highly efficient microwave power system of magnetrons

4424 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 68, NO. 10, OCTOBER 2020

Highly Efficient Microwave Power System ofMagnetrons Utilizing Frequency-Searching

Injection-Locking TechniqueWith No Phase Shifter

Chao Lai, Chaoxia Zhao, Kang Li , Danli Cai, Yi Zhang , Member, IEEE, Yang Yang , Member, IEEE,

Huacheng Zhu , Member, IEEE, Dinesh K. Agrawal, Tania Slawecki , Li Wu, Yanping Zhou ,

Qian Chen, Lin Zhou , and Kama Huang, Senior Member, IEEE

Abstract— A microwave power-combining technique basedon magnetrons has been widely researched to solve theurgent demands for high-power, low-cost microwave sourcesin microwave industrial applications. To achieve high resultantefficiency, conventionally, injection locking with static frequencyis utilized to drag the output frequency of each magnetron sourceto the exact same frequency, and a phase shifter is required toadjust the output phase of each combining source. In this article,we propose a novel phase shifterless microwave power systemwith two magnetrons utilizing a frequency-searching injection-locking technique. Coaxial cables with different physical lengthsare used to realize phase adjustment under different injectionfrequencies. By sweeping the injection frequency in a smallband, the phase difference between combining signals can beadjusted. Thus, high resultant efficiency can be obtained. Whenthe injection frequency is swept, the phase difference between theoutput and injection signals has been analyzed theoretically, thephase difference between the different-length coaxial injectioncables has also been analyzed. Furthermore, the relationshipbetween the sweeping frequency bandwidth and requirementsof the minimum length difference of the coaxial injection cableshas been derived and analyzed. Experimental results show thata high resultant efficiency of 94.6% can be obtained with theproposed phase shifterless system. The system could maintainhigh resultant efficiency even when the free-running frequencyand power of one magnetron are changed by adjusting its anodecurrent. The experiments reveal that the proposed method worksfor different combining sources.

Index Terms— Frequency sweeping, injection locking, mag-netron, microwave power combining, phase shifter.

Manuscript received May 5, 2020; accepted May 11, 2020. Date ofpublication July 15, 2020; date of current version October 5, 2020. This workwas supported by the National Natural Science Foundation of China underGrant 61901286 and Grant 61731013. (Corresponding author: Yi Zhang.)

Chao Lai, Chaoxia Zhao, Kang Li, Danli Cai, Yi Zhang, Yang Yang,Huacheng Zhu, Li Wu, Yanping Zhou, Qian Chen, and Kama Huang are withthe College of Electronics and Information Engineering, Sichuan University,Chengdu 610017, China, and also with the Key Laboratory of Wireless PowerTransmission of Ministry of Education, Chengdu 610064, China (e-mail:[email protected]).

Dinesh K. Agrawal and Tania Slawecki are with the Materials ResearchInstitute, Pennsylvania State University, University Park, State College,PA 16802 USA (e-mail: [email protected]).

Lin Zhou is with the School of Electrical and Electronic Engineering,Nanyang Technological University, Singapore 639798 (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TMTT.2020.3006488

I. INTRODUCTION

M ICROWAVE energy has been increasingly applied toindustrial applications, such as plasma deposition, com-

munication, medical, and material processing [1]–[5], whichresults in the enhanced demands for high-power, low-costmicrowave sources. A magnetron that has efficiency around80% and is available at low cost is very desirable for manyindustrial applications [6]. However, the output power of asingle magnetron cannot meet the rapidly increasing require-ments for high-power industrial applications. The output powerof a single magnetron is limited since its cavity size isdetermined by its operation frequency [7], [8]. At 915 MHz,a continuous-wave (CW) magnetron is capable of generating100-kW microwave power, while at 2.45 GHz, high-powerCW magnetrons can generate up to only 20-kW microwavepower [7], [8]. On the other hand, the lifetime of thehigh-power CW magnetron is another issue. The lifetime ofthe CW magnetrons is strongly related to its cathode. Thelong-life cathode for the low-power CW magnetron is welldeveloped and widely applied, while the cathode for the high-power, high-frequency CW magnetron is still under develop-ment [9], [10]. Therefore, the power combining technique,which combines the microwave power of the well-developed,low-power magnetrons to realize a high-power, low-cost, andlong-lifetime microwave source, is a quite suitable way toaddress such problems [11].

Though the idea of combining power is fascinating,still high-efficiency coherent microwave power combiningby this method is not easily achieved. The main reasonsinclude that every magnetron operates at a slightly differentfrequency [12]–[14]; the output phase of a magnetron israndom every time it starts operating [15]. To address thefrequency discrepancy problem, injection locking is a well-known method, which can lock the output frequency of amagnetron to a stable static injection frequency when theinjection power meets Adler’s condition [16]–[19].

After the output frequency of every magnetron source isdragged and locked to the same injection frequency, phaseshifters are also required in the conventional microwave powercombining system [15], [20]–[23]. The phase shifters are

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https://orcid.org/0000-0002-3024-7616

https://orcid.org/0000-0003-3045-409X

https://orcid.org/0000-0003-0764-1575

https://orcid.org/0000-0002-1737-870X

https://orcid.org/0000-0002-2013-0211

https://orcid.org/0000-0002-3982-8337

https://orcid.org/0000-0001-9653-2551

LAI et al.: HIGHLY EFFICIENT MICROWAVE POWER SYSTEM OF MAGNETRONS 4425

Fig. 1. Schematic of the proposed microwave power-combining system.

used to adjust the signal from every combining branch to besame phase in the combiner to achieve high resultant effi-ciency. Shinohara [15], [20] Shinohara et al. [21] and devel-oped the active phased array with phase-controlled magnetrons(PCMs), whose output frequency is locked to the injectionsignal and output phase is tuned by the phase shifter in theinjection part to achieve spatial microwave power combin-ing for wireless power transmission (WPT) [15], [20], [21].Liu et al. [8], [22] and Huang et al. [23] developed high-power microwave sources with two- and four-way waveguidepower combining systems, which utilized phase shifters toadjust the output phase and injection locking technique.We previously developed a microwave power system withmaster–slave injection-locked magnetrons, but a phase shifteris still applied to adjust the phase of the signal from theslave magnetron [11]. Recently, Liu et al. [24] developed thefirst (to the best of our knowledge) phase shifterless powercombining system of two magnetrons with reported highestresultant efficiency around 86%, which is achieved throughasymmetric injection.

In this article, a novel phase shifterless microwave powercombining system with two magnetrons utilizing a frequency-searching injection-locking technique is proposed. The resul-tant efficiency of the proposed system, whose schematic isshown in Fig. 1, exceeds 94%. The system utilizes twodifferent-length coaxial cables to transmit the very low powerinjection signal. The physical length difference causes a vari-ation of the phase difference in the two cables once the injec-tion frequency varies. Therefore, the phase of the combiningsignals can be adjusted by sweeping the injection frequency.High resultant efficiency can be obtained by searching for theright injection frequency. In this way, the proposed systemrequires no phase shifter to achieve in-phase at the combinerand high resultant efficiency of the whole system. Based on

this idea, the relationship between the frequency sweepingbandwidth and the physical length difference of the coaxialinjection cables has been theoretically derived and analyzed.Experiments of two CW magnetrons have been carried out,and the testing results are in accord with the theoreticalpredictions.

II. THEORETICAL ANALYSIS

A. Injection Locking of Magnetron

Since the magnetrons in the power combining systemhave distinctive free-running frequencies, analyzing the phasechange during the injection of these magnetrons is of greatimportance. Adler’s condition, which is the foundation ofinjection locking of oscillators, can be written as [16]

sinθ =(

1 − ωi

ω f

)2Qext

ρ(1)

where ρ is the injection ratio, Qext is the external Q-factor ofthe magnetron system, ωi and ω f are the angular frequenciesof the injection signal and free-running magnetron, respec-tively, and θ is the phase difference between the output signalof the magnetron and its injection signal.

From Adler’s condition, once the frequency of the injec-tion signal, the frequency of the free-running magnetron, theinjection power, and so on are obtained, the value of sinθcan be calculated. However, the problem is that though thevalue of sinθ could be obtained, and there are two solutionsof θ in [0, 2π]. To solve the problem of multiple solutions,we have to start from the equation before the derived Adler’scondition. Based on the equivalent circuit of the magnetron,the phase difference between the output and injection signalof the magnetron can be written as the following differentialequation [16], [25]:

dθ

dt= ρ

2Qextsin θ + ωi

ω f− 1. (2)

If the magnetron is injection-locked, then the phase differencebetween the injection and output signals becomes constant,indicating that (dθ/dt) equals 0. Equation (2) becomes Adler’scondition.

The differential equation (2) can be solved to get the valueof θ , which can be written as [25]:

θ(ωi ) = 2 arctan

⎛⎝ A

B− F

√(A

B

)2

− 1

⎞⎠ (3)

where

A = ρ

2Qext

B = 1 −(ωi

ω f

)

D =√

A2 − B2

F = 1 − eD(t−t0)

1 + eD(t−t0)

where t is time, and θ(ωi) is the phase difference between theoutput signal of the magnetron and its injection signal. In thisarticle, the injection signal is swept in a certain frequency



band. Therefore, the phase difference (θ) is a function of theinjection frequency (ωi ).

From (3), the phase difference (θ) may vary with time,which is not desired for high resultant efficiency. To avoid that,D in (3) must be real (F approaches −1 as time increases).Therefore, the following equation must be satisfied [16], [25]:

2Qext

∣∣∣∣1 − ωi

ω f

∣∣∣∣ ≤ ρ. (4)

Equation (4) is the well-known Adler’s criterion that, whensatisfied, the output frequency of the magnetron is fully lockedto the frequency of the injection signal. If Adler’s criterion isnot satisfied, sideband frequencies appear, and the phase dif-ference (θ) becomes a function of time [25]. Adler’s criteriondetermines if the injected magnetron could be phase-locked ornot, while (3) calculates the phase difference (θ).

Based on (4), the locking bandwidth (ωBW) can be derivedand expressed as

ωBW = ∣∣ωi − ω f

∣∣ = ρω f

2Qext. (5)

The locking bandwidth defines the boundaries of the injectionsignal’s frequency. When the frequency of the injection signallies in the locking bandwidth, the magnetron can be fullylocked to the injection signal.

B. Resultant Efficiency

When the two magnetrons in the system are both fullylocked to one injection signal source, their output signals arecoherent. The resultant efficiency (η) of two coherent signalscan be written as [8], [11]

η = 1

2+

√Pm1 Pm2

Pm1+Pm2cos

(�ψc

)(6)

where Pm1 and Pm2 are the output powers of magnetron1 andmagnetron2, respectively. �ψc is the phase difference of thetwo combining signals. The highest resultant efficiency isobtained when �ψc equals 2kπ and Pm1 equals Pm2 [8].

The output powers of the magnetrons (Pm1 and Pm2) arecontrolled by their anode currents [14]. The phase difference(�ψc) is related to the travel paths of the signal since boththe two magnetrons are injection-locked to one signal source.For the convenience of analyzing the signals’ phase in thesystem, the reference planes of the phases are described belowand labeled in Fig. 1. The phase of the output signal of thesolid-state source is defined and labeled as ψss. The phase ofthe output and injection signals of magnetron1 is defined andlabeled as ψm1o, and ψm1i , respectively, similarly for ψm2o,and ψm2i of magnetron2. At the output port of the combiner,the phase of the signal from magnetron1 is defined and labeledas ψc1; ψc2 is the phase of the signal from magnetron2 onthe same reference plane. The phase difference caused bythe long- and short-coaxial cables is described as θlc and θsc,respectively.

To analyze the phase difference (�ψc), ψc1 and ψc2 shouldbe analyzed since �ψc equals (ψc2 − ψc1). ψc1 can beexpressed as

ψc1(ωi) = ψss − θlc(ωi )− θm1(ωi )− χ1(ωi) (7)

where χ1(ωi) represents the total phase shift caused by thecirculators, the coaxial to waveguide adaptor, and the coaxialcable after the amplifier in the left branch of the system, asshown in Fig. 1, the subscript “m1” denotes magnetron1, andthe subscript “m2” in the following parts denotes magnetron2.θm1(ωi ) represents the phase difference between the outputsignal of magnetron1 and its injection signal.

In a similar way, ψc2 can be expressed as

ψc2(ωi ) = ψss − θsc(ωi )− θm2(ωi )− χ2(ωi ) (8)

where χ2(ωi ) represents the total phase shift caused by thecirculators, the coaxial to waveguide adaptor, and the coaxialcable after the amplifier in the right branch. θm2(ωi ) rep-resents the phase difference between the output signal ofmagnetron2 and its injection signal.

Based on (7) and (8), �ψc can be expressed as

�ψc(ωi ) = ψc2(ωi )− ψc1(ωi )

= [θlc(ωi )− θ sc(ωi )

] + [θm2(ωi )− θm1(ωi )]

+ [χ1(ωi )− χ2(ωi )]. (9)

Since χ1(ωi ) and χ2(ωi) represent the phase shift of thedevices mirror to each other in the system, and the sweepinginjection frequency is in a very limited band, [χ1(ωi)−χ2(ωi )]can be approximated by a constant C , considering the smalldifference of each component. Then, �ψc(ωi ) can be writtenas

�ψc(ωi ) = [θlc(ωi )− θ sc(ωi )

] + [θm2(ωi )− θm1(ωi )] + C.

(10)

C. Requirements for the Coaxial Injection Cables

The long- and short-coaxial cables in the injection sys-tem play a vital role in achieving high resultant efficiency.The phase difference (θcable) of the injection signal travelingthrough a coaxial cable can be calculated by the followingequation [26]:

θcable(ωi ) = Lωi√εrμr

ν(11)

where L is the length of coaxial cable, ν is the speed of light,and εr and μr are the relative permittivity and permeability ofdielectric materials in the coaxial cable, respectively.

With (11), [θlc(ωi )− θ sc(ωi )] can be written as (�Lωin j

(εrμr )1/2)/ν, Therefore, (10) can be written as

�ψc(ωi )− C = �Lωi√εrμr

ν+ [θm2(ωi )− θm1(ωi )]. (12)

((�Lωi (εrμr )1/2)/ν + [θm2(ωi )− θm1(ωi )]) is represented by

�ψcm(ωi ) for simplification in the following.To obtain high resultant efficiency, one should find the

injection frequency (ωi ) that makes �ψc equal to 2kπ .To guarantee that the method works for different systems[regarding different C in (12)], �ψcm should traverse at least2π as the injection frequency is swept from the lowest to thehighest.

To intuitively show how �ψcm varies with the injectionfrequency (ωi ), Fig. 2 demonstrates the variation of �ψcm



Fig. 2. Phase difference variation with sweeping injection frequency.

TABLE I

SCANNING RANGE OF �ψcm WITH VARIATION

OF CABLE LENGTH DIFFERENCE

with ωi sweeping from 2.44 to 2.46 GHz. The free-running fre-quencies of the two magnetrons are assumed to be 2.451 and2.4519 GHz. θm1(ωi ) and θm2(ωi ) in (10) are calculatedfrom (1). The relative permittivity (εr ) and permeability (μr )of dielectric materials are 2.3 and 1, respectively [26]. Lengthdifference of the coaxial cables in theoretical analysis isincreased from 3 to 15 m with an interval of 3 m.

From Fig. 2, it is easy to find that �ψcm increases with theincreasing of the injection frequencies (ωi). Simultaneously,the slopes of the lines increase with the increasing of thelength difference of the cables (�L). The higher slopes of thelines result in the higher scanning range of �ψcm, as shownin Fig. 2 and Table I. The scanning range of �ψcm increasesfrom 0.57π to 2.97π as �L increases from 3 to 15 m.

Since �ψcm increases with the increasing of the injectionfrequencies (ωi ), the minimum length difference of the coaxialcables can be calculated. To obtain high resultant efficiency,�ψc needs to equal 2kπ , but C in (12) may vary in differentsystems. If �ψcm can traverse over 2π as the injectionfrequency (ωi ) is swept from the lowest to the highest, thenthere must exist at least one injection frequency that makes�ψc to equal 2kπ . High resultant efficiency can then beobtained under that injection frequency. Based on this idea,the minimum length difference of the coaxial cables shouldensure that the traversal range of �ψcm reaches 2π . Since�ψcm monotonically increases with ωi , as shown in Fig. 2,the maximum and minimum of �ψcm can be calculated asfollows:

�ψcmmax = [θm2

(ωimax

) − θm1(ωimax

)] + �Lωimax

√εrμr

ν(13)

�ψcmmin= [

θm2(ωimin

) − θm1(ωimin

)] + �Lωimin

√εrμr

ν(14)

Fig. 3. Required minimum length difference of the coaxial cables (�Lmin)variation with the frequencies of free-running magnetrons.

In (13) and (14), the injection frequency (ωi) range isrelated to the locking bandwidth of the magnetrons, whichcan be calculated based on (5). In our proposed system, bothmagnetron1 and magnetron2 need to be fully locked during thesweeping of the injection frequency. Therefore, the upper andlower boundaries of the injection frequencies can be calculatedby the following equations:ωimax = MIN

{(ω f 1 + ωBW1

2

),(ω f 2 + ωBW2

2

)}= MIN

{(ω f 1 + ρ1ω f 1

4Qext1

),

(ω f 2 + ρ2ω f 2

4Qext2

)}(15)

ωimin = MAX

{(ω f 1 − ρ1ω f 1

4Qext1

),

(ω f 2 − ρ2ω f 2

4Qext2

)}. (16)

To obtain high resultant efficiency, �ψcm should traverseover 2π . Therefore, the following equation should be satisfied:

�ψcmmax−�ψcmmin

≥ 2π. (17)

Based on (17), the minimum length difference of the coaxialcables (�Lmin) is the value that makes (�ψcmmax

−�ψcmmin)

equals 2π . Substituting (13)–(16) into (17), �Lmin can becalculated by the following equation:

�Lmin = ν√εrμr

(ωimax − ωimin

) · (P + Q) (18)

where

P = 2π − [θm2

(ωimax

) − θm2(ωimin

)]Q = θm1

(ωimax

) − θm1(ωimin

).

With (18), the minimum length difference of the coax-ial cables (�Lmin) can be calculated once the frequenciesof the free-running magnetrons in the system are obtained.Fig. 3 demonstrates the requirements of the coaxial cables fordifferent free-running magnetrons’ frequencies. The injectionfrequency is swept from 2.44 to 2.46 GHz, and the injectionratio (ρ) is assumed to be 0.25.

From Fig. 3, it is easy to find that the minimum of�Lmin always occurs when the frequency of free-runningmagnetron1 equals that of magnetron2. As the gap betweenthe frequencies of the two free-running magnetrons increases,�Lmin increases. In addition, �Lmin does not linearlyincrease with the frequency gap of the two magnetrons



Fig. 4. (a) Photograph of the developed experimental system. (b) Blockdiagram of the experimental system.

under free-running conditions. Instead, the increasing speed of�Lmin goes higher as the frequency gap becomes wider.Therefore, two sources with closer free-running frequenciesare beneficial for the proposed power combining system sinceless length difference of cables is required.

III. EXPERIMENTAL SETUP

Based on the theoretical analysis, an experimental system,as shown in Fig. 4 (based on the schematic of Fig. 1), has beenbuilt to verify the abovementioned theoretical predictions. Theinjection signal is generated by a solid-state microwave source(Hittite HMC-T2220) whose output frequency and amplitudecan be tuned. The injection signal is divided and amplifiedby a power divider and microwave amplifiers, respectively.Long- and short-coaxial cables are used to connect the powerdivider and the amplifiers. Since the power of the signals inthese cables is very low, the cables do not require high powercapacity and, thus, can be low loss and low cost [27]. Theamplified injection signals are injected into the magnetronsthrough the circulators. Every two circulators in each branchare applied to separate the injection and reflection signals, andthe reflection power is absorbed by the loads connected to thecirculators. The insertion loss and isolation of the waveguide

Fig. 5. Output spectrum of the free-running magnetrons.

circulators are less than 0.3 dB and more than 20 dB, respec-tively, which ensures the smooth transmission of microwaveand high isolation between the two magnetrons. The mag-netrons in the experimental system are Panasonic 2M244,which can generate around 1000 W of continuous microwavepower. A waveguide excitation cavity was applied to extractmicrowave generated from the magnetron and to form the TE10

mode in the waveguide, thus achieving a good match. Double-directive couplers are applied to measure the output power andspectrum on each branch. The couplers are not essential inpractical applications. The power and spectrum are monitoredby the AV2433 power meter and the ROHDER & SCHWARZFSP spectrum analyzer, respectively. Microwave power iscombined through a T-junction [11] and absorbed by theconnected load.

IV. RESULTS AND DISCUSSION

The spectra of the two magnetrons under free-running con-ditions are measured, as shown in Fig. 5. Notably, the centralfrequencies of the two free-running magnetrons are not coinci-dent under free-running conditions. The central frequencies ofmagnetron1 and magnetgron2 are 2.45125 and 2.45187 GHz,respectively. Thus, the power combining system cannot reachhigh resultant efficiency if free-running magnetrons wereapplied in the combining system. Simultaneously, one canalso note that the spectrum of magnetron2 is slightly differentfrom that of magnetron1. Magnetron2 has a relatively widerbandwidth than magnetron1. The difference of the spectraresults from the different power supply systems used for thetwo magnetrons [17]. The power system for magnetron1 hasa relatively lower ripple, and its anode and filament currentscan both be adjusted.

Once the two magnetrons are injection-locked, the outputfrequencies of them will be pulled and locked to the frequencyof the injection signal. With the injection frequency sweptfrom 2.440 to 2.457 GHz under the injection ratio of 0.27,the output frequencies of both of the two magnetrons arelocked and follow the injection frequencies, as shown in Fig. 6.

The experiments of power combining were carried out afterthe two magnetrons were injection-locked. The variation ofthe resultant efficiency and output power of the magnetrons isdemonstrated in Fig. 7, as the injection frequency is sweptfrom 2.440 to 2.457 GHz. The output power of the twomagnetrons varies a little bit when the injection frequency



Fig. 6. Output spectrum of the injected magnetron.

Fig. 7. Resultant efficiency with sweeping frequency.

changes. The explanation of the output power variation withthe injection is that the output signal amplitude is slightlyrelated to the injection frequency, which can be calculated bythe following equation [25]:

1

Vrf

dVrf

dt+ 1

Q0

(1 − 1

Vrf

)= − ρ

2Qextcos θ (19)

where Vrf is the amplitude of the output rf signal and Q0 isthe unloaded Q-factor of the magnetron system.

Notably, from Fig. 7, the resultant efficiency increases from30% to a maximum of 94.6% and then goes down to 40%. Thehighest resultant efficiency was obtained under the injectionfrequency of 2.448 GHz. The efficiency variation with theinjection frequency reveals that high resultant efficiency canbe obtained when the injection frequency is swept in a certainband.

The output frequency and power of the magnetron arestrongly related to its anode current [14]. In our experimentalsystem, a power supply with an adjustable anode current wasused to power magnetron1. With the rise in the anode current,the output power and frequency both increase monotonically,

Fig. 8. Power-combining with different anode current of Magnetron1, whichis (a) 220 mA, (b) 260 mA, (c) 300 mA, (d) 340 mA, (e) 380 mA, and(f) 410 mA.

TABLE II

OUTPUT POWER VARIATION WITH THE FREE-RUNNING FREQUENCY

as demonstrated in Table II. The output power and frequencyincreased from 504 to 998 W and 2.443225 to 2.451375 GHz,respectively, as the anode current increased from 220 to420 mA. Magnetron2 was also tested by the same powersupply under free-running conditions though, in the combiningexperiments, it was powered by a supply with fixed anodecurrent. Notably, from Table II, the output frequencies werea little bit different even when the anode current was thesame since a magnetron is a vacuum device. The phenomenaindicate that injection locking is essential for coherent powercombining to achieve high resultant efficiency.

With the variation of the anode current of magnetron1,we can examine the power combining system with differentsource frequencies and powers. Fig. 8 shows the experimentalresults of the power combining the system with a fixedanode current of magnetron2 (396 mA) and variable anodecurrent of magnetron1. In Fig. 8(a), the anode current ofmagnetron1 is 220 mA. As the injection frequency is sweptfrom 2.440 to 2.457 GHz, the output power of magnetron1 and



Fig. 9. Variation of the highest resultant power and efficiency with the anodecurrent of magnetron1.

magnetron2 fluctuates between 465 and 548 W and between933 and 1060 W, respectively. The resultant efficiencyincreases from 28% to a maximum of 89% and then decreasesto 25%. In Fig. 8(b), the anode current of magnetron1 is260 mA. The resultant efficiency has a very similar vari-ation tendency, with a maximum efficiency of 90.5%. Theaverage powers of magnetron1 and magnetron2 are 582 and996 W, respectively, and the power fluctuation range ofmagnetron1 and magnetron2 is from 535 to 628 W andfrom 926 to 1070 W, respectively. The anode currents ofmagnetron1 in Fig. 8(c)–(f) are 300, 340, 380, and 410 mA,respectively. The average powers of magnetron1 in thesefigures are 665, 772, 849, and 914 W, respectively. Themeasured resultant efficiencies are 91.8%, 93.8%, 94%, and94.6%. All these subfigures in Fig. 8 have very similarresultant efficiency variation tendency although the frequencyof magnetron1 under free-running conditions increases from2.443225 to 2.45125 GHz as the anode current increases from220 to 410 mA.

The highest obtained resultant efficiency for different anodecurrent of magnetron1 is summarized in Fig. 9. Here, we needto note that the anode current of magnetron2 is fixedat 396 mA, while the anode current of magnetron1 increasedfrom 220 to 410 mA. The highest resultant power andefficiency increased from 1.37 to 1.83 kW and from 89%to 94.6%, respectively. The increase in the resultant effi-ciency results from the shrinkage of the power differencebetween the two magnetrons. From (4), it is easy to find thatthe highest resultant efficiency (when the phase differenceof the combining signals already reaches 2kπ) is obtainedwhen the powers of two magnetrons are the same since((Pm1 Pm2)

1/2)/(Pm1+Pm2) ≤ (1/2) and the condition ofequality is that Pm1 equals Pm2. As |Pm1 − Pm2| goes up,((Pm1 Pm2)

1/2)/(Pm1+Pm2) gets smaller.

V. CONCLUSION

In this article, a novel phase shifterless microwave powersystem with two magnetrons utilizing a frequency-searchinginjection-locking technique was proposed and tested. Twocoaxial injection cables with certain physical length differ-ences were applied in the system to achieve phase tuningby sweeping the injection frequency. The requirements of

minimum length difference of the coaxial cables to achieve2π phase adjustment under different frequencies of mag-netrons were derived and analyzed theoretically. An exper-imental system using two magnetrons with differentfree-running frequencies and power supply systems was inves-tigated. A high resultant efficiency of 94.6% was obtained.The investigation of the power combining experiments withdifferent anode currents applied to magnetron1 revealed thatthe proposed method works for different combing sources.

Currently, the system works for only two combiningsources. The way to realize high-efficiency power combin-ing with a low-cost, simplified multisource system is forfuture research. The tested and proven method herein pro-vides an idea and example for the development of low-cost, high-efficiency microwave power sources. The systemis also capable of providing high microwave power for certainapplications.

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Chao Lai received the B.S. degree in communica-tion and engineering from the School of Informa-tion Science and Technology, Chengdu Universityof Technology, Chengdu, China, in 2018. He iscurrently pursuing the M.S. degree at the Collegeof Electronics and Information Engineering, SichuanUniversity, Chengdu.

His current research interest is microwave powercombining.

Chaoxia Zhao received the B.S. degree in appliedchemistry and the M.S. degree in polymer chem-istry and physics from Sichuan University, Chengdu,China, in 2011 and 2015, respectively, where she iscurrently pursuing the Ph.D. degree at the Collegeof Electronics and Information Engineering.

Her current research interests include microwaveenergy applications and applications of microwaveplasma in wastewater treatment.

Kang Li received the B.S. degree in electronictechnology from the College of Electronic Infor-mation, Shandong Agricultural University, Tai’an,China, in 2017. He is pursuing the M.S. degree at theCollege of Electronics and Information Engineering,Sichuan University, Chengdu, China.

His current research interests include injectionlocking and microwave power combining.

Danli Cai received the B.S. degree in Internet-of-Things Engineering fromthe Jincheng College, Sichuan University, Chengdu, China, in 2017, whereshe is currently pursuing the M.S. degree at the College of Electronics andInformation Engineering.

Her research interest mainly lies in microwave nonreciprocal devices.

Yi Zhang (Member, IEEE) received the B.Sc. degreein electric engineering and information and the Ph.D.degree in radiophysics from Sichuan University,Chengdu, China, in 2013 and 2018, respectively.

From September 2016 to August 2017, he was aVisiting Scholar with Material Research Institute,Pennsylvania State University, State College, PA,USA. From August 2017 to September 2018, he wasa Visiting Scholar with the Department of Electricaland Computer Engineering, Northeastern University,Boston, MA, USA. He is currently a Research

Fellow with Sichuan University. His current research interests include novelmicrowave devices with materials, microwave materials processing, andmicrowave power sources.

Yang Yang (Member, IEEE) received the Ph.D.degree in radiophysics from Sichuan University,Chengdu, China, in 2010.

From 2008 to 2010, he was a Visiting Fellowwith Clemson University, Clemson, SC, USA, wherehe was a Visiting Scholar with the Oak RidgeNational Laboratory, Oak Ridge, TN, USA, in 2010.He is currently an Associate Professor with theCollege of Electronics and Information Engineering,Sichuan University. His current research interestsinclude microwave chemistry and the applicationsof microwave energy.

Huacheng Zhu (Member, IEEE) received theB.S. degree in electric engineering and the Ph.D.degree in information and radiophysics from SichuanUniversity, Chengdu, China, in 2009 and 2014,respectively.

From 2012 to 2013, he was a Visiting Fellowwith the Department of Biological and Environ-mental Engineering, Cornell University, Ithaca, NY,USA, which was supported by the China ScholarshipCouncil. Since 2015, he has been a Faculty Memberwith Sichuan University.

Dinesh K. Agrawal received the master’s degree inphysics from Banaras Hindu University, Varanasi,India, in 1972, the master’s degree in materialsscience from IIT Kanpur, India, in 1975, and thePh.D. degree from Pennsylvania State University,State College, PA, USA, in 1979.

Since 1997, he has been a Senior Scientist and theDirector of the Microwave Processing and Engineer-ing Center, Pennsylvania State University.

Tania Slawecki received the B.A. degree inastronomy/physics from the Lycoming College,Williamsport, PA, USA, in 1987, and the M.S.degree in physics and the Ph.D. degree in materialsscience and engineering from Pennsylvania StateUniversity, State College, PA, USA, in 1989 and1995, respectively.

Since 2004, she has been a Research Associatewith the Materials Research Institute, PennsylvaniaState University.



Li Wu received the B.Sc. degree in electronicinformation engineering from Sichuan University,Chengdu, China, in 2010, and the Ph.D. degree inmicrowave, electromagnetic, and photoelectron fromthe National Polytechnique de Toulouse (INPT),Toulouse, France, in 2016. From 2016 to 2019,she was a Research Fellow at Sichuan University,where she has been an Associate Professor with theCollege of Electronics and Information Engineering,since 2019.

Her current research interests include microwaveplasma discharge theory and its industrial applications, and permittivitymeasurement.

Yanping Zhou received the B.Eng. degree in envi-ronmental engineering from Xi’an Jiaotong Uni-versity, Xi’an, China, in 2008, the M.Sc. degreein environmental science from the Chinese Acad-emy of Sciences, Beijing, China, in 2011, and thePh.D. degree from the School of Materials Scienceand Engineering, Nanyang Technological University,Singapore, in 2016.

She is currently an Associate Professor with theSchool of Electronics and Information Engineering,Sichuan University, Chengdu, China. Her current

research interest includes novel applications of microwave technology.

Qian Chen received the B.S., M.S., and Ph.D.degrees from Sichuan University, Chengdu, China,in 1997, 2002, and 2011, respectively, all in elec-tronic science and technology.

From March 2014 to April 2015, she was a Visit-ing Scholar with Clemson University, Clemson, SC,USA. She is currently an Associate Professor withthe College of Electronics and Information Engi-neering, Sichuan University. Her research interestsinclude microwave measurement and the applicationof microwave energy.

Lin Zhou received the B.S. degree in informa-tion and communication engineering and the Ph.D.degree in radiophysics from Sichuan University,Chengdu, China, in 2013 and 2018, respectively.

He is currently a Research Fellow with NanyangTechnological University, Singapore. His researchis mainly focused on the antenna and metasurfacedesign.

Kama Huang (Senior Member, IEEE) received theM.S. and Ph.D. degrees in microwave theory andtechnology from the University of Electronic Sci-ence and Technology, Chengdu, China, in 1988 and1991, respectively.

Since 1994, he has been a Professor with theDepartment of Radio and Electronics of SichuanUniversity, Chengdu, Sichuan, China, where he hasbeen the Director since 1997. He has authored orcoauthored over 100 articles.


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