Comparison of multistage gyroamplifiers operating in the frequencymultiplication regime with gyroamplifiers operating at a given Cyclotron harmonic
ABSTRACT The operation of gyrodevices at cyclotron harmonics is very attractive because of the possibility to reduce the magnetic field requirement by s times, where s is the cyclotron harmonic number. In recent years, two methods of harmonic operation in multistage gyroamplifiers have been actively studied: operation at a given harmonic in all stages and operation in the frequencymultiplying regime when the input stage operates at a lower harmonic than the output one. The present paper is aimed at making a comparative analysis of these two schemes of operation. To do this, a simple analytical method is developed, which allows one to qualitatively describe saturation effects in both schemes and compare such performance characteristics as the efficiency, gain, and bandwidth in both schemes. The results are of interest for evaluating the pros and cons of both schemes.

Article: Gyromultiplier with sectioned cavity
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ABSTRACT: A novel scheme of a selfexciting singlecavity terahertz gyromultiplier is proposed and theoretically investigated. Simulations predict a possibility to obtain a power of 75 W at the frequency of 1.3 THz from the 80 kV/0.7 A electron beam when operating at the fourth cyclotron harmonic at the relatively low magnetic field of 14 T.Physics of Plasmas 11/2010; 17(11):1107061107064. · 2.25 Impact Factor
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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004957
Comparison of Multistage Gyroamplifiers Operating
in the FrequencyMultiplication Regime With
Gyroamplifiers Operating at a Given
Cyclotron Harmonic
Gregory S. Nusinovich, Fellow, IEEE, Oleksandr V. Sinitsyn, J. Rodgers,
Thomas M. Antonsen, Jr., Senior Member, IEEE, Victor L. Granatstein, Life Fellow, IEEE, and N. C. Luhmann, Jr.
Abstract—The operation of gyrodevices at cyclotron harmonics
is very attractive because of the possibility to reduce the magnetic
field requirement by
times, where
number. In recent years, two methods of harmonic operation in
multistage gyroamplifiers have been actively studied: operation at
a given harmonic in all stages and operation in the frequencymul
tiplying regime when the input stage operates at a lower harmonic
than the output one. The present paper is aimed at making a com
parative analysis of these two schemes of operation. To do this, a
simple analytical method is developed, which allows one to qual
itatively describe saturation effects in both schemes and compare
such performance characteristics as the efficiency, gain, and band
widthinbothschemes.Theresultsareofinterestforevaluatingthe
pros and cons of both schemes.
is the cyclotron harmonic
Index Terms—Cyclotron harmonics, frequency multiplication,
gyroamplifiers, gyrotrons.
I. INTRODUCTION
F
in gyrodevices the magnetic field required for obtaining the cy
clotron radiation at a given frequency is inversely proportional
to the cyclotron resonance harmonic number, , the gyrotron
operation at harmonics
time. In multistage gyroamplifiers, two concepts of harmonic
operation are known: 1) regime of frequency multiplication [1]
(for instance, operation of theinput stage at thefundamental cy
clotron resonance,
, and the output stage operation at the
harmonic
) and 2) operation of all stages at the same
harmonic
[2]. Both concepts have been studied
for a long time. However, so far, no comparative discussion of
their potentials and comparative analysis of these two concepts
have been carried out. Such a comparative analysis is done in
the present paper.
ORMANYapplicationsofgyrodevices,thecriticalissueis
theuseofcryomagnetsorheavyandbulkysolenoids.Since
has been studied for a long
Manuscript received September 23, 2003; revised January 23, 2004. This
work was supported by the Multidisciplinary University Research Initiative on
Innovative Vacuum Electronics sponsored by the Air Force Office of Scientific
Research.
G. S. Nusinovich, O. V. Sinitsyn, J. Rodgers, T. M. Antonsen, Jr., and V. L.
Granatstein are with the Institute for Research in Electronics and Applied
Physics, University of Maryland, College Park, MD 207423511 USA (email:
gregoryn@glue.umd.edu).
N. C. Luhmann, Jr. is with the University of California, Davis, CA 95616
USA.
Digital Object Identifier 10.1109/TPS.2004.827590
Before starting to describe its content, it makes sense to
discuss possible potential applications of both schemes and
their “pros and cons.” Clearly, the frequency multiplication
is a purely nonlinear process, occurring due to the nonlinear
properties of an electron beam, which being modulated at
a given frequency generates in the electron current density,
in addition to this frequency, also its harmonics. Therefore,
frequencymultiplying gyroamplifiers cannot be used in com
munication systems that require a high degree of linearity. [For
example, if the carrier is simultaneously modulated by several
channels as in frequencydivision multipleaccess systems
(FDMA)]. At the same time, however, such devices can be used
in various radar and accelerator applications (see, for instance,
corresponding chapters in [3]) as well as in singlechannel
digital communication systems possibly using codedivision
multiple access (CDMA).
An obvious advantage of frequencymultiplying gyroampli
fiers is operation of the first stage at the fundamental cyclotron
resonance. In comparison with a similar stage operating at
higher harmonic, operation at the fundamental can be realized
at a lower order mode that improves the stability of operation
and simplifies the design and configuration of an input coupler.
Also, the electron beam is more strongly coupled to the
radiofrequency (RF) field at the fundamental resonance than
at harmonics. The difference in the coupling can be roughly
estimated based on the fact that, in accordance with the Schott
formula (see, e.g., [4, eq. (74.8)]), the intensity of cyclotron
radiation, when electron orbital velocities are reasonably small,
is proportional to
. (Here,
normalized to the speed of light.) Therefore, one can use in
frequencymultiplying devices lower power drivers, which
also can operate at lower frequencies. This is a very important
advantage, especially at millimeter (and shorter) wavelengths,
where the availability of drivers of high enough power is a
serious issue.
Since the use of solidstate drivers allows the most compact
solution, it makes sense to briefly discuss the present state
oftheart in solidstate sources in the range of the wavelength
of our interest. As shown in [5], the maximum power of such
sources ranges from 10 to 20 W in the Xband to about 1 W
in the Wband and then falls rapidly at higher frequencies (less
than0.1Wat200GHz).So,thefrequencymultiplicationfactor,
is the electron orbital velocity
00933813/04$20.00 © 2004 IEEE
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958 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
which allows one to use lower frequency for driving the device,
is here important. It should be noted that these power levels can
beincreasedbyusingthetechniqueofspatialpowercombining,
especially in the case of quasioptical power combining [6], [7].
As an illustration of the advances, the work at Sanders can be
mentionedwheretheuseof272individualmillimeterintegrated
circuits (MMICs) arranged in a “tray” architecture resulted in
producing 35 W power at 61 GHz with a 60dB smallsignal
gain[8].Forcomparison,letusmentionasanexamplethatsuch
a vacuum tube as a Wband coupledcavity traveling wave tube
(TWT) (millitron) provides 5kW peak power with a 10% duty
with the bandwidth in excess of 1% [9].
After mentioning these more or less obvious arguments, let
us note that such issues as the preference of one of the dis
cussed above schemes of gyroamplifiers with regard to the ef
ficiency, gain, bandwidth, and other performance characteris
tics is not obvious. Just an analysis of these issues is in the
focus of the present paper. The paper is organized as follows.
Section II briefly describes a general formalism, which we use
in our analysis. Section III contains the results of the analysis
of both schemes of interest for several multistage gyroampli
fiers: twocavity gyroklystrons, threecavity gyroklystrons, and
twostage gyroTWTs. Also, some results of simulations for an
ultrahighgain gyroTWT with distributed losses operating ei
theratthefundamentalorsecondharmonicareincluded.InSec
tion IV, we discuss the results obtained and consider a specific
example illustrating the application of our formalism to pos
sible experimental conditions. Finally, Section V summarizes
our paper.
II. GENERAL FORMALISM
Our consideration is based on the general formalism devel
oped for gyrodevices a long time ago [10]. Recently, the theory
of gyroklystrons based on this formalism was analyzed in de
tail in [11] and [12]. The derivation of corresponding equations
describing the gyroTWT and some results of the studies of
gyroTWTs operating at cyclotron harmonics are given in [13].
Since the equations have been described in detail in those ref
erences, here we will only outline the issues in this formalism,
whicharethemostimportant forourcomparativeanalysis.First
ofall,itshouldbenotedthatthechoiceofthegeneralformalism
was dictated by the fact that it is very difficult, if not impos
sible, to draw any conclusion just from comparison of different
designs, while we will attempt to make these conclusions with
the use of the general formalism.
A. Gyroklystron Formalism
The operation of the gyroklystron can be described by the set
of equations that determine the electron motion in the cavities
and drift regions, and equations that describe the field excita
tion in the cavities. Consideration of electron motion is greatly
simplified in thecase of a socalled point gap model also known
in linearbeam klystrons. (Originally, Yulpatov proposed to use
this model for studying the gyroklystrons [14]; later, Caplan
[15] used it for analyzing the smallsignal gain of a fivecavity
gyroklystron.) This model represents a gyroklystron circuit as
a set of short cavities separated by long drift sections. When
the cavities are short enough, it is possible to neglect there the
effects associated with electron orbital bunching, and hence to
consider there only electron energy modulation by the RF field.
Then,theorbitalelectronbunchingcausedbythechangesinthe
electron cyclotron frequency due to the energy modulation pro
ceeds in the long drift regions. In the framework of this model,
the equations for electron motion can be integrated analytically
even in largesignal regimes.
The field excitation in the cavities can be described by the
balance equations (assuming that we consider the stationary op
eration of a device). For the input cavity, the balance equation
forthenormalizedintensityoftheRFfield,
as [11], [12]
,canbewritten
(1)
Here,
respect to the RF field in the first cavity. (In the framework
of the pointgap model, it can be calculated analytically.) The
first and second terms in the denominator describe the beam
loading and frequency pulling effects, respectively. In the latter,
is the normalized detuning of the
signal frequency
with respect to the coldcavity frequency
. In (1),is the normalized intensity of the field excited in
the input cavity by the drive signal. This intensity relates to the
power of the driver,
, as [11]
is the electron beam susceptibility with
(2)
Here,
cavity having the loaded
is the beam power associated with the electron gyra
tion,
andare the beam voltage and current, respectively,
is the initial orbital velocity of electrons normalized to the
speed of light, and
energy normalized to the rest energy. (The derivation of (2) is
given in [16].)
In (1) and (2), we used the normalized beam current param
eter, which for cylindrical cavities operating at
equal to
is the coupling or externalfactor of the input
denoted by ,
is the initial electron
modes is
(3)
Here,
is the cyclotron resonance harmonic number,
is the normalized axial coordinate, the function
describes the axial distribution of the cavity field,
is the eigenvalue of the operating mode in a cavity of a
radius
, is the radius of electron guiding centers in a thin
annular electron beam, and
is the Bessel function of the first
kind.
The most important for our consideration is the fact that this
normalized beam current parameter is proportional to
This means that, for all other parameters being equal, the nor
malizedcurrentparameterforthecaseofoperationatthesecond
harmonic is
times smaller than for operation at the funda
mental cyclotron resonance. So, in addition to the issue of the
availability of highpower drivers at different frequencies, we
.
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NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCYMULTIPLICATION REGIME 959
get here one more factor, which weakens the RF field amplitude
in the case of operation at harmonics.
Then, the integration of equations for electron motion in the
drift sections and in other cavities yields the expression for the
susceptibility as the function of the bunching parameter,
. Here
length of the drift section. Note that in frequencymultiplying
devices, in order to have the bunches in all beamlets in the
same phase with respect to the RF field of the output stage,
the azimuthal indices of operating modes and the cyclotron har
monic numbers should obey the phase synchronization con
dition
[17], [18]. In the case of multicavity
gyroklystrons, there are more than one bunching parameters,
which account for the bunching caused by the modulation in
corresponding cavities. Then, the efficiency can be calculated
as
is the normalized
(4)
where the orbital efficiency relates to the imaginary part of the
susceptibility of the beam with respect to the output cavity field
as
(5)
As was mentioned above, in the framework of the pointgap
model this susceptibility can be calculated analytically. Corre
spondingly, one can easily find that the maximum orbital effi
ciency of the twocavity gyroklystron is equal to
(6)
(In the case of a larger number of cavities, the expression for
the efficiency is more complicated.) In (6),
frequencymultiplicationfactor,whichisequaltooneinthecase
of operation at the same harmonic in all stages and larger than
one in frequencymultiplying devices.
The gain, in accordance with [11] and [12], can be presented
as a superposition of two terms
is the
(7)
where
depend on the signal frequency variation. For the twocavity
gyroklystron, it is equal to
is the constant part of the gain, which does not
(8)
Correspondingly,
of the gain, which for the twocavity gyroklystron is equal to
is the frequencydependent variable part
(9)
Here,
sence of bunching effects, which for the pointgap model is
equal to
(see [11]), and
parameters now should be given for the second cavity. Also,
is the bunching parameter independent on the
beam loading and frequency pulling effects, while the previ
ously introduced bunching parameter
account. As follows from (1), the parameters are related as
is the susceptibility of the second cavity in the ab
is determined by (3), where all
takes these effects into
(10)
Equations (8) and (9) are written for the largesignal regime of
operation. In the smallsignal regimes, the bunching parameter
is small, and therefore the Bessel functions describing the sat
uration effects can be presented in a simple polynomial form,
viz.
fication of these equations (see [11] and [12]). These equations
become especially simple when the normalized beam current
parameter is small enough, so one can neglect the beam loading
and frequency pulling effects in (1) and below. For our analysis,
it is important that, afterthe polynomial expansionof the Bessel
function, the term in the figure brackets in (9) becomes propor
tional to
, i.e., in the case of small bunching parame
ters the gain of frequencymultiplying devices becomes much
smaller than that in the case of operation at a given harmonic.
This is, of course, obvious because the frequency multiplication
is a purely nonlinear effect.
, that causes a certain simpli
B. GyroTWTs Formalism
There are also some analytical methods to analyze the op
eration of multistage gyroTWTs. The simplest one was pro
posed in [19] and then used in [20] and in later publications.
This model of a gyroTWT, in which there are input and output
waveguides separated by a sever, is based on the following as
sumptions. First, we assume that the input waveguide operates
in the smallsignal regime, which is valid for practically all
known devices. Then, the orbital electron bunching in the drift
region between the input and output waveguides causes the ap
pearance of harmonics of the signal frequency in the electron
current density, whichis the nonlinearprocess thatcan be easily
analyzed analytically as in gyroklystrons. Finally, we assume
that the output waveguideis reasonably short, and hence, can be
considered in a specified current approximation. This approx
imation implies that we consider the excitation of the output
waveguide by a given current, but neglect the effect of the ex
cited wave on the electrons.
Without going into further description of this formalism, let
us present here the expression for the gain
(11)
Here,
second waveguide (this parameter is quite similar to that given
above by (3) for a cavity; see [21] for more details),
normalized amplitude of the wave at the entrance to the input
waveguide, which depends not only on the input power, but
is the normalized beam current parameter for the
is the
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960IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
also on the choice of mode and the resonance harmonic number
[22]:
. As follows from this dependence,
in which the drive power should also be multiplied by the
coupling impedance of a beam to the wave, which is similar to
the last term in the righthand side of (3), the operation in the
input waveguide at the fundamental increases the amplitude
of the excited wave at a given level of the drive power. This
increase becomes even larger when in the input waveguide
operating at the fundamental resonance a lower order mode
is used than in the case of the second harmonic operation.
The bunching parameter in (11) is proportional to the product
of the normalized length of the drift section to the function
describing the wave amplification in the input waveguide [21].
This function describing the wave amplification in the input
waveguide, in the framework of the smallsignal theory, is lin
early proportional to the input wave amplitude,
that, in the case of operation at the fundamental resonance in
the input waveguide, the saturation can be achieved at shorter
lengths of drift sections. Also important is the fact that in the
case of operation in lower order modes at the fundamental
resonance it is easier to provide the stability, since lesser
number of parasitic waves can be excited. (This statement,
of course, does not mean that one cannot provide stability of
the operation at the second cyclotron harmonic; see [23] for a
convincing example of the successful operation at the second
harmonic at the 200kW level in the
The gain given by (11) can again be represented as a sum of
the constant and variable parts. In the smallsignal regime, cor
responding expressions can be properly simplified. Of course,
expanding the Bessel function in (11) as a polynomial, we get
thesameconclusion aboutthegaindependence onthebunching
parameter as we made above for gyroklystrons. So, in general,
for all frequencymultiplying devices, the same dependence of
the output power on the input power,
whichwasobtainedforlinearbeamTWTsalongtimeago[24].
The Bessel function present in (11) also determines the nor
malized intensity of the outgoing radiation
. This means
wave.)
, is valid,
(12)
This intensity, in accordance with the energy conservation law,
determines the orbital efficiency of the device
(13)
which in the framework of the specified current approximation
is small. (As was shown in [21], this approximation is valid for
the normalized lengths of the output waveguide not exceeding
4 and 2 for the normalized beam current parameters
to 0.01 and 0.1, respectively. In longer waveguides, the effect of
the excited wave on the electron motion becomes significant.)
equal
III. RESULTS
A. TwoCavity Gyroklystron
The orbital efficiency of the twocavity gyroklystron (GKL)
was givenabove by (6). As follows from this equation, the max
imum efficiency of the frequencydoubling GKL with
Fig. 1.
the optimal drift section length is shorter in frequencymultiplying GKLs and
vice versa. The solid line corresponds to the case of equal coupling impedances
in input cavities of both schemes; the broken line illustrates the case when
the input cavity operates at the ??
 and ??
schemes, respectively.
Regions of preferable operation in one of two schemes: below the lines
 mode in the 12 and 22
andis approximately equal to 0.12 and the corre
sponding optimum value of the bunching parameter is equal to
. Similarly, the maximum orbital efficiency of the
GKL, in which both cavities operate at the second harmonic
is equal to 0.17 and
lowing, we will designate these schemes of GKLs by their har
monicnumbersineachstage,i.e.,wewillcallthecaseof
and“12 scheme,” the case
and so on.
So,atthefirst glance,the22 schemeoffersmuchhighereffi
ciencythanthe12scheme.Twofacts,however,shouldbetaken
into account here. First, numerical analysis of such schemes of
GKLs with a finite length of output cavities carried out in [25]
shows that the maximum efficiencies of both schemes with an
optimumlengthoftheoutputcavitiesaremuchhigherandmuch
closer: 0.60 and 0.65 for the 12 and 22 schemes, respectively.
Second, it is important to bear in mind that optimal values of
bunching parameters can correspond to different lengths of the
drift sections, because the amplitude of the RF field excited in
the input cavity, as follows from (1) through (3), depends on the
resonant cyclotron harmonic number. Clearly, it is preferable to
have a device in which the drift section length is shorter, be
cause such a device is more compact and its operation is less
sensitive to electron velocity spread and fluctuations in oper
ating parameters. In our case, the preference depends not only
on the harmonic number
, but also on the choice of the op
erating mode. The results are shown in Fig. 1. Here, the solid
and dotted lines show the borders between the region of shorter
drift section length for a 12 scheme (below the line) and a 22
scheme (above the line). The solid line corresponds to the case
when in both schemes the same operating mode is chosen; the
dotted line corresponds to the case when the input cavity oper
atesateither
 ormodeinthe12and 22schemes,
respectively. (It is assumed that
. In the fol
“22 scheme,”
factors of input cavities and
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NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCYMULTIPLICATION REGIME961
the drive power in both cases are the same.) This figure shows
that,withregardtothechoiceofthedriftsectionlength,thepref
erence depends on such parameters as the beam voltage and the
orbitaltoaxial velocity ratio, which determine the electron or
bitalvelocity.Ofcourse,whentheinputcavityinthe12scheme
operates at the lowest order
preferable in a larger region of the parameters. (In accordance
with the above mention phase synchronization condition, when
theinputcavityoperatesatthefundamentalresonanceatamode
with the azimuthal index equal to one, the output cavity, which
operates at the second harmonic, should utilize a mode with the
azimuthal index equal to two, e.g.,
Let us now compare the gain and bandwidth of the two
schemes. The constant part of the gain, which is given by (8),
can be the same in both schemes when the output cavities
operate at the same mode, the drift section length is the
same, and the beam loading in the input cavity is negligibly
small, so the difference in the factor
The variable gain given by (9) is, however, different. As we
already mentioned in the previous section, there is no sense
to discuss the smallsignal gain in frequencymultiplying
devices, because this gain is negative. Therefore, it is better
to compare the variable gain in two schemes in the cases
when the bunching parameter is optimal for each of them.
The results are shown in Fig. 2. Here, (a) and (b) show the
variable gain for 12 and 22 schemes, respectively. The gain
is shown as the function of the normalized signal frequency
detuning,
is the mean frequency of the coldcavity
frequencies) for several values of the staggertuning parameter
mode, this scheme becomes
mode.)
there does not matter.
(here,
. For the 12 scheme,
the parameter
describes the difference in
equal to 0.12, which is the optimal value of
[12];forthe22scheme
shows that in the case of zero detuning, when the variable gain
is maximal,
dB, while
So, the 22 scheme exhibits the gain higher by 6 dB in this case.
Results shown in Fig. 2 allow one also to analyze the effect of
staggertuning on the bandwidth enlargement at the expense of
the gain degradation. Correspondingresults are shown in Fig. 3,
where solid and dotted lines correspond to the 22 and 12
schemes, respectively. These results show that in both schemes
the stagger tuning allows one to increase the bandwidth in
practically the same fashion. In comparison with the case of no
stagger tuning, the maximum increase in bandwidth due to the
stagger tuning is 4.83 [11] and 5.3 [12] times in schemes with
and, respectively. However, this bandwidth
enlargement corresponds to the gain degradation by 16.7 dB
and about 17 dB
gain and bandwidth has been analyzed in more detail in [11]
and [12].
, which
factors of cavities, was taken to be
for this scheme
.ComparisonofFig.2(a)and(b)
dB.
. The tradeoff between the
B. ThreeCavity Gyroklystron
The bunching processes in multicavity gyroklystrons, of
course, are more complicated than those in the twocavity
device considered above. These processes can be characterized
Fig. 2.
(a) 12 scheme with the optimum bunching parameter ?
(b) 22 scheme with ?
? ????.
Modification of the largesignal variable gain with the stagger tuning:
? ????, and
Fig. 3. Gain degradation and bandwidth enlargement with stagger tuning.
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962IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
by a socalled “bunching” efficiency,
orbital efficiency introduced above as
is the harmonic number in the output cavity. This bunching
efficiency is determined by the intensity of the corresponding
harmonic in the electron current density exciting the output
cavity,
. Such harmonics in three and fourcavity GKLs
and their effect on the efficiency, gain, and bandwidth of GKLs
were studied in detail in [11], and [26]–[28]. Therefore, here
we will only outline the issues that are the most important for
our comparative analysis.
As shown in [26], the bunching efficiency in 122 and
222 schemes can exceed 0.6 (so, the orbital efficiency can
be higher than 0.3 in both cases). The 112 scheme yields a
lower bunching efficiency exceeding 0.4 only. It is interesting
to note that in the 122 scheme the efficiency is maximal
when the second cavity is located closer to the input one,
while in the 222 scheme it should be located closer to the
output cavity. (This conclusion is the same as one done in
[29] for a threecavity GKL, in which all cavities operate at
the fundamental resonance.) The optimal values of the first
bunching parameter for 122 and 222 schemes are about 2.5
and 1.5, respectively. The ratio of these values is approximately
the same as the ratio of optimal bunching parameters for
twocavity 12 and 22 schemes considered above. Therefore,
the results presented above (in particular, Fig. 1) are valid for
the present consideration as well. The optimal values of the
second bunching parameter for different schemes are different
[26]. These values can be realized by adjusting not only the
second drift section length, but also the
cavity. (The latter, however, affects the bandwidth.)
The gain of threecavity GKLs can be analyzed in the same
manner as in twocavity devices. Since corresponding expres
sions for the gain now are more complicated, it makes sense
to restrict our consideration by the case of operation close to
the smallsignal one, when the bunching parameter is small,
and hence, the Bessel functions can be represented by poly
nomials. Corresponding expressions readily show that the op
timalpositionofthesecondcavity,whichresultinthemaximum
gain, is 2/3 and 1/2 of the total drift length for 122 and 222
schemes, respectively. For this choice of the second cavity po
sition and negligibly small beam loading in the input cavity, the
constant part of the gain in the 122 scheme is smaller than that
in the 222 scheme by
that the variable gain in frequencymultiplying GKLs contains
also an additional term, which describes the dependence on the
bunching parameter
[cf. (9)]. At small
the difference in gains for 122 and 222 schemes even larger;
however, at large
’s, this term can increase the gain.
The bandwidth of these two schemes can be analyzed using
the expressions for the variable part of the gain, which are quite
similar, but more complicated than (9). Some results, neverthe
less,canbeobtainedanalyticallyeveninthiscase.Suchananal
ysis, for instance, was done for a 122 scheme with two last
cavities having equal
factors, equal values of the normalized
beam current parameter
coldcavity frequencies symmetrically situated with respect to
, i.e.,
cific case the maximum bandwidth can be realized when the pa
, which relates to the
, where
factor of the second
dB. Recall
’s, this term makes
and staggertuned
. It was shown that in this spe
rameter
the stagger tuning parameter
is equal to 7.9. In the case of such a maximal stagger tuning the
bandwidth expressed in terms of the detuning
isequalto13.4,whileintheabsenceofstagger
tuningitwasequalto1.135,i.e.,itincreasesbyalmost12times.
Thisbandwidthenlargementis,however,accompaniedbyabout
32dB gain degradation.
Theseresults canbe comparedwith similarresultsfor a222
scheme, which in terms of normalized parameters is identical
to the 111 scheme studied in [11]. In this case, the optimal
value of
is 0.25 and the maximum bandwidth corresponds
to the staggertuning parameter equal to 6.066. This stagger
tuning increases the normalized bandwidth from 1.2 (when
) to 17.0, i.e., by more than 14 times, at the expense of
the gain degradation by about 36 dB. So, in this sense both
schemes look quite similar. Of course, the gainbandwidth
product, which is often treated as a figureofmerit of various
amplifiers, corresponds to smaller staggertunings than those
that yield the largest bandwidth. (This issue was discussed
elsewhere [11], [12].)
is equal to 0.0504 and
C. TwoStage GyroTWT
Comparison of two concepts of gyroTWTs leads to the con
clusions that are quite similar to those derived for twocavity
GKLs above. The constant part of the gain in the both fre
quencymultiplying gyroTWT and twostage gyroTWT oper
ating at the second harmonic in both stages is the same. The
variablepartofthegain,asfollowsfrom(11),isagaincharacter
ized by the ratio of the Bessel function of the order
to the input wave amplitude. Therefore, the gain degradation
caused by the variable gain is smaller in the case of operation at
a given harmonic in all stages. Also, at small bunching param
eters, the operation at a given harmonic
gain; however, the saturation can be realized faster in the case
of frequencymultiplying devices.
In addition to the comparison based on the use of a very
simplified approach described in Section IIB, we have also
carried out some numerical studies with the use of nonlinear
theory. It was considered a twostage gyroTWT experimen
tally studied at the University of Maryland. We assumed that
the beam parameters such as the beam voltage and current, and
the orbitaltoaxial velocity ratio are fixed: 60 kV, 5 A, and 1.0,
respectively. We also assumed that the lengths of all sections
and their transverse dimensions are the same as in the experi
ment. Then, we did simulations for three cases: 1) operation in
both stages at the fundamental cyclotron resonance (operating
modes are
in both stages); 2) 12 scheme with the op
erating modes
and
respectively; and 3) 22 scheme where the operating mode is
in both stages.
Some results of the simulations are shown in Figs. 4–6.
Fig. 4 shows the case of operation at the fundamental in both
stages. In Fig. 4(a), where vertical lines separate the regions
of the first waveguide, drift section and output waveguide, the
axial dependencies of the output power are shown for several
values of the cyclotron resonance detuning in the output stage
yields a higher
in the first and second stage,
Page 7
NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCYMULTIPLICATION REGIME963
Fig. 4.
operating at the fundamental harmonic in both stages for several values of the
cyclotron resonance detuning in the output stage. (b) Dependence of the output
power on the normalized cyclotron resonance detuning for 50 and 100W input
power levels.
(a) Axial dependence of the output power in a twostage gyroTWT
; this detuning in the
input stage was taken equal to zero because this corresponds
to the maximum growth rate of the wave. (The variation in the
detuning can be realized by a weak tapering of the external
magnetic field.) As one can see, in this case the length of the
output section is long enough to reach saturation when the input
power equals 100 W. Resulting dependencies of the output
power on the detuning in the output stage are shown in Fig. 4(b)
for 50 and 100W input power. For 50W input power, the
output waveguide is not long enough to reach the maximum
power and efficiency. Therefore, the maximum power shown in
Fig. 4(b) for this case is lower than for 100 W. Results shown
in Fig. 4(b) indicate that this device can operate with the gain
exceeding 30 dB.
Fig. 5 shows some results of simulations for the frequency
doubling gyroTWT. Here figures (a) and (b) correspond to 50
Fig. 5.
gyroTWT for several values of the cyclotron resonance detuning in the output
stage for (a) 50 W and (b) 100 W levels of the input power.
Axial dependence of the output power in a frequencydoubling
and 100W input power, respectively. In this case, as follows
fromFig.5(b),toreachmaximumoutputpowereveninthecase
of 100W input power it is necessary to increase the length of
the output stage by about 60%. This will result in about 40kW
output power, while in the11 scheme themaximum output was
higher than 100 kW for the same 100W input power, as shown
in Fig. 4(b).
Finally, Fig. 6 shows some results of simulations for the 22
scheme. In thiscase, as follows from Fig.6(a), to reach the peak
ofthemaximumoutputpowerinthecaseof100Winputpower,
itisnecessarytomorethandoublethelengthoftheoutputwave
guide.Fig.6(b)illustratesthesensitivityoftheoperationofsuch
a device to variation of the external magnetic field. As one can
see, the efficient operation is possible in the range of magnetic
fields about
1% or less of the nominal value. As follows from
Fig. 6(b), the bandwidth corresponding to this nominal value of
the magnetic field,
, is about 1.7%.
Page 8
964IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
Fig. 6.
withbothstagesoperatingatthesecondharmonic.(b)Dependenceoftheoutput
power on the operating frequency in this tube for several values of the external
magnetic field (lengths of the input waveguide, the drift section, and the output
waveguide are equal to 15.2, 9.0, and 14.6 cm, respectively).
(a) Axial dependence of the output power in a twostage gyroTWT
D. UltrahighGain GyroTWT
Regardless of the chosen operating harmonic, the problem
of obtaining highfrequency drive sources capable of saturating
the gyroamplifier in question can be relaxed somewhat if the
gain is high enough. Recently, singlestage gyroTWTs with
distributed loss have been investigated as a means of achieving
gains as high as 70 dB while maintaining operation free of
oscillations from reflective and absolute instabilities [31].
Conceivably, systems with such high gain gyroamplifiers could
achieve output powers as high as several hundred kilowatts in
Wband using lownoise solidstate drivers. In view of the ob
vious advantages of such configurations, not the least of which
is saving the cost of an intermediate vacuum tube amplifier,
there are other issues that require attention. Depending on the
application, signal integrity issues such as linearity, noise, and
phasecharacteristicsmaybeofequalorgreaterimportancethan
Fig. 7.
harmonic gyroTWT configurations with distributed loss applied to the first
15 cm of the interaction circuit followed by 7 cm of high conductivity wall.
The arrows indicate the location of the beam guiding center radius.
Schematic of MAGY model for (a) fundamental and (b) second
cost reduction. It seemed prudent to consider whether similarly
high gains could be achieved in distributed loss harmonic
gyroTWTs as well.
A numerical study using the Maryland Gyrotron code
(MAGY) [30] was conducted to compare the performance of
gyroTWTs with distributed loss operating at the fundamental
and second harmonic of the cyclotron frequency. Fig. 7 shows
a schematic of the two models we considered. The basic
parameters governing synchronism between the fast cyclotron
and EM waves were fixed by comparing a
circuit with radius of 5.5 mm with a
circuit with radius 10.07 mm, both with a length of 22 cm.
Thus, the cutoff frequencies and dispersion characteristics for
the operating modes in both cases are the same, and maximum
gainwas expectedarounda frequencyof35 GHz.Themagnetic
field strength for grazing intersection
as follows:
fundamental
second harmonic
could be determined
(14)
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NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCYMULTIPLICATION REGIME965
TABLE I
MAGY PARAMETERS FOR COMPARISON OF DISTRIBUTED LOSS GYROTWTS
where
ativistic mass factors and
waveguide. Equation (14) shows that this field depends on the
appliedvoltageandtheelectronorbital toaxial velocityratio.In
our simulations, we usedthe electronbeam voltageand electron
velocity pitch fixed at 80 kV and 1.0, respectively. The beam
current and distributed loss rate were left as free parameters and
optimized for each case for maximum gain. Table I summarizes
the model parameters, and Fig. 8 gives the results.
The results show that, while impressive fundamental gain
is realized, the compression slope has fairly dramatic conse
quences on output phase. In applications where linearity is
important, the output power should be “backedoff” below
the peak value to minimize AMPM conversion. We note
that neither model was comprehensively optimized for all
performance figures of merit; rather, the importance was direct
comparison. In the second harmonic case, 45 dBm of drive
power was needed for saturation while in the fundamental
harmonic case the device saturates at less than 20 dBm. This
value (45 dBm) is above the capability of solidstate sources
in the millimeterwave region. The results should not be
interpreted to mean that a distributedloss, second harmonic
gyroTWT could not achieve higher gain and meet this crite
rion. Higher perpendicular velocity (with bandwidth reduction)
and increased gain length may mitigate this shortcoming and
should be investigated further.
is the ratio of the unperturbed total to axial rel
is the cutoff frequency of the
IV. DISCUSSION
Let us illustrate our theoretical analysis by one more ex
ample. Assume (as it was done in [12] for 11 and 12 schemes
of GKLs) that we have an electron beam, which was used in the
experiments with a Wband gyroklystron at NRL [32]: 65kV
beam voltage, 6A beam current, orbitaltoaxial velocity ratio
equal to 1.4. Let us consider the performance characteristics
of twocavity GKLs driven by such a beam for the following
three cases: 1) 11 scheme with both cavities operating at the
mode; 2) 12 scheme with the first cavity operating
at the fundamental at the same
cavity operating at the second cyclotron harmonic at the
mode; and 3) 22 scheme with both cavities operating
at the
mode. In all cases, for the sake of simplicity, we
mode and the second
Fig. 8.
versus drive power for (a) fundamental, and (b) second harmonic gyroTWTs
with distributed loss placed in the interaction circuit to achieve high gain and
stable operation.
MAGY simulation results showing the output power, gain, and phase
will neglect the beam loading and frequency pulling effects.
This allows us to simplify the relation given by (1) between
the normalized field amplitude in the first cavity and the
amplitude of the field excited by a driver there. Then, in order
to estimate the relation between the latter amplitude and the
drive power given by (2), assume that the axial profile of the
input cavity is sinusoidal and the cavity length is equal to one
wavelength; also assume that a thin annular beam is positioned
in the peak of the coupling impedance to the operating mode,
the cavity
factor is equal to 100, and the cavity is critically
coupled, i.e.,
. For this set of parameters, (2)
yields
section length can be found under assumption that for a given
drive power and the case of exact resonance between the
signal frequency and the coldcavity frequency the bunching
parameterhas an optimal value. For instance, if we want tohave
an optimal bunching parameter,
the normalized length of the drift section should be equal to
that corresponds to
similar fashion, we can also evaluate the normalized parameters
. Then, the optimal drift
, for W,
. In a
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966IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004
Fig. 9.
different schemes in the framework of the pointgap model.
Dependence of the output power and gain on the input power for
for two other schemes. In the case of a 12 scheme, it was
assumed that the beam position remains the same as for the
11 scheme; it was also assumed that, to provide the maximum
bandwidth in the case of the staggertuning, in accordance with
[12], we should have the
factor of the second cavity about
2.9 times higher than that for the first one, i.e., it should be
. This value of the
cavities in the third case (22 scheme). Of course, the bunching
parameters were chosen to be optimal for each case.
Resultsshowingthedependenceoftheoutputpowerandgain
on the drive power in all three schemes are shown in Fig. 9.
Since the normalized length of the drift section is rather long,
saturation effects in the schemes without frequency multiplica
tion appear at input power level of about a few watts. Therefore,
the gain curves for 11 and 22 schemes shown in Fig. 9(b) that
start from
W have almost linear slops demonstrating
the saturation. Of course, the 11 scheme has a superior per
formance; it also offers a larger bandwidth, since in this case
the
factors are lower. So, the main obstacles for the use of
factor was also used for both
this simple scheme could be either the necessity to create strong
magnetic fields with the use of large solenoids increasing the
mass and volume of a microwave system, or the tendency to
operate at high power levels, for which it is difficult, if not im
possible, to realize optimal operating conditions in the case of
loworder modes operating at the fundamental resonance. Com
parison of two latter schemes shows that the 22 scheme offers
higher gain and output power than the 12 scheme. To the great
extent, this is the result of describing the saturation effects by
the Bessel functions corresponding to the account of the bal
listic bunching in the drift sections of the pointgap model only.
To consider a more realistic case, we replaced in the formula
for the orbital efficiency given by (6) these functions by the
corresponding polynomials of the bunching parameter
(15)
and determined the coefficients
the conditions that approximations given by (15) yield for the
optimal values of bunching parameters (
) the orbital efficiencies equal to their maximum
valuesfound in [25] (
spondingresultsareshowninFig.10,wherethegainandoutput
powerare shownas functionof theratioof thedrivepowertoits
optimum value (100 W). Now the performance characteristics
of the 12 and 22 schemes are closer, but still the 22 scheme
exhibits a higher gain; for instance, for the optimal drive power
the difference in the gain is a little smaller than 5 dB in favor
of the 22 scheme. Recall that our analysis done in Section III
showed that the tradeoff between the gain and the bandwidth
due to the stagger tuning in both schemes yields approximately
the same results.
So far, we focused our consideration on the analysis of sev
eralimportantcharacteristics(efficiency,power,gain,andband
width) only. For many applications, various signal distortion ef
fects are also important. Some of them can be caused by the
effect of the input signal amplitude modulation on the ampli
tude and phase of the output radiation. Although the effect of
the input signal modulation on the output phase (a socalled
AM/PM conversion, see, e.g., [33]) cannot be studied in the
framework of the pointgap model [11], one can easily use this
model for estimating the effect of the input signal modulation
on the amplitude of the output radiation (AM/AM conversion).
In accordance with [33] and the formalism developed above,
this conversion for a pointgap model can be characterized by
the function
As follows from this simple formula, at small values of the
bunching parameter (smallsignal regime) the 22 scheme oper
ates with 1 dB/dB conversion as any linear amplifier, while the
12 scheme yields 2 dB/dB conversion, which is caused by the
frequencydoublingoperationandcanbebeneficialinsomespe
cific cases of AM signals. Of course, the maximum efficiency
operation corresponds to
Anotherimportantparameterofanyamplifieristhesignalto
noise ratio. As known, the noise of any amplifier is predom
inantly determined by the noise generated in the input stage
and for each scheme from
and
and). Corre
, where.
in any scheme.
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NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCYMULTIPLICATION REGIME967
Fig. 10.
different schemes in the case of corrected polynomial approximation for the
orbital efficiency.
Dependence of the output power and gain on the input power for
by a driver and by an electron beam. (This noise in the fun
damental harmonic GKLs was studied theoretically and exper
imentally in [34] and [35], respectively.) Then, this noise, to
gether with a signal, is amplified in other stages of the device.
In the case of operation at a given harmonic, this process has a
linear nature, but it is a nonlinear process in frequencymulti
plying devices. This nonlinearity weakens the role of noise with
respect to the signal. Therefore, the signaltonoise ratio in fre
quencymultiplying devices can be higher than in the devices
operating without multiplication. This issue was recently ana
lyzed in [36] in detail.
Beforeclosingthissection,letusnotethattheformalismused
and the results obtained are applicable for largeorbit gyrode
vices (LOGs) [37]. As was pointed out in [38], the theory of
these devices can be based on the formalism developed for con
ventional, smallorbit gyrodevices. The peculiarity of LOGs is
the necessity to consider the coupling impedance present in the
definition of the normalized beam current parameter given by
(3)forelectronshavingzeroguidingcenterradius,
immediately yields the rule of mode selection in LOGs [39],
, which is the only case of nonzero coupling impedance
for electrons with
. Note that, in order to enhance radia
tion at high cyclotron harmonics, instead of smoothwall cylin
drical cavities/waveguides, various magnetrontype microwave
circuits are often used in LOGs [40], [41]. In such cases, not
only the numerator, but also denominator in (3) should be prop
erly modified.
.This
V. SUMMARY
A simple theory is developed that allows one to make
a comparative analysis of various schemes of multistage
gyroamplifiers. The results of the analysis done for the fre
quencymultiplying devices and devices, which operate at a
givenharmonicinallstages,showthatsomeoftheperformance
characteristics considered are better for the devices operating
at a given harmonic (gain and efficiency), while others (such
as the bandwidth) can be better in frequencymultiplying
gyroklystrons where the operation of the input stage at the
fundamental resonance allows one to use the low
Without going into details of the analysis performed, it can be
stated that the preference of one of two schemes considered
stronger depends on the practical issues (such as a specific
application, availability of drivers at different frequencies,
ability to provide a stable operation at higher order mode in the
input stage, etc.) than on the details of the saturation effects
and possibilities to increase the bandwidth and gainbandwidth
product by using the stagger tuning (the latter is essentially
the same in both schemes). Thus, it makes sense to continue
analysis of both schemes.
cavities.
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Gregory S. Nusinovich (SM’92–F’00) received the
B.Sc., M.Sc., and Ph.D. degrees from Gorky State
University,Gorky,U.S.S.R.,in1967,1968,and1975,
respectively.
In 1968, he joined the Gorky Radiophysical
Research Institute, Gorky. From 1977 to 1990,
he was a Senior Research Scientist and Head of
the Research Group at the Institute of Applied
Physics, the Academy of Sciences of the U.S.S.R.,
Gorky. From 1968 to 1990, his scientific interests
included developing highpower millimeter and
submillimeterwave gyrotrons. He was a Member of the Scientific Council on
Physical Electronics of the Academy of Sciences of the U.S.S.R. In 1991, he
immigrated to the U.S. and joined the Research Staff at the Institute for Plasma
Research (presently, the Institute for Research in Electronics and Applied
Physics), University of Maryland, College Park. Since 1991, he has also served
as a Consultant to the Science Applications International Corporation, McLean,
VA, the Physical Sciences Corporation, Alexandria, VA, and OmegaP, Inc.,
New Haven, CT. He is the author of Introduction to the Physics of Gyrotrons
(Baltimore, MD: The Johns Hopkins University Press, 2004). He has authored
and coauthored more than 150 papers published in refereed journals. His
current research interests include the study of highpower electromagnetic
radiation from various types of microwave sources.
Dr. Nusinovich is a Member of the Executive Committee of the Plasma Sci
ence and Application Committee of the IEEE Nuclear and Plasma Sciences
Society. He is a Fellow of the American Physical Society (APS). In 1996 and
1999, he was a Guest Editor of the Special Issues of the IEEE TRANSACTIONS
ON PLASMA SCIENCE on HighPower Microwave Generation and on Cyclotron
Resonance Masers and Gyrotrons, respectively. Presently, he is an Associate
Editor of the IEEE TRANSACTIONS ON PLASMA SCIENCE.
Oleksandr V. Sinitsyn was born in Kharkiv,
Ukraine, in 1978. He received the B.S. degree in
radio physics from V. Karazin Kharkiv National
University, Kharkiv, Ukraine, in 2000 and the M.S.
degree in electrical engineering from the University
of Maryland, College Park, in 2002. He is currently
working toward the Ph.D. degree in electrical and
computer engineering at the University of Maryland.
His research interests include theory and design of
highpower microwave sources.
J. Rodgers, photograph and biography not available at the time of publication.
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NUSINOVICH et al.: COMPARISON OF MULTISTAGE GYROAMPLIFIERS OPERATING IN THE FREQUENCYMULTIPLICATION REGIME969
Thomas M. Antonsen, Jr. (M’82–SM’96) was born
in Hackensack, NJ, in 1950. He received the B.S. de
gree in electrical engineering in 1973, and his M.S.
and Ph.D. degrees in 1976 and 1977, all from Cor
nell University, Ithaca, NY.
He was a National Research Council Post Doc
toral Fellow at the Naval Research Laboratory, from
1976 to 1977, and a Research Scientist in the Re
search Laboratory of Electronics at MIT, Cambridge,
from 1977 to 1980. In 1980, he moved to the Univer
sity of Maryland, College Park, where he joined the
faculty of the Departments of Electrical Engineering and Physics in 1984. He
is currently a Professor of physics and electrical and computer engineering. He
has held visiting appointments at the Institute for Theoretical Physics (UCSB),
theEcolePolytechniqueFederaledeLausanne,Switzerland,andtheInstitutede
Physique Theorique, Ecole Polytechnique, Palaiseau, France. He served as the
Acting Director of the Institute for Plasma Research, University of Maryland
from 1998 to 2000. His research interests include the theory of magnetically
confined plasmas, the theory and design of highpower sources of coherent ra
diation,nonlineardynamicsinfluids,andthetheoryoftheinteractionofintense
laser pulses and plasmas. He is the author and coauthor of over 200 journal ar
ticles and coauthor of the book Principles of FreeElectron Lasers.
Dr. Antonsen has served on the editorial board of Physical Review Letters,
The Physics of Fluids, and Comments on Plasma Physics. He was selected as
a Fellow of the Division of Plasma Physics of the American Physical Society
in 1986. In 1999, he was a corecipient of the Robert L. Woods award for Ex
cellence in Vacuum Electronics Technology, and in 2003 he received the IEEE
Plasma Science and Applications Award.
Victor L. Granatstein (S’59–M’64–SM’86–F’02–
LF’02) received the Ph.D. degree in electrical
engineering from Columbia University, New York,
in 1963.
After a year of postdoctoral work at Columbia, he
wasa ResearchScientist at BellTelephoneLaborato
ries from 1964 to 1972, where he studied microwave
scattering from turbulent plasma. In 1972, he joined
the Naval Research Laboratory (NRL), Washington,
DC, as a Research Physicist, and from 1978 to 1983,
heservedasHead ofNRL’sHighPower Electromag
netic Radiation Branch. In August 1983, he became a Professor in the Electrical
Engineering Department, University of Maryland, College Park. From 1988 to
1998, he was Director of the Institute for Plasma Research at the University of
Maryland. He spent semesters in both 1994 and 2003 as a Visiting Professor
at the University of Tel Aviv, Tel Aviv, Israel. He is currently leading studies
of high pulsed power gyroklystron amplifiers for driving highgradient electron
accelerators, efficiency enhancement in high average power gyrotron oscilla
tors for heating and stabilizing magnetic fusion plasmas, advanced concepts in
highpower, wideband millimeterwave amplifiers for radar applications, and
the effect of highpower microwave pulses on electronic components and sys
tems. He has coauthored more than 220 research papers in scientific journals
and has coedited three books. He holds a number of patents on active and pas
sive microwave devices.
Dr.GranatsteinisaFellowoftheAmericanPhysical Society.Hehas received
a number of major research awards including the E.O. Hulbert Annual Science
Awardin1979,theSuperiorCivilianServiceAwardin1980,theCaptainRobert
Dexter Conrad Award for Scientific Achievement (awarded by the Secretary of
the Navy in 1981), the IEEE Plasma Science and Applications Award in 1991,
and the Robert L. Woods Award for Excellence in Electronics Technology in
1998.
N. C. Luhmann, Jr., photograph and biography not available at the time of
publication.
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