# VLoaivvvvt[hr

I moons (28) and (29) may be combined in the i' [lowing extremely useful expression which relates ' t load voltage to the input (generator end) volt-

>iw consider what happens on ail open-circuited 1 w-Joss lii>e one quarter wavelength long. Simple > implex algebra gives the following well-known

(esult:

here, again, a is the attenuation per unit length uf the transmission line, and the j implies that the voltages at the two ends are in phase quadrature. (Tie structure is a lossy, tuned reactive resonator. Siiice the numerator of equation (31) is finite and the denominator is vanishingly small, the voltage landing wave will build up to very large values!

I "he transient build-up to the steady state expressions given above is easy to understand. Initially a i irward wave of voltage is launched from the input end of the line. It propagates toward the open circuit (high impedance) end of the line where it is reflected with zero phase shift. This reflected wave travels back down the line to the low impedance input end, where, not only is it reflected with a 180° phase shift but because it took a haif cycle to travel up and back along (he quarter wave long line, this twice reflected wave is now in phase with the source of original energy being launched into the line.

The voltage wave now being launched into the line will add directly to the twice reflected wave as it travels back toward the open circuit load end. These coherent additions will continue to proceed, sloshing the load voltage higher and higher as a standing wave forms along the line with a voltage minimum at the input end and a voltage maximum at the open circuited load end. This growth process will continue until either a discharge occurs at the toad end (i.e.,- a nonlinearity of the system), or the line's I2R losses are equal to the power being supplied by the source (i.e.,- the generator can't push the system any higher).

What limits the maximum attainable voltage? The power driving the line, the line losses a (, and the breakdown potential of the load geometry (which usually arises from the onset of cold field emission from the electrode).

in common parlance, this is called a Tesla coil. (The secondary of a Tesla coil is a helically distributed quarter-wave resonator, not a lumped tuned circuit. The voltage rise is by standing wave: V,^ = S Vmjri, where S is the VSWR on the transmission line resonator. The actual measured voltage distribution on a Tesla coil or a transmission line resonator follows lite first ninety degrees of a spatial sinusoid, much as it would on a quarter-wave vertical monopole antenna: Vmin at the base and Vjhjx at the top.)

It should be obvious that resonance transformers (or tuned transmissionline resonators) do not have to be shock, excited (link coupled) by spark gap oscillators. They perform equally well when driven by any high power master oscillator - spark gap, vacuum tube or solid state. Junction breakdown and device efficiency become major concerns with the latter two. However, the design philosophy is the same and, historically, the approach has been used with devices operating at frequencies as low as 60 Hz (HV power supplies for X-ray tubes) and as high as several GHz (RF plasma torches). The authors have published fairly extensive engheer-

Figure T. A slow-wave helical transmission line resonator (a Tesla coil).

ing analyses of such circuits in the past.38-19-40'41

There have been many practical applications of the above technique wherever high voltage impulse sources are needed. In particular, the Soviets have advanced the technology to a mature state for use with high current nanosecond pulsed beams.42-43

[it should parenthetically be remarked that, first, what some authors have called Testa Coils in the

### Cavity Resonators

As with both lumped and distributed circuits, it is possible to show that cavity resonators also arc magnifiers. Both Smyihe45 and Condon46 derive expressions for voltage magnification in cavity teso-nators excited by inductive loops and capacitive probes. The former is the ratio of the voltage across the cavity to the emf in the loop, (32) M-|V/{|

and the latter is the ratio of the maximum potential across the cavity to that across the probe. Smythe's past are simply lumped tuned coupled coils. Tcsli was using these prior to 1842. When comparing the voltage rise produced by lumped coupled coils with that obtainable from distributed resonators, Tesla would write, "No such pressures - even in the remotest degree, can be obtained with resonating circuits otherwise constituted with two terminals forming a closed path,"44 The best engineering analysis which we have seen of the coupled coils configuration is given in Smyihe.

Secondly, the "Tesla Coil" so commonly seen today is in fact a link-coupled distributed tuned resonance transformer. It is easily documented that Tesla was using the latter prior to 1898. As Sloan observes, the lumped analysis of this configuration totally fails.

Thirdly, Tesla's most famous high voltage RF experiments, the photographs of which the public at large is so familiar with, employed what he called his "Extra Coil." From his recently published Colorado Springs diary of 1899, it is clear that this structure is actually the slow wave helical transmission line resonator of Figure 7. The structure was excited at its base by a relatively narrow band RF sienal generator,)