D

Na

b

a

ARA

KBPSS

1

imboowndmuircmbttspfsi

1h

Int. J. Electron. Commun. (AEÜ) 67 (2013) 426–432

Contents lists available at SciVerse ScienceDirect

International Journal of Electronics andCommunications (AEÜ)

journa l homepage: www.e lsev ier .com/ locate /aeue

esign and simulation of a �/2-mode spatial-harmonic magnetron

asrin Nasr Esfahania,b,∗, Majid Tayarania, Klaus Schünemannb

Faculty of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran 1684613114, IranInstitut für Hochfrequenztechnik, Technische Universität Hamburg-Harburg, D-21073 Hamburg, Germany

r t i c l e i n f o

rticle history:eceived 6 August 2012ccepted 26 October 2012

eywords:

a b s t r a c t

Design and 3D numerical simulations of a 37.5 GHz spatial-harmonic magnetron (SHM) are presented. Theeffect of geometrical parameters of the side resonators of the anode block on output power are consideredusing the results of a theory based on the single harmonic approximation approach. This theory enablesdetermination of the optimum geometrical parameters of the side resonators. SHM design evaluation

eam–wave interactionarticle-in-cell simulationpace chargepatial-harmonic magnetron

is carried out via numerical simulations performed with a 3D particle-in-cell (PIC) code embedded inCST-Particle Studio. Effect of varying the external quality factor and DC-anode voltage on output power,efficiency and stability of operation are also considered. The presented SHM shows stable operation in the�/2-mode over a range of DC anode voltages extending from 12.4 kV to 13 kV and for an axial magneticflux density equal to 0.53 T. RF output power of the SHM varies from 25 kW to 47 kW over these voltageswith a maximum efficiency of around 18.5%.

© 2012 Elsevier GmbH. All rights reserved.

. Introduction

SHMs were mainly developed to overcome fundamental lim-tations of the �-mode magnetrons (also called n = N/2-mode

agnetrons, where n means mode number and N is the num-er of side resonators) in the millimeter-wave band. Physics ofperation of SHMs is essentially different as compared to thatf the classical �-mode magnetrons. SHMs utilize the first back-ard harmonic of the �/2-mode (also called n = N/4-mode) oreighboring ones (�/2 ± 1 (n = N/4 ± 1)-mode) instead of the fun-amental harmonic of the �-mode which is employed in classicalagnetrons. In comparison with the �-mode operation scenario,

tilizing the first backward harmonic of the �/2- or neighbor-ng mode in the spatial harmonic scenario helps to decrease theequired DC magnetic flux density and to increase the magnetronavity dimensions. Hence SHMs are able to remove two funda-ental drawbacks of the �-mode scenario in the millimeter-wave

and, i.e., the necessity of employing high magnetic flux densi-ies and impractically small anode cavities [1–3]. Besides removinghe above mentioned limitations, SHMs have much better modeeparation in both frequency and phase velocity domain as com-ared to the �-mode magnetrons. Due to this, SHMs do not suffer

rom mode competition even if the anode structure is neithertrapped nor of rising sun type. Another important practical lim-tation of the �-mode scenario in the millimeter-wave band is

∗ Corresponding author.E-mail address: nna@tf.uni-kiel.de (N. Nasr Esfahani).

434-8411/$ – see front matter © 2012 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.aeue.2012.10.013

the short lifetime of the thermionic cathode due to its intensecurrent density. This limitation has been successfully removed inSHMs by introducing a cold cathode instead of a thermionic one[1–3].

Although SHMs demonstrated reliable operation and providepromising solutions for the development of compact, cost-efficient transmitters for environmental monitoring systems,remote sensing of clouds, and precipitation in the millimeter-waveband [1,4], detailed theoretical analysis or numerical simulationsof these devices have been addressed only in a few papers usingin-house developed 2D PIC codes [5–7]. However, in recent yearsand with the advent of multi-physics based codes, the design andtesting of high power microwave sources via numerical algorithmshave become an essential tool for the development of such sources[8–10]. In this paper, design and numerical simulations (using CST-Particle Studio) of a �/2-mode SHM which provide stable operationat 37.5 GHz for DC anode voltages VDC varying from 12.4 kV to 13 kVand for an axial magnetic flux density B equal to 0.53 T are pre-sented. In Section 2, the effects of geometrical parameters of theside resonators on output power are examined using the resultsof a theory based on the single harmonic approximation approach[11,12]. This enables determination of the optimum geometricalparameters of the side resonators. This selection maximizes theoutput power in a wide range of the external quality factor Qext.The remaining part of Section 2 is devoted to describing the details

of the simulation model. In Section 3, considering a distinct oper-ating point and Qext, simulation results consisting of space chargeprofiles, output power, and efficiency are presented. The effect ofvarying Qext and VDC on output power, efficiency, and operation

N. Nasr Esfahani et al. / Int. J. Electron. Co

Fig. 1. A cut-away view of the anode block of the SHM together with its concentricc

sbtc

2

tirdumseitoeaqHSA�smqts�

ttasq

the values of d and � simultaneously. Fig. 2 shows the normalizedvalues of POut/I2 versus N�/� for Qext varying from 200 to 500. As

athode.

tability of the designed SHM are presented in Section 4. As wille discussed in this section, the designed SHM shows stable opera-ion across the whole variation range of VDC and Qext. Finally a briefonclusion is presented in Section 5.

. Design characteristics and simulation model

Fig. 1 depicts a cut-away view of the anode block of the SHMogether with its concentric cathode. This configuration is spec-fied with the following set of geometrical parameters: cathodeadius rc, anode radius ra, number of side resonators N, resonatorepth d, and resonator opening �. As is shown in this figure, SHMstilize the conventional anode structure of the classical �-modeagnetrons, i.e., a coaxial cylindrical cavity loaded with identical

ide resonators. In our simulation model, the geometrical param-ters of the interaction space and the number of side resonators,.e., ra, rc, and N are equal to 2.25 mm, 1.3 mm and 16, respec-ively. These parameters have been selected to be close to thosef an experimental model previously reported in [5]. In this ref-rence, performance of a SHM with the above mentioned ra, rc,nd N was examined for the �/2 − 1-mode (with a resonant fre-uency of about 35 GHz) using an in-house developed 2D PIC code.owever, as had also been mentioned in [5], operation of thisHM in the �/2-mode seriously suffered from mode instabilities.s is presented in the next sub-section, in our simulation modeland d are slightly different from the experimental model pre-

ented in [5].They are selected in a way that they can provideaximum output power for the �/2-mode with a resonant fre-

uency of about 37.5 GHz. As will be shown later, employinghis slight geometrical modification together with an appropriateelection of operating conditions provide stable operation in the/2-mode.

In the next sub-section, the effect of the geometrical parame-ers of the side resonators on output power is considered usinghe results of an approximate theory based on the single harmonicpproximation approach. This theory provides the possibility of

electing optimum resonator opening for a distinct resonant fre-uency.

mmun. (AEÜ) 67 (2013) 426–432 427

2.1. Formulation of output power and optimum geometry

Considering the single harmonic approximation approach, thepower delivered to the RF field, usually called electron power Pe, isgiven by [7],

Pe =A2

−1I2Ql

�Ff n(1)

where A−1 means the amplitude of the synchronous harmonic, i.e.,first backward harmonic, of the electric field of the operating mode(�/2-mode) which is calculated using (2) [12]. I depends on thecurrent density and the field distribution in the interaction spaceand on the operating mode, but weakly depends on the geometricalparameters of the side resonators as long as the resonant frequencydoes not vary considerably [7]. Ql means loaded quality factor, fnis the resonant frequency of the operating mode, and F is a formfactor which is determined using (3).

A−1 = 43�

sin(

3N

4�)

(2)

F = 2U

C2n

(3)

Here U means RF energy stored per unit length of the interactionspace and Cn is the amplitude of the electric field in the interactionspace.

Under steady state conditions, the electron power Pe can beexpressed in terms of losses PLoss and output power POut as follows[11]:

Pe = POut + PLoss (4)

As is known, unloaded quality factor (Qu), Qext, and Ql are definedas

Qu = 2�fnPs

PLoss(5)

Qext = 2�fnPs

POut(6)

Ql = 2�fnPs

POut + PLoss(7)

where Ps is the stored energy. Using Eqs. (4)–(7), it can be concludedthat

POut = �cPe (8)

where

�c = 1 − Ql

Qu(9)

�c is usually called the circuit efficiency. Finally by using Eqs. (1),(8), and (9), POut can be formulated as

POut = Qext

(1 + (Qext/Qu))2

A2−1I2

�Ff n(10)

By solving Maxwell’s equations in the interaction space and sideresonators and satisfying the boundary condition at the interfacebetween the side resonators and the interaction space, we havecalculated fn, F, and Qu. Based on these calculations, the effect ofthe geometrical parameters on POut can be determined using (10).Since here we are interested in having the anode and cathode radiiclose to those of the experimental model presented in [5] and alsoin having a constant resonant frequency of about 37.5 GHz, we vary

has been mentioned earlier, for each value of N�/�, d is selectedin a way that provides fn=37.5 GHz. As can be deduced from this

428 N. Nasr Esfahani et al. / Int. J. Electron. Commun. (AEÜ) 67 (2013) 426–432

0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.850

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nor

mal

ized

(PO

ut/I

2 )

Qext

=200

Qext

=300

Qext

=400

Qext

=500

F

fiNpttsuom

2

bpcoeebtaaaptoimeicms

TC

N /

ig. 2. Normalized values of POut/I2 versus N�/� for Qext varying from 200 to 500.

gure, maximum value of the normalized POut/I2 occurs around�/� = 0.6. For N�/�=0.6, d is selected to be 1.39 mm in order torovide fn = 37.5 GHz. The results of extensive numerical simula-ions based on a 2D code which are presented in [7] also confirmhe presence of an optimum selection around N�/� = 0.6 whichhould provide maximum output power. Hence in our design wese N�/� = 0.6. Table 1 summarizes the geometrical parameters ofur design together with the resonant frequency of the operatingode (�/2-mode) and of the neighboring ones.

.2. Details of the simulation model in CST-Particle Studio

The simulation model of the SHM consisting of a 16-vane anodelock, an impedance matching section (a �g/4 transformer), an out-ut waveguide (WR-28 waveguide), and a cold secondary-emissionathode is shown in Fig. 3. As can be seen in this figure, the cath-de has two end caps which play the important role of confining thelectron clouds in the interaction region. It should be noted that thend caps shown in Fig. 3 are nonmagnetic, consequently the distri-ution of the magnetic field in the interaction space is uniform. Theotal current in the SHMs is mainly due to secondary emission fromcold cathode, which is typically made of a copper core coated withplatinum foil [1–4]. In the SHMs, these cathodes are initialized bybombarding current provided by an auxiliary thermionic cathodelaced outside of the interaction space. An exact incorporation ofhe auxiliary cathode in the simulation model is complicated, andn the other hand, the role of this cathode is only important in thenitializing phase of the secondary-emission cathode [5]. Hence our

odel like the models presented in [5,6] does not incorporate thexact modelling of this auxiliary cathode. In our simulations, the

nitialized bombarding current which is a small amount of the totalurrent in the SHMs, is provided by specifying a relatively small pri-ary emission current to the cold cathode. This primary emission is

ufficient to initiate an avalanche process in the growth of emitted

able 1haracteristics of the designed SHM.

Cathode radius (rc) 1.3 mmAnode radius (ra) 2.25 mmAnode length 6 mmN�/� 0.6Vane depth (d) 1.39 mmNumber of vanes (N) 163(�/2 − 1)-mode resonant frequency 34.7 GHz4(�/2)-mode resonant frequency 37.5 GHz5(�/2 + 1)-mode resonant frequency 38.97 GHz

Fig. 3. Simulation model of the SHM consisting of the anode block, secondary-emission cathode and its end caps, end space, transformer, and output waveguide.

secondary electrons in the interaction space. Due to the maincontribution of secondary emission, accurate modeling of this phe-nomenon is of principal importance for any SHM simulation. Theaverage number of emitted secondary electrons from the cold cath-ode is called secondary emission yield. It depends on the impactenergy and impact angle of the primary electrons. In our simula-tions, the dependence of secondary emission yield on the impactangle and impact energy for platinum has been taken from [5] andthen used to adjust the secondary emission properties of the cath-ode in the simulation model. The maximum of secondary emissionyield for platinum equals 1.8 when the impact energy is around0.9 keV. ımax=1.8 is sufficient to initiate an avalanche process inthe growth of the electron concentration in the interaction spaceby utilizing a low bombarding current.

RF power extraction takes place via a quarter-wavelength trans-former followed by a Ka-band waveguide which is part of thesimulation perimeter as shown in Fig. 3. For SHMs as for otherpulsed magnetrons, the radial electric field between cathode andanode is produced by an additional pulsed power device. How-ever, the simulation model presented here does not incorporatethe details of circuit and switches of the pulsed power device. Thisapproach is typical for 3D PIC simulations of magnetron oscillators[8–10].

Fig. 4 shows the resonant frequencies as a function of the phaseshift between two adjacent vanes for two cases: analytically calcu-lated resonant frequencies assuming perfect magnetic walls (PMC)at the top and bottom of the anode block and neglecting the end-space geometry and end caps; and simulated resonant frequenciesobtained from a realistic model including end spaces, end caps,a quarter-wavelength transformer, and an output waveguide. Theslight discrepancy between the realistic model simulation resultsand the results based on the PMC wall assumption is due to theeffects of the end caps, end spaces, quarter-wavelength trans-former, and output waveguide. Resonant modes are identified fromthe vector electric field profile at each resonant frequency. Theoperating mode, i.e., the �/2-mode (considering N = 16, this modeis also called the n = 4-mode), is found to be resonant at 37.5 GHz.As can be seen in Fig. 4, the resonant frequency of the operation

mode is well separated from the neighboring modes. It is one of theadvantages of the non-�-mode operation scenario. Fig. 5 depictsthe Hull cutoff parabola, the Hartree resonant line of the �/2-mode,

N. Nasr Esfahani et al. / Int. J. Electron. Co

0 45 90 135 180 225 270 315 36015

20

25

30

35

40

45

Phase shift (degree)

Fre

quen

cy (G

Hz)

TheorySimulation (CST)

Fig. 4. Resonant frequencies as a function of the phase shift between two adja-cent vanes for two cases: (1) analytically calculated resonant frequencies assumingperfect magnetic walls (PMC) at the top and bottom of the anode cavity and neglect-ing the end-space geometry and end caps and (2) simulated resonant frequenciesobtained from the realistic model.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20

50

100

150

200

250

Axial magnetic flux density (Tesla)

Vol

tage

(kV

)

Hartree resonant line for /2-modeHartree resonant line for -modeHull cutoff voltageOperation point for /2 modeOperation point for -mode

Ff

aptfias

F

ig. 5. Hull cutoff parabola, Hartree resonant condition, and the operation pointsor the �-mode and the �/2-mode.

nd one of the examined operation points of the SHM. For com-arison purpose, the Hartree resonant line for the �-mode and a

ypical operation point for this mode are also presented in thisgure. Comparing the required B and VDC for both the �/2-modend the �-mode clearly confirms the importance of employing thepatial-harmonic operation scenario in reducing the required VDC

ig. 6. Time-evolved spoke formation plotted in the r� plane at: (a) 0.2 ns, (b) 2 ns, (c) 3.5

mmun. (AEÜ) 67 (2013) 426–432 429

and B values. As is seen in Fig. 5, the operation point of SHMs isusually selected in the region where the Hartree resonant line andthe Hull cutoff parabola are close together. This trend is consis-tent with the theoretical expectations, because, as was mentionedbefore, the wave–particle interaction in the SHM is realized as aninteraction of electrons with the first backward harmonic of the�/2-mode or of neighboring modes, and this harmonic is localizednear the anode structure so that, in order to realize an effectiveelectron–wave interaction, the value of VDC should be close to itscritical value determined by the Hull cutoff parabola.

In the next section, considering the above mentioned simulationmodel, the performance of the designed SHM in the presence of anelectron beam will be simulated using CST-Particle Studio.

3. Simulation results

The nonlinear beam–wave interaction dynamics are examinedthrough the use of a PIC simulator embedded in CST-Particle Stu-dio. Our realistic 3D simulations reflect peculiar features of SHMssuch as secondary emission from the cold cathode. The presentedsimulations also does not suffer from restrictive assumptions onthe number of modes or on the number of spatial harmonics,which are among common restrictions in the numerical simulationmethods based on the mode expansion technique (e.g., see Refs.[5,11]).

Fig. 6(a)–(e) shows the time evolution of the space-charge in theinteraction space plotted in the r� plane at 0.2, 2, 3.5, 4.5, and 6 nsfor VDC=12.6 kV, B = 0.53 T and Qext = 200. Fig. 6(b) shows the forma-tion of four electron bunches. As is known, the number of spokesis equal to the number of complete periods � (where � = mN + n,m and n mean harmonic and mode number, respectively) of thesynchronous harmonic around the anode circumference. Thus for-mation of 4 electron bunches indicates the synchronism betweenthe fundamental harmonic (m = 0) of the n = 4-mode (or �/2-mode)and the space-charge during the first few nanoseconds of simu-lation. As time elapses, the synchronism condition does not holdany longer for the fundamental harmonic of the �/2-mode, and12 electron bunches start to gradually form near the anode sur-face (Fig. 6(c)). After 4.5 ns, the existence of 12 electron bunchesis clearly distinguishable (Fig. 6(d) and (e)). The formation of 12electron bunches indicates that the space-charge shows angularvelocities close to that of the first backward harmonic (m = −1)of the �/2-mode (n = 4-mode).This confirms the �/2-mode oscil-lations. Beyond 3.5 ns, this oscillation mode is dominant for thewhole duration of the simulation. Fig. 7(a) and (b) depicts the max-imum voltage amplitude across the output waveguide broad walls(VOut) as a function of time, and the normalized amplitude of itsfrequency spectrum in dB. As can be deduced from Fig. 7(a), oscil-

lation starts to grow abruptly at about 3.5 ns which coincides withthe time of gradual formation of 12 spokes in the interaction space.The output voltage reaches its saturation value (VOut=5618 V) after9 ns.

ns, (d) 4.5 ns, and (e) 6 ns. The results are for VDC=12.6 kV, B = 0.53 T, and Qext = 200.

430 N. Nasr Esfahani et al. / Int. J. Electron. Commun. (AEÜ) 67 (2013) 426–432

0 2 4 6 8 10-6

-4

-2

0

2

4

6

Time (ns)

Out

put v

olta

ge (k

V)

0 5 10 15 20 25 30 35 40 45-1500

-1000

-500

0

Frequency (GHz)

Nor

mal

ized

vol

tage

mag

nitu

de (d

B)

Fig. 7. (a) Output voltage amplitude measured at the center-line of the output port across the output waveguide broad walls. (b) Normalized amplitude of the frequencyspectrum obtained from a Fourier transform of the time-varying output voltage. The results are for VDC = 12.6 kV, B = 0.53 T, and Qext = 200.

24

68

10

100

150

200

250

300-6-4-20246

Time (ns)

Qext

VO

ut (k

V)

24

68

10

400

450

500

550

600-6-4-20246

Time (ns)

Qext

VO

ut (k

V)

F 00 an

P

wroqt

pb

4

pural1cQta

0

ig. 8. (a) Output voltage for VDC = 12.6 kV, B = 0.53 T, and Qext varying from 100 to 3

The saturated output power is calculated using

Out =(

a

4bZ0

)V2

Out

√(1 −

(fcf

)2)

(11)

here a and b are the output waveguide width and height,espectively, Z0, fc, and f mean wave impedance, cutoff frequencyf output waveguide (WR28-waveguide), and the operation fre-uency. Considering a = 7.11 mm, b = 3.55 mm, and VOut = 5618 V,he saturated output power is equal to 34.6 kW.

In the next section, the effect of varying Qext and VDC on outputower, efficiency, and operation stability of the designed SHM wille considered.

. Examining the effect of Qext and VDC

Fig. 8(a) and (b) shows the output voltage for the simulationserformed for Qext varying from 100 to 600. For all of these sim-lations, VDC and B are set to be equal to 12.6 kV and 0.53 T,espectively. As can be seen in these figures, the output voltagend consequently the output power (see Fig. 10) which is calcu-ated using (11) are steadily increasing while Qext increases from00 to 300. Then increasing Qext from 300 to 500 does not have a

onsiderable effect on output voltage and output power. Finally ifext is increased to values greater than 500, the output power starts

o decrease with a sharp slope. In all of the above mentioned cases,fter a few nanoseconds the �/2-mode with a frequency of about

0

d (b) output voltage for VDC = 12.6 kV, B = 0.53 T, and Qext varying from 400 to 600.

37.5 GHz is the stable and dominant mode of operation and themulti-stability difficulties reported in [5] have not been observedfor this design even when the simulation time is increased to about60 ns. Multi-stability which means a sudden switching of the spacecharge to a different state with an essentially different value ofoutput power can seriously damage the reliability of the mag-netron performance. For Qext greater than 300, a transient modecompetition mainly with a higher order mode (n = 5-mode) witha frequency of about 39 GHz occurs. The AM-modulated appear-ance of the output voltage when Qext is greater than 300 is mainlydue to this transient competition. For these values of Qext, modecompetition continues up to only about 8 ns or even less, and afterat most 8 ns, the �/2-mode becomes the dominant mode of opera-tion. Fig. 9 shows the variation of anode current IAnode versus Qext. Ascan be seen from this figure, the anode current steadily increaseswith increasing Qext, and the efficiency � which is defined as theratio of output power to input power (here, input power is givenby Pin = IAnodeVDC) monotonically decreases (see Fig. 10). The rea-son for the decrease of the efficiency with an increase in Qext canbe deduced from the following discussion: as has been mentionedearlier, � is defined as

� = POut (12)

IAnodeVDC

Considering (8) and (9), � can also be expressed as

� = �c�e (13)

N. Nasr Esfahani et al. / Int. J. Electron. Commun. (AEÜ) 67 (2013) 426–432 431

02

46

810

100

200

300

400

500

6000

5

10

15

20

25

Time(ns)

Qext

Ano

de c

urre

nt(A

)

Fa

w

�

�iQ�t�tcuitwt

Bssop�

FV

02

46

810

12.2

12.4

12.6

12.8

-8-4048

Time (ns)

DC Anode voltage (kV

)

VO

ut (k

V)

Fig. 11. Output voltage for B = 0.53 T, Qext = 200, and for VDC varying from 12.2 kV to13 kV in steps of �VDC = 200 V.

0

5

10

12.2

12.4

12.6

12.8

130

5

10

15

20

25

Time(ns)

DC Anode voltage (kV)

Ano

de c

urre

nt (A

)

ig. 9. Temporal current collected on the anode block for VDC = 12.6 kV, B = 0.53 T,nd Qext varying from 100 to 600.

here �e is

e = Pe

VDCIAnode(14)

e is usually called electronic efficiency. Since both Pe and IAnodencrease with increasing Qext, �e does not vary considerably withext. Hence the variation of � with Qext is mainly determined byc which decreases with increasing Qext. It should be mentionedhat Qext of the designed SHM is controlled by the width of theg/4 transformer (Fig. 3). In our simulations, the width of theransformer has been varied within the range 0.15–0.4 mm whichorresponds to Qext ranging from 600 to 100. Selecting Qext val-es less than 100 or greater than 600 is not recommended, since

t causes power reduction. Moreover, realizing Qext values greaterhan 600 necessitates the use of very small transformer width,hich is either impractical or results in vacuum breakdown due

o the high electric field intensity inside the transformer.Figs. 11 and 12 show the output voltage and anode current for

= 0.53 T, Qext = 200 and for VDC, varying from 12.2 kV to 13 kV inteps of �VDC=200 V. As can be deduced from Fig. 11, for all of these

imulations after at most 8 ns, the �/2-mode is the dominant modef operation except for VDC = 12.2 kV. In case of VDC = 12.2 kV, com-etition between the �/2-mode and a lower order mode (n = 3 (or/2 − 1)-mode) continues up to 10 ns. In this case, we performed

ig. 10. Simulated output power and efficiency versus Qext . Results are forDC = 12.6 kV, B = 0.53 T, and Qext varying from 100 to 600.

Fig. 12. Temporal current collected on the anode block for B = 0.53 T, Qext = 200 andfor VDC varying from 12.2 kV to 13 kV in steps of �VDC = 200 V.

other simulations for 60 ns of simulation time and it was found thatafter 30 ns the dominant operation mode in this case is the 3 (or�/2 − 1)-mode with a frequency of about 34.2 GHz. Thus it can beconcluded that for VDC varying from 12.4 kV to 13 kV, the �/2-mode

Fig. 13. Simulated output power and efficiency versus VDC . Results are for B = 0.53 T,Qext = 200 and for VDC varying from 12.4 kV to 13 kV in steps of �VDC = 200 V.

4 ron. Co

ifev

5

PofsoTo

nomt

R

[

[

[

microwave circuits, diode modeling, solid-state oscillators, and integrated-circuit

32 N. Nasr Esfahani et al. / Int. J. Elect

s the operation mode. Fig. 13 shows output power and efficiencyor different values of VDC. As can be deduced from this figure, thefficiency varies from about 15% to 18.5%, and the output poweraries from 25 kW to 47 kW.

. Conclusions

In this paper, design and 3D simulations of a SHM using CST-article Studio have been presented. The effect of resonator openingn output power has been taken into account and an optimum valueor the resonator opening has been determined. The time evolvedpoke formation, transient mode competition, and the formationf stable oscillations in the interaction space have been examined.he effect of varying Qext and VDC on output power, efficiency andperation stability of the designed SHM has also been considered.

It should be mentioned that the presented 3D simulations doesot suffer from restrictive assumptions on the number of modesr on the number of spatial harmonics, which are among com-on restrictions in the numerical simulation methods based on

he mode expansion technique.

eferences

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Nasrin Nasr Esfahani received the B.Sc. (Hons.) and M.Sc. (Hons.) degrees bothin Electrical Engineering from Iran University of Science and Technology (IUST),Tehran, Iran in 2004 and 2007, respectively. She is currently working towardsthe Ph.D. degree in Electrical Engineering. Her research interests include electro-magnetic theory, passive and active microwave components, metamaterials,andmillimeter-and terahertz-wave vacuum electron devices.

Majid Tayarani received the B.Sc. degree from Iran University of Science and Tech-nology, Tehran, Iran, in 1988, the M.Sc. degree from Sharif University of Technology,Tehran, Iran in 1992, and the Ph.D. degree in communication and systems fromthe University of Electro-Communications, Tokyo, Japan, in 2001. Since 1992, hehas been a member of the faculty with the Department of Electrical Engineering,Iran University of Science and Technology, Tehran, Iran, where he is currently anAssistant Professor. His research interests are qualitative methods in engineeringelectromagnetics, EMC theory, microwave and millimeter-wave linear and non-linear circuit design, microwave measurement techniques, and noise analysis inmicrowave signal sources.

Klaus Schünemann received the Dipl.-Ing. degree in electrical engineering and theDr.-Ing. degree from the Technische Universität Braunschweig, Germany, in 1965and 1969, respectively. From 1983 to 2004, he was a Professor of Electrical Engi-neering and Director of the Arbeitsbereich Hochfrequenztechnik at the TechnischeUniversität Hamburg-Harburg, Hamburg, Germany. He has worked on nonlinear

technologies. His research interests include transport phenomena in submicrondevices, optoelectronics, high-power and high-frequency tubes, submillimeter-wave circuit technology, and millimeter and submillimeter-wave radars for climateand environmental research.