modern Science, 1997.

Tesla - The Scientist Out of Time 2006 - Split, Croatia.

[16] O. Nichelson, Tesla’s Self-Sustaining Electrical Generator, Proc. Tesla Centennial Symposium, 1984.

[17] A. Puharich, Effects of Tesla's Life & Inventions, 1985.

[18] T. W. Barret, Tesla’s nonlinear oscillator-shuttlecircuit (OSC) theory compared with linear, nonlinearfeedback and nonlinear-element electrical engineering circuit theory, Annales de la Fondation Luis de Broglie, Vol. 16, No. 1, 1991.

[19] K. L. Corum, J. F. Corum, RF coils, Helical Resonators and Voltage Magnification by Coherent Spatial Modes, Microwave Review, Vol.7, No.2, 2001.

[20] A. Marinčić, Research of Nikola Tesla in Long Island Laboratory, Int. Scientific Conference in Honor of the 130th Anniversary of the Birth of Nikola Tesla, 1986.

[21] P. C. J. Hill, Some applications of the scattering matrix, Proc. IEE, Vol. 112, No. 1, Jan 1965, pp.1520.

Appendix

Consider a TL of characteristic impedance Z0, concluded with an arbitrary load of impedance ZL such as presented in Fig. 1A. The generator providing power Pg does not have to be matched. Following the well-known scattering matrix theory [21]: b = S×a ,

(1A)

where a = [ai] is the n x 1 matrix of incident waves, b = [bi] the n x 1 matrix of reflected waves and S = [sij] the n x n scattering matrix of the considered n-port passive device. The power Pg of the wave ag from the generator is: 2 1 1 Pg = ag ag∗ = ag .

(2A) 2 2

Pg

~

1 a1

e −γ l ⎤ ⎡ ag + Γ1b1 ⎤ ⎥×⎢ ⎥.

0 ⎦ ⎣ Γ 2 b2 ⎦

1 ∗ 1 b2 b2 = b2 2 2

2

Fig. 1A. Transmission line

Pg

~

a2

a1 1

2 DIRECTIONAL COUPLER

b1 a3

3

ZL b2 a4

4 b4

b3 l=λ

Fig. 2A. Travelling wave ring power multiplier

Next example of power magnification in a linear passive device that can be easily explained by implementation of scattering matrix is a directional coupler connected in the manner depicted in Fig. 2A. All waveguides employed are supposed to be lossless (α = 0), of the same characteristic impedance Z0 and length l equal to an integer of wavelengths of the signal supplied by the generator on end 1.

Again, from (1A) and (2A): ⎡ 0 ⎡ b1 ⎤ ⎢ ⎢b ⎥ ⎢ A ⎢ 2⎥ = ⎢ ⎢ b3 ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎢⎣b4 ⎥⎦ 2 ⎣⎢ j 1 − A

A

0

0

j 1 − A2

j 1 − A2

0

0

A

j 1 − A2 ⎤ ⎡ a ⎤ 1 ⎥ ⎢ ⎥ Γ 2 b2 ⎥⎥ 0 ⎥⎢ ⎥ ⎢ b4 ⎥ A ⎥ ⎢⎢ b ⎥⎥ ⎣ 3 ⎦ 0 ⎦⎥

(6A) where A is a real number and, as before: a1 = ag + Γ1b1 ,

(3A)

(7A)

P4 1 + A 1 , = Pg 1 − A ϒ 0 2

(8A)

ϒ 0 = ϒ l = λ = 1 − Γ1Γ 2 .

(9A)

m=

(4A)

supplied to the load on end 2 of the line and power Pg supplied by the generator as: m=

b2

the power magnification ratio is:

From the matrix equation above, it is easy to calculate ratio of the power P2: P2 =

ZL

l

b1

Then, from (1A) and (2A): ⎡ b1 ⎤ ⎡ 0 ⎢b ⎥ = ⎢ − γ l ⎣ 2 ⎦ ⎣e

a2 2

Z0

where:

The factor A is determined from the factor F as:

F=

P1 P4

= Γ1 =Γ 2 =Γ3 =Γ 4 = 0

1 .

1 − A2

(9A)

−2α l

P2 e = .

2 Pg ϒ

(5A)

Theoretically, for a lossless quarter-wave line and corresponding purely reactive loads on ends such that ϒ = 0 (resonance), this ratio is infinite. If, on the contrary the load is matched to the line (Γ2 = 0) the maximum possible achievement is P2 = Pg. When magnification by standing waves is the prime goal, the distributed resistance (as well as the EM radiation, of course) of the line plays a role of decisive importance.

Consider the case of a 10-dB directional coupler (A = 3/√10) with the matched load on end 2. Then the power circulating in the ring through end 4 is 38 times the power supplied by the generator on end 1, whereas the power supplied to the matched load on end 2 is equal. Two factors are responsible for magnification according to (8A), the first determined by the coupler design and second by matching. By adjusting ϒ0 the magnifying effect can be boosted, which could be related to the peculiar effects mentioned in [12] when a slight mismatch is inserted into the guide.