## Potential Magnification

examples give voltage magnifications on the order of a hundred, The procedure could be formally executed for the Schumann cavity, of course. In all of the above, we have tacitly assumed that the elements are linearand passive. No external pumping is being done as with some possible ionospheric-TWT amplifier effect Thereisyetone more surprise that distributed Linear systems have in store - they can "multiply" power (without violating conservation of energy, of course).

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street lanri Coupler tl» Hi rift r feUiptfor twftjHriUM, Figure 8. A ring power multiplier.

### Traveling Wave Ring Power Multiplier

In addition to quarter wave transmission line resonators and cavity resonators which step up voltage, as just described, there is also a novel technique for stepping up real RF power (within practical limits). This unusual circuit geometry, invented by F.J. Tischer in 1952,47>4g actually makes it possible to obtain practical power level multiplication of 10 to 500 times the transmitter output. (The street lanri Coupler

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tl» Hi rift r feUiptfor twftjHriUM, Figure 8. A ring power multiplier.

Til)Ming'9 vector power (low, v,Rc(t -H'J, is utuully pumped up within this linear, p.i.sivedis-'i diutcd storage network. No physic»! taws are ■> -luted. and energy conservation is preserved in 1 (irocess, of course.)

T1 ■ i losed ring structure shown in Figure 8 is con-it u -I by means of a directional coupler to a second |mi .mission line which is excited by a source and u 11 niiated in a matched load. The distributed net» >ik is characterized by a real Poynting vector r ulating within the ring, where unidirectional p<< 'Hressive traveling waves build up in time. The C ieal explanation advanced by MIT-Lincoln I it' .StanleyMillerisas follows:

1 wave proceedingfrom the source is partially impled into the ring (by the directional cow-¡iter), and propagates around the ring in one direction as shown When this wave passes the • oupling region, a smallfraction is coupled hack into the main transmission line with the remainder proceeding around the ring again At the tame time, more energy is being coupled from the main line into the ring. If the wave pro-needing around the ring and the wave coupled Into the ring are in phase, then it is evident that the wave in the ring can he reinforced and can become quite large in magnitude.^

I lie build up of the power in the ring will continue nth each cycle until the losses around the loop i lus the loss in the termination is equal to the power upplied by the source, Hayes and Surette report « wal ring power levels of 200 kW with a 10 kW

generator at VHF. with a 6-inch coaxial tine resonant ring.50 Miller reported 100 MW levels from a 200 kW source driving a WR 2100 waveguide at 425 MHz, (He also reported spectacular pyrotechnical and acoustical results when a slight mismatch was inserted into the guide.) The primary use ofthe traveling wavering power multiplier has been to test high power RF components which fail due to power dissipation as opposed to voltage breakdown. (The latter could be tested with the transmission line resonance transformers described above.)

It is evident that a great variety of multiply connected geometrical enclosures51 with the appropriate electrical constitutive parameters can serve as ' ring" power multipliers. The necessary condition is that the successively added field components, from the ensuing circulations of field energy, possess the correct phases for constructive interference (coherent buildup). Such multiply connected structures can be operated either in the sinusoidal steady state (cw) or in a pulsed mode. Coherence and synchronization are particularly critical for the operation ofthe latter. We will now present a technique by which the fundamental ring resonator modes of an electrically large resonator may be determined by probing with a DSB (double sideband) modulated carrier of frequency much greater than the self resonant frequency of the resonator. (The theory may be applied to the probing of terrestrial (Schumann type) resonances with a VLF transmitter in place of an ELF source.]

### Multiple Beam Interference

< iptical multiple-beam interference is perhaps best known in the classical case ofthe monochromati-afly illuminated parallel plane geometry, such as dielectric plates and the paralleJ minor geometry of Fabry and Perot. As a result of the multiple reflections, the wave is split into many partial waves reflecting back and forth from the parallel boundaries, which interfere with one another as they linearly superpose together. Knowing the optical path difference between successive partial waves ,uid the phase of the reflections, the plane wave summation is formed and the intensity (the squared modulus ofthe field strength) is determined. The result is the famous expression obtained by G.B. Airy in 1831.

Let us repeat lite analysis for the case of around the circuit propagation for TEM waves. We will consider two cases: the pure carrier wave and the DSB modulated wave. For the sake of simplicity we wili employ an unpretentious ray optic model. The model is appropriate for optical and TEM transmission line ring resonators. It is clear and lends itself to simple prediction and experimentation.

Case I: Monochromatic Carrier

Consider the geometry shown in Figure 9 with fm = 0. The transmitter at wavelength X = c/f is located at the lop of the ring and multiple peripheral propagation paths are shown. In the region near the source, the nlh time around propagating wave can be expressed as

(33) En(X) = E0 A(L>" = E„ff«"1-afiWA

where A(L) is the propagation attenuation, a is the attenuation constant at the wavelength X, and L is die circumference of the ring L = Ce = 2TTR,. For plane wave propagation, A(nL) = e-nuL. The total field observed is

The terms on the right bear resemblance to the geometric series

Consequently, liquation (35 may be expressed at I

(37) E^X)- E„ [I + Ae J^t'= IN where jE-fi = E{, [I + A2 - 2A cosßLJI» 1$ =tan*' [(-A sinpLy( l-A cospL)]

This is a complex phasor associated with the time harmonic field. The lime domain expression fm the field is found from the expression:

EKO = ReiE-rt^'} = Re{|E|{X)| C"'!-The ring is resonant when O = 2rm. The wave power density is proportional to the squared modu lus, so that

*(l - Aie"""1'1 - e'1"1"'1}* A' ] Remembering that cosO = + ePJ, we can write

It is customary to employ the trigonometric identity cos9 = 1-2 sin2(6/2) and write

3 [{1 -A)11 4A sin^nL^X)] which can be manipulated into the form lEftf - EJ J

f40') l "^[i.^A/d -AJ^WK/X)] j which is analogous to Airy's formula for multiple beam interference in a parallel plate geometry. Fabry called the Airy denominator factor F = 4A/ (t-AP the finesse. Bom and Wolf52 define the finesse a5 3= V2k-/F. When a polychromatic beam is present, as in a spectrum analyzer, the latter expression has the advantage of being equal to the ratio of the line separation (At) to the half power width of the line (Bf).5-1 Finesse is a measure ofthe resolution of an interferometer. The term [1 +■ F

sin'ft)]1 A f(£) is called the Airy function and it Figure 9. Ray optic model Of VLF transglobal propagating , ., , .. ...

TEM waves anda DSBiransmrtier The p^crtn is choose represents the transmitted power density, figure an f0 appropriate for propagation around U-.e ring, and then 10 is a plot of lEf/IEJ1 verses the phase shift to tune ihe modulating frequency to produce beats betwean arising from the path-length di fferencc between suc-the USB and the LSB such that envelope resonances are - , ,,, observed. « tays = 2jtA (IX).

where c is the speed of light. Again. A(L) is ihe amplitude attenuation per global traverse. Algebraic manipulation leads to the expression

1'' C) - L cJccfi'M^/c)j or. equ î valent! y, as (46) |fif -:

where u is measured at the carrier frequency. This expression reduces to Equation (41) when the modulation frequency is zero. Equation (46) is a rather remarkable resull. Admittedly, it is rigorous only in the ring optics and I f-Vt transmission line case, but it does describe the ring resonator physics. It implies that ring resonances (and power multiplication) can occur in a "more-or-less" carrier independent manner. That is, holding the carrier frequency mn&Uinl mil varying the modulaliiit frequency will lead In field slrertgLh maxima will-It ever fni = Vinc/1 I he greatest line Me of ilw resonator will occur when ihe carrier frequency il tuned such that nf„L/c = nii That is theease whenever f0 = nc/L.

[Terrestrial resonances might be observed (depewk ing on propagation losses) whenever the bi»l frequency is fb = 2 fm = 7.5>i for n = 0,1.2.5, J Equation (12) of Appendix ! gives the spalial distance between intensity maxima, A, for if,=7.5 ILt as A = 40,000 km=the circumference of the eanh if the beat frequency is tuned to 7.5 Hz. the travelling modulation envelope peak of the DSB wawi will propagate all the way around the globe Und return to the position of the source exactly in synchronism with the next peak ol'ihe modulaimi envelope. Although the result is "more-or-less" independent of the carricr frequency, critical u> detecting the phenomenon is the use of a carriei frequency low enough so that great circle glohiil propagation can occur (VLr), and it is necessary to tune the beat frequency for ring resonance ]

### Example

Suppose f„ = 5 kHz (i.e., A. = 60 km). Further, suppose lhat we tune the modulation oscillator to fm - 3.75 Hz. This will produce two signals: fj = 5.00375 kill (i.e.. X, = 59.955 km) f; =4.99625 kHz (i.e., = 60.045 km). And the beat frequency is Af = f[ - f, = 7.5 Hz.

I he result is that the observed fringe spacing, given by Equation (1.12) is exactly the circumference of the earth: A = 40,000 km = Ct = 27tRc = 2n( 6.37 x 106) meters. If the beat frequency is tuned to 1JS i Iz, the travelling modulation envelope peak of the 5.0 kHz DSB wave in Figure 9 propagates all Ihe way around the globe and returns to the position of the source exactly in .synchronism with the next peak of the modulation envelope.

### Calculation of Field Strength at VLF

While (here exists extensive theoretical literature on the computation of electric field strengths at VI .F, we s lia II employ an empirically determined engineering VLF formula. The most famous VLF empirical formula was obtained by Dr. Louis W. Austin and Cohen for the U.S. Bureau of Standards,5'58 Although now considered more-or-less as a "museum piece", the empirical formula gives daylight VLF signal strength for ranges of2,000 to 10,000 km with fair accuracy. According to Austin, when the transmitted power is known, the daytime field strength may be calculated as"

A V siflli where d is the range in km, X is the wavelength in km, and 6 is Ihe angular separation between the transmitter and ihe receiver. The formula can be used for wavelengths from 100 down to 10 km, corresponding to frequencies from 3 to 30 kHz. At ranges beyond 10.000 km, the Austin-Cohen formula gives values well below what is actually observed, it is particularly poor in the region of the anlipode, and consequently it cannot be used to examine transglobal propagation.

yvw to the following concluiion».

• I to icrrestrul globe retonitei it multiple« of 7,5 11/ (Sm Figure 11 above.) Tto received pulse train his echoes which occur

0 HK4S4 seconds after the transmitted pulses. (See < - lercnces 62 and 63.)

• t It w conclusions are more-or-less independent of .«ii itr frequency for f„ s 2SkHz. Above2S ItHi

1 hr global response "washes out". (See Figure 11 ■fccive.)

I nit's exact words from May of 1900 were, ■ i - -rid say that the frequency should be smaller '■.m twenty thousand per second, though

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