ieee transactions on electron devices, vol. 62, no. 12 ...pz/twt15.pdf · 4286 ieee transactions on...

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 62, NO. 12, DECEMBER 2015 4285

Harmonic Content in the Beam Current in aTraveling-Wave Tube

C. F. Dong, Peng Zhang, Member, IEEE, David Chernin, Y. Y. Lau, Fellow, IEEE, Brad W. Hoff, Member, IEEE,D. H. Simon, Patrick Wong, Geoffrey B. Greening, and Ronald M. Gilgenbach, Life Fellow, IEEE

Abstract— In a klystron, charge overtaking of electrons leadsto an infinity of ac current on the electron beam. This paperextends the klystron theory of orbital bunching to a traveling-wave tube (TWT). We calculate the harmonic content of thebeam current in a TWT that results from an input signal of asingle frequency. We assume that the electron orbits are governedby Pierce’s classical three-wave, linear theory. The crowding ofthese linear orbits may lead to charge overtaking and, therefore,harmonic generation on the beam current, as in a klystron.We analytically calculate the buildup of harmonic content asa function of tube length from the input, and compare theresults with the CHRISTINE code. Good agreement is found.Also found is the surprisingly high level of harmonic contents inthe electron beam current, even when the TWT operates in thesmall signal regime. A dimensionless bunching parameter for aTWT, X = (2 Pin/(PbC))1/2, is identified, which characterizesthe harmonic content in the ac beam current, where Pin is theinput power of the signal, Pb is the dc beam power, and C isPierce’s gain parameter.

Index Terms— Current modulation, frequency multiplier,harmonic generation, traveling-wave tube (TWT).

I. INTRODUCTION

IN A traveling-wave tube (TWT), it is recognized that thelinear theory of Pierce provides an adequate description ofthe electron-circuit interaction over approximately 85 % of thetube length, even when the TWT is driven to saturation [1].Because Pierce’s theory is in the linear (small signal) regime,we are not aware of any analytic formulation that calculatesthe harmonic content in the ac current that would buildupover this 85% of tube length. This paper provides such aformulation.

Manuscript received August 31, 2015; revised October 9, 2015; acceptedOctober 12, 2015. Date of current version November 20, 2015. This workwas supported in part by the Air Force Office of Scientific Research underGrant FA9550-15-1-0097, in part by the Office of Naval Research underGrant N00014-13-1-0566, in part by the L-3 Communications ElectronDevice Division, and in part by the Air Force Research Laboratory underGrant FA9451-14-1-0374. The review of this paper was arranged by EditorM. Thumm. (Corresponding author: Y. Y. Lau.)

C. F. Dong is with the Department of Climate and Space Sciences andEngineering, University of Michigan, Ann Arbor, MI 48109 USA (e-mail:[email protected]).

P. Zhang, Y. Y. Lau, D. H. Simon, P. Wong, G. B. Greening, andR. M. Gilgenbach are with the Department of Nuclear Engineering andRadiological Sciences, University of Michigan, Ann Arbor, MI 48109 USA(e-mail: [email protected]; [email protected]; [email protected];[email protected]; [email protected]; [email protected]).

D. Chernin is with Leidos Corporation, Reston, VA 20190 USA (e-mail:[email protected]).

B. W. Hoff is with the Air Force Research Laboratory, Kirkland Air ForceBase, Albuquerque, NM 87117 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2015.2490584

The generation of harmonics in the small signal regimeseems contradictory at first sight. Our experience with klystrontheory [2], however, demonstrates that harmonic generationdoes indeed occur in the small signal regime [3], [4], and thatthe amplitudes and phases of the harmonic currents can becalculated accurately [4], [5]. Consider a klystron operatingin the low current, small signal, and ballistic regime driven byan input signal of frequency, ω0. An electron, A, leaves theinput gap with velocity vA at time tA. Another electron, B ,leaves the input gap with velocity vB at a later time tB .If vB > vA, electron B will catch up with electron A at somedownstream location and at some later time. At that particularposition, and at that instant of time, the ac current is infinite,because charge overtaking occurs [2]. This means that the accurrent necessarily contains an infinite number of harmonicfrequencies in its Fourier representation, despite the fact thatelectrons A and B are initially subjected to a small signalvelocity modulation (and zero electric field within the drifttube if one ignores the space charge effect as in the ballisticregime) [2]. This charge overtaking, and the resultant harmoniccontent at ω = nω0, n = 1, 2, 3, . . ., can be calculatedexactly and analytically. The key to this success is that, oncethe (1-D) electron orbit is known, the nonlinear current thatresults from the crowding of the electron orbits can be cal-culated exactly using charge conservation [5]. In effect, wesolve the linearized force law to obtain the electron motion, butsolve the nonlinear continuity equation exactly for the densityand current that results from the crowding of these orbits.If we include ac space charge effects within the drift tube,electrons A and B will be acted upon by the small signal acelectric field associated with the fast and slow space chargewave. Again, we solve the linearized force law to obtain theelectron motion, but solve the nonlinear continuity equationexactly for the density and current that results. This procedurewas verified to be valid [3] by comparing the analytic theorywith particle simulation at the fundamental frequency, from thehighly nonlinear, ballistic regime to the linear, space-chargedominated regime in a klystron. The fact that all harmoniccontents are correctly accounted for in such a formulation,even when charge overtaking occurs, is demonstrated explicitlywith examples [5].

Returning to the buildup of harmonic ac current in aTWT, we adopt the analogous treatment as in the klystron.We assume that the electron orbit can be described by Pierce’sclassical three-wave theory of TWT [1]. These three wavesare the forward propagating circuit wave and the fast and slowspace charge wave. They are waves in the small signal regime,

0018-9383 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

4286 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 62, NO. 12, DECEMBER 2015

therefore, containing only the fundamental frequency, ω0, ofthe input signal. These three waves in the TWT are entirelyanalogous to the two waves (fast and slow space charge waves)in the klystron drift tube. Despite the spatial amplification ofone of the three waves in the TWT, we assume that a lineardescription of these three waves suffices. From the linearizedelectron orbits constructed from these three waves, we solvethe nonlinear continuity equation (charge conservation law)exactly. This gives the harmonic content in the beam currentthat can build up kinematically, which we find to be quitesignificant even in the small signal regime.

For the TWT model, we consider a nonrelativistic,monoenergetic electron beam. It is subjected to an ac electricfield, E10e jω0t , at the input. At ω = ω0, we assume thatPierce’s dimensionless parameters [1], C , b, QC, and d ,are known constants. The launching loss is accounted forin the evolution of the three waves [1], [2]. We expressthe electron orbits as a linear combination of these threelinear waves. We then calculate the current modulation thatresults from the crowding in such orbits, using the Lagrangiandescription. If charge overtaking occurs among these orbits,our formulation automatically and completely accounts forit [5]. The results from our analytic formulation are comparedwith those obtained from the 1-D large signal TWT simulationcode, CHRISTINE [6], for a high perveance, C-band TWTexample. Excellent agreement is found for two values of inputpower.

II. FORMULATION

Consider a monoenergetic, nonrelativistic electron beamof drift velocity v0, carrying a dc beam current I0 andconfined by an infinite axial magnetic field. The beam interactswith a TWT slow-wave structure. To calculate the currentmodulation including the harmonic content that results froman input signal, we follow the conventional klystron theory andconsider an electron that arrives at the input (located at z = 0)at time t = t0. The unperturbed trajectory of this electron, inthe absence of an input signal, is given by

z = z0(t, t0) = v0(t − t0). (1)In the presence of a finite input signal at frequency ω0,we write z = z0(t, t0) + z1(t, t0), where the perturbationdisplacement z1 may be written as

z1(t, t0) = Re[Z1(t, t0)] (2)whose complex amplitude Z1 evolves according to Pierce’sthree-wave small signal theory [7]

d3 Z1dt3

+ jω0C(b − jd)d2Z1dt2

+ 4QC3ω20d Z1dt

+ jω30C3[4QC(b − jd) + (1 + Cb)2]Z1 = 0. (3)In (3), C , b, Q, and d are, respectively, Pierce’s dimensionlessparameters [1] that characterize the TWT gain, detuning,space charge effect, and cold-tube circuit loss, respectively.

The initial conditions to (3) are

Z1(t = t0) = 0 (4a)Ż1(t = t0) = 0 (4b)Z̈1(t = t0) = − e

mE10e

jω0t0 . (4c)

Equations (4a) and (4b) state that there is no initial displace-ment or initial velocity perturbation of this electron when itdeparts the input at z = 0 at time t = t0. Equation (4c) statesthat the electron is acted upon by the ac input electric fieldof amplitude E10 and frequency ω0, and e is the magnitudeof the electron charge. Equations (1)–(4) are written in theLagrangian variables (t , t0). In the following, we transformthe solution to the Eulerian variables (z, t) to recover Pierce’sclassical three-wave solution that includes the launchingloss.

The solution to the third-order ordinary differentialequation (3) subject to the initial condition (4) is

Z1(t, t0) = − eE10mω20C

2e jω0t0 ×[α1eCω0δ1(t−t0) + α2eCω0δ2(t−t0)

+ α3eCω0δ3(t−t0)] (5)where δ1, δ2, and δ3 are the three complex roots to the cubiccharacteristic equation

δ3+ j (b − jd)δ2 + 4QCδ + j [4QC(b − jd) + (1+ Cb)2] = 0(6a)

or

(δ2 + 4QC)(δ + jb + d) = − j (1 + Cb)2 (6b)and the mode amplitudes α1, α2, and α3 are obtained from thesolution to

α1 + α2 + α3 = 0 (7a)α1δ1 + α2δ2 + α3δ3 = 0 (7b)α1δ

21 + α2δ22 + α3δ23 = 1. (7c)

Note that (6b) is simply Pierce’s TWT dispersion relation [1].One may verify that v1(z, t), the linearized electron velocityperturbation in the Eulerian description in Pierce’s three-wavetheory, is given by ∂ Z1(t, t0)/∂ t , evaluated at t0 = t − z/v0.Note further that, from (7), α1, α2, and α3 depend onlyon δ1, δ2, and δ3, which depend only on the dimensionlessPierce parameters C , b, Q, and d according to (6). Thesolution (5) accounts for the launching loss for the amplifyingwave [1], [2], since the input wave is shared by the three (3)waves, as in Pierce’s theory [see (7)].

The electron arrives at the downstream position z = L attime t , given by L = v0(t − t0) + z1(t, t0), which yields thefollowing implicit relationship between the departure time (t0)and the arrival time (t):

ω0(t − t0) = ω0 Lv0

− ω0z1(t, t0)v0

≡ ω0 Lv0

+ XRe[e jω0t0 R(t − t0)] (8)

DONG et al.: HARMONIC CONTENT IN THE BEAM CURRENT IN A TWT 4287

Fig. 1. Evolution of RF power on the circuit wave for 1-mW drive withC = 0.1194 and d = 0. The three curves assume, from top to bottom,(b, QC) = (1, 0), (0, 0), and (0, 0.296). Excellent agreement between theanalytic theory and CHRISTINE results is found.

where

X = eE10mω0v0C2

(9)

R(t − t0) = α1eCω0δ1(t−t0) + α2eCω0δ2(t−t0) + α3eCω0δ3(t−t0).(10)

In deriving (8), we have used (2) and (5). Note that X isthe dimensionless bunching parameter which measures thestrength of the input electric field. It may be expressed interms of the input power, Pin, and the dc beam power,Pb = I0Vb = I0(mv20/2e), upon using Pierce’s couplingimpedance K , and its relation to C3

K = E210

2Pink2= E

210

2Pin(ω0/v0)2(11)

C3 = K I04Vb

. (12)

Equation (9) then yields the bunching parameter X in variousforms

X = eE10mω0v0C2

= 1C2

(vwv0

)=

√2

C

(PinPb

)(13)

where vw = eE10/mω0 is the characteristic electron wigglingvelocity associated with the ac electric field at the input. Thelast form of (13) is particularly convenient.

The arrival time t may be explicitly solved in terms of thedeparture time t0 to any desired order of accuracy [4], in apower series in X . To the zeroth (lowest) order, we have,from (8)

ω0(t(0) − t0) = ω0 L

v0. (14)

To the kth order, we have [4]

ω0(t(k) − t0) = ω0 L

v0+ XRe[e jω0t0 R(t(k−1) − t0)]

k = 1, 2, 3, . . . (15)

Fig. 2. Harmonic content as a function of z, in log scale, for 1-mW drive.From top to bottom: (b, QC) = (0, 0), (1, 0), and (0, 0.296).

We found that taking k = 4 suffices in all of our numericalcomputation. Henceforth, we assume that the arrival time, t , isan explicit function of the departure time, t0. Note from (15)that (t − t0) is a periodic function of t0 with period 2π/ω0,irrespective of the order k.

Having obtained the arrival time t (at z = L) of anelectron, whose departure time (at z = 0) is t0, we may


Fig. 3. Harmonic content at z = 4 cm for 1-mW drive. From top to bottom:(b, QC) = (0, 0), (1, 0), and (0, 0.296).

calculate total current at z = L, denoted as IL (t), by chargeconservation [2], [5]

IL(t)dt = I0dt0. (16)Since IL (t) is a periodic function of t of period 2π/ω0, wemay also represent it in terms of a Fourier series

IL(t) =∞∑

n=−∞Ĩne

jnω0t . (17)

Fig. 4. Evolution of RF power on the circuit wave for 54-mW drive withC = 0.1194 and d = 0. The three curves assume, from top to bottom,(b, QC) = (1, 0), (0, 0), and (0, 0.296). Excellent agreement between theanalytic theory and the CHRISTINE results is found.

The ac current at the nth harmonic is then given by, uponusing (16) and (17)

Ĩn = ω02π

∫ 2π/ω00

dt IL (t)e− j nω0t

= I02π

∫ 2π0

d(ω0t0)e− j nω0t0− j nω0(t−t0) (18)

in which ω0(t − t0) in the last expression may be calculatedfrom (15) to any desired order k (k = 4 in our numericalcalculations) [4], [5]. It is easy to show from (18) thatĨ−n = Ĩ ∗n for all integer n, where the asterisk denotes thecomplex conjugate. Equation (17) then gives the total currentat z = L at time t

IL (t) ≡ I (L, t) = I0 + Re∞∑

n=1Ine

jnω0t (19)

where In = 2 Ĩn is the complex amplitude of the ac current atthe nth harmonic, and Ĩn is given by (18).

Finally, the evolution of the small signal electricfield, E1(z, t), on the circuit may be deduced from the lin-earized force law, written in accordance with Pierce’s theoryof TWT [7]

d2 Z1dt2

+ ω204QC3 Z1 = −e

mE1(z, t) (20)

where the right-hand side (RHS) represents the force on theelectron by the operating circuit mode, and the middle term(proportional to Q) represents the force due to all residualmodes. This term, proportional to Q, is said to representthe ac space charge effect in the TWT literature [1], [2].Since E1(0, t) = E10e jω0t , we obtain from (20), upon


Fig. 5. (a) Harmonic content as a function of z, in linear scale, for 54-mW drive. From top to bottom: (b, QC) = (0, 0), (1, 0), and (0, 0.296). (b) Harmoniccontent as a function of z, in logarithmic scale, for 54-mW drive. From top to bottom: (b, QC) = (0, 0), (1, 0), and (0, 0.296).

using (5) and (14)

∣∣∣∣ E1(z, t)E1(0, t)∣∣∣∣2

=∣∣∣∣∣

3∑i=1

αiδ2i e

Cδi (ω0z/v0)+ 4QC3∑

i=1αi e

Cδi (ω0z/v0)

∣∣∣∣∣2

(21)

which expresses the RF power gain of the circuit wave as itpropagates along the tube, according to the small signal theory.

III. EXAMPLE

Consider a C-band TWT operating at ω0 = 2π × 4.5 GHz,Vb = 2.776 kV, I0 = 0.17 A, C = 0.1194, K = 111.2 ohm,


Fig. 6. Harmonic content at z = 4 cm for 54-mW drive. From top to bottom:(b, QC) = (0, 0), (1, 0), and (0, 0.296).

v0 = 5.93 × 107 m/s, Pb = Vb I0 = 417.9 W,and I0/V

3/2b = 1.16 microperveance. Note that this value

of C = 0.1194 is very high. We consider the evolution ofthe current modulation and of the RF electric field up toz = L = 5 cm. We assume that there is no cold-tube loss,so that d = 0. The remaining parameters are b, QC, andthe input power Pin. We shall consider two values of inputpower: 1) Pin = 1 mW and 2) Pin = 54 mW, corresponding

to X = 0.006331 and 0.04652, respectively. The tube isoperating entirely (0 < z < L) in the small signal regimefor Pin = 1 mW. Pin = 54 mW is the value of input powerthat produces a 1 dB compression (reduction) of the gain inthe case QC = 0, and so the TWT may be considered to beoperating in the slightly nonlinear regime at z = L in thiscase.

A. Pin = 1 mWIn this case, (13) gives X = 0.006331. The tube operates in

the strictly linear regime for z < 5 cm. The evolution of the RFpower for (b, QC) = (0, 0), (1, 0), and (0, 0.296) accordingto (21) is shown in Fig. 1 (solid lines). For comparison,the CHRISTINE [6] results are shown by the dotted lines.Excellent agreement between the analytic theory and theCHRISTINE is found for all z up to 5 cm. The evolution of theharmonic contents for these three cases of (b, QC) is shownin Fig. 2. Fig. 3 shows negligible harmonic content, |In/I0|at z = 4 cm, for this very low drive case, as expected. Thedetailed agreement shown in Figs. 2 and 3, even for very lowlevels of harmonic ac current, may be considered as validationof both the analytic theory and the CHRISTINE code.

B. Pin = 54 mWIn this case, X = 0.04652. The evolution of the RF power

for (b, QC) = (0, 0), (1, 0), and (0, 0.296) accordingto (21) is shown in Fig. 4 (solid lines). For comparison, theCHRISTINE simulation results are shown by the dotted lines.Excellent agreement between (21) and CHRISTINE is noted.Since CHRISTINE is a nonlinear code, Fig. 4 suggested thatthe linear theory applies up to z = 4.5 cm. The evolution of theharmonic contents for these three cases of (b, QC) is shownin Fig. 5(a) in linear scale, and in Fig. 5(b) in logarithmicscale. It is seen that the harmonic content in the ac currentis quite significant even when the TWT operates in the linearregime. These harmonic ac currents are only due to kinematicbunching (i.e., orbital crowding) in the electron orbits, andis predicted reasonably well by the analytic theory for zup to 4 cm. The harmonic content relative to the dc current,|In/I0| at z = 4 cm, is shown in Fig. 6. Fig. 6 clearlydemonstrates that even when the linear theory applies, theharmonic content is quite sizable, and that it can be fairlypredicted by our analytic theory.

IV. CONCLUDING REMARKS

By considering the crowding of the linearized electronorbits, we compute the harmonic content in the ac currentthat accompanies these orbits, as a result of charge conser-vation. Therefore, only the kinematic bunching is considered.Surprisingly high levels of harmonic ac current on the electronbeam were found, even when the tube operates in the smallsignal regime. The theory is applicable even if charge overtak-ing has occurred. The approach is entirely analogous to thatestablished for electron bunching in the klystron theory. Ourresults reduce to Pierce’s classical three-wave theory, includingthe effects of the launching loss, for the fundamental frequencycomponent. The harmonic content evaluated from our analytic


theory is in good agreement with the CHRISTINE simulationresults.

In our theory, we only calculated the generation of harmonicac currents on the electron beam. All harmonic componentsin the ac current are due only to the input wave of a singlefrequency, ω0. These harmonic ac currents do not have anyintrinsic gain mechanism. This is not necessarily true in awideband TWT, where the second harmonic, ω = 2ω0,may also be within the amplification band. Should this bethe case, the harmonic ac current calculated here may thenbe considered as a seed to the second harmonic generation(without any 2ω0 input). Note that the amplitude and phaseof this 2ω0 output are completely controlled by the inputsignal, which is at the fundamental frequency ω0. This secondharmonic may also be suppressed by injecting a 2ω0 signalwith an appropriate amplitude and phase [8]. The new insight,together with higher harmonic generation, will be explored infuture study.

Our formulation has been restricted to a nonrelativisticelectron beam. For a relativistic or even mildly relativisticbeam, care must be exercised in the formulation of thelinearized force law, in the construction of Pierce’s parame-ters, C and QC, and in the relation between the couplingimpedance K and C3.

REFERENCES

[1] J. R. Pierce, Traveling Wave Tubes. New York, NY, USA: Van Nostrand,1950.

[2] G. W. Gewartowski and H. A. Watson, Principles of Electron Tubes.Princeton, NJ, USA: Van Nostrand, 1966.

[3] M. Friedman, J. Krall, Y. Y. Lau, and V. Serlin, “Externally modulatedintense relativistic electron beams,” J. Appl. Phys., vol. 64, no. 7,pp. 3353–3379, Oct. 1988.

[4] Y. Y. Lau, D. P. Chernin, C. Wilsen, and R. M. Gilgenbach, “Theory ofintermodulation in a klystron,” IEEE Trans. Plasma Sci., vol. 28, no. 3,pp. 959–970, Jun. 2000.

[5] C. B. Wilsen, Y. Y. Lau, D. P. Chernin, and R. M. Gilgenbach, “A note oncurrent modulation from nonlinear electron orbits,” IEEE Trans. PlasmaSci., vol. 30, no. 3, pp. 1176–1178, Jun. 2002.

[6] T. M. Antonsen and B. Levush, “Traveling-wave tube devices withnonlinear dielectric elements,” IEEE Trans. Plasma Sci., vol. 26, no. 3,pp. 774–786, Jun. 1998.

[7] I. M. Rittersdorf, T. M. Antonsen, D. Chernin, and Y. Y. Lau, “Effects ofrandom circuit fabrication errors on the mean and standard deviation ofsmall signal gain and phase of a traveling wave tube,” IEEE J. ElectronDevices Soc., vol. 1, no. 5, pp. 117–128, May 2013.

[8] A. Singh, J. E. Scharer, J. H. Booske, and J. G. Wohlbier, “Second-and third-order signal predistortion for nonlinear distortion suppressionin a TWT,” IEEE Trans. Electron Devices, vol. 52, no. 5, pp. 709–717,May 2005.

C. F. Dong received the M.Eng. degree innuclear engineering and radiological sciences andthe Ph.D. degree in space sciences from the Uni-versity of Michigan, Ann Arbor, MI, USA, in 2014and 2015, respectively.

He is currently a Visiting Scholar with the SpaceSciences Laboratory, University of California atBerkeley, Berkeley, CA, USA, and will join thePrinceton Plasma Physics Laboratory in 2016, as aPost-Doctoral Fellow.

Peng Zhang (S’07–M’12) received the Ph.D. degreein nuclear engineering and radiological sciencesfrom the University of Michigan, Ann Arbor, MI,USA, in 2012, where he is currently an AssistantResearch Scientist.

His current research interests include nanoelec-tronics, electromagnetic fields and waves, lasers, andplasmas.

Dr. Zhang was a recipient of the Richardand Eleanor Towner Prize for Outstanding Ph.D.Research.

David Chernin received the Ph.D. degree in appliedmathematics from Harvard University, Cambridge,MA, USA, in 1976.

He has been with Leidos, Reston, VA, USA, andits predecessor company SAIC since 1984, where hehas conducted research on beam-wave interactionsand other topics in the physics of particle accelera-tors and vacuum electron devices.

Dr. Chernin is a member of the American PhysicalSociety.

Y. Y. Lau (M’98–SM’06–F’08) received the B.S.,M.S., and Ph.D. degrees from the MassachusettsInstitute of Technology, Cambridge, MA, USA, inelectrical engineering.

He is currently a Professor with the University ofMichigan, Ann Arbor, MI, USA, specialized in RFsources, heating, and discharge.

Prof. Lau received the IEEE Plasma Science andApplications Award. He is an APS Fellow.

Brad W. Hoff (S’04–M’10) received the Ph.D.degree in nuclear engineering from the Universityof Michigan, Ann Arbor, MI, USA, in 2009.

He is currently a Senior Research Physicistwith the Directed Energy Directorate, Air ForceResearch Laboratory, Kirtland Air Force Base, Albu-querque, NM, USA. His current research interestsinclude high-power microwave sources and applica-tions of additive manufacturing to directed energytechnology.

D. H. Simon photograph and biography not available at the time of publica-tion.

Patrick Wong photograph and biography not available at the time ofpublication.


Geoffrey B. Greening received the B.S.E. andM.S.E. degrees in nuclear engineering and radiolog-ical sciences (plasma option) from the Universityof Michigan, Ann Arbor, MI, USA, in 2010 and2012, respectively, where he is currently pursuingthe Ph.D. degree with the Plasma, Pulsed Power,and Microwave Laboratory under the supervision ofProf. R. M. Gilgenbach.

His current research interests include high-powermicrowave source development.

Ronald M. Gilgenbach (S’73–M’74–SM’92–F’06–LF’15) received the B.S. and M.S. degrees fromthe University of Wisconsin, Madison, WI, USA, in1972 and 1973, respectively, and the Ph.D. degreefrom Columbia University, New York, NY, USA, in1978.

He is currently the Department Chair and the Col-legiate Professor with the University of Michigan,Ann Arbor, MI, USA.

Dr. Gilgenbach is a fellow of the APS DPP.

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 600 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages false /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/Description >>> setdistillerparams> setpagedevice

ieee transactions on electron devices, vol. 62, no. 12 ...pz/twt15.pdf · 4286 ieee transactions on...

Documents