Investigation of the bandpass properties of the local impedance of slow wave structures

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The properties of the local coupling impedance that determines the efficiency of the electron–wave interaction in periodic slow-wave structures are investigated. This impedance is determined (i) through the char- acteristics of the electromagnetic field in a slow-wave structure and (ii) through the parameters of a two-port chain simulating the structure. The continuous behavior of the local coupling impedance in the passbands of slow-wave structures, at the boundaries of the passbands, and beyond the passbands is confirmed with the help of a waveguide–resonator model.

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Investigation of the bandpass properties of the local impedance of slow wave structures

  1. 1. 1250ISSN 1064-2269, Journal of Communications Technology and Electronics, 2008, Vol. 53, No. 10, pp. 1250–1258. © Pleiades Publishing, Inc., 2008.Original Russian Text © S.V. Mukhin, D.Yu. Nikonov, V.A. Solntsev, 2008, published in Radiotekhnika i Elektronika, 2008, Vol. 53, No. 10, pp. 1324–1332.INTRODUCTIONThe interaction of an electron flow and the electro-magnetic field in the resonator slow-wave structures(SWSs) of power traveling-wave tubes (TWTs) is ofteninvestigated with the use of RLC circuits or multiportchains (excited by the current induced by the electronflow) equivalent to SWSs. This representation makes itpossible to analyze the interaction characteristicswithin and, sometimes, beyond the passbands of SWSsand to perform calculation of a device in a reasonabletime. However, the choice of equivalent circuits ofSWSs and determination of the parameters of these cir-cuits are a special problem necessitating more rigoroustheoretical methods.Direct simultaneous numerical solution of the inho-mogeneous Maxwell equations and the equations ofelectromagnetics according to the available Karat,Mafia, and Magic computer codes allows the analysisand simulation of certain variants of devices. However,the application of these equations for calculation ofpower TWTs is impeded because real resonator SWSsare complex and calculations necessitate substantialcomputational resources.The theory of excitation of periodic waveguidesuses a solution to the inhomogeneous Maxwell equa-tions for the fields excited by the electron beam’s cur-rent. The solution is decomposed in the eigenmodes ofa periodic waveguide, and the quasi-static electromag-netic field is separated in the decomposition. Therefore,this theory facilitates the solution of the above prob-lems [1]. In this approach, it suffices to take intoaccount only (i) the fields of the waves that are synchro-nous with the electron flow; (ii) the quasi-static field ofthe spatial charge; and, sometimes, (iii) a small numberof asynchronous waves that provide for dynamic cor-rections to this field. The equations of wave excitationcontain parameters that can be calculated from theexact 3D simulation of an SWS or can be determinedapproximately with the help of other methods for cal-culating cold SWSs without electron beams [2]. Usu-ally, the phase velocity and the coupling impedance ofthe operating spatial field harmonic are involved in theexcitation equations. However, it is impossible to usethese parameters when a TWT operates at frequenciesnear the passband of an SWS and when a transition tothe stopband occurs. The reason is that the couplingimpedance becomes infinite at the passband boundary.It is known that this difficulty can be eliminated if weconsider the total excited field of forward and counter-propagating SWS waves that, near the passband bound-ary, have spatial harmonics synchronous with the beam.The most general approach is based on the second-order finite-difference equation of SWS excitation forthe total field [3, 4]. This equation contains, instead ofthe standard coupling impedance for spatial harmonics,the local coupling impedance that characterizes thefield intensity in interaction gaps. The analysis [4, 5] ofthe general properties of this impedance has shown thatit is continuous at the boundaries of the passbands of anSWS. A simple example of calculation of the localimpedance is presented in [5].Here, we demonstrate that the local coupling imped-ance can be determined through both the characteristicsof the electromagnetic field of an SWS and the param-eters of a two-port chain simulating this structure. Awaveguide–resonator model (WRM) of an SWS is usedfor investigation of the frequency properties of the localcoupling impedance for looped SWSs applied in powerTWTs. Such SWSs include interdigital lines, loopedwaveguides, chains of coupled resonators with cou-pling slots turned through 180°, and similar structuresInvestigation of the Bandpass Properties of the Local Impedanceof Slow-Wave StructuresS. V. Mukhin, D. Yu. Nikonov, and V. A. SolntsevReceived April 8, 2008Abstract—The properties of the local coupling impedance that determines the efficiency of the electron–waveinteraction in periodic slow-wave structures are investigated. This impedance is determined (i) through the char-acteristics of the electromagnetic field in a slow-wave structure and (ii) through the parameters of a two-portchain simulating the structure. The continuous behavior of the local coupling impedance in the passbands ofslow-wave structures, at the boundaries of the passbands, and beyond the passbands is confirmed with the helpof a waveguide–resonator model.PACS numbers: 41.20.Jb, 84.40.DcDOI: 10.1134/S1064226908100136MICROWAVEELECTRONICS
  2. 2. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008INVESTIGATION OF THE BANDPASS PROPERTIES 1251in which the flux of the electromagnetic-wave energyrepeatedly crosses the axis of a structure. We show thatthe local coupling impedance is continuous and finitewithin, at the boundaries of, and beyond the passbandsof the considered SWSs.1. LOCAL COUPLING IMPEDANCEIN THE DIFFERENCE EQUATIONOF EXCITATION OF PERIODIC WAVEGUIDESConsider sum (x, y, z, t) of the electric fields of theforward (+s) and counterpropagating (–s) modes of aperiodic waveguide with period L:(1)According to the Floquet theorem, the eigenmodefield in a periodic waveguide has the form(2)where hs are wave numbers, is the amplitude of thechosen field component at point (x0, y0, z0) at which thedistribution function of this component is unity, (x,y, z) are distribution functions periodic in z, and thetime factor exp(–iωt) is omitted. According to the the-ory of excitation of periodic waveguides [1, 3], theexcitation coefficients satisfy the equations(3)where Ns = – is the normof an eigenmode, S(z) is the cross section of a structure,and (x, y, z) is the density of the exciting current at fre-quency ω.A second-order equation coupling the total fields inthree period-spaced sections z, z ± L can be obtainedwith the use of first- and second-order finite differencesintroduced according to the relationships(4)The calculation of the differences from (1)–(3) yields [3](5)EE x y z, ,( ) Cs z( )Es x y z, ,( ) C s– z( )E s– x y z, ,( ).+=E s± x y z, ,( ) E s±0e s± x y z, ,( ) ihsz±( ),exp=E s±0e s±dC s±dz-----------1Ns------ j x y z, ,( )E s+− x y z, ,( ) S,dS z( )∫±=EsH s–[ ]{S z( )∫ E s– Hs[ ]}z0dSj∆±E E x y z L±, ,( ) E x y z, ,( ),–=∆2E ∆+E ∆–E– E x y z L+, ,( )= =– 2E x y z, ,( ) E x y z L–, ,( ),+∆±C+s± C+s z L±( ) C+s z( ),–=∆±C s–± C s– z L±( ) C s– z( ).–=∆2E 2E 1 ϕscos–( )+ G,=where ϕs = hsL is generally the period complex phaseshift and excitation function has the form(6)We investigate SWS excitation within passbandsand stopbands with the use of a 1D model of the inter-action between the electron flow and the field. In thiscase, the exciting current is aligned with the longitu-dinal axis of the SWS. Function ψ(x, y) of the currentdistribution over cross section Se of the beam is spec-ified as(7)and normalized with the relationship(8)Then, J(z) is the HF current of the beam and S =1/ is the effective area of its crosssection.In the 1D model, the field and other quantities areaveraged according to the relationship(9)In [3, 5], the difference equation of excitation for thefield averaged according to this relationship is derivedin a different way. Here, we present this equation for thecase of the discrete interaction between the electronbeam and the field in SWS gaps. There are structures inwhich full period L contains one interaction gap (e.g.,comb structures, diaphragmatic waveguides). In alooped SWS, there are two interaction gaps spaced bythe step D = L/2. It can be assumed that the field inter-acting with the electron beam is in phase along theSWS axis. By averaging field (2), we obtain for the qthstep(11)where (z) is the real distribution function averagedover the section. This function is the same (e.g.,because of the structure’s symmetry) for the forwardand counterpropagating waves.By averaging (5), we can show [3, 4] that the aver-aged total longitudinal field has the following form forthe qth step:(12)GG ∆+C+s iϕs( )exp ∆–C+s iϕs–( )exp–( )E+s=+ ∆+C s– i– ϕs( )exp ∆–C s– iϕs( )exp–( )E s– .j x y z, ,( ) J z( )ψ x y,( )z0=ψ x y,( ) SdSe∫ 1.=ψ2x y,( ) SdSe∫( )1–Ez z( ) ψ x y,( )Ez x y z, ,( ) S.dSe∫=E s z,± z( ) E s±0e s± z( ) ihsz±( )exp== E s±0e z( ) iqϕs±( ),expeEq z( ) e z( )Eq,=
  3. 3. 1252JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008MUKHIN et al.where Eq satisfies the difference equation(13)In this equation,(14)is the specific coupling resistance at point (x0, y0, z0),while(15)is the effective width of the equivalent plane gap. Thewidth is chosen such that the voltage across the gap isequal to the step voltage:(16)where zq is the coordinate of the qth gap’s center.The quantity(17)has the meaning of the current induced in the qth-stepinterval.The difference equation for the voltage is obtainedfrom (13) through simple multiplication by –d:(18)The local coupling impedance of the gap is intro-duced according to the relationship(19)This impedance enters the right-hand side of differ-ence equation (18) and characterizes the intensity of thefield excited by the current. Note that the local imped-ance takes into account the distributions of the field and∆2Eq 2Eq 1 ϕscos–( )+ iRs0ϕsJqd.sin–=Rs0 2E+s0E s–0Ns-------------------–=d e z( ) zdzq D/2–zq D/2+∫=Uq Eqd– Eq e z( ) zdzq D/2–zq D/2+∫– E z( ) z,dzq D/2–zq D/2+∫–= = =Jq1d--- J z( )e z( ) zdzq D/2–zq D/2+∫=∆2Uq 2Uq 1 ϕscos–( )+ iRs0Jqd2ϕs.sin=Zs Rs0d2ϕs.sin=of the HF current flowing across effective width d of thegap and allows the uniform analysis of the SWS excita-tion within, at the boundary of, and beyond the pass-band because this impedance has singularities at theboundary. Generally, local coupling impedance Zs canbe calculated via the above relationships upon process-ing of the results of 3D simulation of the SWS fields orvia simpler models.2. LOCAL COUPLING IMPEDANCEOF A TWO-PORT CHAINLet us represent an SWS as a chain of two-portsconnected in series (Fig. 1) and consider the behaviorof Zs(ω) and dispersion characteristics ϕs(ω). This rep-resentation is widely used in numerous studies on thetheory and calculation of TWTs. In order to express Zsthrough the two-port parameters, we derive a differenceequation of excitation of form (18) by directly using theequivalent circuit from Fig. 1. The problem of excita-tion by a given current is solved under the assumptionthat the motion of electrons in the gap is known anddoes not depend on the voltage; i.e., it is assumed thatelectron-beam current J(z) is specified. In this case, theexcitation currents of the two-port chain coincide withthe specified induced currents (see Fig. 1). Then, thecurrents and voltages within one period are coupledaccording to the following formulas:(20)Using the second-order finite differences ∆2U =Uq + 1 – 2Uq + Uq – 1 and taking into account the reciproc-Uk A11Uk 1+ A12Ik 1+ ,+=Uk 1– A11Uk A12 Ik Jk––( )+ ,=Ik Jk++ A21Uk 1+ A22Ik 1+ ,+=Ik 1– A11Uk A22 Ik Jk+–( ),+=Jk Jk–Jk+.+=^^q = QAAAAq = k + 1Uk + 1q = kIk + 1ZQUkIkJk–Jk+Ik–1q = 1 q = k–1Uk–1JkZ1 AFig. 1. Two-port chain modeling an SWS section of Q steps: Uq is the voltage across the interaction gap between the (q – 1)th andqth two-ports, and Z1 and ZQ are the impedances of the loads at the beginning and end of the chain, respectively.
  4. 4. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008INVESTIGATION OF THE BANDPASS PROPERTIES 1253ity conditions A11A22 – A12A21 = 1 for two-ports and thedispersion equation(21)we obtain from (20) an equation similar to (18):(22)The comparison of (18) and (22) yields the elemen-tary expressionZs = iA12. (23)It should be emphasized that the two-ports formingthe chain (see Fig. 1) describe SWS cells of a complexshape and the corresponding transmission matricesmay contain coefficients Aij that are expressed accord-ing to complicated formulas, but satisfy the reciprocitycondition.It is a common practice in the theory of TWTs to usecoupling resistance Ks, m of the mth spatial harmonic ofthe sth wave. This resistance is related to Zs as follows [5]:(24)where ϕs, m = 2ϕs + 2πm is the phase shift of the mth-harmonic field by full period L and |es, m| are the ampli-tudes of spatial harmonics.When, in the nonlinear theory of TWTs, an SWS ismodeled as a two-port chain, the voltage across the qthgap is often represented in the form(25)where mutual impedances Zqk determine the contribu-tion of the kth-gap current to the voltage across the qthgap. These impedances are calculated through the two-port parameters and impedances Z1 and ZQ of the inputand output loads with the help of the recurrence recal-culation formulas. Using these formulas and formingthe second-order finite difference ∆2U = Uq + 1 – 2Uq +Uq – 1 with allowance for (25), we can show that repre-sentation of Uq in form (25) exactly satisfies second-order difference equation (22) with specified boundaryconditions that are determined by loads Z1 and ZQ.Therefore, expression (25) can be regarded as the rep-resentation of a solution to Eq. (22) through the discretesource function (the discrete Green’s function) deter-mined by the set of impedances Zqk. System of linearequations (22) can be solved with the help of othermethods, in particular, the sweep method. In the nonlin-ear theory of TWTs, the method of solution is chosenaccording to the possibilities of minimizing the compu-tation time and necessary computational resources.ϕscosA11 A12+2----------------------,=∆2Uk 2Uk 1 ϕscos–( )+ A12Jk.–=Ks m, Zses m,2ϕs m,2ϕssin-----------------------,=Uq U1 i q 1–( )ϕs( )exp=+ UQ i q Q–( )ϕs–( )exp Zqk Jk,k 1=Q∑+3. ANALYSIS OF THE BANDPASS PROPERTIESOF THE LOCAL COUPLINGIMPEDANCE WITH THE USEOF THE WAVEGUIDE–RESONATOR MODELAccording to the WRM proposed in [6], an SWS isreplaced with an equivalent structure made fromwaveguide sections. In contrast to [7], in the WRM,waveguide sections may be oriented along and acrossthe SWS axis in accordance with the direction of theenergy flux on a specific SWS section. The applicationof the WRM in theVEGA code for TWT simulation hasshown that it is possible to describe the SWS propertiesqualitatively well. This description can likewise bequantitatively adequate if the waveguide dimensionsare appropriately chosen with the use of referencepoints. In this study, the properties of the local couplingimpedance and the properties of dispersion in the SWSpassbands and stopbands are considered with the helpof an elementary WRM consisting of two or threewaveguide sections (Fig. 2). In each section, only onewave (with wave number h and impedance Z) is takeninto account.For a rectangular waveguide, this wave is the H10mode with the components of the electric field(26)where k = ω/c and Z0 = are the wave numberand impedance in free space, respectively; kc = π/a isthe critical wave number of the H10 waveguide mode;h = ; and the x, y, and z axes are oriented alongthe wide wall with dimension a, the narrow wall withdimension b, and the longitudinal axis of the consid-ered waveguide section, respectively. Generally, theWRM can be constructed with the use of waveguidesections of arbitrary cross sections: H-shaped,Π-shaped, or of other shapes.Such a WRM approximately describes the mainSWS properties within at least the first two passbandsand stopbands for a looped SWS (a looped waveguide,an interdigital line, a chain of coupled resonators withcoupling slots turned through the angle 180°) and for ahelical twisted waveguide.We determine impedance Z of a rectangularwaveguide as the ratio of x-averaged voltage U to sur-face current J flowing along the waveguide’s axis [8]:Hz C kcx ihz( ),expcos=Hx ihkc----C kcx ihz( ),expsin–=Ey iCkkc----Z0 kcsin x ihz( ),exp=Ex Ez 0, Hy 0,= = =µ0/ε0k2kc2–
  5. 5. 1254JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008MUKHIN et al.(27)If loss is disregarded, impedance Z is real in thewaveguide’s passband when k > kc and imaginarybeyond the passband when k < kc. In the latter case, theimpedance is inductive for a decaying wave (h =).Then, the transmission matrix of the jth waveguidesection of geometric length lj and electric length θj = hjljhas the form(28)The transmission matrix of the SWS section betweenneighboring interaction gaps (see Fig. 2) is a product ofthree matrices: Ä = Ä1Ä2Ä3. The first and thirdwaveguide sections are identical. Then, Ä1 = Ä3 and weobtain the following expressions for the components:ZUJ----1a--- Eyb xd0a∫–Hx xd0h∫------------------------- Z0ba---kh---.= = =i kc2k2–Ajθj( )cos iZj θj( )sin–iZj----- θj( )sin– θj( )cos.=A11 A22 θ1 θ2coscos12---Z1Z2-----Z2Z1-----+⎝ ⎠⎛ ⎞ θ1 θ2,sinsin–= =A12 iZ1 θ1 2 θ1 θ2coscosZ1Z2----- θ1 θ2sinsin–⎝ ⎠⎛ ⎞sin–=(29)where θ1 denotes the total electric length of the first andthird waveguide sections.Along with (21), (23), and (27), expressions (29),which are obtained with the use of the WRM, allow theanalysis of the dispersion and the local coupling imped-ance in a frequency band. Let us consider looped SWSswith allowance for the geometric rotation of the fieldphase in the neighboring gaps. In a linear WRM that isa two-port chain (see Fig. 1), this change of the phasecan be taken into account through introduction ofinduced currents of opposite directions in the neighbor-ing gaps. The dispersion is calculated for the first spa-tial harmonic, which is used in such TWTs, accordingto the formulas(30)where ϕs is the phase shift on step interval D, ϕs, 0 =ϕs – π ≤ 0 is the phase shift of the fundamental spatialharmonic with allowance for the geometric rotation,ϕs, 1 = ϕs, 0 + 2π is the phase shift of the first spatial har-monic, and hs, 1 = ϕs, 1/D is its wave number.– iZ2 θ1 θ2,sincos2A21iZ1----- θ1 2 θ1 θ2coscosZ2Z1----- θ1 θ2sinsin–⎝ ⎠⎛ ⎞sin–=–iZ2----- θ1 θ2,sincos2ϕscos A11, ϕs 1, ϕs π,+= =cVs 1,---------hs 1,k--------ϕs 1,kD---------,= =Electronbeamq + 1ql123DFig. 2. Schematic of a looped SWS with an electron flow: (2) waveguide section (of the length l2 = D) corresponding to a couplingslot and (1, 3) waveguide sections of the total length l-D with interaction gaps.
  6. 6. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008INVESTIGATION OF THE BANDPASS PROPERTIES 1255Local coupling impedance Zs is calculated with theuse of expression (23), where A12 is calculated from for-mula (29), or through direct multiplication of threematrices, Ä = Ä1Ä2Ä1. The correctness of single-modedescription (26) of waveguide sections and their imped-ance (27) is checked through calculation of the trans-mission matrix of three series-connected sections bymeans of the ISFEL-3D code. As an example, couplingresistance K and slowing factor n calculated through theparameters of the transmission matrix (solid line) andwith the help of the ISFEL-3D code (dots) are depictedin Fig. 3.Consider certain variants that characterize thebehavior of the properties of looped SWSs.A Homogeneous Looped Waveguideof a Constant SectionIn this case, from (29) and (30), we obtain(31)Here, λπ is the low-frequency boundary of the mainpassband. This boundary corresponds to ϕs, 1 = π and isdetermined by the critical frequency of a loopedwaveguide:A11 θ1 θ2+( ), ϕscos θ1 θ2+ hl,= = =ϕs 1, θ π,+=cVs 1,---------ϕs 1,kD---------lD---- 1λλπ-----⎝ ⎠⎛ ⎞2–λ2D-------+= ==cV1 π,----------2lλπ----- 1λλπ-----⎝ ⎠⎛ ⎞2–λλπ-----+⎝ ⎠⎛ ⎞ .(32)High-frequency boundary λ2π of the main passbandcorresponds to ϕs, 1 = 2π and is absent in an ideal reflec-tion-free WRM. However, in a real structure, thisboundary exists because of reflections from thewaveguide’s bends and from the junctions ofwaveguide sections.We apply relationship (31) to estimate the maximumpossible width of the main passband of the structure.Setting in (31) ϕs, 1 = 2π and taking into account (32),we obtain(33)i.e., we always have λ2π < λπ.Consider the possibility of equal slowing factors atthe edges of the main passband. This situation is real-ized when(34)In this case, the width of the main passband is oneoctave,λπ λc, h 0,cV1 π,----------λπ2D-------.= = =1λ2π2-------1λπ2-----–12l( )2-----------,cV1 2π,------------ 2λ2πλπ-------cV1 π,----------;= =cV1 2π,------------cV1π--------, λπ 2λ2π, lλπ2 3----------,= = =D3lc/V1π-------------.=f 2π 2 f π, ∆f / f 0 2f 2π f π–f 2π f π+------------------- 66.6%,= = =3.02.82.52.32.01.8654321012010080604020n K, Ωλ, cm2211Fig. 3. (Curves 1) Coupling impedance K and (curves 2) slowing factor n calculated (solid line) on the basis the parameters of thetransmission matrix and (dots) with the use of the ISFEL-3D code.
  7. 7. 1256JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008MUKHIN et al.and the slowing factor is determined from the relation-shipThe corresponding dispersion characteristic isdepicted in Fig. 4a (curve 1). Dispersion curves for thelooped waveguide are calculated for various l/λπ =with disregard for reflections and presented in Fig. 4a inthe normalized slowing-factor and frequency variablesn/nπ = V1π/V1 and ω/ωπ = λπ/λ, respectively.From (23), (27), (29), and (31), we can find anelementary expression for the local coupling imped-ance of the homogeneous looped waveguide for thecase Z2 = Z1:(35)cV1------cV1π--------13------- 1λλπ-----⎝ ⎠⎛ ⎞2–λλπ-----+ .=kπl2π-------Zs Z1 ϕssin Z01kh--- lh,sin= =which coincides with the expression obtained in [5].The frequency dependences of the local couplingimpedance that are depicted in Fig. 4b show that itchanges continuously when the passband-to-stopbandtransition occurs. Depending on the waveguide’sdimensions, the maximum of Zs may be located in dif-ferent bands. The absolute value of Zs is independent ofthe shape of the looped waveguide: In particular, Z01 == for a rectangular waveguide and, for anH-shaped waveguide or a slot line, Z01 may substan-tially increase, because a Ӷ λc/2.An Inhomogeneous Looped Waveguide Formedfrom Sections with Equal Critical frequencies and Dif-ferent Impedances: kc2 = kc1 = kc, Z2 ≠ Z1.This case corresponds to a looped waveguide suchthat only width b of the narrow wall may be variable ifrectangular-waveguide sections are considered. Thiscase also describes the properties of an interdigitalSWS.Then, we haveand the dispersion equation takes the form(36)In this case, there are gaps in a passband at ϕs, 1 = 2π,the width of the first passband decreases simulta-neously with the slowing factor in this passband, andboth normal and anomalous dispersions are possible(Fig. 5a). The local coupling impedance remains a con-tinuous function of frequency when the transition to thestopband occurs. The presence of the stopband is due tothe superposition of reflections from the junctions ofwaveguide sections, i.e., due to the longitudinal reso-nance (the Bragg resonance) in the SWS (Fig. 5b). Thedispersion in the SWS remains unchanged when thefirst and second waveguide sections are interchanged,because β is the same for Z1 Z2. However, the fre-quency dependences and the value of the local couplingimpedance change because the exciting electron-beamcurrent is introduced into different waveguide sections.This effect was investigated earlier during the analysisof the frequency dependences of the coupling imped-ance [9].An Inhomogeneous Looped Waveguide Formedfrom Sections with Different Critical Frequencies andDifferent Impedances: kc2 ≠ kc1 and Z2 = Z1.This case is the most general and describes the tran-sition from an interdigital SWS with a transmitting cou-ba---Z02bλc------12---Z1Z2-----Z2Z1-----+⎝ ⎠⎛ ⎞ 12---Z01Z02-------Z02Z01-------+⎝ ⎠⎛ ⎞ β const 1,≥= = =ϕscos θ1( ) θ2( )coscos β θ1( )sin θ2( ).sin–=2.01.51.00.501.81.61.41.21.0n/nπϕ = πϕ = 2πϕ = 3π123(a)2.01.51.00.5ω/ωπ040302010Zs/Z02(b)123ω/ωπFig. 4. Frequency dependences of (a) the normalized slow-ing factor of the first spatial harmonic and (b) the local cou-pling impedance for a homogeneous looped waveguide cal-culated at l/λπ = (1) 0.29, (2) 0.5, and (3) 0.75.
  8. 8. JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008INVESTIGATION OF THE BANDPASS PROPERTIES 1257pling slot to an SWS that is a coupled-resonator chainwith an evanescent coupling slot.The values ϕs = 0, ϕs, 1 = π (transverse resonance)and ϕs = π, ϕs, 1 = 2π (longitudinal resonance) corre-spond to the boundary frequencies of the first passband.We have performed calculations for a WRM formedfrom rectangular-waveguide sections. Width a2 of thecoupling slot was decreased in comparison with thewidth of the coupling slot used for Fig. 4 (curve 2), theremaining parameters being retained.Accordingly, crit-ical frequency ωc2 of the second waveguide sectiondescribing the slot was increased; i.e., ωc2 > ωc1. Thecalculation results presented in Fig. 6 show that thepassband decreases as ωc2 grows, but, in this case, thelocal coupling impedance changes continuously whenthe transition to the stopband occurs.CONCLUSIONSThe local coupling impedance determines the effi-ciency of the electron–wave interaction in a periodicSWS with allowance for the forward and counterprop-agating waves of the structure that are synchronouswith the electron flow near the cutoff frequencies. Thisimpedance is determined (i) in terms of the characteris-tics of the electromagnetic field with the use of 3D andsimpler models of SWSs and (ii) in terms of the param-eters of the two-port chain modeling an SWS. It hasbeen confirmed that the local coupling impedance is2.01.51.00.502.01.51.00.5n/nπϕ = 2πϕ = 3πϕ = π1, 31, 3(a)(b)15129630 2.01.51.00.5ω/ωπ123ω/ωπZs/Z02Fig. 5. Frequency dependences of (a) the normalized slow-ing factor of the first spatial harmonic and (b) the local cou-pling impedance for an inhomogeneous looped waveguideformed from waveguide sections with equal critical fre-quencies and different impedances. The results are obtainedfor l/λπ = 0.5; l/D = 1.875; and Z1/Z2 = (1) 2/3, (2) 1, and(3) 3/2.1.00.5 1.501.31.00.80.5n/n2π(a) ϕ = 2πϕ = 3πϕ = π123412341.51.00.50ω/ωπ1512963(b)1342Zs/Z02 ω/ωπFig. 6. Frequency dependences of (a) the normalized slow-ing factor of the first spatial harmonic and (b) the local cou-pling impedance for an inhomogeneous rectangular loopedwaveguide. The results are obtained for l/λc1, 3 = 0.5, l/D =1.875, and varying widths of the coupling slot: a2/a1, 3 = (1)0.5, (2) 0.8, (3) 0.9, and (4) 1. Curves 1 correspond to a nar-row coupling slot that is evanescent for all frequencies ofthe main passband of a homogeneous looped waveguide.Curve 4 is obtained for this waveguide.
  9. 9. 1258JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 53 No. 10 2008MUKHIN et al.continuous within, at the boundaries of, and beyondSWS passbands. Therefore, the linear and nonlinearinteractions between the electron flow and the fieldwithin and beyond SWS passbands can be uniformlydescribed in a wide frequency range. The obtainedresults can serve as the basis for 1D–3D simulation ofpower TWTs with periodic SWSs.ACKNOWLEDGMENTSThis study was supported by the Russian Founda-tion for Basic Research, project no. 07-02-00947.REFERENCES1. L. A. Vainshtein and V. A. Solntsev, Lectures on Micro-wave Electronics (Sovetskoe Radio, Moscow, 1973) [inRussian].2. R. A. Silin, Periodic Waveguides (Fazis, Moscow, 2002)[in Russian].3. V. A. Solntsev and S. V. Mukhin, Radiotekh. Elektron.(Moscow) 36, 2161 (1991).4. V. A. Solntsev, Lectures on Microwave Electronics andRadiophysics (X Winter School–Seminar for Engineers)(GosUNTs “Kolledzh”, Saratov, 1996) [in Russian].5. V. A. Solntsev and R. P. Koltunov, Radiotekh. Elektron.(Moscow) 53, 738 (2008) [J. Commun. Technol. Elec-tron. 53, 700 (2008)].6. S. V. Mukhin, O. E. Lomakin, and V. A. Solntsev,Radiotekh. Elektron. (Moscow) 33, 1637 (1988).7. R. A. Silin and I. P. Chepurnykh, Radiotekh. Elektron.(Moscow) 35, 939 (1990).8. I. V. Lebedev, Microwave Engineering and Devices(Vysshaya Shkola, Moscow, 1970), Vol. 1 [in Russian].9. S. V. Mukhin, D. Yu. Nikonov, and V. A. Solntsev, inThes. LIV Sci. Session of the Popov Society Dedicated tothe Radio Day, Moscow, Russia, 1999 (Ross. Nauch.-Tekh. Obshch. Radiotekh. Elektron. Svyazi, Moscow,1999), p. 42.

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