NONLINEAR DISTORTION AND SUPPRESSION IN TRAVELING WAVE TUBES: INSIGHTS AND METHODS. John G. Wöhlbier

Transcription

1 NONLINEAR DISTORTION AND SUPPRESSION IN TRAVELING WAVE TUBES: INSIGHTS AND METHODS by John G. Wöhbier A dissertation submitted in partia fufiment of the requirements for the degree of Doctor of Phiosophy (Eectrica Engineering) at the UNIVERSITY OF WISCONSIN MADISON 23

2 c Copyright by John G. Wöhbier 23 A Rights Reserved

3 To Piper. I am so ucky to have you. i

4 ii ACKNOWLEDGMENTS Professor John Booske has provided exceent advising. His zea for research is infectious, his intuitions are inspiring, and his wiingness to take a chance is refreshing. Professor Ian Dobson has provided exceent advising. His taents for computations and exposition are superior, his genuine and honest nature is wecome, and his wiingness to take a chance is refreshing. Mark Converse, my cose graduate coeague for the ast four years, provided countess usefu discussions that heped to figure out just what it was that we were doing. I aso thank him for taking a chance on the code msuite by using it in his Ph.D. work. Aarti Singh has provided invauabe feedback in trying to appy my theories. I sincerey hope that her effort to earn how to do some of the cacuations pays off. Her babysitting was deepy appreciated by mysef, Piper, and especiay Eveyn. I thank my great friend Gen Luckjiff for putting the idea of going back to graduate schoo into my head, and for his encouragement of the pursuit of Mathematics. Finay, I am indebted to my ovey wife Piper. Her unsefish nature has aowed me to pour countess hours into competing my Ph.D. The gift of our daughter Eveyn, and hence the creation of our famiy, has made me one hundred-fod happier than any academic degree ever coud. Striking a baance between famiy time and work has been very difficut for me, and I can ony hope that I have done right by my famiy.

5 iii TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES vi vii ABSTRACT xiv 1 Introduction and background Introduction Literature review TWT modeing using Euerian coordinates TWT modeing using Lagrangian coordinates TWT transfer functions Harmonic injection in TWTs Steady state TWT modes Introduction TWT modes MUSE LATTE S-MUSE Numerica exampe Circuit power versus axia position Constant of the motion Eectron overtaking Choosing simuation frequencies Discussion Noninearities Dimension scaing of LATTE and MUSE Reation to method of coective variabes

6 iv Page 3 Anaysis of S-MUSE Introduction Series soution to S-MUSE Practica computation of (3.11) Computation of mode ampitudes Generation and growth rates of noninear distortions Introduction Theory Appications On the mechanisms of phase distortion in a TWT Introduction Phase distortion mechanisms Simuation resuts Anaytic resuts Ampitude-phase mode & S-MUSE Parametric dependence of phase distortion Insights into TWT inearization Concusions Anaytic formuas for equation (5.8) Series term x (1) Series term x (2) Series term x (3) Effect of average beam veocity in LATTE On the physics of harmonic injection in a TWT Introduction Theory Anaytic soution structure Eectron beam diagnostics Appications Fundamenta and harmonic frequencies Intermoduation frequencies Concusions Formuas for mode ampitudes

7 v Page First order (inear) soution Second order soution Third order soution Concusions APPENDICES Appendix A: Normaization, attenuation, and sowy varying enveopes in MUSE, S-MUSE, and LATTE Appendix B: Matched input impedance Appendix C: Induced norms Appendix D: Derivation of moda ampitudes Appendix E: Appication of MUSE and LATTE methods to kystron modeing. 146 LIST OF REFERENCES

8 vi LIST OF TABLES Tabe Page H Parameters (Constant Pitch Section) Simuation Frequencies and Dispersion Parameters Growth rates for two drive frequencies and noninear products up to order five for the bandwidth.8 9. GHz. Resuts for input powers of 3 db sat, 1 db sat and db sat are given. µ Chr is growth rate fit to Christine 1-d power versus axia distance data at an axia position in the sma-signa regime, i.e., after the power curves have reached their asymptotic exponentia growth state, but prior to saturation of any of the power curves (see Fig. 4.2). The percent difference coumns compare µ Chr to formua (4.9) using % diff. = µ Chr Eq. (4.9) /µ Chr Ku-band TWT eectron beam and circuit parameters Ku-band TWT dispersion parameters XWING TWT eectron beam and circuit parameters XWING TWT dispersion parameters XWING TWT eectron beam and circuit parameters XWING TWT dispersion parameters

9 vii LIST OF FIGURES Figure Page 2.1 Power versus axia distance for LATTE and MUSE db difference of drive frequencies from LATTE versus axia distance db difference of harmonics and sum frequency from LATTE versus axia distance db difference 3IMs from LATTE versus axia distance Constant of the motion versus axia distance Energy densities in circuit and beam versus axia distance for LATTE and MUSE. Energy density in space charge fied not shown Disk orbits versus axia distance Jacobian vs. ψ for severa axia positions Power versus axia distance for LATTE and MUSE Power versus axia distance for partiay inearized equations Input (a) and output (b) spectra of a simuation with 42 frequencies Circuit power versus distance for fourteen tones with P in = 3 db sat Power versus axia distance for 3 harmonicay reated drive frequencies predicted by Christine 1-d. 1 GHz is a second order product of 2 GHz and 3 GHz and exhibits first its α = 1 maximum growth rate, then its α = 2 maximum growth rate produced by 2 GHz and 3 GHz

10 viii Figure Page 5.1 AM/AM and AM/PM curves at 14 GHz generated by LATTE, MUSE, and S- MUSE for the TWT parameters in Tabes 5.1 and 5.2. The vertica ines at P in = 19.5 dbm and P in = 26.5 dbm correspond to 1 db gain compression (3.8 db backed off from saturation) and 1 db backed off from saturation respectivey, as predicted by LATTE. The simuations to generate the resuts accounted for circuit frequencies up to the third harmonic and eectron beam frequencies up to the tenth harmonic Output phase versus input power curves generated by MUSE simuations with varying frequencies incuded in the simuation. The egend indicates which frequencies were incuded in the simuation generating the trace. The maximum power represented on the graph corresponds to the 1 db gain compression point as seen in Fig For the input powers in this figure LATTE and MUSE have neary identica phase predictions accounting for dc through the tenth harmonic, as seen in Fig MUSE computations of the hot phase veocity (5.4) at the fundamenta frequency with varying frequencies incuded in the simuation. The egend indicates which frequencies were incuded in the simuation generating the trace. The input power P in = 19.5 dbm corresponds to the 1 db gain compression point shown in Fig For this input power the phase difference as predicted by LATTE is neary identica to MUSE when accounting for dc through the tenth harmonic Output phase for LATTE simuations with and without remova of the average beam veocity reduction as described in Section 5.8. One hundred space charge harmonics were used to compute v from (5.49) Average eectron beam veocities computed by LATTE and MUSE. LATTE traces were computed by (5.49), MUSE traces are the dc frequency of the veocity ṽ (z). Shown are computations with and without the veocity adjusted to remove the average change in the dc component. The input power used to generate the traces is P in = 23 dbm, which is the maximum power appearing in Fig Average eectron beam veocity computed by MUSE, and hot phase veocity at the fundamenta computed by LATTE and MUSE for P in = 2 dbm. The ranges of vaues on both axes are 3% of the vaue of the respective curve at z = 5 cm

11 ix Figure Page 5.7 Simuation and anaytic predictions of S-MUSE output phase. For the anaytic formua to match the simuation the contributions from the 5IM term need to be incuded. The maximum power represented on the graph corresponds to the 1 db gain compression point as seen in Fig The simuation accounts for circuit frequencies up to the third harmonic and eectron beam frequencies up to the tenth harmonic Anaytic and simuation predictions of S-MUSE hot phase veocity at the fundamenta frequency. Incusion of the 5IM contribution to the anaytic soution (5.8) is required to match the simuation resut. A of the compex exponentias from the inear portion of the soution are incuded to get the correct behavior of the hot phase veocity for z < 4 cm. The simuation incudes circuit frequencies up to the third harmonic and eectron beam frequencies up to the tenth harmonic Comparison of S-MUSE simuation and the AP mode output spectra for two input tones. The input power is P in = 3 dbm and the moduation frequency is ω m /2π = 1. MHz Comparison of S-MUSE simuation and the AP mode output spectra for two input tones. The input power is P in = 3 dbm and the moduation frequency is ω m /2π = 1. MHz Comparison of S-MUSE simuation and the AP mode output spectra for two input tones. The input power is P in = 23 dbm and the moduation frequency is ω m /2π = 1. MHz Comparison of S-MUSE simuation and the AP mode output spectra for two input tones. The input power is P in = 23 dbm and the moduation frequency is ω m /2π = 1. MHz AM/AM and AM/PM distortion for the Ku-band design at 14 GHz for five vaues of cod circuit phase veocity at the second harmonic. The egend represents the five vaues ranging from the minimum parameter vaue (min) to the maximum parameter vaue (max) AM/AM and AM/PM distortion for the Ku-band design at 14 GHz for five vaues of cod circuit interaction impedance at the second harmonic. The egend represents the five vaues ranging from the minimum parameter vaue (min) to the maximum parameter vaue (max)

12 x Figure Page 5.15 AM/AM and AM/PM distortion for the Ku-band design at 14 GHz for five vaues of space charge reduction factor at the second harmonic. The egend represents the five vaues ranging from the minimum parameter vaue (min) to the maximum parameter vaue (max) AM/AM and AM/PM distortion for the C-band design at 2 GHz for five vaues of cod circuit phase veocity at the second harmonic. The egend represents the five vaues ranging from the minimum parameter vaue (min) to the maximum parameter vaue (max) AM/AM and AM/PM distortion for the C-band design at 2 GHz for five vaues of cod circuit interaction impedance at the second harmonic. The egend represents the five vaues ranging from the minimum parameter vaue (min) to the maximum parameter vaue (max) AM/AM and AM/PM distortion for the C-band design at 2 GHz for five vaues of space charge reduction factor at the second harmonic. The egend represents the five vaues ranging from the minimum parameter vaue (min) to the maximum parameter vaue (max) Output phase versus input power for severa vaues of dc beam votage for the Ku-band TWT design. The range of the bias votages spans 48 V, ess than 1% of the design beam votage Sma signa gain of XWING TWT parameters as a function of frequency. Curve was computed with the S-MUSE mode Output power at (a) fundamenta (2 GHz) and (b) second harmonic (4 GHz) as a function of injected harmonic power P 2 () and injected harmonic phase ϕ 2 () for second harmonic injection. For both figures the fundamenta input power and phase are P 1 () = 5 dbm, ϕ 1 () =.. With no harmonic injection the harmonic is 9.23 db beow the fundamenta at the TWT output

13 xi Figure Page 6.3 Magnitude and phase of (6.1) and component magnitudes of (6.1) for second harmonic 4. GHz with second harmonic injection to achieve second harmonic canceation. The driven mode dominates the soution prior to z = 15 cm, and the noninear mode dominates the soution after z = 15 cm. This can be seen from the component magnitudes, as we as the 18 phase change of the tota soution at z = 15 cm. Fundamenta and second harmonic input powers and phases are 5., 8.86 dbm and., respectivey. Votage phase is with respect to the cod circuit wave at 4 GHz Fundamenta and harmonic output power versus (a) harmonic input phase and (b) harmonic input power for fundamenta input power which produces saturated output with no harmonic injection. Fundamenta input power and phase are 2. dbm and. respectivey for (a) and (b). In (a) harmonic input power is dbm, and in (b) harmonic input phase is Output power (a) and votage phase (b) for fundamenta through fourth harmonic with second harmonic injection to cance the second harmonic. The abrupt phase change of 18 in the second harmonic is evidence that even for saturated operation the second harmonic soution is comprised of two modes as in the approximate anaytic soution (6.1). Fundamenta input power and phase of 2. dbm and. produce saturation at z = 15 cm in absence of harmonic injection. Second harmonic input power and phase are dbm and respectivey. Votage phase is with respect to cod circuit wave at the respective frequency Beam current moduation (a) magnitude and (b) phase at fundamenta and second harmonic for second harmonic injection to cance the second harmonic at z = 15 cm predicted by LATTE. Harmonic beam current moduation changes modes at about z = 13 cm as evidenced by magnitude dip and phase change. However, the modes do not cance to produce zero beam current second harmonic moduation ( db) at any point aong the TWT. Fundamenta and second harmonic input power and phase are 2., 57.5 dbm and., 91. respectivey, we beow powers which produce saturation effects. Beam current moduation magnitudes are in db with respect to 1 A, and beam current moduation phases are with respect to cod circuit waves at the respective frequencies

14 xii Figure Page 6.7 Fundamenta and second harmonic output power versus second harmonic input power for harmonic input phase equa to (a) 47.9 and (b) In (a) harmonic input phase is set to minimize output harmonic for injected harmonic power of 15 dbm, and in (b) harmonic input phase is set to cance output harmonic for injected harmonic power of dbm. Fundamenta input power and phase of 13. dbm and. produce saturated output power of 54.1 dbm at the fundamenta and 4.6 dbm at the harmonic with no harmonic injection Output phasor picture produced by anaytic S-MUSE soution for second and third harmonic injection. Phasor A represents the second harmonic mode due to noninear product of fundamenta with itsef, phasor B represents the injected second harmonic mode, and phasor C represents the mode due to the noninear product of the third harmonic with the fundamenta. Phasor B + C cances phasor A Votage phase of the second harmonic with second and third harmonic injection for fundamenta input powers in the inear regime, output 3 db compressed, and output saturated. Phase is with respect to cod circuit phase veocity at 3. GHz. The second harmonic is canceed at z = 15 cm. A traces show change in phase at canceation point, but characters are different due to different reative inputs. Fundamenta, second, and third harmonic power and phase inputs are: inear 1., 13.3, 9.6 dbm, 2., 55., 15. ; 3 db compressed 23., 11.86, 1.25 dbm,., 15., 13. ; saturated 28., , dbm, 4., 55., Magnitude of (6.1) and component magnitudes of (6.1) for 3IM frequency 1.8 GHz with second harmonic injection to cance the 3IM frequency. The canceing mode (noninear difference product of 3.8 GHz and 2. GHz) dominates the soution prior to z = 15 cm, and the noninear mode dominates the soution after z = 15 cm. Fundamenta (1.9, 2. GHz) and second harmonic (3.8 GHz) input powers and phases are.,., dbm and., 3., respectivey Magnitude of (6.1) and component magnitudes of (6.1) for 3IM frequency 1.8 GHz with 3IM injection to cance the 3IM frequency. The driven mode dominates the soution prior to z = 15 cm, and the noninear mode dominates the soution after z = 15 cm. Fundamenta (1.9, 2. GHz) and 3IM (1.8 GHz) input powers and phases are.,., dbm and., 3., respectivey

15 xiii Figure Page 6.12 Output spectrum (a) near fundamentas and (b) near second harmonics with and without second harmonic injection when second harmonic is out of the inear gain bandwidth. Note additiona intermoduation frequencies (e.g., at 7.7 GHz) due to injection of the second harmonic. Fundamenta inputs 3.9, 4. GHz have input power 5. dbm and respective phases of. and 3.. Injected harmonic 7.8 GHz has input power and phase of 1.6 dbm and Note that there is aso partia suppression of the second harmonic at 7.8 GHz Output spectrum (a) near fundamentas and (b) near second harmonics with and without difference frequency injection. Note additiona intermoduation frequencies due to injection of the difference frequency. Fundamenta inputs 1.9, 2. GHz have input power. dbm and respective phases of. and 3.. The injected difference frequency 1. MHz has input power and phase of 17.4 dbm and Output spectrum (a) near fundamentas and (b) near second harmonics with and without 3IM and second harmonic injection. Note additiona intermoduation frequencies due to the injection of the signas. Fundamenta inputs 1.9, 2. GHz have input power. dbm and respective phases of. and 3.. The injected 3IM and second harmonic 2.1, 4. GHz have input powers and phases of 3., 22. dbm and 13., Output spectrum (a) near fundamentas and (b) near second harmonics with and without second harmonic injection of both fundamentas. Note additiona intermoduation frequencies due to the injection of the harmonic signas. Fundamenta inputs 1.9, 2. GHz have input power. dbm and respective phases of. and 3.. The injected harmonics 3.8, 4. GHz have input powers and phases of 23.25, 19.4 dbm and 66.5, Output spectrum (a) near fundamentas and (b) near second harmonics with and without injection of both 3IMs (1.8, 2.1 GHz) and both second harmonics (3.8, 4. GHz). Note additiona intermoduation frequencies due to injection the of the signas. Fundamenta inputs 1.9, 2. GHz have input power. dbm and respective phases of. and 3.. The injected 3IM and second harmonics 1.8, 2.1, 3.8, 4. GHz have input powers and phases of 26.7, 3., 25., 22. dbm and 15., 13., 14., 35.. The arge dynamic range of the figure is so that a of the spectra components are shown

16 xiv ABSTRACT Traveing Wave Tubes (TWTs) are microwave and miimeter wave ampifiers used in radar, sateite communications, and eectronic countermeasure appications. On a sateite, the TWT provides the fina communications signa boost before transmitting the signa back to earth. In eectronic countermeasures, TWTs boost signas that are sent out to deny detection by an enemy radar. The TWT, as with any ampifier, has a ess than idea behavior due to the ampifier noninearities. The non-ideaities typicay resut in decreased ampifier efficiencies or reduced bandwidth. Such compromising behavior can be extremey expensive where a 1% increase in efficiency of a TWT coud save $1,, over the operating ifetime of a communications sateite. In this dissertation new advances in noninear modeing of TWTs are presented. These advances incude new techniques for cacuating properties of noninear behavior, and new insights into the physica processes responsibe for the noninear distortions. In particuar, the physics of intermoduation distortion, phase distortion, and harmonic injection are studied in detai. The new ideas on intermoduation distortion and phase distortion presented in the thesis revise ong-standing assumptions about TWT noninearity, and shoud utimatey pay a roe in improving TWT designs through improved understanding. The new ideas and expanations of harmonic and signa injection may enabe new technoogies that woud increase efficiency, bandwidth, and inearity, and provide increased functionaity and higher data rates with arge cost savings for eectronic countermeasure and sateite communications markets.

17 1 Chapter 1 Introduction and background 1.1 Introduction Traveing Wave Tubes (TWTs) are microwave and miimeter wave ampifiers that are used extensivey in communications, radar, and eectronic countermeasure appications. They continue to find wide-spread appication due to their inherenty wide bandwidths and their high frequency, high power operating points. A compromising feature of the TWT is the device noninearity. The noninearity manifests as a saturating mechanism and spectra distortion; both of these effects imit TWT efficiency. For eectronic countermeasures, power produced in the harmonics of the fundamenta imits the output powers obtainabe at the fundamenta, and this imits the obtainabe efficiency at the fundamenta. For digita communications appications, the distortions increase bit error rates, and this imits data rates. Reducing noninear distortions in TWTs woud increase efficiency and bandwidths of eectronic countermeasure systems, and woud increase data rates and efficiency in digita communications appications. There are severa exampes of noninear effects that have ong pagued TWTs and have been the focus of much research, but are sti not competey understood in terms of physica mechanisms. Among these are harmonic injection, cross-moduation, and phase distortion. Moreover, due to increased sophistication of moduation techniques in digita communications, there are new probems associated with TWT noninearities which aso ack expanations and remedies. For exampe, there is no genera description based on TWT physics for how statistics of the input signa transate to statistics of the output signa. Since improved understanding inevitaby eads to improved techniques and designs, wireess communications systems and eectronic countermeasure systems certainy stand to benefit from improved expanations and insights into noninear TWT physics. The methods avaiabe to study noninear TWT physics incude experimentation, modeing, simuation, and anaysis; this thesis focuses on modeing, simuation, and anaysis. The two primary methods for modeing TWT behavior are physics based modes and generic input-output modes. There are a wide variety of avaiabe physics based modes. On one end of the spectrum are steady-state modes which assume a Fourier series form for the RF input and are usuay systems of noninear ordinary differentia equations. On the other end of the spectrum are eectromagnetic partice-in-ce (PIC) modes that invove soving eectron

18 2 beam equations and Maxwe s equations (partia differentia equations) in the time-domain. Modes in this spectrum can be 1-d, 2-d, 3-d, or some mixture, i.e., different dimensions for the eectron beam and eectromagnetic fieds. We have deveoped three new 1-d noninear steady-state TWT modes, the MUSE mode, the S-MUSE mode, and LATTE [82]. This suite of modes provides a foundation for simuation and anaysis that has ed to a new understanding of many aspects of noninear TWT physics. The LATTE mode captures the argest amount of noninear physics, incuding RF power saturation, and is ideay suited for simuation. By virtue of its construction, the MUSE mode fais to predict the same RF power saturation as LATTE, and is therefore not considered to be as accurate at LATTE. However, the MUSE mode has a mathematica structure that aows certain physica aspects of the device physics to be studied which cannot easiy be probed in LATTE or any other TWT mode past or present. The S-MUSE mode is derived from the MUSE mode by dropping certain noninear terms, and can be considered an approximation to the LATTE soutions appicabe prior to power saturation. However, the S-MUSE mode is anayticay sovabe and the structure of the soutions describes much about the underying noninear physics of the TWT. Therefore, athough the MUSE and S-MUSE modes sacrifice certain physica predictive capabiities and have a more imited range of use, they are advantageous in other respects incuding speed of computation, anaytic tractabiity, and access to physics not avaiabe to other modes. In this thesis we appy LATTE, MUSE, and S-MUSE, derived and compared with the arge signa code Christine 1-d [4, 5] in Chapter 2, to some important noninear probems in TWT physics. Christine 1-d predictions have been extensivey vaidated with experiments, and are therefore regarded as an acceptabe vaidation benchmark for the new modes. In Chapter 3 the anaytic soutions to S-MUSE are computed, and technica detais are discussed. In Chapter 4 a process for generation of harmonic and intermoduation distortions is given, aong with a formua to compute exponentia growth rates of the distortions. In Chapter 5 we study phase distortion in the TWT. Using the modes we offer a new view of the physica mechanisms of phase distortion that is counter to a decades od view, and appy the new insights to severa phase distortion reated probems. In Chapter 6 we study the theory and simuation of harmonic injection, and more generay signa injection, for shaping output spectra. The materia gives unique insights into many aspects of many different signa injection schemes. Such insights wi probaby pay a roe in the deveopment of new inearizer technoogies. Finay, in Chapter 7 we give an overview of the entire thesis and describe its impact on the fied of vacuum eectronics. 1.2 Literature review In this section we review the TWT iterature on the foowing topics: 1. physica TWT modeing using Euerian coordinates, 2. physica TWT modeing using Lagrangian coordinates,

19 3 3. TWT transfer functions, 4. harmonic injection in TWTs. Uness otherwise specified the TWT modes discussed beow are steady-state frequency domain modes, i.e., inputs may be written as a Fourier series TWT modeing using Euerian coordinates Linear Euerian theory The cassic inear TWT mode was pubished by J.R. Pierce in 1947 [6]. The anaysis uses a one-dimensiona transmission ine to represent the sow-wave circuit, and modes the eectron beam as a one dimensiona fuid. The fuid equations for the eectron beam are a Newton s Law reation and an equation of continuity, both expressed in Euerian coordinates. The theory, known as Pierce theory or sma-signa theory, is widey used and the detais have been presented in severa text books, for exampe [7, 39, 41, 48, 54, 61]. The inear theory is a singe frequency theory and does not predict saturation of the circuit fied Noninear Euerian theory Severa authors have modeed TWT noninearities using the Euerian eectron beam equations that Pierce inearized to get the sma-signa theory. Briouin [11] deveops a singe frequency noninear TWT theory based on the noninearized Pierce equations. He first re-derives the sma signa resuts. However, he incudes reations for energy density and energy fux, as we as forma inequaities for the imitations of the inear equations. He then considers arge ampitude regimes of the noninear equations. Casses of stabe waves, shock waves, and osciating waves of moderatey arge ampitude are considered. Putz [64] is the first author to attempt an extension of the sma signa theory to incude mutipe frequencies. The foowing is a summary of [64] from Curtice [19]: Putz s method of anaysis is to find, to a reasonabe approximation, the powerseries expansion for the eectric fied of the sow-wave structure. Starting with an anaysis of eectron bunching owing to an assumed heica fied, the fundamenta aternating current can be evauated and expressed in a power series. This power series, together with the reationship between beam current and eectric fied on a traveing-wave structure, eads to a power-series expansion for the eectric fied, the first term of which is merey the usua sma-signa resut. Severa successive approximations are used to obtain a sef-consistent soution which, however, ignores space-charge forces in the beam and any transverse motions of the eectrons. The anaysis of bunching is done with a singe-wave soution (the growing wave) for each signa. This approximation is accurate in tubes of high gain, but wi be

20 4 erroneous in cases where the other waves cannot be negected; e.g. it is not cear how to anayse a t.w.t. with a umped attenuator. However, if there is at east 2dB gain foowing the attenuator, the growing wave wi adequatey describe the circuit fied, and [intermoduation] and [cross moduation] effects wi resut principay from the noninear effects after the attenuator. If the gain after the attenuator is insufficient, new effects occur as Ober has shown. Sobo [73] aso attempts a mutifrequency anaysis. From E-Shandwiy [29]: Sobo used the same procedure [as DeGrasse] to derive four sets of equations (in addition he negected the second harmonic that was considered by DeGrasse), and gave numerica soution ony for the ongitudina beam parametric ampifier case with the pump frequency at twice the signa frequency. DeGrasse [26] uses a method simiar to Sobo [73]. Again from E-Shandwiy [29]: DeGrasse gave an anaysis of inear O-type ampifiers with two input signas.... Five sets of equations which describe the operation of the traveing-wave ampifier were derived. One set was for each input signa and one set for the difference, the sum and the second harmonic of one of the input signas. To sove the equations, DeGrasse assumed that both of the input signas propagate independenty according to Pierce s inear theory. Knowing the soution for the two input signas the other components coud be obtained. Other intermoduation components such as those at 2f 2 f 1 and 2f 1 f 2 which coud be important were not considered. Curtice [19] extends Putz [64] to incude space charge and arger vaues of the Pierce gain parameter C. A series expansion of the beam current gives a series expansion for the circuit fied, the second term of which contains the first-order noninear effects of t.w.t. operation. As in [64] the noninear term is normaized into a normaized distortion factor. This factor is reated to the cross-moduation factor, which is then reated to output ampitude and phase moduation, and intermoduation power ratio. The normaized distortion factor is computed versus TWT parameters. The anaysis however is imited to the weaky saturated condition. The group of Datta et a. have pubished a coection of artices using a third order noninear mode based on Euerian eectron beam equations. Foowing work by Paschke [57, 58, 59] Datta et a. use successive approximations to obtain anaytic soutions to the noninear system. Whie a 1-d anaysis, their mode incudes a reevant physics, athough noninearities above the third order are discarded [23]. However, they note that for their resuts to be fairy acceptabe [2] the beam pasma frequency must be much beow the operating frequency, referring to [59] for justification of this fact. In [2] they introduce the mode, sove it for a singe frequency, and compare their resuts to a Lagrangian mode for a particuar TWT. Interestingy, the Euerian mode demonstrates saturation effects for the TWT parameters presented. In [23] they provide a sighty more detaied derivation and

21 5 soution of their mode again for a singe frequency. In this artice they compare their mode to a Lagrangian mode for two different TWTs and estabish a regime of correctness. In [22] they again study their third-order mode, but now consider the generation of harmonics and their contro. The effects of dispersion parameters, circuit oss, and harmonic injection on the amount of harmonic content present in the output are studied. For the generation of a harmonic from a drive frequency they consider ony the second order noninearity, but the third-order noninearities are incuded in the computation of the fundamenta frequency. However, in the case of second harmonic injection they incude second and third order noninearities for both the fundamenta and harmonic signas. Reference [24] is simiar to [22] athough they do not treat harmonic injection. Lasty, in [21] they study harmonic injection to reduce third order intermoduation frequencies using the same formaism as their other papers. Finay, a paper containing the derivations of the Mutifrequency Spectra Euerian (MUSE) mode, S-MUSE mode, and a Lagrangian disk mode LATTE, a presented in this thesis, appears in [82]. It is worth noting that the MUSE mode is the ony exact steady-state Euerian mode in the iterature in that no approximations on the noninearities are made. As such, there does not seem to be an anaytic soution to the MUSE mode and numerica integration must be used to sove it. However, numerica integration of the MUSE mode seems to be easier to perform than impementation of the anaytic soutions given by Datta et a., especiay as the number of frequencies increases TWT modeing using Lagrangian coordinates The impetus for origina work using Lagrangian coordinates for the eectron beam was that eectrons overtake one another at or even consideraby before the point aong the tube where the imiting power eve is obtained, [56] in which case Euerian functions become muti-vaued. Codes using such formuations have been quite successfu in accuratey predicting TWT behavior. Since these modes must be numericay integrated, parametric studies of various physica phenomena are more chaenging and time consuming than anaogous parametric studies of an anaytic soution, even if the anaytic soution is for an approximate mode Singe frequency The origina paper using Lagrangian coordinates to mode the eectron beam was by Nordsieck [56]. This work ignores space charge effects and circuit oss. Severa authors foowed Nordsieck with simiar Lagrangian modes incuding more physics. From Tien [75]: Pouter [62] has extended Nordsieck equations to incude space charge, finite C and circuit oss, athough he has not perfecty taken into account the space charge and the backward wave. Recenty Tien, Waker, and Woontis [77] have pubished a sma C theory in which eectrons are considered in the form of

22 6 uniformy charged discs and the space charge fied is cacuated by computing the force exerted on one disc by the others. Resuts extended to finite C, have been reported by Rowe [66], and aso by Tien and Waker [76]. Rowe, using a space charge expression simiar to Pouter s, computed the space charge fied based on the eectron distribution in time instead of the distribution in space. This may ead to appreciabe error in his space charge term, athough its infuence on the fina resuts cannot be easiy evauated. Tien [75] takes the mode by Tien, Waker, and Woontis [77] and extends it to finite C. Additionay, a method for cacuating a backward wave contribution is provided and the effect of the backward wave is studied. This anaysis however ignores circuit oss. For a summary of these papers and those that foowed, the reader is referred to the book by Rowe [67]. Muti-dimensionaity of the eectron beam is aso covered in [67] Mutipe frequencies To extend the 1-d singe frequency modes to hande input signas with mutifrequency content, and the associated intermoduation frequencies due to the TWT noninearity, modes using Fourier series for circuit quantities were deveoped. Among the origina modes were E-Shandwiy [29] and Giaroa [4]. References [71, 78, 27] aso deveop such modes. The papers by Srivastava and Joshi [74] and Datta et a. [25] are simiar to the earier papers in how they treat the eectron beam, but each handes the circuit and space charge fieds differenty. More recenty, another coection of such modes have appeared in the iterature. The modes compute the RF quantities using eectromagnetic fied representations and Maxwe s equations rather than the equivaent circuit modes. Antonsen and Levush present a 1-d frequency domain mode [4, 5] which is extended to 3-d by Chernin et a. [17]. Freund et a. formuate 3-d modes in both the time domain [37, 36, 34, 35] and the frequency domain [32, 33]. The time domain modes are restricted to singe or mutifrequency sinusoids with a sowy varying enveope TWT transfer functions From the perspective of a system engineer, the TWT can be characterized by inputoutput transfer functions. Most commony these incude an AM/AM curve reating the output power to the input power, and an AM/PM curve reating the output phase to the input power. The character of these curves captures some of the noninearity of the TWT. Putz [65] considers noninear effects in TWTs based on anaysis of a physica mode, using graphica methods, and using a power series transfer function. In particuar, crossmoduation effects are considered with the output of the physica mode as we as rues of thumb; harmonic outputs are predicted with graphica methods; intermoduation products

23 7 are predicted using a mixture of empirica formuas and graphica methods. Harmonic injection is discussed noting that the harmonic and drive frequencies mix to produce a signa at the drive frequency that adds to the origina drive signa. By substituting a mutifrequency signa into a power series transfer characteristic, he derives expressions for the noninearities present in the output signas at the drive, harmonic, and intermoduation frequencies. Based on these formuas he discusses how the output votage at intermoduation frequencies depends on products of the input votages making the intermoduation frequencies. He aso discusses the presence of cross-moduation terms and how they indicate a inear proportionaity of the change in output votage (for sma input votages) to interfering signa power. He then compares the power series outputs to the equation derived from the physica mode (presumaby from [64]) and notes how the forms agree. Lasty, based on the physicay derived equation, he caims that noninear effects wi be east for TWTs with highest vaues of C and owest vaues of circuit oss. Saeh [69] proposes empirica frequency independent ampitude-phase and quadrature modes as we as a frequency dependent quadrature mode. For mutipe phase moduated carriers he soves the frequency-independent quadrature mode for intermoduation frequencies. The two carrier case gives a compete anaytic time domain soution whie the many carrier soution contains an integra that must be evauated numericay. He proposes the frequency-dependent quadrature mode as an extension to the frequency-independent mode, but does no studies using the mode. The introductory paragraphs of [69] give a nice summary of ampitude-phase and quadrature modes prior to [69]. Abuema atti [3] gives a frequency-dependent quadrature mode for the TWT basing it on a frequency-independent quadrature mode, and gives an exampe of its use in computing intermoduation products of two frequencies. Guida [42, 43] provides methods for computing intermoduation power for given input signas using specific TWT input-output modes. Reference [42] summarizes previous methods for doing such cacuations. Severa authors study the physica mechanisms behind the AM/AM and AM/PM transfer curves with arge signa codes. The AM/AM mechanism is due to the saturation of the device, whie the AM/PM mechanism is shown to be reated to decrease of eectron veocities and the phase of the eectron bunch with respect to the votage wave. Ezura and Kano [3] study the dependence of the AM/PM curves on TWT parameters, ampitudes of circuit waves, and average eectron veocities theoreticay and experimentay. Hirata and Kanai [46] and Hirata [45] study the reation of AM/PM conversion to the generation of intermoduation products. It is found that the in-phase component of the fundamenta charge density moduation with respect to the circuit votage dominates the generation of intermoduation products. Carter et a. [12] deveoped a code for predicting intermoduation distortions based on the singe carrier transfer characteristics of a TWT. Athough detais of their input-output mode are not given, it is frequency-independent in that it assumes TWT characteristics do not vary appreciaby over the band of frequencies under consideration. They compare this mode to the output of a mutifrequency arge signa TWT mode for the case of two carriers, three carriers, and eight carriers. To achieve the effect of uncorreated carriers, severa runs of the arge signa code with random initia phases are averaged. The resuts

24 8 are generay favorabe and a differences are attributed to the frequency-independence of the input-output mode Harmonic injection in TWTs By injecting harmonics of drive signas the TWT output spectrum can be advantageousy shaped. In the case of singe frequency drive, harmonic injection can suppress the output harmonic and boost the fundamenta reative to the harmonic. In the case of mutifrequency drive, harmonic injection can reduce the intermoduation spectrum Singe drive frequency Mende [55] states that enhancement by injecting harmonics... was discovered quite some time ago when it was observed that the wrong type of second-harmonic input woud seriousy degrade the power output at the fundamenta frequency.... This process is is one of canceation, whereby the injected second-harmonic signa is such that it is 18 out of phase with the second harmonic signa generated by the noninear processes inherent in the interaction mechanism. Eary artices discussing harmonic injection mechanisms and hardware impementations incude [65, 44, 38]. We have provided a refinement to the mechanistic picture given in these references. Later, Datta et a. [22] give numerica studies of harmonic injection where they compare predictions of Euerian and Lagrangian modes Mutipe drive frequencies Sauseng et a. [7] show that intermoduation distortions can be reduced with harmonic injection by 6 db to 18 db beow the fundamentas. More recenty, Wirth et a. [79] have done simiar experiments in which they were abe to attain 24 db of suppression. Datta et a. [21] have done simuations demonstrating this phenomenon.

25 9 Chapter 2 Steady state TWT modes 2.1 Introduction In this chapter we derive a new 1-d noninear mutifrequency Euerian TWT mode, the MUSE (Mutifrequency Spectra Euerian) mode. We aso derive a Lagrangian disk mode LATTE (Lagrangian TWT Equations) from the same initia equations for comparison purposes as we as to demonstrate the theoretica reation between MUSE and a disk mode. A simpified MUSE mode S-MUSE more suitabe for anaysis is aso derived. These three modes are compared to each other and Christine 1-d for a set of TWT parameters which are based on the Hughes (now Boeing) 8537H L-band TWT design. The comparison to Christine 1-d is particuary usefu since this code is widey known and used, and it has been vaidated against experiment for the Hughes TWT [2, 68]. The noninearities of MUSE and LATTE are compared, and an exampe of how MUSE can examine fundamenta distortion mechanisms is provided. We aso study how the dimensions of the MUSE mode and LATTE scae with number of frequencies, an important issue for assessing the use of MUSE as a numerica too. Resuts from a simuation with 42 frequencies are provided. Lasty, we discuss the reation of the MUSE mode to the method of coective variabes in free eectron aser theory [1]. Section 2.2 presents the modes to be considered. We derive the MUSE mode and discuss its numerica soution, derive the disk mode LATTE, and derive the S-MUSE mode. The modes are compared to each other and Christine 1-d in Section 2.3. In particuar we ook at circuit power versus axia distance, a constant of the motion and the issue of eectron overtaking. Section 2.4 discusses the noninearities in MUSE, the dimensiona dependence of MUSE and LATTE on simuation parameters, and the reation of MUSE to the coective variabe theory of free eectron asers.

26 1 2.2 TWT modes MUSE Derivation For the MUSE mode the heix is modeed as a ossess transmission ine 1 and Euerian equations are used for the eectron beam. In particuar, the time domain mode equations are V z I z E z v t + v v z ρ t + v ρ z = h 1 I t = h 2 V t A ρ t (2.1) (2.2) = ρ ɛ (2.3) = e m e h 1 I t + e m e R E (2.4) = ρ v z. (2.5) where z is axia distance, t is time, V is transmission ine votage, I is transmission ine current, E is the space charge eectric fied, v is eectron beam veocity, and ρ is the voume charge density of an eectron beam with cross sectiona area A. The denotes convoution and this aows for frequency dependence of circuit and beam parameters. The functions h 1, h 2 and R are the inverse Fourier transforms { } K(z, h 1 (z, t) = F 1 f ω ) (2.6) ṽ ph (z, f ω ) { } h 2 (z, t) = F 1 1 (2.7) K(z, f ω )ṽ ph (z, f ω ) { } R(z, t) = F 1 R(z, f ω ) (2.8) where the functions K(z, f ω ), ṽ ph (z, f ω ) and R(z, f ω ) are frequency domain circuit interaction impedance [39], cod circuit phase veocity, and space charge reduction factor [48] respectivey. The inverse transforms are aperiodic functions of t and are functions of z to aow for spatia variation of circuit parameters. In the remainder of the thesis notation of the z dependence is suppressed. The constants e, m e, and ɛ are eectron charge, eectron mass, and permittivity of free space respectivey. 1 See Appendix A for formuations of the modes with circuit oss incuded.

27 11 Then For reasons that wi be made cear ater we first make the coordinate transformation [ ] [ ] [ ] z 1 z = ω ψ. (2.9) u ω t V z I z E z v v z v ρ z = ω V u ψ ω h 1 I ψ = ω h 2 V ψ ω I u ψ + Aω = ω u E = ω e m e h 1 I ρ ψ (2.1) (2.11) ψ + ρ (2.12) ɛ ψ + e ( R E + ω 1 v ) v (2.13) m e u ψ = ω ( 1 v u ) ρ ψ ρ ( v z + ω u v ψ ). (2.14) We assume a inputs to the system (signas at z = ) are periodic in t with fundamenta frequency ω. This impies that soutions as functions of (z, t) are periodic in t with fundamenta period 2π ω and that soutions as functions of (z, ψ) are periodic in ψ with fundamenta period 2π. For a function x(z, ψ) periodic in ψ we use the Fourier series reations x(z, ψ) = = x (z) = 1 2π x (z)e if ψ 2π (2.15) x(z, ψ)e if ψ dψ (2.16) where the f are integers indexed by. The set of frequencies {f } is chosen to be the frequencies with nonzero Fourier coefficients, thus {f } is the drive frequencies together with the frequencies produced from noninear interactions. We index the frequencies so that f = f and f m > f n for m > n. Since our functions are rea vaued, x = x. (2.17) Computing Fourier coefficients of (2.1) (2.14) gives the MUSE mode: dṽ dz dĩ dz dẽ dz = if ω Ṽ if ω K(f ω ) Ĩ (2.18) u ṽ ph (f ω ) = if ω Ṽ if ω Ĩ + if ω A ρ (2.19) K(f ω )ṽ ph (f ω ) u = if ω Ẽ + ρ (2.2) u ɛ

28 12 ṽ m m,n fm+fn=f ṽ m m,n fm+fn=f dṽ n dz d ρ n dz = if ω e K(f ω ) m e ṽ ph (f ω ) Ĩ + e m e R(f ω )Ẽ + if ω ṽ m,n fm+fn=f = if ω ρ if ω u if n ω u ṽ m ṽ n (2.21) ṽ m ρ n m,n fm+fn=f m,n fm+fn=f dṽ m dz ρ n (2.22) where. We have used that for x(z, ψ) and y(z, ψ) periodic, mutipication becomes convoution: x(z, ψ)y(z, ψ) x m (z)ỹ n (z). (2.23) F m,n f =fm+fn The summation notation shoud be read as sum over integers m and n such that f m + f n = f Method of numerica soution For practica impementation one negects higher frequencies and imits to M M. Then the MUSE mode has 5(2M + 1) compex equations. During integration of the MUSE mode one needs to sove (2.21) and (2.22) for the derivatives dṽ and d ρ dz dz.2 Equations (2.21) and (2.22) for M M are the inear systems Sw v = b v (2.24) Sw ρ = b ρ (2.25) where w v, w ρ, b v, and b ρ are 2M + 1 vectors and S is a (2M + 1) (2M + 1) matrix. The th entries of w v and w ρ are dṽ and d ρ respectivey, the dz dz th entries of b v and b ρ are equa to the right hand sides of (2.21) and (2.22) respectivey, and the th row and n th coumn entry of S is ṽ m where f m + f n = f. We choose the reation between the initia vaue of the circuit current and the initia vaue of the circuit votage as Ĩ () = Ṽ() K(f ω ). (2.26) 2 For better numerica performance one shoud sove a normaized system of equations. See Appendix A for normaized versions of the TWT modes.