Theory Of Traveling-wave Tubes Research Articles

Modification of Pierce’s Classical Theory of Traveling-Wave Tubes

For a realistic case of an electron beam interacting with the electromagnetic fields supported by a thin perfectly conducting tape helix, we derive the exact dispersion relation, from which an unambiguous determination of Pierce's space-charge parameter Q is made. In the process of doing so, we discover that the circuit mode in the equivalent three-wave theory of Pierce must be modified at high beam current, an aspect not included in Pierce's original analysis. We quantify this circuit mode modification by a new parameter that we call q, introduced here for the first time in traveling-wave tube theory. In a realistic example, we find that the effect of q is equivalent to a modification of the circuit phase velocity by as much as two percent, which is a significant effect equivalent-to a detune of two percent.

Analytic Exploration of the Accuracy of Pierce’s Three-Wave Beam-Wave Interaction Theory of Traveling-Wave Tubes

As it is well known that the simplified classical Pierce's three-wave theory can provide scaling insights into traveling-wave tube (TWT) operation for its close-form solutions, and thus becomes a useful guide for TWT design. However, the classical Pierce's three-wave theory shows poor agreement with Lagrangian theory, so determining how to achieve a more accurately Pierce's three-wave theory from Pierce's four-wave theory becomes an open problem. In this paper, a more accurately revised Pierce's three-wave dispersion relation is deduced by using an approximate treatment on the nonlinear term of the dispersion relation of Pierce's four-wave beam-wave interaction theory. Meanwhile, the boundary conditions of classical Pierce's three-wave theory are revised by theoretical analysis. Thus, the revised Pierce's three-wave theory is established. The Pierce small-signal theories are compared to each other and Lagrangian theory on a set of TWT parameters which are based on a single pitch section of C- and Ku- bands TWTs. It is found that the revised Pierce's three-wave theory agrees extremely well with Lagrangian theory and gains more accuracy than the classical Pierce's three-wave theory in the small-signal beam-wave interaction region. Finally, the phase difference between Pierce's four-wave theory and revised Pierce's three-wave theory is discussed.

Study of a high power sine waveguide traveling wave tube amplifier centered at 8 GHz

Performance of a 20-stage X-band sine waveguide amplifier, driven by a 40 A, 100 kV, cylindrical electron beam, is studied using numerical simulation and interpreted using Pierce's classical traveling wave tube theory. For an input signal power level of 1.8 kW, particle-in-cell simulations predict gain and bandwidth values exceeding 14 dB and 13%, respectively. For an input signal power level of 7.2 kW, particle-in-cell simulations predict gain and bandwidth values exceeding 12 dB and 15%, respectively, with output power levels exceeding 110 kW at peak gain. Also given are: an assessment of the space charge factor (Pierce's QC parameter) for the complex circuit using simulation data, and an evaluation of the harmonic contents in the beam current.

Sub-wavelength waveguide loaded by a complementary electric metamaterial for vacuum electron devices

We report the electromagnetic properties of a waveguide loaded by complementary electric split ring resonators (CeSRRs) and the application of the waveguide in vacuum electronics. The S-parameters of the CeSRRs in free space are calculated using the HFSS code and are used to retrieve the effective permittivity and permeability in an effective medium theory. The dispersion relation of a waveguide loaded with the CeSRRs is calculated by two approaches: by direct calculation with HFSS and by calculation with the effective medium theory; the results are in good agreement. An improved agreement is obtained using a fitting procedure for the permittivity tensor in the effective medium theory. The gain of a backward wave mode of the CeSRR-loaded waveguide interacting with an electron beam is calculated by two methods: by using the HFSS model and traveling wave tube theory; and by using a dispersion relation derived in the effective medium model. Results of the two methods are in very good agreement. The proposed all-metal structure may be useful in miniaturized vacuum electron devices.

Electrical Engineering Hall of Fame: John R. Pierce [Scanning our Past

The recipient of the Edison Medal in 1963 was John R. Pierce (Fig. 1). He was cited "for his pioneer work and leadership in satellite communications." The citation also noted "his stimulus and contributions to electron optics, traveling-wave tube theory, and the control of noise in electron streams." He later became one of a select few to receive both prestigious awards when the IEEE presented its Medal of Honor to him in 1975 (Fig. 2). He spent most of his professional career in research and research management at the Bell Telephone Laboratories. He was a prolific author of technical books and articles and received more than 90 patents for inventions.

Wave coupling in sheet- and multiple-beam traveling-wave tubes

To increase the power level of the sources of coherent electromagnetic radiation at frequencies from 100 GHz up to the terahertz range it makes sense to develop devices with a spatially extended interaction space. Sheet-beam and multiple-beam devices belong to the category. In the present paper the small-signal theory of traveling-wave tubes with sheet-beam and multiple sheet-beam configurations is developed. It is shown that in such tubes the wave coupling on electron beams may occur even in small-signal regimes. The wave coupling and its role for amplification of forward and excitation of backward waves in such amplifiers is studied. Also the effect of transverse nonuniformity of the electromagnetic field on the device operation is analyzed and illustrated by several examples.

Inclusion of space-charge effects with Maxwell's equations in the single-particle analysis of free-electron lasers

In free-electron lasers, the motion of electrons is governed by the Lorentz force equations in the presence of a radiation field. The variation of the radiation field then follows from Maxwell equations in which the modulated electron current acts as a source. A self-consistent analysis, which completely incorporates these two concepts, is presented to describe the behavior of the radiation field and the modulation of the electron beam. In this analytical study, we consider the space-charge effect as well as high interaction strength in the small-signal limit so that the result can be compared directly with the traveling-wave tube theory. It is found that the well-known three-wave solution is essentially applicable to the electron beam only. The variation of the radiation field is much more complicated. According to the analysis, there are only three controlling parameters: the pumping strength, the electron density, and the electron energy detuning. For different choices of those parameters, the field can be in a stable regime where its growth is limited or in an unstable regime where it grows exponentially. The boundary between these two regimes is defined quantitatively. The effect of the plasma resonance is observed at high electron densities as a natural result of maximum single-pass gains. The traveling-wave tube is then analyzed as a special example.

A two wave theory of traveling-wave tubes and backward-wave oscillations

Starting from Pierce's small-signal TWT equations, a two-wave theory for complicated TWT's and BWO's is developed. Explicit gain expressions for TWT's with severs and velocity tapers are given. A detailed treatment is given of a BWA structure with a single-step velocity taper in the center. The BWO start oscillation current is calculated as a function of the taper strength. The effciency of a step velocity taper as a remedy for backward-wave oscillations in the higher passbands of a TWT is then analyzed. It is shown that a suitable taper can increase the start oscillation current by more than a factor of 10 without adverse effects on the TWT performance if the backward waye coupling impedance is low. A tube with a weak linear taper is analyzed by a perturbation technique and it is found that there is no significant difference between the start oscillation currents for tubes with weak linear or abrupt tapers of comparable strengths.

A theoretical description for the multiple-signal operation of a TWT

A multiple-large-signal theory for traveling-wave tubes (TWT's) is developed. The equations associated with the growth of the signal waves along the length of the tube are integrated by means of digital computation methods. The assumptions made during the theoretical formulation are more restrictive than those made by El-Shandwily in his multiple-large-signal theory, however, by restricting the generality of the theory judiciously, the computer time normally required to make calculations of this type is substantially reduced, Calculations have been presented for only a few special cases. Computed results are presented here for the growth of up to 12 input signals and some of the more imporant intermodulation products. Agreement between theoretical predictions and experimental results are shown for the case of two input signals. In addition some important empirical relations are developed which can be used to approximately predict the large signal performance of a TWT and to further reduce the computer time.

IEEE Awards

Three physicists were honored by the Institute of Electrical and Electronics Engineers during its recent convention in New York City. John R. Pierce of Bell Telephone Laboratories received the Edison Medal for his work in satellite communications, electron optics, traveling wave tube theory, and the control of noise in electron streams. Arthur L. Schawlow of Stanford University won the Liebmann Memorial Prize for his contributions in the field of optical masers. Donald L. White, also of Bell Laboratories, was given the Baker Prize for his paper entitled “The Depletion Layer Transducer” that appeared in the July 1962 issue of the IEEE Transactions on Ultrasonics Engineering.

A Small-RF-Signal Theory for an Electrostatically Focused Traveling-Wave Tube

This paper describes a small-RF-signal theory for a new type of electrostatically focused traveling-wave tube. Consideration is given to the effect on the gain of large periodic variations of dc electron velocity, resulting from electrostatic focusing. LlewellynPeterson equations which apply to dc voltage distributions, similar to those used in the electrostatically focused traveling-wave tube, are used for analysis. The limitation of the Llewellyn-Peterson equations to one-dimensional problems is overcome by use of correction factors derived from Pierce's theory for traveling-wave tubes. Generalized expressions taking into account the periodic variations of dc electron velocity are derived for the gain parameter C and spacecharge parameter QC. For uniform dc electron velocity, such as that found in a magnetically focused tube, these expressions reduce to the forms derived by Pierce. In addition to their suitability for analysis of electrostatically focused traveling-wave tubes, the LlewellynPeterson equations provide a method of analyzing magnetically focused traveling-wave tubes alternative to those used previously. It is shown that the gain of the electrostatically focused tube can be calculated by use of the gain and space-charge parameters of the electrostatically focused tube and graphical data available for magnetically focused tubes. Depending upon the voltage distribution in the region of electron flow, the gain and bandwidth of a travelingwave RF circuit employing an electron beam having a periodically varying dc velocity can be greater or smaller than the gain and bandwidth of the same RF circuit employing a uniform dc velocity electron beam.

Theory of the Crestatron: A Forward-Wave Amplifier

In the past a considerable amount of experimental evidence presented by various workers has indicated that gain apparently occurs in a traveling-wave type device even though the voltage may be so high that growing waves cease to exist. This means, in terms of Pierce's traveling-wave tube theory, that the growth constant of the growing wave x1 is zero in this regime of operation. A theory explaining this phenomenon has been worked out and both small-signal and large-signal calculations have been carried out to investigate the characteristics of this type of operation. The gain occurs in this region due to a beating effect produced between the three small-signal forward waves described in traveling-wave tube theory as they travel along the RF structure. The maximum achievable gain is determined by the injection velocity and not by the length of the tube as in the normal case. Based on the above theory a device named the Crestatron, which utilizes this new mode of operation, has been built and tested to verify the theory, and it has been found that moderate gain (10-20 db) and high operating efficiency coupled with a very short length (4-6 wave-lengths) characterize this mode of operation.

Potential-minimum noise in the microwave diode

An analysis of the potential-minimum region in the microwave diode is presented which predicts the amount of noise convection current at the potential minimum under conditions present in guns of low-noise traveling-wave tubes and klystrons. Contrary to previous work by one of the authors, this analysis partially includes the effects of finite transit angle between the cathode and potential minimum. The results indicate that under certain conditions of space-charge limitation, the noise convection current may be larger than full shot noise, whereas under other conditions reduction by as much as a factor of four may be obtained. When the results of this work are applied to the noise-figure theory of traveling-wave tubes and klystrons, it is found that their minimum obtainable noise figure may be in the vicinity of 4 db rather than the previously predicted 6 db with reasonable cathode current densities. For very large current densities or low operating frequencies, the present theory indicates that there is no theoretical limit to the minimum noise figure of such devices, in agreement with the previous work.

An electromagnetic difference-equation of importance in the theory of traveling-wave tubes

Auf Grund der Sechspoltheorie der elektrischen Nachrichtentechnik wurde an anderer Stelle [1, S. 23] eine elektromagnetische Differenzengleichung zweiter Ordnung abgeleitet. Dieses auf formalem Wege gewonnene Resultat ist fur die Theorie der Traveling-Wave-Tubes von Bedeutung. Mit Hilfe der Maxwell-Lorentzschen Theorie wird nun gezeigt, dass die genannte Differenzengleichung unter viel allgemeineren Bedingungen zu Recht besteht. Der gleiche Weg fuhrt zu einer Methode, die auftretenden Parameter explizite zu berechnen.

Backward-Wave Oscillators

The Pierce traveling-wave tube theory is modified to apply to the backward-wave oscillator. Theoretical dependence of both starting current and frequency upon space charge and circuit loss is calculated, as is the dependence of efficiency upon these parameters. Good experimental confirmation is obtained with two tubes, one of which was of adjustable length. Pushing, pulling, stability, frequency gaps, and spurious oscillations are described and explained. Lack of oscillation at low-voltage end of tuning range of some tubes is tentatively assigned to a velocity-distribution effect.