Exceptional points of degeneracy in traveling wave tubes
aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec EXCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES
ALEXANDER FIGOTIN
Abstract.
Traveling wave tube (TWT) is a powerful vacuum electronic device used to amplifyradio-frequency (RF) signals with numerous applications, including radar, television and telephonesatellite communications. TWT design in a nutshell comprises of a pencil-like electron beam (e-beam) in vacuum interacting with guiding it slow-wave structure (SWS). In our studies here thee-beam is represented by one-dimensional electron flow and SWS is represented by a transmissionline (TL). The interaction between the e-beam and the TL is modeled by an analytic theory thatgeneralizes the well-known Pierce model by taking into account the so-called space-charge effectsparticularly electron-to-electron repulsion (debunching). Many important aspects of the analytictheory of TWTs have been already analyzed in our monograph on the subject. The focus ofthe studies here is on degeneracies of the TWT dispersion relations particularly on exceptionalpoints of degeneracy and their applications. The term exceptional point of degeneracy (EPD)refers to the property of the relevant matrix to have nontrivial Jordan block structure. Usingspecial parameterization particularly suited to chosen EPD we derive exact formulas for the relevantJordan basis including the eigenvectors and the so-called root vector associated with the Jordanblock. Based on these studies we develop constructive approach to sensing of small signals. Introduction
There is growing interest to electromagnetic system exhibiting Jordan eigenvector degeneracy,which is a degeneracy of the system evolution matrix when not only some eigenvalues coincide butthe corresponding eigenvectors coincide also. The degeneracy of this type is sometimes referredto as exceptional point of degeneracy (EPD), [Kato, II.1]. A particularly important class ofapplications of EPDs is sensing, [CheN]. [PeLiXu], [Wie], [Wie1].In our prior work in [FigSynbJ], [FigSynbJ, FigPert] we advanced and studied simple circuitsexhibiting EPDs and their applications to sensing. Operation of electric circuits though is limitedto frequencies up to hundreds of MHz, and to overcome this limitation other physical systems thatcan operate at higher frequencies must be considered. Our prior studies of traveling wave tubes(TWT) in [FigTWTbk, 4, 7, 13, 14, 54, 55] demonstrate that TWTs always have EPDs. Operatingfrequencies of TWTs can go up to hundreds of GHz and even into THz frequency range, [BoosVE],[Burt] and for this reason they are the primary subject of our studies here.We start with a concise review of the basics of traveling wave tubes. Traveling wave tube (TWT)utilizes the energy of the electron beam (e-beam) as a flow of free electrons in a vacuum andconverts it into an RF signal, see Fig. 1. To facilitate energy conversion and signal amplification,the electron beam is enclosed in the so-called slow wave structure (SWS), which supports wavesthat are slow enough to effectively interact with the e-beam. As a result of this interaction,the kinetic energy of electrons is converted into the electromagnetic energy stored in the field,[Gilm1], [Tsim], [Nusi, 2.2], [SchaB, 4]. Consequently, the key operational principle of a TWTis a positive feedback interaction between the slow-wave structure and the flow of electrons . Thephysical mechanism of radiation generation and its amplification is the electron bunching causedby the acceleration and deceleration of electrons along the e-beam.
Key words and phrases.
Traveling wave tube, TWT, exceptional point of degeneracy (EPD), Jordan block,perturbations, instability, sensitivity.
Figure 1.
The upper picture is a schematic presentation of a traveling wave tube.The lower picture shows an RF perturbation in the form of a space-charge wave thatgets amplified exponentially as it propagates through the traveling wave tube.A schematic sketch of typical TWT is shown in Fig. 1. Such a typical TWT consists of avacuum tube containing an e-beam that passes down the middle of an SWS such as an RF circuit.It operates as follows. The left end of the RF circuit is fed with a low-powered RF signal to beamplified. The SWS electromagnetic field acts upon the e-beam causing electron bunching and theformation of the so-called space-charge wave . In turn, the electromagnetic field generated by thespace charge wave induces more current back into the RF circuit with a consequent enhancement ofelectron bunching. As a result, the EM field is amplified as the RF signal passes down the structureuntil a saturation regime is reached and a large RF signal is collected at the output. The role of theSWS is to provide slow-wave modes to match up with the velocity of the electrons in the e-beam.This velocity is usually a small fraction of the speed of light. Importantly, synchronism is requiredfor effective in-phase interaction between the SWS and the e-beam with optimal extraction of thekinetic energy of the electrons. A typical simple SWS is the helix, which reduces the speed ofpropagation according to its pitch. The TWT is designed so that the RF signal travels along thetube at nearly the same speed as electrons in the e-beam to facilitate effective coupling. Technicaldetails on the designs and operation of TWTs can be found in [Gilm1], [Nusi, 4] [PierTWT], [Tsim].As for a rich and interesting history of traveling wave tubes, we refer the reader to [MAEAD] andreferences therein.An effective mathematical model for a TWT interacting with the e-beam was introduced byPierce [Pier51, I], [PierTWT]. The Pierce model is one-dimensional; it accounts for the waveamplification, energy extraction from the e-beam and its conversion into microwave radiation inthe TWT [Gilm1], [Gilm], [Nusi, 4], [SchaB, 4], [Tsim]. This model captures remarkably wellsignificant features of the wave amplification and the beam-wave energy transfer, and is still usedfor basic design estimates. In our paper [FigRey1], we have constructed a Lagrangian field theoryby generalizing and extending the Pierce theory to the case of a possibly inhomogeneous MTLcoupled to the e-beam. This work was extended to an analytic theory of multi-stream electronbeams in traveling wave tubes in [FigTWTbk].According to our analytic theory the TWT dispersion relations always have EPDs which canbe effectively found [FigTWTbk, 4, 7, 13, 14, 54, 55]. We study here the simplest TWT modelthat generalizes the Pierce model by integrating into it the space-charge effects. We introducefor this model a special parameterization that allow for (i) explicit representation of chosen EPDassociated with a Jordan block of size 2; (ii) exact formulas for the Jordan basis including theeigenvectors and the so-called root vector associated with the Jordan block.The paper is organized as follows. In Section 2 we review our analytic model of TWT introducedand studied in [FigTWTbk, 4, 24]. In Section 3 we carry out detailed studies of the relevant TWT
XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 3 matrix and its Jordan form. In Section 4 we (i) carry out the perturbation analysis of the relevantTWT matrix; (ii) develop constructive approach for using the TWT EPD for sensing of smallsignals.While quoting monographs we identify the relevant sections as follows. Reference [X,Y] refersto Section/Chapter “Y” of monograph (article) “X”, whereas [X, p. Y] refers to page “Y” ofmonograph (article) “X”. For instance, reference [2, VI.3] refers to monograph [2], Section VI.3;reference [2, p. 131] refers to page 131 of monograph [2].2.
An analytic model of the traveling wave tube
We concisely review here an analytic model of the traveling wave tube introduced and studiedin our monograph [FigTWTbk, 4, 24]. According to this model an ideal TWT is represented bya single-stream e-beam interacting with single transmission line. This model is a generalization ofthe Pierce model [Pier51, I], [PierTWT] and its parameters are as follows. The main parameterdescribing the single-stream e-beam is e-beam coefficient β = σ B π R ω = e m R σ B ˚ n, ω = 4 π ˚ ne m , (2.1)where − e is electron charge with e > m is the electron mass, ω p is the e-beam plasma frequency, σ B is the area of the cross-section of the e-beam, the constant R sc is the plasma frequency reductionfactor that accounts phenomenologically for finite dimensions of the e-beam cylinder as well asgeometric features of the slow-wave structure, [FigTWTbk, 41, 63], and ˚ n is the density of thenumber of electrons. The single-stream e-beam has steady velocity ˚ v > C >
L >
0. The coupling constant0 < b ≤ w and the single TL principal coefficient θ defined by w = 1 √ CL , θ = b C . (2.2)Following to [FigTWTbk, 3] we assume that0 < ˚ v < w. (2.3)2.1. TWT system Lagrangian and evolution equations.
Following to developments in [FigTWTbk]we introduce the
TWT principal parameter ¯ γ = θβ . This parameter in view of equations (2.1)and (2.2) can be represented as follows γ = θβ = b C σ B π R ω = b C e m R σ B ˚ n, θ = b C , β = e m R σ B ˚ n. (2.4)The TWT-system Lagrangian L TB is defined by [FigTWTbk, 4, 24] L TB = L Tb + L B , (2.5) L Tb = L ∂ t Q ) − C ( ∂ z Q + b∂ z q ) , L B = 12 β ( ∂ t q + ˚ v∂ z q ) − πσ B q , where q ( z, t ) and Q ( z, t ) are charges associated with the e-beam and the TL defined as timeintegrals of the corresponding e-beam currents i ( z, t ) and TL current I ( z, t ), that is q ( z, t ) = Z t ( z, t ′ ) d t ′ , .Q ( z, t ) = Z t I ( z, t ′ ) d t ′ . (2.6) XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 4
The corresponding Euler-Lagrange equations are the following system of second-order differentialequations L∂ t Q − ∂ z (cid:2) C − ( ∂ z Q + b∂ z q ) (cid:3) = 0 , (2.7)1 β ( ∂ t + ˚ v∂ z ) q + 4 πσ B q − b∂ z (cid:2) C − ( ∂ z Q + b∂ z q ) (cid:3) = 0 , β = σ B π R ω . (2.8)The Fourier transformation (see Section A) in time t and space variable z of equations (2.7) and(2.8) yields (cid:0) k C − − ω L (cid:1) ˆ Q + k C − b ˆ q = 0 , (2.9)4 πσ B " − ( ω − ˚ vk ) R ω ˆ q + k bC − h b ˆ q + ˆ Q i = 0 , (2.10)where ω and k = k ( ω ) are the frequency and the wavenumber, respectively, and functions ˆ Q =ˆ Q ( ω, k ) and ˆ q = ˆ q ( ω, k ) are the Fourier transforms of the system vector variables Q ( t, z ) and q ( t, z ), see Appendix A. In this case, the general TWT-system eigenmodes are of the form Q ( z, t ) = ˆ Q ( k, ω ) e − i( ωt − kz ) , q ( z, t ) = ˆ q ( k, ω ) e − i( ωt − kz ) , (2.11)where as it turns outˆ q ( k, ω ) = a u − ˚ v ) , ˆ Q ( k, ω ) = a bw ( u − ˚ v ) ( u − w ) , u = ωk . (2.12)In equation (2.12) a is a constant of proper physical dimensions and, importantly, u is the complex-valued characteristic velocity satisfying characteristic equation (2.18) provided below.Notice that dividing equations (2.9)-(2.10) by k yields (cid:0) k C − − u L (cid:1) ˆ Q + C − b ˆ q = 0 , u = ωk , (2.13)4 πσ B " u ω − ( u − ˚ v ) Ω ˆ q + bC − h b ˆ q + ˆ Q i = 0 , Ω p = R sc ω p . (2.14)Concise matrix form of equations (2.13)-(2.14) is the following eigenvalue type problem for k and x assuming that ω is fixed M kω x = 0 , M kω = " − Lω + k C bk Cbk C b k C + πσ (cid:16) − ( k ˚ v − ω ) Ω (cid:17) , x = (cid:20) ˆ Q ˆ q (cid:21) , (2.15)The problem (2.15) is equivalent to another eigenvalue type problem for u and x assuming that ω is fixed M uω x = 0 , M uω = " u − w bu bu b γ ˘ ω + b u − b ( u − ˚ v ) γu , x = (cid:20) ˆ Q ˆ q (cid:21) , (2.16) u = ωk , ˘ ω = ω Ω p . where matrix M uω encodes the information about TWT eigenmodes and we refer to it as TWTprincipal matrix . As we show below this eigenvalue problem can be recast in terms of the theoryof matrix polynomials reviewed in Section B.
XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 5
Notice the first equation in (2.16) is also kind of eigenvalue problem where being given w , ˚ v , b , γ and ˘ ω we need to find u and two-dimensional nonzero vector x that solve it. The problem offinding such u is evidently reduced evidently to the characteristic equationdet { M uω } = 0 , (2.17)where M uω is TWT principal matrix defined in equations (2.16). After elementary algebraictransformations of equation (2.17) we find that it is equivalent to the following equation D ( u, γ ) = γw − u + ( u − ˚ v ) u = 1˘ ω , ˘ ω = ω Ω p , u = ωk , (2.18)and we refer to function D ( u, γ ) as the characteristic function . We refer to solutions of thecharacteristic equation (2.18) as characteristic velocities . Since the dimensionless frequency ˘ ω in characteristic equation (2.18) is real, the characteristic equation is equivalent to the followingsystem of equations: ℑ { D ( u, ¯ γ ) } = 0 , ℜ { D ( u, ¯ γ ) } = 1˘ ω ≥ . (2.19)With the above equations in mind, we denote the set of all characteristic velocities u by U +TB = U +TB (¯ γ ) U +TB = U +TB (¯ γ ) = { u : ℜ { D ( u, ¯ γ ) } ≥ } (2.20)with superscript “+” being a reminder of ℜ { D ( u, ¯ γ ) } ≥ D ( u, ¯ γ ) defined by equation (2.9) can be represented in theform D ( u, γ ) = γ D T ( u ) + D B ( u ) = γw − u + ( u − ˚ v ) u , < ˚ v < w, γ = θβ, (2.21)where D T ( u ) and D B ( u ) are respectively characteristic functions of the TL and e-beam definedby D T ( u ) = u − ∆ T ( u ) = 1 w − u , D B ( u ) = u − ∆ − ( u ) = ( u − ˚ v ) u . (2.22)Functions ∆ T ( u ) and ∆ B ( u ) are defined in turn by∆ T ( u ) = u w − u , ∆ B ( u ) = 1( u − ˚ v ) , u = ωk . (2.23)Using dimensionless variables˘ ω = ω Ω p , Ω p = R sc ω p , ˇ γ = γ ˚ v , ˇ u = u ˚ v = ωk ˚ v , χ = w ˚ v . (2.24)we can recast eigenmode equation (2.16) as follows M uω x = 0 , M uω = " u − χ b ˇ u b ˇ u (cid:16) u + γ (cid:16) ω − (ˇ u − ˇ u (cid:17)(cid:17) b , x = (cid:20) ˆ Q ˆ q (cid:21) . (2.25)It is convenient to recast the first equation into the standard matrix polynomial in ˇ u by multiplyingit by ˇ u . That action after elementary algebraic transformations yields M uω x = 0 , M uω = D ′ b (cid:20) ˇ u − χ − χ γ ˇ ω − − ˇ u + u ˇ ω − − + ˇ γ − ω − − (cid:21) D b , x = (cid:20) ˆ Q ˆ q (cid:21) , (2.26)where D ′ b = " − χ b ( ˇ ω − − ) ˇ γ , D b = (cid:20) b (cid:21) . (2.27) XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 6
Matrix polynomial equation (2.26) can in turn be readily recast into the following equivalent form M uω x = 0 , M uω = (cid:20) ˇ u − χ − χ γ ˇ ω − − ˇ u + u ˇ ω − − + ˇ γ − ω − − (cid:21) , x = (cid:20) ˆ Qb ˆ q (cid:21) . (2.28)We refer to matrix M uω defined in equations (2.28) TWT matrix polynomial . The theory ofmatrix polynomials is reviewed in Section B. From now on the matrix monic polynomial in ˇ u equation (2.28) will be our preferred form. The corresponding characteristic equation equivalentto det { M uω } = 0 and related to it characteristic function D (ˇ u, ˇ γ ) and the corresponding equationare as follows D (ˇ u, ˇ γ ) = ˇ γχ − ˇ u + (ˇ u − ˇ u = 1˘ ω . (2.29)The conventional plot of the characteristic function D ( u, γ ) helps to visualize solutions u tothe characteristic equations (2.18), that is, D ( u, ¯ γ ) = ω , when the solutions are real-valued.But a non-real solution u to the characteristic equation, that is when ℑ { u } 6 = 0 is “invisible”in the conventional plot of the characteristic function D ( u, ¯ γ ). To remedy this, we extend theconventional following to [FigTWTbk, 8]. Suppose that for D > D ( u, ¯ γ ) = D has acomplex-valued solution u , ℑ { u } 6 = 0. We then represent such complex u in the plot of D ( u, ¯ γ )as a point ( ℜ { u } , D ) , where D ( u, ¯ γ ) = D > . (2.30)We distinguish such points ( ℜ { u } , D ) by plotting curves comprised from them as solid (brown).We refer to the so extended plot as the plot of the characteristic function with instability branches .We also refer to the endpoints ( u , D ( u , ¯ γ )) of the instability branches as the characteristic func-tion instability nodes . When generating plots and numerical examples we often use the followingdata ˚ v = 1 , w = 1 . , γ = 3 . (2.31)Figures 2, 3, and 4 show plots of the characteristic function D ( u, γ ) defined in equations (2.18)with its instability branches.2.2. Dispersion relations.
Following to [FigTWTbk, 4] we define for any characteristic velocity u from characteristic velocity set U +TB (¯ γ ) as in equations (2.20) the TWT frequency function Ω ( u )so that it is the frequency corresponding to u in the characteristic equation (2.18), namelyΩ ( u ) = Ω p p D ( u, ¯ γ ) > , Ω p = R sc ω p , u ∈ U +TB (¯ γ ) , D ( u, ¯ γ ) > . (2.32)Notice that by the definition of the characteristic velocity u , the value of D ( u, ¯ γ ) must be realand positive, and the square root p D ( u, ¯ γ ) is assumed to be positive as well. Consequently,Ω ( u ) is real-valued. Also note that if u is a real-valued characteristic velocity, then for anyreal u belonging to its sufficiently small vicinity, we have D ( u, ¯ γ ) >
0, implying that it is also acharacteristic velocity. Consequently, Ω ( u ) defined by equation (2.32) is a real analytic functionin the vicinity of u . We may also view the equationΩ ( u ) = Ω p p D ( u, ¯ γ ) = ω (2.33)as an equivalent form of the characteristic equation (2.18) for u in the vicinity of u . Equation(2.33) can be viewed also as a form of the dispersion relations and refer to it velocity dispersion XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 7
Figure 2.
Single-stream e-beam coupled to a single TL. Plot of the characteristicfunction D ( u, γ ) = γw − u + ( u − ˚ v ) u with its instability branches for ˚ v = 1, w = 1 . γ = 3 (horizontal axis – ℜ { u } , vertical axis – D ). The instability branches arerepresented by D ( ℜ { u } , γ ), where u is complex characteristic velocity, ℑ { u } 6 = 0,satisfying equations (2.9), that is, D ( u, γ ) = ω . Dash-dot (black) lines representthe plot of D ( u, γ ) for real u , solid (brown) lines represent unstable branches with ℑ { u } 6 = 0. The characteristic function instability nodes are represented by solid(brown) diamond dots. Vertical dash-dot (black) straight lines represent asymptotesassociated with real-valued poles of function γ D T ( u ) = γw − u and consequently ofthe characteristic function D ( u, γ ). relation relation . Using the frequency function Ω ( u ), we naturally define the TWT wavenumberfunction K ( u ) by the following equation: K ( u ) = Ω ( u ) u , u ∈ U +TB (¯ γ ) , Ω ( u ) > . (2.34)To visualize features of the TWT instability in its dispersion relation we proceed as follows[FigTWTbk, 7]. We represent the set of all oscillatory and unstable modes of the TWT-systemgeometrically by the set Π TB of the corresponding modal points ( k ( ω ) , ω ) and ( ℜ { k ( ω ) } , ω ) inthe kω -plane. We name the set Π TB as the dispersion-instability graph . To distinguish graphicallypoints ( k ( ω ) , ω ) associated oscillatory modes when k ( ω ) is real-valued from points ( ℜ { k ( ω ) } , ω )associated unstable modes when k ( ω ) is complex-valued with ℑ { k ( ω ) } 6 = 0, we assign them colorsas follows: (i) blue color is assigned to points ( k ( ω ) , ω ) when k is real-valued; (ii) brown color isassigned to points ( ℜ { k ( ω ) } , ω ) when k ( ω ) is complex-valued with ℑ { k ( ω ) } 6 = 0. We remindonce again that every brown point ( ω, ℜ { k ( ω ) } ) represents exactly two complex conjugate unstablemodes associated with ±ℑ { k ( ω ) } . We introduce then two subsets of the set Π TB , namely, Π TBo
XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 8
Figure 3.
Single-stream e-beam coupled to a single TL. Plot of the characteristicfunction D ( u, γ ) = γw − u + ( u − ˚ v ) u with its instability branches for ˚ v = 1, w = 1 . γ = 3 (horizontal axis – ℜ { u } , vertical axis – D ). The instability branches arerepresented by D ( ℜ { u } , γ ), where u is complex characteristic velocity, ℑ { u } 6 = 0,satisfying equations (2.9), that is, D ( u, γ ) = ω . Dash-dot (black) lines representthe plot of D ( u, γ ) for real u , solid (brown) lines represent unstable branches with ℑ { u } 6 = 0, and dashed (brown, blue) lines represent, respectively, plots of functions γ D T ( u ) = γw − u and D B ( u ) = ( u − ˚ v ) u . The characteristic function instability nodesare represented by solid (brown) diamond dots. Vertical dash-dot (black) straightlines represent asymptotes associated with real-valued poles of function γw − u andconsequently of the characteristic function D ( u, γ ).and Π TBu representing, respectively, all oscillatory and unstable modes. Evidently, the sets Π
TBo and Π
TBu are disjoint, and they partition the set Π TB , that is, Π TB = Π TBo ∪ Π TBu . According tothe color assignments, the points of sets Π
TBo and Π
TBu are, respectively, blue and brown. Fig. 5shows a typical dispersion-instability graph for the data as in equation (2.31).2.3.
Nodal velocities, nodal function and equation.
We consider here the nodal velocities which are the phase velocities that signify the onset of the TWT instability. According to ourtheory developed in [FigTWTbk, 13, 30] the nodal velocities are solutions to the nodal equation R ( u ) = − ∂ u D B ( u ) ∂ u D T ( u ) = ˚ v (˚ v − u ) ( w − u ) u = γ, (2.35)where functions D T ( u ) and D B ( u ) are defined by equations (2.22) and γ is TWT principal param-eter defined equations (2.4). We refer to function R ( u ) in the nodal function . Figure 6 illustratesgraphically features of the nodal function R ( u ) for a TWT composed of a single-stream e-beam XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 9 (a) (b)
Figure 4.
Single-stream e-beam coupled to a single TL. Zoomed fragments ofthe plot in Fig. 3 of the characteristic function D ( ℜ { u } , γ ) with its instabilitybranches (horizontal axis – ℜ { u } , vertical axis – D ) for ˚ v = 1, w = 1 . γ = 3:(a) negative u and ℜ { u } ; (b) positive u and ℜ { u } . The instability branches arerepresented by D ( ℜ { u } , γ ), where u is complex characteristic velocity, ℑ { u } 6 = 0,satisfying equations (2.9), that is, D ( u, γ ) = ω . Dash-dot (black) lines representthe plot of D ( u, γ ) for real u , solid (brown) lines represent unstable branches with ℑ { u } 6 = 0, and dashed (brown, blue) lines represent plots of functions γ D T ( u ) = γw − u and D B ( u ) = ( u − ˚ v ) u , respectively. The characteristic function instability nodesare represented by solid (brown) diamond dots. Vertical dash-dot (black) straightlines represent the asymptotes associated with real-valued poles of function γw − u and consequently of the characteristic function D ( u, γ ).and a single TL. Equation (2.35) can be recast in terms of dimensionless variables as follows: R (ˇ u ) = (1 − ˇ u ) ( χ − ˇ u ) ˇ u = ˇ γ, ˇ γ = γ ˚ v , ˇ u = u ˚ v , χ = w ˚ v , (2.36)where ˇ γ is the dimensionless TWT principal parameter. If u is the nodal velocity satisfyingequation (2.36), then the corresponding instability node frequency ω as well as instability nodewavenumber k can be expressed in terms of the frequency function Ω ( u ) defined by equation(2.32), that is, ω = Ω ( u ) = Ω p p D ( u , ¯ γ ) , k = ω u , (2.37)where Ω ( u ) is defined by equations (2.32).Observe that the nodal equation (2.35) has exactly three real-valued solutions: (i) one positive u + (¯ γ ) >
0; (ii) two negative u − (¯ γ ) < u − (¯ γ ) <
0. Notice also since in equation (2.32) γ > u + (¯ γ ) < < u + (¯ γ ) < . (2.38)2.4. Nodal points as points of degeneracy of the dispersion relations.
The diamond(brown) dots Fig. 5 identify the three points of the degeneracy of the dispersion relations. Vicinitesof these points demonstrate graphically the transition to instability when points on dot-dashed
XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 10
Figure 5.
Single-stream e-beam coupled to a single TL. Dispersion-instabilitygraph (horizontal axis –
ℜ { k } , vertical axis – ω ) for ˚ v = 1, w = 1 . γ = 3. Solid(brown) curves represent unstable branches with ℑ { k } 6 = 0, dash-dotted (black) linesrepresent oscillatory branches with ℑ { k } = 0, dashed lines represent the dispersionrelations for uncoupled TL (blue, converging to the origin) and for an uncouplede-beam (red). The instability nodes are represented by solid (brown) diamond dots.(a) (b) Figure 6.
Single e-beam coupled to a single TL. Plot (a) of the nodal function R ( u ) = v ( v − u ) ( w − u ) u and (b) its fragment (horizontal axis – u , vertical axis – R )for ˚ v = 1, w = 1 .
1. Solid (black) line represents function R ( u ), and dashed (blue)line represents asymptotics function R ( u ) as u → ∞ , the dash-dot (green) linerepresents constant γ = 3, and solid (brown) diamond dots represent solutions forthe instability node equation (2.35), that is, R ( u ) = γ = 3 . (blue) lines associated with oscillatory modes with real phase velocities u lines and real wavenum-bers k merge with points on solid (brown) lines associated with complex-conjugate pairs of unstablemodes with non-real real phase velocities at nontrivial kω -nodes.Importantly according to our studies in [FigTWTbk, 13, 30] the nodal velocities are in fact thepoints of degeneracy of the dispersion relation. In addition to that, any nodal velocity u has to bea real number which is an extreme point of characteristic function D ( u, ¯ γ ). Consequently, u hasto be an extreme point function Ω ( u ) defined by equation (2.32). Based on this, we can assume XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 11 that ∂ u Ω ( u ) | u = 0 , ∂ u Ω ( u ) (cid:12)(cid:12) u = 0 , (2.39)that is, u is a point of degeneracy of second-order of function Ω ( u ).To assess the analytic properties of solutions for the characteristic equation (2.18) in the vicinityof u , we introduce the following “small” dimensionless parameters, which are useful for our studiesof local extrema of function Ω ( u ) defined by equations (2.32): δ u = u − u u , δ ω = ω − ω ω , δ k = k − k k , κ = ℜ { δ k } , (2.40) η = (cid:2) ξ − δ ω (cid:3) = (cid:20) ω − ω ω (cid:21) , ξ = ω ω , ω = u ∂ u Ω ( u ) (cid:12)(cid:12) u = 0 , (2.41)where the square roots in equation (2.41) are naturally defined up to a factor ±
1. Notice thatparameter ξ can positive or negative depending on the sign of ω . Observe also that δ k can beexpressed in terms of η and δ u by the following formulas: δ k = k − k k = 1 + δ ω δ u − ξ η δ u − , (2.42) k = ωu , k = ω u . (2.43)Then using the analyticity of function Ω ( u ), we introduce its power seriesΩ ( u ) = ω + ω δ u " X n ≥ Ω n δ n − u , δ u = u − u u , (2.44) ω = u ∂ u Ω ( u ) (cid:12)(cid:12) u = 0 , Ω n = u n ω ∂ nu Ω ( u ) | u n ! , n ≥ , where ω and ω have the physical dimensions of frequency, whereas variable δ u and coefficientsΩ n are dimensionless. Notice that coefficients ω , ω and Ω n are all real.Using dimensionless variables (2.42) and series (2.44) we can recast equation (2.32) as follows: δ ω = ξ δ u " X n ≥ Ω n δ n − u . (2.45)The analysis of equations (2.40)-(2.45) carried out in [FigTWTbk, 13, 54] provides the followingpower series approximation that relates δ ω and κ = ℜ { δ k } : δ ω = ω − ω ω = (cid:26) ξ κ + · · · , if ξ − δ ω > ξ ξ +Ω κ + · · · , if ξ − δ ω < , κ = ℜ { δ k } = ℜ (cid:26) k − k k (cid:27) . (2.46)Solutions to characteristic equation (2.18) or the equivalent to it equation (2.33) can be transformedinto dispersion relation ω = ω ( ℜ { k } ), which is represented graphically in Fig. 7 for the data asin equation (2.31). 3. Analysis of the TWT at the EPD
The analysis of the TWT at the EPD we carry out here is based on the TWT properties reviewedin Section 2. In particular, we take a close look into the spectral properties of the TWT at the
XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 12
Figure 7.
A fragment of the dispersion-instability graph in a vicinity of nodalvelocity u + (¯ γ ) computed based on asymptotic formulas (2.46) for ˚ v = 1, w = 1 . γ = 3 (horizontal axis – ℜ { δ k } , vertical axis – δ ω ). Solid (brown) lines representunstable branches with ℑ { k } 6 = 0, and dotted (blue) lines represent oscillatorybranches with ℑ { k } = 0.EPD associated with the nodal velocity u + (¯ γ ) considered in Section 2.3. These properties arerelated to the TWT monic matrix polynomial M uω defined in equations (2.28), that is M uω = (cid:20) ˇ u − χ − χ γ ˇ ω − − ˇ u + u ˇ ω − − + ˇ γ − ω − − (cid:21) . (3.1)for ˇ u = u + (¯ γ ), and to the corresponding companion matrix we consider in Section 3.2. Forthe reader’s convenience the basics of the theory of matrix polynomials and the correspondingcompanion matrices are reviewed in Section B.In turns out that it is advantageous for our purposes here to use the nodal velocity p = u + (¯ γ ) > new independent variable in place of parameter ¯ γ . Using the fact that u + (¯ γ ) is a monoton-ically decreasing function of ¯ γ we can uniquely recover ¯ γ from p based on equation (3.2). Wenaturally refer to p and the nodal velocity for: (i) it is a point of degeneracy for the dispersionrelations associated with the nodal velocity u + (¯ γ ) and (ii) it is also a point of local minimum ofthe characteristic function D (ˇ u, ˇ γ ) defined by equation (2.29), see Figs. 4 (b), 5 and 6 (a) forgraphical illustration.3.1. Nodal velocity as the TWT parameter.
Plugging ˇ u = p in the nodal equation (2.36) weobtain the following representation of ˇ γ in terms p :ˇ γ = ˇ γ e = ˇ γ e ( p, χ ) = (1 − p ) ( p − χ ) p > , < p < . (3.3)Notice that equation (3.3) can be viewed as the inversion of equation (3.2) that defines the nodalvelocity p = u + (¯ γ ). Under assumption that χ is given and fixed function ˇ γ e ( p, χ ) in equation(3.3) relates the nodal velocity p to the TWT principle parameter ˇ γ . Notice that ˇ γ e ( p, χ ) is amonotonically decreasing function of p as illustrated by Figure 8. The function monotonicity canbe established by an examination of the sign of its derivative, which is ∂ p (ˇ γ e ( p, χ )) = ( χ − p ) (3 χ p + p − χ ) p < , for 0 < p < < χ. (3.4) XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 13
Since factor 3 χ p + p − χ in the right-hand side of equation (3.4) is a monotonically growingfunction of p it satisfies the following inequalities − χ < χ p + p − χ ≤ − χ < , for 0 < p < < χ, (3.5)implying that ˇ γ e ( p, χ ) is indeed a monotonically decreasing function of p .Typical values of the TWT principal parameter ˇ γ are small and according to [FigTWTbk,Remark 62.1] they vary between 2 · − and 0 . γ according to relation(3.3) correspond to values of the nodal velocity p that are close to 1, and the following asymptoticformula holds ˇ γ e ( p, χ ) = (cid:0) χ − (cid:1) (1 − p ) + O ((1 − p )) , p → − . (3.6)Figure 8 shows the graph of function ˇ γ e ( p, χ ) as in equation (3.3) for small values of ˇ γ .(a) (b) Figure 8.
Plots of function ˇ γ e ( p, χ ) as in (3.3) for χ = 1 . . < p < . < p < D and the corresponding characteristic equation: D e (ˇ u, p ) = ( p −
1) ( p − χ ) (ˇ u − χ ) p + (ˇ u − ˇ u = 1ˇ ω . (3.7)In particular, plugging in ˇ u = p in D (ˇ u, p ) we obtain D e ( p, p ) = (1 − p ) ( χ − p ) p > , for 0 < p < < χ. (3.8)Hence we can introduce now a positive frequency ˇ ω e = ˇ ω e ( p, χ ) by the following equality1ˇ ω = D e ( p, p ) = (1 − p ) ( χ − p ) p , (3.9)or equivalentlyˇ ω e = ˇ ω e ( p, χ ) = 1 p D e ( p, p ) = p p (1 − p ) ( χ − p ) > , for 0 < p < < χ. (3.10)We refer to ˇ ω e = ˇ ω e ( p, χ ) as EPD frequency . In view of equations (3.7) and (3.4) the EPDfrequency ˇ ω e ( p, χ ) defined by equations (3.10) corresponds to the nodal velocity p . Figure 9 is a XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 14 graphical representation of function ˇ ω e ( p, χ ) as in equation (3.10) for different ranges of values ofthe nodal velocity p . (a) (b) Figure 9.
Plots of function ˇ ω e ( p, χ ) as in equation (3.10) for χ = 1 . < p < .
95; (b) 0 . < p < ω e ( p, χ ) defined by equations (3.10) we obtain the following equivalentform of the characteristic equation (3.7) D e (ˇ u, p ) = D e (ˇ u, p ) − ω ( p, χ ) = 1ˇ ω − ω ( p, χ ) , where (3.11) D e (ˇ u, p ) = (ˇ u − p ) [( χ p + p − χ ) ˇ u + 2 χ p ( p −
1) ˇ u − p χ ]ˇ u p (ˇ u − χ ) . It is evident that for ˇ ω = ˇ ω e velocity ˇ u = p is a solution to equation (3.11) of multiplicity 2. Werefer to equation (3.11) as the EPD form of the characteristic equation . An algebraic factorizedform of rational function D e (ˇ u, p ) defined by equations (3.11) is D e (ˇ u, p ) = ( χ p + p − χ ) (ˇ u − p ) (ˇ u − λ + ( p )) (ˇ u − λ + ( p ))ˇ u p (ˇ u − χ ) , (3.12)where λ ± ( p ) = pχ h χ (1 − p ) ± p p − pχ (1 − p ) i p − (1 − p ) χ . (3.13)We infer based on the above analysis that for any given p and χ the EPD values of the originalTWT dimensionless parameters areEPD: ˇ u = p, ˇ γ = ˇ γ e ( p, χ ) = (1 − p ) ( p − χ ) p , ˇ ω = ˇ ω e ( p, χ ) = p p (1 − p ) ( χ − p ) . (3.14)3.2. Companion matrix spectral analysis.
According to our review on matrix polynomialsand associated with them companion matrices in Section B the companion matrix associated withmatrix polynomial M uω as in equation (3.1) takes the form C = χ χ − ˇ γ ˇ ω − − − ˇ γ − ω − − − ω − − . (3.15) XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 15
If ˇ u is a characteristic velocity and consequently an eigenvalue of companion matrix C then itsunique eigenvector Y (ˇ u ) is defined by the following expression Y (ˇ u ) = χ ˇ u − χ χ ˇ u ˇ u (ˇ u − χ ) . (3.16)Notice that expression (3.1) for the eigenvector Y (ˇ u ) of companion matrix C depends remarkablyon parameter χ and the value of the corresponding eigenvalue ˇ u of C only.The value C e of companion matrix C at the EPD point ˇ u = p can be obtained by plugging inits expression (3.1) the EPD values of ˇ γ = ˇ γ e and ˇ ω = ˇ ω e defined in equations (3.14) resulting in C e = χ χ (1 − p ) ( p − χ ) p − (1 − p ) χ (1 − p ) χ ( χ − p ) − p p − (1 − p ) χ p p − (1 − p ) χ . (3.17)Elementary but tedious analysis shows the Jordan canonical form J e of companion matrix C e is J e = p p λ + ( p ) 00 0 0 λ − ( p ) , λ ± ( p ) = pχ h χ (1 − p ) ± p p − pχ (1 − p ) i p − (1 − p ) χ . (3.18)Notice that expressions for eigenvalues λ ± ( p ) are consistent with equations (3.12) and (3.13) asone may expect.The Jordan basis of the generalized eigenspace of the EPD companion matrix C e is formed bythe eigenvector Y e ( p ) and the so-called root vector Y ′ e ( p ) which are as follows Y e ( p ) = χ p − χ χ pp ( p − χ ) , Y ′ e ( p ) = ∂ p Y e ( p ) = uχ u − χ , (3.19)where root vector Y ′ e ( p ) satisfies the following relations( C e − p I ) Y ′ e ( p ) = Y e ( p ) , ( C e − p I ) Y ′ e ( p ) = 0 . (3.20)The fact that Y ′ e ( p ) = ∂ p Y e ( p ) is of course not incidental. It can be argued based on the fact thatthe eigenvector Y (ˇ u ) of companion matrix C defined by equation (3.16) depends on parameter χ and the value of the corresponding eigenvalue ˇ u of C only. But regardless to the argument onecan verify the validity of equations (3.20) by tedious by straightforward evaluation.Then using (i) the Jordan basis formed by vectors Y e ( p ) and Y ′ e ( p ) defined by equations (3.19)and (ii) the general expression (3.16) for the eigenvector Y (ˇ u ) of companion matrix C we obtaincomplete Jordan basis for the EPD companion matrix C e formed by columns of the followingmatrix Y = [ Y e ( p ) , Y ′ e ( p ) , Y ( λ + ( p )) , Y ( λ − ( p ))] . (3.21)Consequently, we have C e = Y J e Y − , (3.22)where matrices J e and Y are defined respectively by equations (3.18) and (3.21).As to physical significance of the EPD phase velocity p and the EPD frequency ˇ ω e they can bedetected and identified by their intrinsic association with the onset of instability. Indeed, graduallyincreasing frequency ω of probing excitation of the TWT one can detect its value ˇ ω e when the XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 16 instability sets up. At this point one can also assess the value of the EPD phase velocity p bycomparing time dependent input and output signals. Using then equations (3.14) and assumingthat p and ˇ ω e are known we recover the TWT model parameters as follows: χ = p (cid:18) p (1 − p ) ˇ ω e (cid:19) , ˇ γ = (cid:2) p − ˇ ω (1 − p ) (cid:3) (1 − p ) ˇ ω . (3.23)4. Using an EPD for enhanced sensing of small signals
Based on our studies in Section 3 we develop here an approach for using the EPD of the TWTfor enhanced sensing of small signals. This approach has some similarity to what we advanced in[FigPert] for simple circuits with EPDs but it is naturally somewhat more complex. We remind thatour primary motivation for considering the TWT is that it can operate at much higher frequenciescompare to frequencies for lumped circuits.Since TWTs are used mostly as amplifiers with the gain varying exponentially with their lengthone might entertain an idea that the exponential amplification can be exploited for sensing ofsmall signals. The problem with this idea though is that the origin of TWT amplification is aninstability and that is hardly compatible with enhanced sensing of small signals. In addition tothat, in the case of exponential amplification a variety of noises that naturally occur in any TWTcan obscure the small sensor signal.An EPD in a TWT is in fact also associated with an instability but the EPD regime is at leastmarginally stable. In particular, if we choose TWT regime to be near the EPD rather than exactlyat it the TWT operation can be stable as we showed in [FigPert]. This approach is of course atrade off allowing to buy the stability in exchange for reduced value of the enhancement factor forthe small sensed signal. But even with such a trade off in place one can get more than 100 foldenhancement [FigPert].4.1.
Mathematical model for sensing.
We start with equations (2.2) and (2.4) that relate theTWT system model parameters χ and ˇ γ to the primary physical quantities, namely χ = w ˚ v = 1˚ v √ CL , ˇ γ = γ ˚ v = K γ C , K γ = b ˚ v e m R σ B ˚ n. (4.1)We proceed then with an assumption that the e-beam parameters ˚ v , σ B and ˚ n are chosen, fixedand maintain their values through the process of sensing. We suppose further that: (i) it is eitherthe distributed capacitance C or the distributed inductance L utilized for sensing; (ii) parameter C or L which is selected for sensing is slightly altered by the small sensor signal . The relevantalteration caused by sensed signal is assessed by measuring the relevant characteristic velocities ˇ u of the TWT. These velocities ˇ u are solutions to the EPD form (3.11) of the characteristic equation,that is (ˇ u − p ) [( χ p + p − χ ) ˇ u + 2 χ p ( p −
1) ˇ u − p χ ]ˇ u p (ˇ u − χ ) = 1ˇ ω − ω ( p, χ ) , (4.2)where parameters p and ˇ ω e are related to parameters ˇ γ and χ by equations (3.23). More precisely,with the equation (4.2) in mind we probe the TWT at frequency ˇ ω of our choosing and thenmeasure the phase velocities ˇ u of the excited eigenmodes of the TWT system. Since accordingto our TWT model these velocities ˇ u satisfy characteristic equation (4.2) we can relate them tovalues of parameters p and χ . Having found p and χ we can recover then the values of C and L based on equations (3.23) and (4.1) as we show below.To have a clarity on what kind of the TWT states are considered. We remind that the TWTsignificant properties are encoded in its companion matrix C defined by equation (3.15). Thismatrix in turn is determined by parameters ˇ γ , χ and frequency ˇ ω . Instead of parameter ˇ γ we can XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 17 use the nodal velocity p related to it by equation (3.3). Then based on the mentioned quantitieswe introduce the following definitions of the TWT configuration and TWT state. Definition 1 (TWT configuration and state) . TWT configuration is defined as a pair of twodimensionless TWT parameters: (i) χ = w ˚ v ; (ii) the nodal velocity p = p ( χ, ˇ γ ) satisfying equation(3.14). TWT state is defined as a triple of o dimensionless parameters: χ , p and ˇ ω . Frequency ˇ ω can be viewed as a parameter that selects the corresponding four TWT eigenmodes associated withthe TWT companion matrix C defined by equation (3.15). The selection is facilitated physicallyby exciting/probing the TWT at frequency ˇ ω . An EPD state is defined by a particular choice ofits parameters, namely χ , p and ˇ ω = ˇ ω e ( p, χ ) where the EPD frequency ˇ ω e ( p, χ ) is defined byequations (3.10). Work point state is defined by the following choice of its parameters χ , p andˇ ω = ˇ ω w < ˇ ω e ( p, χ ). We refer to ˇ ω w as work point frequency . There is a flexibility in choosing ˇ ω w tobe proximate to the EPD frequency ˇ ω e ( p, χ ) when at the same time to maintain certain distancefrom it to provide for the stability of the TWT operation as explained in Remark 2. There areexactly two eigenmodes with their phase velocities ˇ u close to p that are of particular significance. Remark . It turns out that the ideal EPD state is intrinsically only marginally stableand the purpose of the work point is to overcome this problem. The stability of the work pointis achieved by making a deliberate small departure from the EPD frequency ˇ ω e . The departureis achieved by appropriate selection of the probing frequency ˇ ω = ˇ ω w so that ˇ ω w < ˇ ω e . Anadditional benefit and utility of the work point frequency ˇ ω w is that it lifts the characteristicvelocity degeneracy causing the velocity split. This velocity split can be measured and used todetermine the small sensed signal, see Theorem 3, Fig. 10 and equations (4.23) and (4.24).Suppose that the TWT configuration before sensing is defined by parameters χ and p and theTWT configuration altered by the small sensed signal is defined by parameters χ ′ and p ′ . Weproceed then with introducing a larger set of parameters associated with the EPD state of theTWT before it it receives the small sensed signal:EPD state: C, L, p, χ, ˇ ω = ˇ ω e ( p, χ ) . (4.3)Using the above EPD state as a reference point we introduce a larger set of parameters for alteredEPD state associated with the small sensed signal:altered EPD state: C ′ = C (1 + δ C ) , L ′ = L (1 + δ L ) , p ′ = p (1 + δ p ) , (4.4)ˇ( χ ) ′ = χ (1 + δ χ ) , ω = ˇ ω ′ e = ˇ ω e ( p ′ , χ ′ ) = ˇ ω e (1 + δ ω ) , δ ω = ˇ ω ′ e − ˇ ω e ˇ ω e , where as a matter of computational convenience we use parameter χ rather than χ . In relations(4.3)-(4.7) the relative variation coefficients δ ∗ are assumed to be small and satisfy the followingrelations | δ C | , | δ L | , | δ χ | , | δ p | ≪ | δ ω | ≪ | δ w | ≪ . (4.5)Using once again the EPD state as a reference point we introduce parameters of the work pointstate and its altered version as follows:work point state: C, L, p, χ, ˇ ω = ˇ ω w = ˇ ω e (1 + δ w ) , δ w = ˇ ω w − ˇ ω e ˇ ω e , (4.6)altered work point state: C ′ = C (1 + δ C ) , L ′ = L (1 + δ L ) , p ′ = p (1 + δ p ) , (4.7) (cid:0) χ (cid:1) ′ = χ (1 + δ χ ) , ˇ ω = ˇ ω w = ˇ ω e (1 + δ w ) , ˇ ω ′ e = ˇ ω e ( p ′ , χ ′ ) = ˇ ω e (1 + δ ω ) , δ ω = ˇ ω ′ e − ˇ ω e ˇ ω e . XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 18
Relative incremental frequency differences δ w and δ ω employ effectively the EPD frequency ˇ ω e as anatural frequency unit. Notice that parameters in relations (4.3)-(4.7) are not independent but rather parameters χ , p and consequently ˇ ω e can be expressed in terms of the primary physical parameters C and L .Namely, according to relations (3.14) and (4.1) we have χ = 1˚ v √ CL , ˇ γ = K γ C = (1 − p ) ( p − χ ) p , ˇ ω e = p p (1 − p ) ( χ − p ) . (4.8)Since in view of made assumptions the e-beam parameters ˚ v and K γ can be considered to beconstants the first two equations in (4.8) allow to express χ and p as a functions of C and L .Consequently the third equation in (4.8) determines ˇ ω e as a function of C and L also.We proceed now with the derivation of linear approximations, that is the differentials, for δ χ , δ p and δ ω in terms of δ C and δ L . The assumed smallness of δ C , δ L and δ ω imply the smallness of δ χ , δ p and δ ′ ω . Hence the work point and the altered work point states are all close to the EPD state. Using the proximity of all these states we obtain the following first order approximation tothe characteristic equation (4.2) for the altered state(ˇ u − p ′ ) ∼ = S (ˇ ω ′ e − ˇ ω w ) = S ˇ ω e ( δ ω − δ w ) , (4.9)where δ ω = ˇ ω ′ e − ˇ ω e ˇ ω e , δ w = ˇ ω w − ˇ ω e ˇ ω e , ˇ ω e = ˇ ω e ( p, χ ) , ˇ ω ′ e = ˇ ω e ( p ′ , χ ′ ) , (4.10) S = S ( p, χ ) = 2 p ( χ − p ) [(1 − p ) ( χ − p )] p (4 χ − χ p − p ) > , < p < , χ > . (4.11)We refer to equation (4.9) as EPD approximation to the characteristic equation , and we refer toquantities δ ω and δ w ∆ ′ and ∆ w respectively as relative EPD increment and relative work pointincrement . The EPD approximation to the characteristic equation which is a quadratic equationselects two characteristic velocities out of the total of four by their property to be proximate to theEPD velocity p . Equations (4.9) readily imply the following statement.
Theorem 3 (velocity split near the EPD) . . The velocities ˇ u ± that solve the quadratic in ˇ u equation (4.9) are ˇ u ± = p ′ ± p S (ˇ ω ′ e − ˇ ω w ) = p ′ ± p S ˇ ω e ( δ ω − δ w ) , (4.12) where factor S > satisfies relations (4.11) and frequency ˇ ω e > satisfies the last equation in(4.8). The parameters χ and p that determine S and ˇ ω e are associated with the EPD state as in(4.3). Since S > velocities ˇ u ± defined by equation (4.12) are real if and only if ˇ ω w < ˇ ω ′ e . (4.13) Equations (4.12) imply the following representation for the velocity split ˇ u + − ˇ u − at the EPD: ˇ u + − ˇ u − = 2 p S (ˇ ω ′ e − ˇ ω w ) = 2 p S ˇ ω e ( δ ω − δ w ) , (4.14) δ ω = ˇ ω ′ e − ˇ ω e ˇ ω e , δ w = ˇ ω w − ˇ ω e ˇ ω e , where frequencies ˇ ω e , ˇ ω ′ e and ˇ ω w are associated respectively with the EPD state, the altered stateand the work point state defined by equations (4.3), (4.6) and (4.7).If the following inequalities hold | δ ω | < | δ w | and − δ w > , (4.15) XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 19
Figure 10.
An illustrative plot for the approximate characteristic equation similar(4.9) for EPD state represented by black solid parabola and the altered state repre-sented by blue dashed parabola (horizontal axis – u , vertical axis – ω ). The blackhorizontal line corresponds to the work frequency ˇ ω = ˇ ω w . The points of intersectionof the horizontal line and blue dashed parabola shown as blue disk points representsolutions ˇ u ± to equation (4.12). The points of intersection of the horizontal line andblack solid parabola are shown as brown diamond points represents solutions ˇ u ± toequation similar (4.12) corresponding to p rather than p ′ . The EPDs correspond theto vertexes of the parabolas shown as black square points. Notice that the EPDfrequencies ˇ ω e and ˇ ω ′ e associated with the parabolas vertices satisfy the frequencystability conditions (4.17). then δ ω − δ w = ˇ ω ′ e − ˇ ω w ˇ ω e > readily implying the inequality (4.13).Remark . We refer to inequalitiesˇ ω w < ˇ ω e , ˇ ω ′ e , (4.17)that appear in the statement of Theorem 3 as the frequency stability conditions . These conditionsare relevant to the stability for they imply the real-valuedness of the velocities ˇ u ± . Note thatinequalities (4.15) imply frequency stability conditions (4.17).Note also that according to equation (4.14) the velocity split ˇ u + − ˇ u − , a quantity that can bemeasured, is proportional to √ ˇ ω ′ e − ˇ ω w . The square root operation applied to the small frequencydifference ˇ ω ′ e − ˇ ω w effectively “magnifies” it. This is a typical manifestation of the proximity tothe EPD. The rise of the square root operation can be traced to the EPD approximation to thecharacteristic equation (4.9) which is a quadratic in ˇ u equation .The EPD approximation to the characteristic equation (4.9) and its solutions ˇ u ± are illustratedby Fig. 10.4.2. Sensing algorithm.
Our approach to utilize the TWT for sensing is as follows. First of all,we assume that (i) the small sensed signal alters either distributed capacitance C or distributedinductance L ; (ii) the physical alteration of the TWT indicated in (i) is described mathematicallyby relations (4.3)-(4.5). Second of all, we assume that prior to any measurements the values ofparameters C , L , χ and p that determine the current state of the TWT are established and known.These values constitute a reference point for an assessment of the results of sensing. Third of all,we assume that the relationship between δ C or δ L and the small sensed signal is known and is anintegral part of the sensor. The consecutive steps of our sensing algorithm are as follows:
XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 20 (i) Apply the small sensed signal to the relevant component of the TWT. Probe then theTWT by an excitation at the work point frequency ˇ ω = ˇ ω w .(ii) Observe the two TWT eigenmodes and assess the corresponding to them characteristic ve-locities ˇ u + and ˇ u − that are close to the EPD velocity p and satisfy the EPD approximation(4.9) to the characteristic equation.(iii) Compute the velocity split ˇ u + − ˇ u − . Depending on what of the parameters C or L wasutilized for sensing find the corresponding value δ C or δ L using respectively formulas (4.23)and (4.24) below.(iv) Based on assumed to be known relationship between δ C or δ L and the small sensed signalrecover the value of the small sensed signal.Fig. 10 illustrates graphically the proposed sensing approach. The sensing approach utilizingthe EPD of the TWT is conceptually similar to the one for circuits advanced in [FigPert] butit is naturally more complex. The reason for complexity is that the TWT as a physical systemis naturally a more complex system compare to the simple circuits with EPDs we advanced in[FigPert].In the light of Theorem 3 and Remark 4 let us take a closer look at the relative EPD increment δ ω and the relative work point increment δ w defined by equations (4.10). Both relative increments δ ω and δ w in equations (4.10) use the EPD frequency ˇ ω e = ˇ ω e ( p, χ ) as a reference point and effec-tively as a natural frequency unit. The stability of the TWT operation requires the characteristicvelocities ˇ u ± defined by equations (4.12) to be real-valued. This requirement is fulfilled always ifand only if the expression under square root in the right-hand side of equation (4.9) is non-negative.That in turn leads to δ ω − δ w = ˇ ω ′ e − ˇ ω w ˇ ω e > , implying ˇ ω w < ˇ ω e , (4.18)since δ ω can be zero and factors ˇ ω e , S > | δ ω | = (cid:12)(cid:12)(cid:12)(cid:12) ˇ ω ′ e − ˇ ω e ˇ ω e (cid:12)(cid:12)(cid:12)(cid:12) ≪ | δ w | = (cid:12)(cid:12)(cid:12)(cid:12) ˇ ω w − ˇ ω e ˇ ω e (cid:12)(cid:12)(cid:12)(cid:12) ≪ . (4.19)The point of inequalities (4.19) as a requirement is to assure that the altered EPD frequency ˇ ω ′ e satisfies the frequency stability conditions (4.17) as soon as the EPD frequency ˇ ω e satisfies them.Consequently relations (4.19) assure the effectiveness and robustness of described above approachto sensing.We proceed now with relating the relative increments δ ω and δ w defined by equations (4.10) toquantities δ C and δ L that are directly effected by the small sensor signal. Using equations (4.8) andassuming that δ C and δ L are small after tedious but elementary evaluations we find the followingfirst order approximations δ χ ∼ = − ( δ C + δ L ) , δ p ∼ = − (1 − p ) [( χ + p ) δ C + 2 χ δ L ](4 − p ) χ − p . (4.20)Using representation (4.8) for ˇ ω e ( p, χ ) and equations (4.20) (4.20) under assumption that δ C and δ L are small we obtain the following first order approximation δ ω = ˇ ω e ( p ′ , χ ′ ) − ˇ ω e ( p, χ )ˇ ω e ( p, χ ) ∼ = − p δ C + χ δ L χ − p ) . (4.21)Fig. 11 is a graphical representations of function the increment ˇ ω e ( p ′ , χ ′ ) − ˇ ω e ( p, χ ) and its firstorder approximation as in equation (4.21) for ranges of values of relative variation coefficient δ C when δ L = 0. Fig. 12 similarly is a graphical representation of function the increment ˇ ω e ( p ′ , χ ′ ) − XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 21 ˇ ω e ( p, χ ) and its first order approximation as in equation (4.21) for two ranges of values of therelative variation coefficient δ L when δ C = 0. Figures 11 and 12 indicate that for the chosen valuesof parameters when the value of relative variation coefficient δ C or δ L is under 5% the simpleformula (4.21) yields pretty accurate values compare to the exact value of the increment. If valuesrelative variation coefficient δ C or δ L values are below 1% the approximation values are nearly thesame as the exact values. (a) (b) Figure 11.
Plots of the increment ˇ ω e ( p ′ , χ ′ ) − ˇ ω e ( p, χ ) and its first order approx-imation as in equation (4.21) for ˚ v = 1 . K γ = 1 . C = 1 . L = 1 . δ L = 0 and:(a) 0 < δ C < .
3; (b) 0 < δ C < . Figure 12.
Plots of the increment ˇ ω e ( p ′ , χ ′ ) − ˇ ω e ( p, χ ) and its first order approx-imation as in equation (4.21) for ˚ v = 1 . K γ = 1 . C = 1 . L = 1 . δ C = 0 and:(a) 0 < δ L < .
3; (b) 0 < δ L < . δ ˇ ω based on the measured velocityshift ˇ u + − ˇ u − as follows δ ˇ ω ∼ = − p δ C + χ δ L χ − p ) = (ˇ u + − ˇ u − ) S ˇ ω e + δ w = (ˇ u + − ˇ u − ) S ˇ ω e + ˇ ω w ˇ ω e − . (4.22)Consequently, if the distributed capacitance C was used for sensing and hence δ L = 0 we obtainfrom equations (4.22) the value of δ C , that is δ C ∼ = 2 ( χ − p ) p " − ˇ ω w ˇ ω e − (ˇ u + − ˇ u − ) S ˇ ω e . (4.23) XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 22
Similarly, if the distributed inductance L was used for sensing and hence δ C = 0 we obtain fromequations (4.22) the value of δ L , that is δ L ∼ = 2 ( χ − p ) χ " − ˇ ω w ˇ ω e − (ˇ u + − ˇ u − ) S ˇ ω e . (4.24) Appendix A. Fourier transform
Our preferred form of the Fourier transforms as in [Foll, 7.2, 7.5], [ArfWeb, 20.2]: f ( t ) = Z ∞−∞ ˆ f ( ω ) e − i ωt d ω, ˆ f ( ω ) = 12 π Z ∞−∞ f ( t ) e i ωt d t, (A.1) f ( z, t ) = Z ∞−∞ ˆ f ( k, ω ) e − i( ωt − kz ) d k d ω, (A.2)ˆ f ( k, ω ) = 1(2 π ) Z ∞−∞ f ( z, t ) e i( ωt − kz ) dz d t. This preference was motivated by the fact that the so-defined Fourier transform of the convolutionof two functions has its simplest form. Namely, the convolution f ∗ g of two functions f and g isdefined by [Foll, 7.2, 7.5],[ f ∗ g ] ( t ) = [ g ∗ f ] ( t ) = Z ∞−∞ f ( t − t ′ ) g ( t ′ ) d t ′ , (A.3)[ f ∗ g ] ( z, t ) = [ g ∗ f ] ( z, t ) = Z ∞−∞ f ( z − z ′ , t − t ′ ) g ( z ′ , t ′ ) d z ′ d t ′ . (A.4)Then its Fourier transform as defined by equations (A.1) and (A.2) satisfies the following properties: [ f ∗ g ( ω ) = ˆ f ( ω ) ˆ g ( ω ) , (A.5) [ f ∗ g ( k, ω ) = ˆ f ( k, ω ) ˆ g ( k, ω ) . (A.6) Appendix B. Matrix polynomial and its companion matrix
An important incentive for considering matrix polynomials is that they are relevant to thespectral theory of the differential equations of the order higher than 1, particularly the Euler-Lagrange equations which are the second-order differential equations in time. We provide hereselected elements of the theory of matrix polynomials following mostly to [GoLaRo, II.7, II.8],[Baum, 9]. General matrix polynomial eigenvalue problem reads A ( s ) x = 0 , A ( s ) = ν X j =0 A j s j , x = 0 , (B.1)where s is complex number, A k are constant m × m matrices and x ∈ C m is m -dimensional column-vector. We refer to problem (B.1) of funding complex-valued s and non-zero vector x ∈ C m aspolynomial eigenvalue problem.If a pair of a complex s and non-zero vector x solves problem (B.1) we refer to s as an eigen-value or as a characteristic value and to x as the corresponding to s eigenvector . Evidently thecharacteristic values of problem (B.1) can be found from polynomial characteristic equation det { A ( s ) } = 0 . (B.2)We refer to matrix polynomial A ( s ) as regular if det { A ( s ) } is not identically zero. We denote by m ( s ) the multiplicity (called also algebraic multiplicity ) of eigenvalue s as a root of polynomialdet { A ( s ) } . In contrast, the geometric multiplicity of eigenvalue s is defined as dim { ker { A ( s ) }} , XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 23 where ker { A } defined for any square matrix A stands for the subspace of solutions x to equation Ax = 0. Evidently, the geometric multiplicity of eigenvalue does not exceed its algebraic one, seeCorollary 7.It turns out that the matrix polynomial eigenvalue problem (B.1) can be always recast as thestandard “linear” eigenvalue problem, namely( s B − A ) x = 0 , (B.3)where mν × mν matrices A and B are defined by B = I · · · I · · ·
00 0 . . . · · · ...... ... . . . I
00 0 · · · A ν , A = I · · · I · · ·
00 0 0 · · · ...... ... . . . I − A − A · · · − A ν − − A ν − , (B.4)with I being m × m identity matrix. Matrix A , particularly in monic case, is often referred to as companion matrix . In the case of monic polynomial A ( λ ), when A ν = I is m × m identity matrix,matrix B = I is mν × mν identity matrix. The reduction of original polynomial problem (B.1) toan equivalent linear problem (B.3) is called linearization .The linearization is not unique, and one way to accomplish is by introducing the so-called known“ companion polynomia l” which is mν × mν matrix C A ( s ) = s B − A = s I − I · · · s I − I · · ·
00 0 . . . · · · ...... ... ... s I − I A A · · · A ν − sA ν + A ν − . (B.5)Notice that in the case of the EL equations the linearization can be accomplished by the relevantHamilton equations.To demonstrate the equivalency between the eigenvalue problems for mν × mν companion poly-nomial C A ( s ) and the original m × m matrix polynomial A ( s ) we introduce two mν × mν matrixpolynomials E ( s ) and F ( s ). Namely, E ( s ) = E ( s ) E ( s ) · · · E ν − ( s ) I − I · · · − I . . . · · · ...... ... . . . · · · − I , (B.6)det { E ( s ) } = 1 , where m × m matrix polynomials E j ( s ) are defined by the following recursive formulas E ν ( s ) = A ν , E j − ( s ) = A j − + sE j ( s ) , j = ν, . . . , . (B.7) XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 24
Matrix polynomial F ( s ) is defined by F ( s ) = I · · · − s I I · · · − s I . . . · · · ...... ... . . . I
00 0 · · · − s I I , det { F ( s ) } = 1 . (B.8)Notice, that both matrix polynomials E ( s ) and F ( s ) have constant determinants readily implyingthat their inverses E − ( s ) and F − ( s ) are also matrix polynomials. Then it is straightforward toverify that E ( s ) C A ( s ) F − ( s ) = E ( s ) ( s B − A ) F − ( s ) = A ( s ) 0 · · · I · · ·
00 0 . . . · · · ...... ... . . . I
00 0 · · · I . (B.9)The identity (B.9) where matrix polynomials E ( s ) and F ( s ) have constant determinants can beviewed as the definition of equivalency between matrix polynomial A ( s ) and its companion poly-nomial C A ( s ).Let us take a look at the eigenvalue problem for eigenvalue s and eigenvector x ∈ C mν associatedwith companion polynomial C A ( s ), that is( s B − A ) x = 0 , x = x x x ...x ν − ∈ C mν , x j ∈ C m , ≤ j ≤ ν − , (B.10)where ( s B − A ) x = sx − x sx − x ...sx ν − − x ν − P ν − j =0 A j x j + ( sA ν + A ν − ) x ν − . (B.11)With equations (B.10) and (B.11) in mind we introduce the following vector polynomial x s = x sx ...s ν − x s ν − x , x ∈ C m . (B.12)Not accidentally, the components of the vector x s in its representation (B.12) are in evident relationwith the derivatives ∂ jt ( x e st ) = s j x e st . That is just another sign of the intimate relations betweenthe matrix polynomial theory and the theory of systems of ordinary differential equations [Hale,III.4]. XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 25
Theorem 5 (eigenvectors) . Let A ( s ) as in equations (B.1) be regular, that det { A ( s ) } is notidentically zero, and let mν × mν matrices A and B be defined by equations (B.2). Then thefollowing identities hold ( s B − A ) x s = ... A ( s ) x , x s = x sx ...s ν − x s ν − x , (B.13)det { A ( s ) } = det { s B − A } , det { B } = det { A ν } , (B.14) where det { A ( s ) } = det { s B − A } is a polynomial of the degree mν if det { B } = det { A ν } 6 = 0 .There is one-to-one correspondence between solutions of equations A ( s ) x = 0 and ( s B − A ) x = 0 .Namely, a pair s, x solves eigenvalue problem ( s B − A ) x = 0 if and only if the following equalitieshold x = x s = x sx ...s ν − x s ν − x , A ( s ) x = 0 , x = 0; det { A ( s ) } = 0 . (B.15) Proof.
Polynomial vector identity (B.13) readily follows from equations (B.11) and (B.12). Identi-ties (B.14) for the determinants follow straightforwardly from equations (B.12), (B.15) and (B.9).If det { B } = det { A ν } 6 = 0 then the degree of the polynomial det { s B − A } has to be mν since A and B are mν × mν matrices.Suppose that equations (B.15) hold. Then combining them with proven identity (B.13) we get( s B − A ) x s = 0 proving that expressions (B.15) define an eigenvalue s and an eigenvector x = x s .Suppose now that ( s B − A ) x = 0 where x = 0. Combing that with equations (B.11) we obtain x = sx , x = sx = s x , · · · , x ν − = s ν − x , (B.16)implying that x = x s = x sx ...s ν − x s ν − x , x = 0 , (B.17)and ν − X j =0 A j x j + ( sA ν + A ν − ) x ν − = A ( s ) x . (B.18)Using equations (B.17) and identity (B.13) we obtain0 = ( s B − A ) x = ( s B − A ) x s = ... A ( s ) x . (B.19) XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 26
Equations (B.19) readily imply A ( s ) x = 0 and det { A ( s ) } = 0 since x = 0. That completes theproof. (cid:3) Remark . Notice that according to Theorem 5 the character-istic polynomial det { A ( s ) } for m × m matrix polynomial A ( s ) has the degree mν , whereas inlinear case s I − A for m × m identity matrix I and m × m matrix A the characteristic polynomialdet { s I − A } is of the degree m . This can be explained by observing that in the non-linear caseof m × m matrix polynomial A ( s ) we are dealing effectively with many more m × m matrices A than just a single matrix A .Another problem of our particular interest related to the theory of matrix polynomials is eigen-values and eigenvectors degeneracy and consequently the existence of non-trivial Jordan blocks,that is Jordan blocks of dimensions higher or equal to 2. The general theory addresses this problemby introducing so-called “Jordan chains” which are intimately related to the theory of system ofdifferential equations expressed as A ( ∂ t ) x ( t ) = 0 and their solutions of the form x ( t ) = p ( t ) e st where p ( t ) is a vector polynomial, see [GoLaRo, I, II], [Baum, 9]. Avoiding the details of Jordanchains developments we simply notice that an important to us point of Theorem 5 is that there isone-to-one correspondence between solutions of equations A ( s ) x = 0 and ( s B − A ) x = 0, and ithas the following immediate implication. Corollary 7 (equality of the dimensions of eigenspaces) . Under the conditions of Theorem 5 forany eigenvalue s , that is det { A ( s ) } = 0 , we have dim { ker { s B − A }} = dim { ker { A ( s ) }} . (B.20) In other words, the geometric multiplicities of the eigenvalue s associated with matrices A ( s ) and s B − A are equal. In view of identity (B.20) the following inequality holds for the (algebraic)multiplicity m ( s ) m ( s ) ≥ dim { ker { A ( s ) }} . (B.21)The next statement shows that if the geometric multiplicity of an eigenvalue is strictly less thanits algebraic one than there exist non-trivial Jordan blocks, that is Jordan blocks of dimensionshigher or equal to 2. Theorem 8 (non-trivial Jordan block) . Assuming notations introduced in Theorem 5 let us supposethat the multiplicity m ( s ) of eigenvalue s satisfies m ( s ) > dim { ker { A ( s ) }} . (B.22) Then the Jordan canonical form of companion polynomial C A ( s ) = s B − A has a least one nontrivialJordan block of the dimension exceeding 2.In particular, if dim { ker { s B − A }} = dim { ker { A ( s ) }} = 1 , (B.23) and m ( s ) ≥ then the Jordan canonical form of companion polynomial C A ( s ) = s B − A hasexactly one Jordan block associated with eigenvalue s and its dimension is m ( s ) . The proof of Theorem 8 follows straightforwardly from the definition of the Jordan canonicalform and its basic properties. Notice that if equations (B.23) hold that implies that the eigenvalue0 is cyclic (nonderogatory) for matrix A ( s ) and eigenvalue s is cyclic (nonderogatory) for matrix B − A provided B − exists. We remind that an eigenvalue is called cyclic (nonderogatory) if itsgeometric multiplicity is 1. A square matrix is called cyclic (nonderogatory) if all its eigenvaluesare cyclic [BernM, 5.5]. XCEPTIONAL POINTS OF DEGENERACY IN TRAVELING WAVE TUBES 27
Appendix C. Notations • C is a set of complex number. • ¯ s is complex-conjugate to complex number s • C n is a set of n dimensional column vectors with complex complex-valued entries. • C n × m is a set of n × m matrices with complex-valued entries. • R n × m is a set of n × m matrices with real-valued entries. • dim ( W ) is the dimension of the vector space W . • ker ( A ) is the kernel of matrix A , that is the vector space of vector x such that Ax = 0. • det { A } is the determinant of matrix A . • χ A ( s ) = det { s I ν − A } is the characteristic polynomial of a ν × ν matrix A . • I ν is ν × ν identity matrix. • M T is a matrix transposed to matrix M . • EL stands for the Euler-Lagrange (equations).
Data Availability:
The data that supports the findings of this study are available within thearticle.
Acknowledgment:
This research was supported by AFOSR grant
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